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+arXiv:2301.00276v1  [cs.IT]  31 Dec 2022
+1
+Impact of Phase-Shift Error on the Secrecy
+Performance of Uplink RIS Communication
+Systems
+Abdelhamid Salem, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, and Chan-Byoung Chae,
+Fellow, IEEE
+Abstract
+Reconfigurable intelligent surface (RIS) has been recognized as a promising technique for the sixth gen-
+eration (6G) of mobile communication networks. The key feature of RIS is to reconfigure the propagation
+environment via smart signal reflections. In addition, active RIS schemes have been recently proposed to
+overcome the deep path loss attenuation inherent in the RIS-aided communication systems. Accordingly, this
+paper considers the secrecy performance of up-link RIS-aided multiple users multiple-input single-output (MU-
+MISO) communication systems, in the presence of multiple passive eavesdroppers. In contrast to the existing
+works, we investigate the impact of the RIS phase shift errors on the secrecy performance. Taking into account
+the complex environment, where a general Rician channel model is adopted for all the communication links,
+closed-form approximate expressions for the ergodic secrecy rate are derived for three RIS configurations,
+namely, i) passive RIS, ii) active RIS, iii) active RIS with energy harvesting (EH RIS). Then, based on the
+derived expressions, we optimize the phase shifts at the RIS to enhance the system performance. In addition,
+the best RIS configuration selection is considered for a given target secrecy rate and amount of the power
+available at the users. Finally, Monte-Carlo simulations are provided to verify the accuracy of the analysis,
+and the impact of different system parameters on the secrecy performance is investigated. The results in this
+Abdelhamid Salem is with the department of Electronic and Electrical Engineering, University College London, London, UK, (emails:
+a.salem@ucl.ac.uk).
+Kai-Kit Wong is with the department of Electronic and Electrical Engineering, University College London, London, UK, Kai-Kit
+Wong is also affiliated with Yonsei University, Seoul, Korea (email: kai-kit.wong@ucl.ac.uk).
+Chan-Byoung Chae is with Yonsei University, Seoul, Korea (e-mail: cbchae@yonsei.ac.kr).
+The work is supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/V052942/1. For the
+purpose of open access, the authors will apply a Creative Commons Attribution (CCBY) licence to any Author Accepted Manuscript
+version arising.
+
+2
+paper show that, an active RIS scheme can be implemented to enhance the secrecy performance of RIS-aided
+communication systems with phase shift errors, especially when the users have limited transmission power.
+Index Terms
+Reconfigurable intelligent surface, Physical layer security, MU-MISO, MRC.
+I. INTRODUCTION
+Reconfigurable intelligent surface (RIS), also known as intelligent reflecting surface (IRS), has been
+proposed recently as a promising technique to extend the coverage and improve the spectral efficiency
+of wireless communication networks [1], [2]. Specifically, RIS is composed of reflecting elements, each
+of which independently imposes a phase shift on the incident signals. By tuning the phase shifts of the
+reflecting elements, RIS can convert the propagation environments into smart ones and thus enhance
+the received signals quality [1], [2]. Due to these advantages, RIS techniques have been extensively
+considered in the literature. For instance, in [3], the fundamental capacity limit of RIS-aided multiple-
+input multipleoutput (MIMO) communication systems has been considered. The achievable ergodic rate
+of a RIS-assisted MIMO system which comprises links of a Rician channel was derived in [4]. In [5], a
+closed-form asymptotic ergodic sum rate of a RIS-assisted MIMO communication system was derived
+under the assumption that the number of base station (BS) antennas tends to infinity. In [6], the up-link
+achievable rate in RIS-aided massive MIMO systems has been analyzed and optimized. The authors
+in [7], [8] analyzed the achievable rate of RIS-assisted multiple users (MU) up-link massive MIMO
+system under Rician fading channels. In [9], [10], a closed-form expression of ergodic achievable rate
+for RIS-aided massive MIMO systems with zero forcing (ZF) detector has been derived. In addition, a
+closed-form analytical expression for the symbol error probability and the upper bound on the channel
+capacity of a RIS communication system have been derived in [11]. The work in [12] considered the
+impact of hardware impairments on a general RIS MU-MISO system with Rayleigh fading channels.
+The ergodic capacity of RIS MIMO networks over Rayleigh-Rician channels was considered in [13].
+However, the practical implementation of passive RIS-aided communication systems may face several
+challenges. For instance, the transmitted signal propagates through the RIS experiences a double-fading
+attenuation, e.g, source-RIS and RIS-destination links. This issue has been tackled in the literature by
+increasing the number of passive RIS elements [14]. However, this solution leads to an increase in
+
+3
+the size of the RIS module, which is impractical in some scenarios. To tackle this issue the authors
+in [15] proposed RIS with active elements. The main idea of active RIS is to adjust the phase shifts
+and also amplify the reflected signal attenuated from the first link with extra power consumption.
+Theoretical comparison between the active RIS-assisted system and the passive RIS-aided system has
+been presented in [16]. The results in [16] show that the active RIS has better performance than
+passive RIS. The use of active RIS elements to overcome the double-fading problem has been also
+investigated in [17], where the results illustrated that using active elements results in a severe reduction
+in the physical size of RIS to achieve a certain performance. To reduce the power consumption of
+active RIS, a sub-connected architecture has been proposed in [18]. The energy efficiency in an active
+RIS-aided MU-MISO down-link system has been investigated in [19].
+Although fixed embedded batteries can be used to power the RIS, these batteries cannot be relied on
+for long time and uninterrupted operations. In addition, wired charging might not be possible to use if
+the RIS is deployed in inaccessible places. Therefore, equipping RIS elements with energy harvesting
+(EH) modules can solve these issues. Accordingly, a self-sustainable RIS approach was proposed and
+studied in the resent researches on RIS. In this regard, in [20] time switching (TS) and power splitting
+(PS) EH protocols for the RIS to harvest sufficient amount of energy from an access point have
+been proposed and investigated. The work in [21] considered a self-sustainable RIS-aided MU-MISO
+communication systems, in which the RIS collected energy from the radio frequency (RF) transmitter
+using the PS protocol. In [22], a novel transmission policy for a communication network assisted by
+self-sustainable RIS has been proposed, where the RIS harvests energy from an energy transmitter to
+support its operation. In [23], self-sustainable RIS with the PS protocol to assist broadcasting network
+was studied. In [24], self-sustainable RIS-aided communication between a gateway and a device was
+studied, in which the RIS harvested energy prior communication.
+Moreover, due to the broadcast nature of wireless channels, confidential messages are vulnerable
+to eavesdropping attacks. For the provision of secure transmission, physical layer security (PHYSec)
+has been proposed from the information theory perspective [25], [26]. PHYSec exploits the nature of
+wireless channels to enhance the system security [25], [26]. PHYSec of RIS systems has also been
+studied in the literature. In [27], the secrecy throughput maximization problem has been formulated
+and solved to enhance the secrecy performance of the RIS-assisted MIMO systems. In [28], a novel
+active RIS design to enhance the security of wireless transmission was proposed. PHYSec of RIS-
+
+4
+aided wireless networks has been considered in [29] to achieve secure transmission between a source
+and a legitimate user in the presence of a malicious eavesdropper. In [30], RIS has been used to
+perform secure transmission from a multiple antennas transmitter to a multiple antennas legitimate
+receiver. Further work in [31] considered the secrecy transmission in a RIS-aided multiple antennas
+communication, where the secrecy rate was improved by optimizing the RIS location. In [32], an active
+RIS-aided multiple antennas PHYSec transmission scheme was considered, where the active RIS was
+designed to amplify the signal actively.
+Accordingly, this paper investigates the impact of phase shift error on the secrecy performance of
+up-link RIS-aided MU-MISO systems in the presence of multiple eavesdroppers. The BS receives
+the users messages only through the RIS, while eavesdroppers can receive the signals from both the
+direct and reflected links. Under Rician fading channels and phase shift errors, the ergodic secrecy
+rate is analyzed for three RIS configurations, namely, 1) passive RIS, 2) active RIS, and 3) EH RIS.
+Based on the derived rate expressions, the phase shifts at the RIS are optimized to enhance the system
+performance. Then, the best RIS configuration selection is considered based on the target secrecy rate
+and amount of power available at the users. For clarity we list the main contributions of this work as
+follows:
+1) We investigate the impact of RIS phase shift error on the secrecy performance of up-link MU-
+MIMO systems in the presence of multiple passive eavesdroppers.
+2) New closed-form explicit analytical expressions for the ergodic secrecy rate are derived for
+the RIS-assisted MU-MIMO systems, when the RIS is passive, active and EH node under Rician
+fading channels. This channel model is more general but also very challenging to be considered
+mathematically. The derived secrecy rate expressions are simple, explicit and in closed form, and
+provide several important practical design insights.
+3) Based on the derived expressions, a genetic algorithm (GA)-based approach is used to obtain the
+optimal phase shifts. Also, a simple suboptimal technique is proposed to enhance the secrecy rate for
+a legitimate user.
+4) Given a target secrecy rate, we calculate the required user power, and we present steps to select
+best RIS configuration which depend mainly on the available power at the users.
+5) Finally, Monte-Carlo simulations are performed to validate the analytical expressions. Then, the
+impact of several system parameters on the secrecy performance are investigated.
+
+5
+The results in this work show that active RIS is an efficient scheme to achieve secure communication
+in the presence of phase shift errors at the RIS, especially when there is no sufficient amount of power
+at the users.
+Next, Section II presents the RIS-aided uplink MU-MISO system model. In Section III, we derive
+the ergodic secrecy rate of the passive RIS model. Section IV presents the ergodic secrecy rate of the
+active RIS scheme. Section V derives the ergodic secrecy rate of the EH RIS scheme. Section VII
+depicts our numerical results. Our main conclusions are summarized in Section VIII.
+II. SYSTEM MODEL
+Consider a typical up-link RIS-aided MU-MISO communication system consisting of a multiple
+antennas BS, an RIS and K single-antenna users in the presence of J single antenna passive eaves-
+droppers. The BS is equipped with N antennas, and the RIS is equipped with M reflecting elements,
+as shown in Fig. 1.
+UE 1
+UE k
+UE K
+.
+.
+.
+.
+Eave J
+.
+.
+Eave 1
+BS
+RIS
+Figure 1: An RIS-aided uplink MU-MISO system with N BS antennas, M RIS elements, K users and
+J eavesdroppers.
+The BS and RIS are connected to control and adjust the phase shifts of the the RIS elements. It is
+assumed that the eavesdroppers can hear the signals from the direct and reflected links, and trying to
+eavesdrop a specific confidential message in the system. On the other side, the direct links between the
+users and BS are assumed to be blocked, which justifies the use of the RIS. It is known that, the RIS is
+most likely to be installed on the buildings, and thus it can create channels dominated by line-of-sight
+(LoS) path along with scatters. Accordingly, a Rician fading model is considered for the RIS channels.
+The channel matrix between the RIS and the BS is denoted by G ∈ CN×M , and the channel vector
+
+6
+between user k and the RIS is presented by hr,k ∈ CM×1. The mathematical expressions of the channel
+matrix G and the channel vector hr,k can be expressed, respectively, as
+G =
+��
+ρb
+ρb + 1
+¯G +
+�
+1
+ρb + 1
+˜G
+�
+,
+hr,k =
+��
+ρk
+ρk + 1
+¯hr,k +
+�
+1
+ρk + 1
+˜hr,k
+�
+(1)
+where ρb and ρk are the Rician factors, ¯G and ¯hr,k are the LoS components and ˜G and ˜hr,k are the
+NLoS components, in which
+¯G = aN (φa
+r, φe
+r) aH
+M (φa
+t , φe
+t) ,
+¯hr,k = aM (φa
+kr, φe
+kr)
+(2)
+where φa
+kr, φe
+kr denote the azimuth and elevation angles of arrival (AoA) from user k to the RIS ,
+respectively, φa
+t , φe
+t are the azimuth and elevation angles of departure (AoD) at the BS from the RIS,
+respectively, φa
+r, φe
+r are the azimuth and elevation AoA from the RIS to the BS, respectively. The kth
+element of the vector aX can be written as [aX (φ1, φ2)]k = ej2π d
+λ (xk sin φ1 sin φ2+yk cos φ2), where λ is the
+wavelength, d is the elements/antennas spacing, and xk = (k − 1) mod√x, yk = k−1
+√
+X . On the other
+hand, the channel vector between the RIS and eavesdropper j is presented by hej,r ∈ C1×M , and
+the channel from user k to eavesdropper j is hej,k ∈ C1×1. The direct channel fading is assumed to
+be Rayleigh fading due to extensive scatterers, while for the RIS-related channels, is assumed to be
+Rician fading. Thus the expression of hej,ris given by
+hej,r =
+��
+ρej,r
+ρej,r + 1
+¯hej,r +
+�
+1
+ρej,r + 1
+˜hej,r
+�
+(3)
+where ρej,r is the Rician factor, ¯hej,r and ˜hej,r are the LoS of NLoS components, respectively.
+The channel state information (CSI) of the eavesdroppers is assumed to be unknown at the BS/RIS
+(only statistical information can be known), and the eavesdroppers are non-colluding. Therefore, the
+ergodic secrecy rate can be calculated by [33]
+ˆRs =
+�
+ˆRbk − ˆRej,k
+�+
+(4)
+where [l]+= max (0, l), ˆRbk = E {Rbk}, Rbk is the up-link rate of user k, and ˆRej,k = max E
+�
+Rej,k
+�
+,
+Rej,k is the rate at eavesdropper j.
+
+7
+In the following sections, we consider the secrecy performance of the three RIS configurations.
+III. PASSIVE RIS
+As we have mentioned earlier, passive RIS reflects the users messages constructively to the BS with
+passive elements. Thus, the received signal at the BS can be expressed as
+yb =
+K
+�
+k=1
+�
+pk Luk,bG˜Θhr,kxk + nb
+(5)
+where Luk,b = d−αr
+uk,rd−αb
+r,b
+is the large scale fading, duk,r is the distance between user k and RIS, dr,b
+is the distance between RIS and the BS, αr and αb are the path-loss exponents, nb is the additive
+wight Gaussian noise (AWGN) at the BS, nb ∼ CN (0, σ2
+bI), ˜Θ = ¯ΘΘ where Θ = diag (θ), and
+θ = [θ1, ......, θM]Tis the RIS reflection coefficients with θm = ejϕm, where ϕm∈[0, 2π) is the phase
+shift of element m. However, in practical systems, phase shift errors can exist due to imperfect channel
+knowledge and finite precision in phase adjustment. Thus, we define ¯Θ = [ej ¯
+ϕ1, ...., ej ¯
+ϕM] as the phase-
+shift errors at the RIS. The phase-error is modeled according to Von-Mises (VM) distribution with
+zero-mean and a characteristic function (CF) E [ej ¯
+ϕm] = I1(κ)
+I0(κ) = ρ (κ), where κ is the concentration
+parameter and Ii is the modified Bessel function of the first kind and order i. By applying the receive
+beamforming vector wk at the BS, the received signal of user k is
+yb,k =
+�
+pk Luk,bwkG¯ΘΘhr,kxk+
+K
+�
+i=1
+i̸=k
+�
+pi Lui,bwkG¯ΘΘhr,ixi + wknb.
+(6)
+On the other hand, the received signal at eavesdropper j to detect user k signal is
+yej,k = √pkxk
+��
+d−αe
+ej,k hej,k +
+�
+Luk,ejhej,r ¯ΘΘhr,k
+�
++
+K
+�
+i=1
+i̸=k
+√pixi
+
+
+
+�
+d−αe
+ej,i hej,i+
+K
+�
+i=1
+i̸=k
+�
+Lui,ejhej,r ¯ΘΘhr,i
+
+
+ + nej
+(7)
+where d−αe
+ej,k is the distance between user k and eavesdropper j, αe is the path-loss exponent, Luk,B =
+d−αr
+uk,rd−e
+ej,r and d−e
+ej,r denotes the distance between the RIS and eavesdropper j.
+
+8
+To calculate the ergodic secrecy rate, the ergodic up-link rate for user k and ergodic rate at the
+eavesdropper j should be derived, which will be considered in the following sub-sections.
+A. Ergodic Up-link rate of user k
+To calculate the ergodic user rate, maximum ratio combining (MRC) is adopted at the BS. The
+beamforming matrix is given by W = (GΘH)H, and thus wk = hH
+r,kΘHGH. The signal to interference
+plus noise ratio (SINR) at the BS to decode user k signal can be written as
+γbk =
+pk Luk,b
+��hH
+r,kΘHGHGΘ¯Θhr,k
+��2
+K�
+i=1
+i̸=k
+pi Lui,b
+��hH
+r,kΘHGHGΘ¯Θhr,i
+��2 +
+��hH
+r,kΘHGH��2 σ2
+b
+.
+(8)
+Lemma 1. The ergodic up-link rate of user k in passive RIS-aided MU-MISO systems under Rician
+fading channels and with phase shift error can be calculated by
+E {Rbk} ≈ log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk
+K�
+i=1
+i̸=k
+pi Lui,bςi + υkσ2
+b
+
+
+
+
+
+
+
+(9)
+where
+ξk = E
+���hH
+r,kΘHGHGΘ¯Θhr,k
+��2�
+=
+1
+(ρb + 1)2 (ρk + 1)2
+�
+a1N2 + a2NM2 + a3NM + a4N
+�
+a1 =
+�
+ρ (κ)2 (ρk + ρb + 1)2 +
+�
+1 − ρ (κ)2�
+ρkρ2
+b + ρ2
+b
+�
+M2
++
+��
+(2ρk + 3ρb + 2 − ρkρb) ρ (κ)2 + (1 + ρk) ρb
+�
+ρbρk |fk|2 + (ρk + ρb + 2)2
+− ρ (κ)2 (ρk + ρb + 1)2 − 2ρ (κ)2 ρkρb − 2
+�
+M
++ρ (κ)2 ρ2
+bρ2
+k |fk|4 + 2
+��
+1 − ρ (κ)2�
+(ρk + ρb) + 2
+�
+ρbρk |fk|2,
+a2 =
+�
+−ρ (κ)2 ρkρb (1 + ρk) M2�
++ (ρk + ρb + 1) ρbρr + (ρk + ρb + 1)2 − (ρk + 1) ρ2
+b,
+a3 =
+��
+(ρk + 1) ρ (κ)2�
++ (ρk + 1)
+�
+ρbρk |fk|2 − 2ρbρkρ (κ)2 + 2ρbρk + 2ρk + 2ρb − 1,
+a4 = 2ρbρr |fk|2 �
+1 + ρ (κ)2�
+and
+
+9
+ςi = E
+���hH
+r,kΘHGHGΘ¯Θhr,i
+��2�
+=
+1
+(ρb + 1)2 (ρk + 1) (ρi + 1)
+�
+b1N2 + b2NM2 + b3NM
+�
+b1 =
+�
+ρi + 1 − ρ (κ)2 ρi
+�
+M2ρ2
+b
++M
+��
+ρi + 1 − ρ (κ)2 ρi
+�
+ρ2
+bρk |fk|2 + ρ (κ)2 ρ2
+bρi |fi|2 + (ρk + 2ρb + 1)
+�
+ρi + 1 − ρ (κ)2 ρi
+�
++ ρ (κ)2 ρi
+�
++
+�
+2ρb |fi|2 + ρk
+��¯hH
+k ¯hi
+��2 + 2ρbρkRe
+�
+f ∗
+kfi¯hH
+i ¯hk
+��
+ρ (κ)2 ρi
++
+�
+ρ (κ)2 ρbρi |fi|2 + 2ρi
+�
+1 − ρ (κ)2�
++ 2
+�
+ρbρk |fk|2
+b2 =
+�
+(ρb + 1) ρk + (ρb + 1)2 − ρ2
+b
+�
+(ρi + 1) − (ρb + 1) ρbρiρ (κ)2 − 1
+b3 = (ρi + 1) ρbρk |fk|2 + (ρk + 1) ρ (κ)2 ρbρi |fi|2
+and
+υk = E
+���hH
+r,kΘHGH��2�
+=
+Luk,b
+(ρb + 1) (ρk + 1)
+�
+ρbρk |fk|2 + (ρb + ρk + 1) M
+�
+Proof: The proof is provided in Appendix A.
+B. Ergodic Rate at Eavesdropper j
+The SINR at eavesdropper j to decode user k signal can be expressed as
+γej,k =
+pk
+���d
+− αr
+2
+uk,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,k + d
+− αe
+2
+ej,k hej,k
+���
+2
+K�
+i=1
+i̸=k
+pi
+���d
+− αr
+2
+ui,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,i + d
+− αe
+2
+ej,i hej,i
+���
+2
++ σ2ej
+.
+(10)
+Lemma 2. The ergodic rate at eavesdropper j in up-link passive RIS-aided MU-MISO systems under
+Rician fading channels and with phase shift error can be calculated by
+E
+�
+Rej,k
+�
+= log2
+
+
+
+
+
+
+
+1 +
+pk xk
+K�
+i=1
+i̸=k
+pi yi + σ2
+ej
+
+
+
+
+
+
+
+(11)
+where
+xk =
+�
+d−αr
+uk,rd−αe
+ej,r
+�
+ρej
+ρej +1
+ρk
+ρk+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρk+1M +
+ρk
+ρk+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρk+1M
+�
++ d−αe
+ej,r
+�
+,
+and
+
+10
+yi = d−αr
+ui,r d−αe
+ej,r
+�
+ρej
+ρej +1
+ρi
+ρi+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρi+1M +
+ρi
+ρi+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρi+1M + d−αe
+ej,i
+�
+.
+Proof: The proof is provided in Appendix B.
+Finally, the ergodic secrecy rate in passive RIS scheme is presented in the next theorem.
+Theorem 1. The ergodic secrecy rate in passive RIS-aided MU-MISO systems under Rician fading
+channels and with phase shift error can be calculated by
+ˆRs =
+
+
+log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk
+K�
+i=1
+i̸=k
+pi Lui,bςi + υkσ2
+b
+
+
+
+
+
+
+
+− log2
+
+
+
+
+
+
+
+1 +
+pk xk
+K�
+i=1
+i̸=k
+pi yi + σ2ej
+
+
+
+
+
+
+
+
+
+.+
+(12)
+IV. ACTIVE RIS
+As we mentioned earlier, active RIS can adjust the phase shifts and also amplify the reflected signal
+to compensate the attenuation from the first link with extra power consumption. The signal reflected
+by the active IRS can be written as
+yr = ˜Θ
+K
+�
+k=1
+�
+pk d−αr
+uk,rhr,ixi + ˜Θnr
+(13)
+where nr is the noise at RIS elements nr ∼ CN (0, σ2
+rI). In this case ˜Θ = ¯ΘΘ where Θ = diag (θ),
+and θ = [θ1, ......, θM]T with θm = ̺mejϕm, ̺m > 1 and ϕm∈[0, 2π) represents the amplification factor
+and phase shift coefficient, respectively, at element m. For simplicity, we assume that ̺m = ̺ and then
+define Θ = ̺diag {ejϕ1, ...., ejϕM}. The active RIS amplification power can be expressed as
+Pr =
+� K
+�
+k=1
+pk
+dαr
+uk,r
+E
+����˜Θhr,i
+���
+2�
++ E
+����˜Θnr
+���
+2��
+=
+� K
+�
+k=1
+pk
+dαr
+uk,r
+M̺2 + M̺2σ2
+r
+�
+(14)
+where E
+���˜Θnr
+���
+2
+= M̺2σ2
+r, and E
+���˜Θhr,i
+���
+2
+=
+̺2
+ρi+1
+�
+ρiE
+�¯hH
+r,i¯hr,i
+�
++ E
+�
+˜hH
+r,i˜hr,i
+��
+=
+̺2
+ρi+1 (ρiM + M) =
+M̺2. Thus, the amplification factor for each element on the active RIS is given by
+̺ =
+�
+�
+�
+�
+�
+Pr
+M
+� K
+�
+k=1
+pk
+dαr
+uk,r + σ2
+r
+�.
+(15)
+
+11
+By applying the receive beamforming vector wk at the BS, the received signal of user k is
+yb,k =
+�
+pk Luk,bwkG¯ΘΘhr,kxk+
+K
+�
+i=1
+i̸=k
+�
+pi Lui,bwkG¯ΘΘhr,ixi +
+�
+d−αr
+r,b wkG¯ΘΘnr + wknb.
+(16)
+On the other hand, the received signal at eavesdropper j to detect user k signal is
+yej,k = √pkxk
+��
+d−αe
+ej,k hej,k +
+�
+Luk,ejhej,r ¯ΘΘhr,k
+�
++
+K
+�
+i=1
+i̸=k
+√pixi
+
+
+
+�
+d−αe
+ej,i hej,i+
+K
+�
+i=1
+i̸=k
+�
+Lui,ejhej,r ¯ΘΘhr,i
+
+
+ +
+�
+d−αe
+ej,r hej,r ¯ΘΘnr + nej.
+(17)
+A. Ergodic Up-link rate of user k
+Applying MRC beamforming at the BS, the SINRs at the BS to decode user k signal can be expressed
+as
+γbk =
+pk Luk,b
+��hH
+r,kΘHGHGΘ¯Θhr,k
+��2
+K�
+i=1
+i̸=k
+pi Lui,b
+��hH
+r,kΘHGHGΘ¯Θhr,i
+��2 + d−αr
+r,b
+��hH
+r,kΘHGHG¯ΘΘ
+��2 σ2
+r +
+��hH
+r,kΘHGH��2 σ2
+b
+.
+(18)
+Lemma 3. The ergodic up-link rate of user k in active RIS-aided MU-MISO systems under Rician
+fading channels and with phase shift error can be calculated by
+E {Rbk} ≈ log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk̺4
+K�
+i=1
+i̸=k
+pi Lui,bςi̺4 + ̺4d−αr
+r,b σ2
+rνk + ̺2υkσ2
+b
+
+
+
+
+
+
+
+(19)
+where
+νk = E
+��hH
+r,kΘHGHG¯ΘΘ
+��2 =
+1
+(ρb + 1)
+�
+(ρk + 1)
+(X1 + X2)
+(20)
+and X1 = E
+�
+|∆1,1|2�
++ E
+�
+|∆1,2|2�
++ E
+�
+|∆1,3|2�
++ E
+�
+|∆1,4|2�
++ E
+�
+∆1,1∆∗
+1,4
+�
+
+12
+E
+�
+|∆1,1|2�
+= ρ2
+bρk
+���
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+���2 × M,
+E
+�
+|∆1,2|2�
+= ρbρk
+��aH
+M (φa
+kr, φe
+kr) ΘHaM (φa
+r, φe
+r)
+��2 NM,
+E
+�
+|∆1,3|2�
+= ρbρkMN
+�
+ρ (κ)2 M +
+�
+1 − ρ (κ)2�
+M
+�
+, E
+�
+|∆1,4|2�
+= ρk
+�
+N2M + NM2�
+,
+E
+�
+∆1,1∆∗
+1,4
+�
+= ρbρk
+�
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+�
+× (aM (φa
+r, φe
+r) Θ) ρkaH
+M (φa
+kr, φe
+kr) ΘHNΘ,
+and X2 = E
+�
+|∆2,1|2�
++ E
+�
+|∆2,2|2�
++ E
+�
+|∆2,3|2�
++ E
+�
+|∆2,4|2�
++ E
+�
+∆2,1∆∗
+2,4
+�
+E
+�
+|∆2,1|2�
+= ρ2
+b
+��ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b) aM (φa
+r, φe
+r) Θ
+��2
+F ,
+E
+�
+|∆2,2|2�
+= ρb
+��ΘHaM (φa
+r, φe
+r)
+��2 NM,
+E
+�
+|∆2,3|2�
+= ρbMN
+���aH
+M (φa
+r, φe
+r) Θ¯Θ
+��2�
+= ρbM2N,
+E
+�
+|∆2,4|2�
+= ρ2
+k
+�
+N2M + NM2�
+,
+E
+�
+∆2,1∆∗
+2,4
+�
+= ρk
+�
+ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+�
+(aM (φa
+r, φe
+r) Θ) ρkΘNΘH.
+Proof: The proof is provided in Appendix C.
+B. Ergodic Rate at Eavesdropper j
+The SINR at eavesdropper j to decode user k signal in this scenario can be written as
+
+13
+γej,k =
+pk
+���d
+− αr
+2
+uk,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,k + d
+− αe
+2
+ej,k hej,k
+���
+2
+K�
+i=1
+i̸=k
+pi
+���d
+− αr
+2
+ui,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,i + d
+− αe
+2
+ej,i hej,i
+���
+2
++ d−αe
+ej,r
+��hej,r ¯ΘΘ
+��2 σ2r + σ2ej
+.
+(21)
+Lemma 4. The ergodic rate at eavesdropper j in up-link active RIS-aided MU-MISO systems under
+Rician fading channels and with phase shift error can be calculated by
+E
+�
+Rej,k
+�
+= log2
+
+
+
+
+
+
+
+1 +
+pk xj
+K�
+i=1
+i̸=k
+piyi + zjσ2
+r + σ2
+ej
+
+
+
+
+
+
+
+(22)
+where
+xj =
+�
+d−αr
+uk,rd−αe
+ej,r ̺2 �
+ρej
+ρej +1
+ρk
+ρk+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρk+1M +
+ρk
+ρk+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρk+1M
+�
++ d−αe
+ej,r
+�
+,
+yk = d−αr
+ui,r d−αe
+ej,r ̺2 �
+ρej
+ρej +1
+ρi
+ρi+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρi+1M +
+ρi
+ρi+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρi+1M + d−αe
+ej,i
+�
+which have been derived in Appemdix B, and
+zj = d−αe
+ej,r E
+���hej,r ˜Θ
+���
+2
+= d−αe
+ej,r
+̺2
+ρej,r+1
+�
+ρej,rE
+�
+¯hH
+ej,r¯hej,r
+�
++ E
+�
+˜hH
+ej,r˜hej,r
+��
+= d−αe
+ej,r
+̺2
+ρej,r+1
+�
+ρej,rM + M
+�
+= d−αe
+ej,r M̺2.
+The ergodic secrecy rate in active RIS scheme is presented in the following Theorem.
+Theorem 2. The ergodic secrecy rate in active RIS-aided MU-MISO systems under Rician fading
+channels and with phase shift error can be calculated by
+ˆRs =
+
+
+log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk̺4
+K�
+i=1
+i̸=k
+pi Lui,bςi̺4 + ̺4d−αr
+r,b σ2rνk + ̺2υkσ2
+b
+
+
+
+
+
+
+
+− log2
+
+
+
+
+
+
+
+1 +
+pk xj
+K�
+i=1
+i̸=k
+piyi + zjσ2r + σ2ej
+
+
+
+
+
+
+
+
+
++
+.
+(23)
+V. EH RIS
+Following the recent works in [20], [21], [22], [23], [24], in this section, the RIS is an energy
+constrained node and it can harvest RF energy to support its operation. Thus, in this scenario the
+
+14
+whole operation time block, T, is split into two time periods, the energy transfer (ET) slot and the
+information transfer (IT) slot. During the ET slot, the BS transmits energy signals to the RIS to support
+its operation. During the IT slot, the users deliver their messages to the BS through the RIS. We denote
+τT as the time duration for the ET, and (1 − τ) T as the time duration for IT. The received signals at
+the RIS in the first sub-slot is expressed as
+yr =
+�
+PbGpWpxp + nr
+(24)
+where Pb is the BS power, Gp =
+��
+ρp
+ρp+1 ¯Gp +
+�
+1
+ρp+1 ˜Gp
+�
+is the BS-RIS channel in the ET slot, Wp
+is the precoding matrix and xp is the energy signals vector. Using the maximum ratio transmission
+(MRT) scheme, the harvested power at the RIS can be expressed as Pr = ηeff τPb∥Gp∥2
+F
+1−τ
+, which can
+be written as Pr =
+ηeff τPbTr(GbGH
+b )
+1−τ
+where ηeff is the efficiency of EH. Since GbGH
+b
+has Wishart
+distribution, the average harvested power can be written as
+Pr = ηeffτPbE
+�
+Tr
+�
+GbGH
+b
+��
+1 − τ
+= ηeffτPbNM
+1 − τ
+.
+(25)
+By substituting (25) into (15), the amplification factor for each element on the RIS in this case is given
+by
+ˆ̺ =
+�
+�
+�
+�
+�
+ηeffτPbNM
+M (1 − τ)
+� K�
+k=1
+pk
+dαr
+uk,r + σ2
+r
+�.
+(26)
+A. Ergodic Up-link rate of user k
+Applying MRC beamforming at the BS, the SINR at the BS to decode user k signal can be expressed
+as
+γbk =
+pk Luk,b
+��hH
+r,kΘHGHGΘ¯Θhr,k
+��2
+K�
+i=1
+i̸=k
+pi Lui,b
+��hH
+r,kΘHGHGΘ¯Θhr,i
+��2 + d−αr
+r,b
+��hH
+r,kΘHGHG¯ΘΘ
+��2 σ2r +
+��hH
+r,kΘHGH��2 σ2
+b
+.
+(27)
+Lemma 5. The ergodic up-link rate of user k in EH RIS-aided MU-MISO systems under Rician fading
+channels and with phase shift error can be calculated by
+
+15
+E {Rbk} ≈ (1 − τ) log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξkˆ̺4
+K�
+i=1
+i̸=k
+pi Lui,bςiˆ̺4 + ˆ̺4d−αr
+r,b σ2rνk + ˆ̺2υkσ2
+b
+
+
+
+
+
+
+
+.
+(28)
+Proof: This expression can be obtained by following same derivation in Appendix C.
+B. Ergodic Rate at Eavesdropper j
+The SINR at eavesdropper j to decode user k signal is given by
+γej,k =
+pk
+���d
+− αr
+2
+uk,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,k + d
+− αe
+2
+ej,k hej,k
+���
+2
+K�
+i=1
+i̸=k
+pi
+���d
+− αr
+2
+ui,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,i + d
+− αe
+2
+ej,i hej,i
+���
+2
++ d−αe
+ej,r
+��hej,r ¯ΘΘ
+��2 σ2
+r + σ2
+ej
+.
+(29)
+Lemma 6. The ergodic rate at eavesdropper j in up-link EH RIS-aided MU-MISO systems under
+Rician fading channels and with phase shift error can be calculated by
+E
+�
+Rej,k
+�
+= (1 − τ) log2
+
+
+
+
+
+
+
+1 +
+pk ˆxj
+K�
+i=1
+i̸=k
+piˆyi + ˆzjσ2r + σ2ej
+
+
+
+
+
+
+
+(30)
+where
+ˆxj =
+�
+d−αr
+uk,rd−αe
+ej,r ˆ̺2 �
+ρej
+ρej +1
+ρk
+ρk+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρk+1M +
+ρk
+ρk+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρk+1M
+�
++ d−αe
+ej,r
+�
+,
+ˆyi = d−αr
+ui,r d−αe
+ej,r ˆ̺2 �
+ρej
+ρej +1
+ρi
+ρi+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρi+1M +
+ρi
+ρi+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρi+1M + d−αe
+ej,i
+�
+ˆzj = d−αe
+ej,r M ˆ̺2,
+which have been derived in the previous section.
+Finally, the ergodic secrecy rate in EH RIS scheme is presented in the next Theorem.
+Theorem 3. The ergodic secrecy rate of user k in EH active RIS-aided MU-MISO systems under
+Rician fading channels and with phase shift error can be calculated by
+
+16
+ˆRs
+=
+
+
+(1 − τ) log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk ˆ̺4
+K�
+i=1
+i̸=k
+pi Lui,bςiˆ̺4 + ˆ̺4d−αr
+r,b σ2
+rνk + ˆ̺2υkσ2
+b
+
+
+
+
+
+
+
+− (1 − τ) log2
+
+
+
+
+
+
+
+1 +
+pk ˆxj
+K�
+i=1
+i̸=k
+piˆyi + ˆzjσ2r + σ2ej
+
+
+
+
+
+
+
+
+
++
+.
+(31)
+VI. SYSTEM DESIGN
+In this section, based on the derived analytical expressions, we first design the phase shifts of the RIS
+configurations considered in this work. Then, the best RIS configuration selection scheme is presented.
+A. Phase Shift Optimization
+The secrecy rate expressions presented in Theorems, 1, 2 and 3, show that the secrecy performance
+relies on the phase shifts of the RIS elements. In this work, it is assumed that the CSI of the
+eavesdroppers is unknown at the BS/RIS (only channel distribution known). Therefore, to enhance
+the system performance, the RIS phase shifts can be optimized by maximizing the achievable ergodic
+sum rate. Since the phase shift at each unit of the RIS lies in the range of [0; 2π), the phase shift
+optimization problem can be formulated as
+max
+Θ
+K�
+i=1
+ˆRbi
+s.t
+θm ∈ [0, 2π) ,
+∨m.
+(32)
+Due to the complicated formula of the ergodic sum rate, it is difficult to optimize (32) based on
+the conventional techniques. However, GA-based methods can be employed to solve this optimization
+problem. Due to the page limitation, we refer readers to [6] for more details about the GA methods.
+
+17
+As an efficient suboptimal solution, the RIS phase shifts can be aligned to user k, who transmits
+the confidential message. This presents a simple sub-optimal solution for enhancing the secrecy rate
+[6]. Accordingly, the phase shifts should be
+θm = −2π d
+λ (xmtk + ymlk) , tk = sin φa
+kr sin φe
+kr − sin φa
+t sin φe
+t, lk = cos φe
+kr − cos φe
+t.
+(33)
+B. RIS Configuration Selection Scheme
+Based on the required secrecy rate (rs) and amount of the power available at user k, and the RIS,
+we can decide which system configuration, i.e., passive RIS, active RIS or EH RIS, should be selected.
+A) If user k has sufficient amount of power to achieve the target secrecy rate, in this case passive
+RIS can be implemented. Based on the secrecy rate expression provided in Theorem 1, the required
+user k power, pk, to achieve the target secrecy rate, rs, can be obtained by solving
+rs = log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk
+K�
+i=1
+i̸=k
+pi Lui,bςi + υkσ2
+b
+
+
+
+
+
+
+
+− log2
+
+
+
+
+
+
+
+1 +
+pk xk
+K�
+i=1
+i̸=k
+pi yi + σ2
+ej
+
+
+
+
+
+
+
+(34)
+which can be found as
+pk = p1 − p2
+p3 − p4
+(35)
+where p1 =
+K
+�
+i=1
+i̸=k
+pi Lui,bςi+υkσ2
+b
+K
+�
+i=1
+i̸=k
+pi Lui,bςi+υkσ2
+b
+, p2 =
+2rs
+K
+�
+i=1
+i̸=k
+pi yi+2rsσ2
+ej
+K
+�
+i=1
+i̸=k
+pi yi+σ2ej
+, p3 =
+2rs xk
+K
+�
+i=1
+i̸=k
+pi yi+σ2ej
+and p4 =
+Luk,bξk
+K
+�
+i=1
+i̸=k
+pi Lui,bςi+υkσ2
+b
+.
+B) If user k has limited amount of power, e.g., the user power, pk, is less than the power required
+in (35). In this case active RIS can be implemented to provide the target secrecy rate. Based on the
+secrecy rate expression provided in Theorem 2, the required RIS power, ̺ or Pr ,to achieve the target
+secrecy rate, rs, can be obtained by solving
+
+18
+rs = log2
+
+
+
+
+
+
+
+1 +
+pk Luk,bξk̺2
+K�
+i=1
+i̸=k
+pi Lui,bςi̺2 + ̺2d−αr
+r,b σ2rνk + υkσ2
+b
+
+
+
+
+
+
+
+− log2
+
+
+
+
+
+
+
+1 +
+pk̺2x1 + pkx2
+K�
+i=1
+i̸=k
+pi̺2y1i+
+K
+�
+i=1
+i̸=k
+piy2i + z1̺2σ2r + σ2ej
+
+
+
+
+
+
+
+(36)
+where x1 = d−αr
+uk,rd−αe
+ej,r
+�
+ρej
+ρej +1
+ρk
+ρk+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρk+1M +
+ρk
+ρk+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρk+1M
+�
+, x2 =
+d−αe
+ej,r ,
+y1i = d−αr
+ui,r d−αe
+ej,r
+�
+ρej
+ρej +1
+ρi
+ρi+1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej +1
+1
+ρi+1M +
+ρi
+ρi+1
+1
+ρej +1M +
+1
+ρej +1
+1
+ρi+1M
+�
+, y2i = d−αe
+ej,i ,
+and z1 = d−αe
+ej,r M. After some simplifications, the last equation can be expressed as
+̺4 (q1 − q3) + ̺2 (q2 − q4 − q5 + q7) + (q8 − q6) = 0
+(37)
+where
+q1 =
+K�
+i=1
+i̸=k
+pi Lui,bςi
+K
+�
+i=1
+i̸=k
+piy1i+d−αr
+r,b σ2
+rνk
+K�
+i=1
+i̸=k
+piy1i+
+K
+�
+i=1
+i̸=k
+pi Lui,bςiz1σ2
+r+d−αr
+r,b σ2
+rνkz1σ2
+r+
+K�
+i=1
+i̸=k
+pi Lui,bςipkx1+
+d−αr
+r,b σ2
+rνkpkx1,
+q2 =
+
+υkσ2
+b
+K
+�
+i=1
+i̸=k
+piy1i + υkσ2
+bz1σ2
+r + υkσ2
+bpkx1
+
+ ,
+q3 =
+K�
+i=1
+i̸=k
+piy1ipk Luk,bξk+z1σ2
+rpk Luk,bξk+
+K
+�
+i=1
+i̸=k
+piy1i
+K�
+i=1
+i̸=k
+pi Lui,bςi+z1σ2
+r
+K
+�
+i=1
+i̸=k
+pi Lui,bςi+
+K
+�
+i=1
+i̸=k
+piy1id−αr
+r,b σ2
+rνk+
+z1σ2
+rd−αr
+r,b σ2
+rνk,
+q4 =
+K�
+i=1
+i̸=k
+piy2ipk Luk,bξk+σ2
+ejpk Luk,bξk+
+K
+�
+i=1
+i̸=k
+piy2i
+K
+�
+i=1
+i̸=k
+pi Lui,bςi+σ2
+ej
+K�
+i=1
+i̸=k
+pi Lui,bςi+
+K�
+i=1
+i̸=k
+piy2id−αr
+r,b σ2
+rνk+
+σ2
+ejd−αr
+r,b σ2
+rνk,
+q5 =
+
+
+K�
+i=1
+i̸=k
+piy1iυkσ2
+b + z1σ2
+rυkσ2
+b
+
+, q6 =
+K�
+i=1
+i̸=k
+piy2iυkσ2
+b + σ2
+ejυkσ2
+b,
+q7 =
+K�
+i=1
+i̸=k
+pi Lui,bςi̺22rspkx2+̺2d−αr
+r,b σ2
+rνk2rspkx2+
+K
+�
+i=1
+i̸=k
+pi Lui,bςi2rsσ2
+ej+d−αr
+r,b σ2
+rνk2rsσ2
+ej+
+K
+�
+i=1
+i̸=k
+pi Lui,bςi2rs
+K�
+i=1
+i̸=k
+piy2i + d−αr
+r,b σ2
+rνk2rs
+K�
+i=1
+i̸=k
+piy2i,
+
+19
+q8 = υkσ2
+b2rspkx2 + υkσ2
+b2rsσ2
+ej + υkσ2
+b2rs
+K
+�
+i=1
+i̸=k
+piy2i.
+Thus, from (15), the RIS power should be higher than or equal to
+Pr = M
+
+− (q2 − q4 − q5 + q7) ±
+�
+(q2 − q4 − q5 + q7)2 − 4 (q1 − q3) (q8 − q6)
+2 (q1 − q3)
+
+
+� K
+�
+k=1
+pk
+dαr
+uk,r
++ σ2
+r
+�
+.
+(38)
+C) If user k and the RIS have limited amount of power, e,g., user k power, pk, is less than the
+required power in (35) and the RIS power, Pr, is less than the required power in (38). In this case EH
+RIS can be implemented to provide the target secrecy rate. Based on (25) and (38), the required BS
+power, Pb, to charge the RIS and achieve the target secrecy rate, rs, can be obtained by
+Pb = M (1 − τ)
+ηeffτNM
+
+− (q2 − q4 − q5 + q7) ±
+�
+(q2 − q4 − q5 + q7)2 − 4 (q1 − q3) (q8 − q6)
+2 (q1 − q3)
+
+
+×
+� K
+�
+k=1
+pk
+dαr
+uk,r
++ σ2
+r
+�
+.
+(39)
+VII. NUMERICAL RESULTS
+In this section, we present simulation and numerical results to assess the accuracy of the derived
+expressions and the secrecy performance of the RIS schemes considered in this paper. Monte-Carlo
+simulations with 105 independent trials are excuted. The locations of the BS and the RIS are (0 m, 0
+m), (20 m, 20 m), respectively, while the users are scattered on the corners of a square. Specifically,
+the coordinates for the users square are (30 m, 5 m), (35 m, 5 m), (30 m,−5 m), and (35 m,−5 m),
+respectively, while the eavesdroppers are distributed in a circle centered at (20 m, 0 m) with radius
+of 10 m. Unless otherwise specified, the simulation settings are assumed as follows: K = J = 4,
+N = 10, M = 5, the users power pi = 2W, the active RIS power Pr = 7W , the BS power in EH
+RIS scenario Pb = 50W, and the nodes have same noise variance, σ2 = −70 dBm. In addition, the
+path-loss exponent is 2.7, the Rician factors ρ = 0.5. The values of the AoA and AoD of the BS and
+the RIS are uni-formally distributed in (0, 2π), and the concentration parameter of RIS phase error
+κ = 2.
+
+20
+Firstly, in Fig. 2, we illustrate the ergodic secrecy rate versus the transmission user power, pk, for
+the three considered RIS schemes. Fig. 2a shows the secrecy rate with phase shift errors and Fig. 2b,
+presents the secrecy rate for the ideal scenario, when there is no phase error at RIS. It is clear from
+this figure that the analytical results are in good agreement with the simulated results, which confirms
+the validity of the analysis presented in this paper. It is also evident that for the given parameters
+values, the secrecy rate loss due to the imperfect phase shift at the RIS is about 0.75 bits/s/Hz. In
+addition, passive RIS achieves the lowest secrecy rate, but with small amount of power consumption.
+The secrecy rate gain of active RIS above passive RIS is about 0.8 bits/s/Hz for a given user power.
+Furthermore, high secrecy rates can be achieved and controlled by implementing EH RIS. However,
+in this case the BS should transmit high power in the EH phase to provide sufficient amount of energy
+at the RIS to achieve higher secrecy rates.
+0
+10
+20
+30
+40
+50
+60
+pk (W)
+0
+0.5
+1
+1.5
+2
+2.5
+Secrecy Rate (bits\sec\Hz)
+Active RIS
+Passive RIS
+Analytical
+EH RIS
+(a) Secrecy rate versus user, k , power with phase shift error.
+0
+10
+20
+30
+40
+50
+60
+pk (W)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+Secrecy Rate (bits\sec\Hz)
+Active RIS
+Passive RIS
+Analytical
+EH RIS
+(b) Secrecy rate versus user, k , power with no phase shift error.
+Figure 2: Secrecy rate versus user, k , power with and without phase shift error.
+To explain the impact of the phase errors at the RIS on the secrecy performance, in Fig. 3, we plot
+the secrecy rate versus the concentration parameter of the phase error, κ. Additionally, the results of
+ideal RIS are also presented in this figure. It can be observed from these results that the secrecy rate
+enhances as the concentration parameter, κ, increases. In addition, at high concentration parameter
+values, κ −→ ∞, the secrecy rate achieved by imperfect RIS saturates to that achieved by ideal RIS.
+This can be explained by the fact that the phase error at the RIS is assumed to follow a Von Mises
+distribution, thus high concentration parameter values make the error fluctuate in a smaller range, and
+
+21
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Secrecy Rate (bits\sec\Hz)
+Ideal RIS without Error
+ RIS with Error
+ RIS with Error
+ RIS with Error
+Ideal RIS without Error
+Active RIS
+Passive RIS
+EH RIS
+Ideal RIS without Error
+Figure 3: Secrecy rate versus concentration parameter, κ, of RIS phase error.
+when κ −→ ∞, the error at the RIS tends to zero. Accordingly, the secrecy rate of imperfect RIS
+converges to the ideal RIS case as κ −→ ∞, as explained in Fig. 3.
+Furthermore, Fig. 4 shows the secrecy rate versus the number of BS antennas N for the all RIS
+schemes. It is evident and as expected, increasing the number of BS antennas N enhances the secrecy
+performance for the all RIS schemes. It should be pointed out that the number of BS antennas, N, has
+impact only on the received signal at the BS, thus increasing N results in enhancing the rate of the
+legitimate users. However N dose not have any impact on the rate at the eavesdroppers. Having said
+that in EH RIS, increasing N also increases the amount of the harvested energy at the RIS. Thus, in
+EH RIS, N has impact on both achievable rates at the BS and the eavesdroppers.
+In Fig. 5, we depict the secrecy rate versus the number of RIS elements, M, for the all considered RIS
+schemes. To obtain clear insights and results, in this figure the noise variance at the nodes is assumed
+to be σ2 = −20 dBm. Notably and as expected, increasing M results in enhancing the secrecy rate
+for the all considered scenarios. In addition, as we can notice from the analytical expressions of the
+secrecy rate presented in this paper, the number of RIS elements M has impact on both the achievable
+rate at the BS and the eavesdroppers, e.g., adding more RIS elements increases the rate at the BS and
+the eavesdroppers. However this improvement in the rate is essential at the BS, because the RIS phase
+shifts are designed to be toward the BS direction. Furthermore, in the EH RIS scheme, increasing the
+
+22
+5
+10
+15
+20
+N
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+Secrecy Rate (bits\sec\Hz)
+Passive RIS
+Analytical
+Active RIS
+EH RIS
+Figure 4: Secrecy rate versus number of BS antennas, N, with phase shift error.
+20
+40
+60
+80
+100
+120
+140
+M
+0
+1
+2
+3
+4
+5
+6
+Secrecy Rate (bits\Sec\Hz)
+Active RIS
+Passive RIS
+EH RIS
+Analytical
+Figure 5: Secrecy rate versus number of RIS elements, M, with phase shift error.
+number of the RIS elements, M, leads to an increase in the amount of the harvested energy at the RIS
+and thus Pr will be high when the number of elements M is very large.
+In order to illustrate the RIS configuration selection scheme, in Fig. 6 we plot the user power
+versus the target secrecy rate for different values of the concentration parameter of RIS phase error,
+κ = 2 and 8. Firstly, in Figs. 6a and 6b, we consider two examples, when the target secrecy rate is
+assumed to be rs = 0.75 (bits/s/Hz) and rs = 1.2 (bits/s/Hz) for κ = 2 and 8. As we can see from
+
+23
+the results in Fig. 6a, when rs = 0.75 (bits/s/Hz), passive RIS can achieve the target secrecy rate
+with total transmission power is PT = pk = 50W, (neglecting the small amount of power consuming
+at passive RIS elements), and in the active RIS scheme the user transmission power can be reduced
+to around pk = 7W and thus the total transmission power is PT = pk + Pr = 14W, while EH
+RIS scheme can achieve the target secrecy rate with the smallest amount of the user power which is
+about pk = 2.95W, but with the highest total transmission power PT = pk + Pb = 52.95W. Similar
+observations can be noticed from the second scenario when rs = 1.2 (bits/s/Hz), passive RIS achieves
+the target secrecy rate with the highest user power, while EH RIS achieves, rs, with the smallest user
+power but with very high total consumption power, and the active RIS scheme works between these
+two regions. In addition, the concentration parameter of RIS phase error, κ, has essential impact on
+the required user power. By comparing Figs 6a and 6b, one can notice that as κ increases the required
+user power to achieve the target secrecy rate decreases. For instance when the target secrecy rate is
+rs = 0.75 (bits/s/Hz), the required user power in the passive RIS scheme is about 50W when κ = 2,
+and 20W when κ = 8. This is due to the fact explained in Fig. 3.
+Then, in Figs. 6c and 6d, we present the RIS configuration selection scheme when the available
+user power is pk = 20W for κ = 2 and 8. In the first case when κ = 2, if the target secrecy rate
+is rs ≤ 0.45 (bits/s/Hz), passive RIS can be selected, and active RIS can be implemented if the
+target secrecy rate is rs ≤ 1.17 (bits/s/Hz), while EH RIS can be selected if rs ≤ 1.87 (bits/s/Hz).
+These secrecy rate regions of the RIS schemes become wider as the concentration parameter of RIS
+phase error, κ, increases. In Fig. 6d when κ = 8, passive RIS can be selected to achieve secrecy
+rates up to rs ≤ 0.77 (bits/s/Hz), and active RIS can be selected to perform secrecy rates less
+than or equal to rs ≤ 1.635 (bits/s/Hz), whilst EH RIS can be used to achieve secrecy rates up to
+rs ≤ 2.48 (bits/s/Hz).
+
+24
+0
+0.5
+1
+1.5
+2
+2.5
+Target Secrecy Rate (bits/s/Hz)
+0
+10
+20
+30
+40
+50
+60
+User Power (W)
+Passive RIS
+Active RIS, Pr=7W
+EH RIS, Pb=50W
+ Target Secrecy Rate 
+rs=1.2 (bits/s/Hz)
+ Target Secrecy Rate 
+rs=0.75 (bits/s/Hz)
+(a) The user power versus target secrecy rate when κ = 2 .
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Target Secrecy Rate (bits/s/Hz)
+0
+10
+20
+30
+40
+50
+60
+User Power (W)
+Passive RIS
+Active RIS, Pr=7W
+EH RIS, Pb=50W
+ Target Secrecy Rate 
+rs=1.2 (bits/s/Hz)
+ Target Secrecy Rate 
+rs=0.75 (bits/s/Hz)
+(b) The user power versus target secrecy rate when κ = 8.
+0
+0.5
+1
+1.5
+2
+2.5
+ Target Secrecy Rate (bits/s/Hz)
+0
+10
+20
+30
+40
+50
+60
+User Power (W)
+Passive RIS
+Active RIS, Pr=7W
+EH RIS, Pb=50W
+ 
+  Only EH RIS
+can be selected
+ 
+  Active RIS
+can be selected
+ Passive RIS 
+can be selected
+Available 
+ power pk
+(c) RIS configuration selection scheme when pk = 20W and κ = 2.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Target Secrecy Rate (bits/s/Hz)
+0
+10
+20
+30
+40
+50
+60
+ User Power (W)
+Passive RIS
+Active RIS, Pr=7W
+EH RIS, Pb=50W
+Available 
+ power pk
+ Passive RIS 
+can be selected
+ 
+  Active RIS
+can be selected
+ 
+  Only EH RIS
+can be selected
+(d) RIS configuration selection scheme when pk = 20W and, κ = 8.
+Figure 6: The user power versus target secrecy rate for different values of the concentration parameter
+of RIS phase error, κ.
+VIII. CONCLUSIONS
+In this paper the impact of phase shift error on the secrecy performance of up-link RIS-aided MU-
+MISO systems was considered. Under Rician fading channels and phase shift errors the ergodic secrecy
+rate for, passive RIS, active RIS, and EH RIS have been analyzed. Then, the phase shifts at the RIS
+have been optimized based on the derived rate expressions. In addition, according to the target secrecy
+rate and amount of power available at the users, the best RIS configuration selection scheme has been
+considered. The results presented in this work demonstrated that an active RIS scheme can enhance
+the secrecy performance of imperfect RIS elements, especially when the users have limited amount of
+
+25
+power. Furthermore, increasing the number of BS antennas, the concentration parameter of RIS phase
+error, and the number of RIS elements lead to the enhancement of the secrecy performance.
+APPENDIX A
+By using Jensen inequality, the ergodic rate can be expressed as
+E {Rbk} ≈ log2
+
+
+
+
+
+
+
+1 + E
+
+
+
+
+
+
+
+
+
+
+
+
+
+pk Luk,b
+��hH
+r,kΘHGHGΘ¯Θhr,k
+��2
+K�
+i=1
+i̸=k
+pi Lui,b
+��hH
+r,kΘHGHGΘ¯Θhr,i
+��2 +
+��hH
+r,kΘHGH��2 σ2
+b
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+(40)
+Due to the paper length limitation, in this Appendix we will explain how to calculate the average
+of the first term, similarly and by following similar steps we can find the average of the other terms.
+The first term is
+E
+�
+Pk Luk,b
+��hH
+r,kΘHGHGΘ¯Θhr,k
+��2�
+= Pk Luk,bE
+���hH
+r,kΘHGHGΘ¯Θhr,k
+��2�
+(41)
+where
+hH
+r,kΘHGHGΘ¯Θhr,k = hH
+r,kΘH
+�
+ρb
+ρb + 1
+¯GH ¯G +
+√ρb
+ρb + 1
+¯GH ˜G +
+√ρb
+ρb + 1
+˜GH ¯G +
+1
+ρb + 1
+˜GH ˜G
+�
+Θ¯Θhr,k
+=
+1
+ρb + 1hH
+r,kΘH �
+ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G
+�
+Θ¯Θhr,k =
+1
+ρb + 1hH
+r,kA¯Θhr,k
+(42)
+where A = ΘH �
+ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G
+�
+Θ. Now (42) can be expressed as
+hH
+r,kΘHGHGΘ¯Θhr,k =
+1
+(ρb + 1) (ρk + 1)
+�√ρk¯hH
+r,k + ˜hH
+r,k
+�
+A¯Θ
+�√ρk¯hr,k + ˜hr,k
+�
+=
+1
+(ρb + 1) (ρk + 1)
+
+
+ρk¯hH
+r,kA¯Θ¯hr,k
+�
+��
+�
+∆1
++ √ρk¯hH
+r,kA¯Θ˜hr,k
+�
+��
+�
+∆2
++ √ρk˜hH
+r,kA¯Θ¯hr,k
+�
+��
+�
+∆3
++ ˜hH
+r,kA¯Θ˜hr,k
+�
+��
+�
+∆4
+
+
+
+(43)
+
+26
+The channels are independent and have zero mean. Thus by removing the zero expectation terms,
+we can get
+E
+���hH
+r,kΘHGHGΘ¯Θhr,k
+��2�
+=
+1
+(ρb + 1)2 (ρk + 1)2E
+
+
+
+�����
+4
+�
+i=1
+∆i
+�����
+2
+
+
+=
+1
+(ρb + 1)2 (ρk + 1)2
+�
+4
+�
+i=1
+E
+�
+|∆i|2�
++ 2E {∆1∆∗
+4}
+�
+(44)
+Now the first term
+∆1 = ρk¯hH
+r,kΘH �
+ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G
+�
+Θ¯Θ¯hr,k
+=
+
+
+��ρbρk¯hH
+r,kΘH ¯GH ¯GΘ¯Θ¯hr,k
+�
+��
+�
+∆1,1
++ √ρbρk¯hH
+r,kΘH ¯GH ˜GΘ¯Θ¯hr,k
+�
+��
+�
+∆1,2
++√ρbρk¯hH
+r,kΘH ˜GH ¯GΘ¯Θ¯hr,k
+�
+��
+�
+∆1,3
++ ρk¯hH
+r,kΘH ˜GH ˜GΘ¯Θ¯hr,k
+�
+��
+�
+∆1,4
+
+
+
+(45)
+The average of the first term
+E
+�
+|∆1|2�
+= E
+�
+|∆1,1|2�
++ E
+�
+|∆1,2|2�
++ E
+�
+|∆1,3|2�
++ E
+�
+|∆1,4|2�
++ 2E
+�
+∆1,1∆H
+1,4
+�
+(46)
+where ∆1,1 = ρbρk¯hH
+r,kΘH ¯GH ¯GΘ¯Θ¯hr,k, which can be written as
+∆1,1 = ρbρkaH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aM (φa
+r, φe
+r) Θ¯ΘaM (φa
+kr, φe
+kr) ,
+∆1,1 = ρbρk
+� M
+�
+m=1
+aH
+M,m (φa
+kr, φe
+kr) e−jϕmaH
+M,m (φa
+r, φe
+r)
+� � M
+�
+m=1
+aM,m (φa
+kr, φe
+kr) ejϕmej ¯
+ϕmaM,m (φa
+r, φe
+r)
+�
+.
+(47)
+The average can now be written as
+
+27
+E
+�
+|∆1,1|2�
+= ρ2
+bρ2
+k
+� M
+�
+m=1
+aH
+M,m (φa
+kr, φe
+kr) e−jϕmaH
+M,m (φa
+r, φe
+r)
+�2
+×
+
+M + ρ (κ)2
+�����
+M
+�
+m1=1
+M
+�
+m2̸=m1
+�
+aM,m1 (φa
+kr, φe
+kr) ejϕm1aM,m1 (φa
+r, φe
+r)
+� �
+aM,m2 (φa
+kr, φe
+kr) ejϕm2aM,m2 (φa
+r, φe
+r)
+�H
+�����
+2
+
+(48)
+E
+�
+|∆1,1|2�
+= ρ2
+bρ2
+k |fk|2 ��
+1 − ρ (κ)2�
+M + ρ (κ)2 |fk|2�
+(49)
+where fk =
+M
+�
+m=1
+fk,m, fk,m = aH
+M.m (φa
+r, φe
+r) ejϕmaM,m (φa
+kr, φe
+kr). The second term,
+∆1,2 = √ρbρkaH
+M (φa
+kr, φe
+kr) ΘHaM (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) ˜GΘ¯ΘaM (φa
+kr, φe
+kr)
+= √ρbρkf ∗
+k
+M
+�
+m=1
+N
+�
+n=1
+aH
+N,n (φa
+b, φe
+b) ˜gnmejϕmej ¯
+ϕmaM,m (φa
+kr, φe
+kr) ,
+(50)
+E
+�
+|∆1,2|2�
+= ρbρ2
+kNM |fk|2 .
+(51)
+The third term
+∆1,3 = √ρbρkaH
+M (φa
+kr, φe
+kr) ΘH ˜GHaN (φa
+b, φe
+b) aH
+M (φa
+r, φe
+r) Θ¯ΘaM (φa
+kr, φe
+kr)
+= √ρbρk
+M
+�
+m=1
+N
+�
+n=1
+aH
+N,n (φa
+b, φe
+b) ˜gH
+nme−jϕmaM,m (φa
+kr, φe
+kr)
+M
+�
+m=1
+ej ¯
+ϕmfk,m,
+(52)
+E
+�
+|∆1,3|2�
+= ρbρ2
+k
+�
+NMρ (κ)2 |fk|2 +
+�
+1 − ρ (κ)2�
+NM2�
+.
+(53)
+The forth term
+∆1,4 = ρkaH
+M (φa
+kr, φe
+kr) ΘH ˜GH ˜GΘ¯ΘaM (φa
+kr, φe
+kr)
+
+28
+= ρk
+M
+�
+m1=1
+aH
+M,m1 (φa
+kr, φe
+kr) e−jϕm˜gH
+nm1
+M
+�
+m2=1
+˜gnm2ejϕmej ¯
+ϕmaM,m2 (φa
+kr, φe
+kr) ,
+(54)
+E
+�
+|∆1,4|2�
+= ρkNM
+�
+Mρ (κ)2 + 1 − ρ (κ)2�
++ NM2.
+(55)
+The last term
+E
+�
+∆1,1∆∗
+1,4
+�
+= N |fk|2 �
+Mρ (κ)2 + 1 − ρ (κ)2�
+.
+(56)
+Similarly, following the same way we can find the average of the other terms.
+APPENDIX B
+Using Jensen inequality, the ergodic rate can be written as
+E
+�
+Rej,k
+�
+≈ log2
+
+
+
+
+
+
+
+1 + E
+
+
+
+
+
+
+
+
+
+
+
+
+
+pk
+���d
+− αr
+2
+uk,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,k + d
+− αe
+2
+ej,k hej,k
+���
+2
+K�
+i=1
+i̸=k
+pi
+���d
+− αr
+2
+ui,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,i + d
+− αe
+2
+ej,i hej,i
+���
+2
++ σ2
+ej
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(57)
+The average of the first term, after removing the zero expectation terms can be calculated by,
+E
+����d
+− αr
+2
+uk,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,k + d
+− αe
+2
+ej,k hej,k
+���
+2�
+= d−αr
+uk,rd−αe
+ej,r E
+���hej,rΘ¯Θhr,k
+��2�
++ d−αe
+ej,r
+(58)
+where
+E
+���hej,rΘ¯Θhr,k
+��2�
+= E
+�����
+��
+ρej
+ρej + 1
+�
+ρk
+ρk + 1
+¯hejΘ¯Θ¯hr,k +
+�
+ρej
+ρej + 1
+�
+1
+ρk + 1
+¯hejΘ¯Θ˜hr,k
++
+�
+ρk
+ρk + 1
+�
+1
+ρej + 1
+˜hejΘ¯Θ¯hr,k +
+�
+1
+ρej + 1
+�
+1
+ρk + 1
+˜hejΘ¯Θ˜hr,k
+������
+2
+
+
+(59)
+E
+���hej,rΘ¯Θhr,k
+��2�
+=
+ρej
+ρej + 1
+ρk
+ρk + 1E
+��¯hejΘ¯Θ¯hr,k
+��2 +
+ρej
+ρej + 1
+1
+ρk + 1E
+���¯hejΘ¯Θ˜hr,k
+���
+2
+
+29
++
+ρk
+ρk + 1
+1
+ρej + 1E
+���˜hejΘ¯Θ¯hr,k
+���
+2
++
+1
+ρej + 1
+1
+ρk + 1E
+���˜hejΘ¯Θ˜hr,k
+���
+2
+(60)
+Now
+¯hejΘ¯Θ¯hr,k =
+� M
+�
+m=1
+aM,m (φa
+kr, φe
+kr) ejϕmej ¯
+ϕmaM,m
+�
+φa
+ejr, φe
+ejr
+��
+(61)
+E
+��¯hejΘ¯Θ¯hr,k
+��2 = E
+�����
+M
+�
+m=1
+aM,m (φa
+kr, φe
+kr) ejϕmej ¯
+ϕmaM,m
+�
+φa
+ejr, φe
+ejr
+������
+2
+= M+ρ (κ)2
+M
+�
+m1=1
+M
+�
+m2̸=m1
+�
+aM,m1 (φa
+kr, φe
+kr) ejϕm1aM,m1 (φa
+r, φe
+r)
+� �
+aM,m2 (φa
+kr, φe
+kr) ejϕm2aM,m2 (φa
+r, φe
+r)
+�H
+= M + ρ (κ)2 ξ
+(62)
+where ξ =
+M
+�
+m1=1
+M
+�
+m2̸=m1
+(aM,m1 (φa
+kr, φe
+kr) ejϕm1aM,m1 (φa
+r, φe
+r)) (aM,m2 (φa
+kr, φe
+kr) ejϕm2aM,m2 (φa
+r, φe
+r))
+H.
+Similarly, the second term,
+¯hejΘ¯Θ˜hr,k = aM
+�
+φa
+ejr, φe
+ejr
+�
+Θ¯Θ˜hr,k =
+M
+�
+m=1
+aMm
+�
+φa
+ejr, φe
+ejr
+�
+ejϕmej ¯
+ϕm �
+˜hr,k
+�
+m
+(63)
+E
+���¯hejΘ¯Θ˜hr,k
+���
+2
+= E
+�����
+M
+�
+m=1
+aMm
+�
+φa
+ejr, φe
+ejr
+�
+ejϕmej ¯
+ϕm �
+˜hr,k
+�
+m
+�����
+2
+E
+���¯hejΘ¯Θ˜hr,k
+���
+2
+= M+
+E
+�
+M
+�
+m1=1
+M
+�
+m2̸=m1
+�
+aMm1
+�
+φa
+ejr, φe
+ejr
+�
+ejϕm1ej
+¯
+ϕm1 �
+˜hr,k
+�
+m1
+� �
+aMm2
+�
+φa
+ejr, φe
+ejr
+�
+ejϕm2ej
+¯
+ϕm2 �
+˜hr,k
+�
+m2
+�H�
+= M
+(64)
+other terms,
+
+30
+˜hejΘ¯Θ¯hr,k =
+M
+�
+m=1
+˜hej,mejϕmej ¯
+ϕmaM,m (φa
+kr, φe
+kr)
+(65)
+E
+���˜hejΘ¯Θ¯hr,k
+���
+2
+= M+
+E
+�
+M
+�
+m1=1
+M
+�
+m2̸=m1
+��
+˜hej
+�
+m1 ejϕm1ej
+¯
+ϕm1aMm1 (φa
+kr, φe
+kr)
+� ��
+˜hej
+�
+m2 ejϕm2ej
+¯
+ϕm2aMm2 (φa
+kr, φe
+kr)
+�H�
+= M
+(66)
+and
+˜hejΘ¯Θ˜hr,k =
+M
+�
+m=1
+�
+˜hej
+�
+m ejϕmej ¯
+ϕm �
+˜hr,k
+�
+m
+(67)
+E
+���˜hejΘ¯Θ˜hr,k
+���
+2
+= E
+�����
+M
+�
+m=1
+�
+˜hej
+�
+m ejϕmej ¯
+ϕm �
+˜hr,k
+�
+m
+�����
+2
+= M
+(68)
+Now, we are ready to write the average of the first term,
+E
+���hej,rΘ¯Θhr,k
+��2�
+=
+ρej
+ρej + 1
+ρk
+ρk + 1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej + 1
+1
+ρk + 1M
++
+ρk
+ρk + 1
+1
+ρej + 1M +
+1
+ρej + 1
+1
+ρk + 1M
+(69)
+Similarly we can find the average of the second term as,
+E
+����d
+− αr
+2
+ui,r d
+− αe
+2
+ej,r hej,rΘ¯Θhr,i + d
+− αe
+2
+ej,i hej,i
+���
+2�
+= d−αr
+ui,r d−αe
+ej,r E
+���hej,rΘ¯Θhr,i
+��2�
++ d−αe
+ej,i
+(70)
+E
+���hej,rΘ¯Θhr,i
+��2�
+=
+ρej
+ρej + 1
+ρi
+ρi + 1
+�
+M + ρ (κ)2 ξ
+�
++
+ρej
+ρej + 1
+1
+ρi + 1M
++
+ρi
+ρi + 1
+1
+ρej + 1M +
+1
+ρej + 1
+1
+ρi + 1M
+(71)
+
+31
+APPENDIX C
+Using Jensen inequality, the ergodic rate can be written as
+E {Rbk} ≈ log2 (1 + E {γbk})
+(72)
+We will follow similar steps as in Appendix A,
+hH
+r,kΘHGHG¯ΘΘ =
+1
+ρb + 1hH
+r,kA¯Θ
+(73)
+where A = ΘH �
+ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G
+�
+Θ. Last expression can be written as
+hH
+r,kΘHGHG¯ΘΘ =
+1
+(ρb + 1)
+�
+(ρk + 1)
+�√ρk¯hH
+r,k + ˜hH
+r,k
+�
+A¯Θ
+=
+1
+(ρb + 1)
+�
+(ρk + 1)
+
+
+√ρk¯hH
+r,kA¯Θ
+�
+��
+�
+∆1
++ ˜hH
+r,kA¯Θ
+� �� �
+∆2
+
+
+
+(74)
+The average can be written as,
+E
+���hH
+r,kΘHGHG¯ΘΘ
+��2�
+=
+1
+(ρb + 1)2 (ρk + 1)
+E
+
+
+
+
+
+ρr
+�������
+¯hH
+r,kA¯Θ
+� �� �
+∆1
+�������
+2
++
+�������
+˜hH
+r,kA¯Θ
+� �� �
+∆2
+�������
+2
+
+
+
+
+(75)
+∆1 = √ρk¯hH
+r,kΘH �
+ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G
+�
+Θ¯Θ
+=
+
+
+ρb
+√ρk¯hH
+r,kΘH ¯GH ¯GΘ¯Θ
+�
+��
+�
+∆1,1
++ √ρb
+√ρk¯hH
+r,kΘH ¯GH ˜GΘ¯Θ
+�
+��
+�
+∆1,2
++√ρb
+√ρk¯hH
+r,kΘH ˜GH ¯GΘ¯Θ
+�
+��
+�
+∆1,3
++ √ρk¯hH
+r,kΘH ˜GH ˜GΘ¯Θ
+�
+��
+�
+∆1,4
+
+
+
+(76)
+E
+�
+|∆1|2�
+= E
+�
+|∆1,1|2�
++ E
+�
+|∆1,2|2�
++ E
+�
+|∆1,3|2�
++ E
+�
+|∆1,4|2�
++ 2E
+�
+∆1,1∆H
+1,4
+�
+(77)
+
+32
+where
+∆1,1 = ρb
+√ρk
+�
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+� �
+aM (φa
+r, φe
+r) Θ¯Θ
+�
+(78)
+E
+�
+|∆1,1|2�
+= ρ2
+bρk
+���
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+���2
+× E
+���aM (φa
+r, φe
+r) Θ¯Θ
+��2�
+(79)
+E
+�
+|∆1,1|2�
+= ρ2
+bρk
+���
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+���2 × M
+(80)
+The second term can be expressed as,
+∆1,2 = √ρb
+√ρkaH
+M (φa
+kr, φe
+kr) ΘHaM (φa
+r, φe
+r)
+M
+�
+m=1
+N
+�
+n=1
+aH
+N,n (φa
+b, φe
+b) ˜gnmejϕmej ¯
+ϕm
+(81)
+E
+�
+|∆1,2|2�
+= ρbρk
+��aH
+M (φa
+kr, φe
+kr) ΘHaM (φa
+r, φe
+r)
+��2 NM
+(82)
+The third term can be written as
+∆1,3 = √ρb
+√ρk
+M
+�
+m=1
+N
+�
+n=1
+aH
+M,m (φa
+kr, φe
+kr) ˜gH
+nme−jϕmaN,n (φa
+b, φe
+b)
+M
+�
+m=1
+aH
+M,m (φa
+r, φe
+r) ej ¯
+ϕmejϕm
+(83)
+E
+�
+|∆1,3|2�
+= ρbρkMN
+
+E
+�����
+M
+�
+m=1
+aH
+M,m (φa
+r, φe
+r) ej ¯
+ϕmejϕm
+�����
+2
+
+(84)
+E
+�
+|∆1,3|2�
+= ρbρkMN
+�
+M + ρ (κ)2
+M
+�
+m1=1
+M
+�
+m2̸=m1
+aH
+M,m1 (φa
+r, φe
+r) ejϕm1aM,m2 (φa
+r, φe
+r) e−jϕm2
+�
+(85)
+E
+�
+|∆1,3|2�
+= ρbρkMN
+�
+ρ (κ)2 M +
+�
+1 − ρ (κ)2�
+M
+�
+(86)
+
+33
+The forth term can be represented as,
+∆1,4 = √ρk
+N
+�
+n=1
+M
+�
+m1=1
+aH
+M,m1 (φa
+kr, φe
+kr) e−jϕm˜gH
+nm1
+M
+�
+m2=1
+˜gnm2ejϕmej ¯
+ϕm
+(87)
+E
+�
+|∆1,4|2�
+= ρk
+�
+N2M + NM2�
+(88)
+Now the last term can be written as
+E
+�
+∆1,1∆∗
+1,4
+�
+= ρbρk
+�
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+�
+×
+�
+aM (φa
+r, φe
+r) Θ¯Θ
+�
+ρk¯hH
+r,kΘH ˜GH ˜GΘ¯Θ
+(89)
+E
+�
+∆1,1∆∗
+1,4
+�
+= ρbρk
+�
+aH
+M (φa
+kr, φe
+kr) ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+�
+(aM (φa
+r, φe
+r) Θ) ρk¯hH
+r,kΘHNΘ
+(90)
+We will repeat similar steps for ∆2,
+∆2 =
+
+
+ρb˜hH
+r,kΘH ¯GH ¯GΘ¯Θ
+�
+��
+�
+∆2,1
++ √ρb˜hH
+r,kΘH ¯GH ˜GΘ¯Θ
+�
+��
+�
+∆2,2
++√ρb˜hH
+r,kΘH ˜GH ¯GΘ¯Θ
+�
+��
+�
+∆2,3
++ ˜hH
+r,kΘH ˜GH ˜GΘ¯Θ
+�
+��
+�
+∆2,4
+
+
+
+(91)
+E
+�
+|∆2|2�
+= E
+�
+|∆2,1|2�
++ E
+�
+|∆2,2|2�
++ E
+�
+|∆2,3|2�
++ E
+�
+|∆2,4|2�
++ 2E
+�
+∆2,1∆H
+2,4
+�
+(92)
+where
+∆2,1 = ρb˜hH
+r,kΘH ¯GH ¯GΘ¯Θ
+(93)
+
+34
+E
+�
+|∆2,1|2�
+= ρ2
+b
+��ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b) aM (φa
+r, φe
+r) Θ
+��2
+F
+(94)
+and
+∆2,2 = √ρb˜hH
+r,kΘHaM (φa
+r, φe
+r)
+M
+�
+m=1
+N
+�
+n=1
+aH
+N,n (φa
+b, φe
+b) ˜gnmejϕmej ¯
+ϕm
+(95)
+E
+�
+|∆2,2|2�
+= ρb
+���˜hH
+r,kΘHaM (φa
+r, φe
+r)
+���
+2
+NM
+(96)
+while
+∆2,3 = √ρb
+M
+�
+m=1
+N
+�
+n=1
+˜hH
+r,k,nm˜gH
+nme−jϕmaN,n (φa
+b, φe
+b)
+M
+�
+m=1
+aH
+M,m (φa
+r, φe
+r) ej ¯
+ϕmejϕm
+(97)
+E
+�
+|∆2,3|2�
+= ρbMN
+
+E
+�����
+M
+�
+m=1
+aH
+M,m (φa
+r, φe
+r) ej ¯
+ϕmejϕm
+�����
+2
+ = ρbMN
+�
+ρ (κ)2 M +
+�
+1 − ρ (κ)2�
+M
+�
+(98)
+Finally,
+∆2,4 = ρk
+N
+�
+n=1
+M
+�
+m1=1
+˜hH
+r,k,nm1e−jϕm˜gH
+nm1
+M
+�
+m2=1
+˜gnm2ejϕmej ¯
+ϕm
+(99)
+E
+�
+|∆2,4|2�
+= ρ2
+k
+�
+N2M + NM2�
+(100)
+and
+E
+�
+∆2,1∆∗
+2,4
+�
+= E
+�
+ρbρk
+�
+˜hH
+r,kΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+� �
+aM (φa
+r, φe
+r) Θ¯Θ
+�
+ρk˜hH
+r,kΘH ˜GH ˜GΘ¯Θ
+�
+= ρk
+�
+ΘHaH
+M (φa
+r, φe
+r) aH
+N (φa
+b, φe
+b) aN (φa
+b, φe
+b)
+�
+(aM (φa
+r, φe
+r) Θ) ρkΘNΘH
+(101)
+
+35
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+

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+arXiv:2301.11398v1  [math.SP]  26 Jan 2023
+ˇSmigoc’s glue for universal realizability in the
+left half-plane∗
+Jaime H. Alfaro, Ricardo L. Soto†
+Dpto. Matem´aticas, Universidad Cat´olica del Norte, Casilla 1280
+Antofagasta, Chile.
+Abstract
+A list Λ = {λ1, λ2, . . . , λn} of complex numbers is said to be real-
+izable if it is the spectrum of a nonnegative matrix. Λ is said to be
+universally realizable (UR) if it is realizable for each possible Jordan
+canonical form allowed by Λ. In this paper, using companion matrices
+and applying a procedure by ˇSmigoc, is provides a sufficient condi-
+tion for the universal realizability of left half-plane spectra, that is,
+Λ = {λ1, . . . , λn} with λ1 > 0, Re λi ≤ 0, i = 2, . . . , n. It is also shown
+how the effect of adding a negative real number to a not UR left half-
+plane list of complex numbers, makes the new list UR, and a family
+of left half-plane lists that are UR is characterized.
+AMS classification:
+15A18, 15A20, 15A29
+Key words: Nonnegative matrix; companion matrix; Universal realizabil-
+ity; ˇSmigoc’s glue.
+1
+Introduction
+A list Λ = {λ1, λ2, . . . , λn} of complex numbers is said to be realizable if it
+is the spectrum of an n-by-n nonnegative matrix A, and A is said to be a
+realizing matrix for Λ. The problem of the realizability of spectra is called the
+∗Supported by Universidad Cat´olica del Norte-VRIDT 036-2020, N´ucleo 6 UCN
+VRIDT 083-2020, Chile.
+†E-mail addresses: rsoto@ucn.cl (R. L. Soto), jaime.alfaro@ucn.cl (J. H. Alfaro).
+1
+
+nonnegative inverse eigenvalue problem (NIEP). From the Perron-Frobenius
+Theorem we know that if Λ = {λ1, λ2, . . . , λn} is the spectrum of an n-
+by-n nonnegative matrix A, then the leading eigenvalue of A equals to the
+spectral radius of A, ρ(A) =: max
+1≤i≤n |λi| . This eigenvalue is called the Perron
+eigenvalue, and we shall assume in this paper, that ρ(A) = λ1.
+A matrix is said to have constant row sums, if each one of its rows sums
+up to the same constant α. The set of all matrices with constant row sums
+equal to α, is denoted by CSα. Then, any matrix A ∈ CSα has the eigenvector
+eT = [1, 1, . . . , 1], corresponding to the eigenvalue α. The real matrices with
+constant row sums are important because it is known that the problem of
+finding a nonnegative matrix with spectrum Λ = {λ1, . . . , λn}, is equivalent
+to the problem of finding a nonnegative matrix in CSλ1 with spectrum Λ (see
+[3]). We denote by ek, the n-dimensional vector, with 1 in the kth position
+and zeros elsewhere. If Λ = {λ1, . . . , λn}, then sk(Λ) =
+n
+�
+i=1
+λk
+i , k = 1, 2, . . . .
+A list Λ = {λ1, λ2, . . . , λn} of complex numbers, is said to be diagonaliz-
+ably realizable (DR), if there is a diagonalizable realizing matrix for Λ The
+list Λ is said to be universally realizable (UR), if it is realizable for each possi-
+ble Jordan canonical form (JCF) allowed by Λ. The problem of the universal
+realizability of spectra, is called the universal realizability problem (URP).
+The URP contains the NIEP, and both problems are equivalent if the given
+numbers λ1, λ2, . . . , λn are distinct. In terms of n, both problems remain
+unsolved for n ≥ 5. It is clear that if Λ is UR, then Λ must be DR. The
+first known results on the URP are due to Minc [7, 8]. In terms of the URP,
+Minc [7] showed that if a list Λ = {λ1, λ2, . . . , λn} of complex numbers is the
+spectrum of a diagonalizable positive matrix, then Λ is UR. The positivity
+condition is necessary for Minc’s proof, and the question set by Minc himself,
+whether the result holds for nonnegative realizations was open for almost 40
+years. Recently, two extensions of Minc’s result have been obtained in [1, 4].
+In [1], Collao et al. showed that a nonnegative matrix A ∈ CSλ1, with a pos-
+itive column, is similar to a positive matrix. Note that if A is nonnegative
+with a positive row and AT has a positive eigenvector, then AT is also similar
+to a positive matrix. Besides, if Λ is diagonalizably realizable by a matrix
+A ∈ CSλ1 having a positive column, then Λ is UR. In [4], Johnson et al. in-
+troduced the concept of ODP matrices, that is, nonnegative matrices with all
+positive off-diagonal entries (zero diagonal entries are permitted) and proved
+2
+
+that if Λ is diagonalizably ODP realizable, then Λ is UR. Note that both
+extensions contain, as a particular case, Minc’s result in [7]. Both extensions
+allow us to significantly increase the set of spectra that can be proved to be
+UR, as for instance, certain spectra Λ = {λ1, . . . , λn} with s1(Λ) = 0, which
+is not possible from Minc’s result. In particular, we shall use the extension
+in [1] to generate some of our results.
+Remark 1.1 In [1], Section 2, Theorem 2.1 and Corollary 2.1, there is an
+error in assuming that if A is nonnegative with a positive row, then AT, which
+has a positive column, is similar to a positive matrix. The reason is that we
+cannot guarantee that AT has a positive eigenvector.
+Regarding non-positive universal realizations, we mention that in [10,
+2] the authors proved, respectively, that lists of complex numbers Λ =
+{λ1, . . . , λn}, of Suleimanova type, that is,
+λ1 > 0, Re λi ≤ 0, |Re λi| ≥ |Im λi| , i = 2, 3, . . . , n,
+or of ˇSmigoc type, that is,
+λ1 > 0, Re λi ≤ 0,
+√
+3 |Re λi| ≥ |Im λi| , i = 2, 3, . . . , n,
+(1)
+are UR if and only if they are realizable if and only if
+n
+�
+i=1
+λi ≥ 0.
+Outline of the paper: The paper is organized as follows: In Section 2, we
+present the mathematical tools that will be used to generate our results. In
+Section 3, we study the URP for a left half-plane list and we give a sufficient
+condition for it to be UR. In Section 4, we discuss the effect of adding a
+negative real number −c to a left half-plane list Λ = {λ1, −a±bi, . . . , −a±bi},
+which is not UR (or even not realizable), or we do not know whether it is,
+and we show how Λ ∪ {−c} becomes UR. We also characterize a family of
+left half-plane lists that are UR. In Section 5, we show that the merge of two
+lists diagonalizably realizable Γ1 ∈ CSλ1 and Γ2 ∈ CSµ1 is UR. Examples
+are shown to illustrate the results.
+2
+Preliminaries
+Throughout this paper we use the following results: The first one, by ˇSmigoc
+[9], gives a procedure that we call ˇSmigoc’s glue technique, to obtain from two
+3
+
+matrices A and B of size n-by-n and m-by-m, respectively, a new (n+m−1)-
+by-(n + m − 1) matrix C, preserving in certain way, the corresponding JCFs
+of A and B. The second one, by Laffey and ˇSmigoc [6] solves the NIEP for
+lists of complex numbers on the left half-plane, that is, lists with λ1 > 0,
+Re λi ≤ 0, i = 2, . . . , n. Moreover, we also use Lemma 5 in [6].
+Theorem 2.1 [9] Suppose B is an m-by-m matrix with a JCF that contains
+at least one 1-by-1 Jordan block corresponding to the eigenvalue c:
+J(B) =
+� c
+0
+0
+I(B)
+�
+.
+Let t and s, respectively, be the left and the right eigenvectors of B associated
+with the 1-by-1 Jordan block in the above canonical form. Furthermore, we
+normalize vectors t and s so that t
+Ts = 1. Let J(A) be a JCF for the n-by-n
+matrix
+A =
+�
+A1
+a
+bT
+c
+�
+,
+where A1 is an (n − 1)-by-(n − 1) matrix and a and b are vectors in C
+n-1.
+Then the matrix
+C =
+�
+A1
+at
+T
+sb
+T
+B
+�
+has JCF
+J(C) =
+�
+J(A)
+0
+0
+I(B)
+�
+.
+Theorem 2.2 [6] Let Λ = {λ1, λ2, . . . , λn} be a list of complex numbers with
+λ1 ≥ |λi| and Re λi ≤ 0, i = 2, . . . , n. Then Λ is realizable if and only if
+s1 = s1(Λ) ≥ 0,
+s2 = s2(Λ) ≥ 0,
+s2
+1 ≤ ns2.
+Lemma 2.1 [6] Let t be a nonnegative real number and let λ2, λ3, . . . , λn be
+complex numbers with real parts less than or equal to zero, such that the list
+{λ2, λ3, . . . , λn} is closed under complex conjugation. Set ρ = 2t−λ2−· · ·−λn
+and
+f(x) = (x − ρ)
+n
+�
+j=2
+(x − λj) = xn − 2txn−1 + b2xn−2 + · · · + bn.
+(2)
+Then b2 ≤ 0 implies bj ≤ 0 for j = 3, 4, . . . , n.
+4
+
+3
+Companion matrices and the ˇSmigoc’s glue.
+We say that a list Λ = {λ1, λ2, . . . , λn} of complex numbers is on the left half-
+plane if λ1 > 0, Re λi ≤ 0, i = 2, 3, . . . , n. In this section we give a sufficient
+condition for a left half-plane list of complex numbers to be UR. Of course,
+it is our interest to consider lists of complex numbers containing elements
+out of realizability region of lists of ˇSmigoc type. Our strategy consists in to
+decompose the given list Λ = {λ1, λ2, . . . , λn} into sub-lists
+Λk = {λk1, λk2, . . . , λkpk}, λ11 = λ1, k = 1, 2, . . . , t,
+with auxiliary lists
+Γ1
+=
+Λ1
+Γk
+=
+{s1(Γk−1), λk1, λk2, . . . , λkpk},
+k = 2, , . . . , t,
+each one of them being the spectrum of a nonnegative companion matrix
+Ak, in such a way that it be possible to apply ˇSmigoc’s glue technique to
+the matrices Ak, to obtain an n-by-n nonnegative matrix with spectrum Λ
+for each possible JCF allowed by Λ. In the case s1(Λ) > 0, with λi ̸= 0,
+i = 2, . . . , n, we may choose, if they exist, sub-lists Γk being the spectrum
+of a diagonalizable nonnegative companion matrix with a positive column.
+Then, after ˇSmigoc’s glue, we obtain a diagonalizable nonnegative n-by-n
+matrix A with spectrum Λ and a positive column, which is similar to a
+diagonalizable positive matrix. Thus, from the extension in [1], Λ is UR.
+Next we have the following corollary from Theorem 2.1:
+Corollary 3.1 Let Λ = {λ1, λ2, . . . , λn} be a realizable left half-plane list of
+complex numbers. Suppose that for each JCF J allowed by Λ, there exists a
+decomposition of Λ as
+Λ
+=
+Λ1 ∪ Λ2 ∪ · · · ∪ Λt, where
+Λk
+=
+{λk1, λk2, . . . , λkpk}, k = 1, 2, . . . , t, λ11 = λ1,
+with auxiliary lists
+Γ1
+=
+Λ1,
+Γk
+=
+{s1(Γk−1), λk1, λk2 . . . , λkpk}, k = 2, . . . , t,
+being the spectrum of a nonnegative companion matrix Ak with JCF J(Ak)
+as a sub-matrix of J, k = 1, 2, . . . , t.
+Then Λ is universally realizable.
+5
+
+Proof.
+Since each matrix Ak, k = 1, 2, . . . , t, is nonnegative companion
+with JCF J(Ak) being a submatrix of J, then, from ˇSmigoc’s glue applied to
+matrices Ak, we obtain an n-by-n nonnegative matrix with spectrum Λ and
+JCF J. As J is any JCF allowed by Λ, then Λ is UR.
+The following result is well known and useful.
+Lemma 3.1 Let A be a diagonalizable irreducible nonnegative matrix with
+spectrum Λ = {λ1, . . . , λn} and a positive row or column. Then A is similar
+to a diagonalizable nonnegative matrix B ∈ CSλ1, with a positive row or
+column.
+Proof. If A is irreducible nonnegative, it has a positive eigenvector xT =
+[x1, . . . , xn]. Then if D = dig{x1, . . . , xn}, the matrix
+B = D−1AD =
+�xj
+xi
+ai,j
+�
+∈ CSλ1
+is nonnegative with a positive row or column.
+Suppose all lists Γk in Corollary 3.1, can be taken as the spectrum of a
+diagonalizable nonnegative companion matrix Ak with a positive column (the
+last one). Then, since the glue of matrices Ak gives an n-by-n diagonalizable
+irreducible nonnegative matrix A with a positive column and spectrum Λ, A
+is similar to a diagonalizable positive matrix with spectrum Λ and therefore
+Λ is UR. This is what the next result shows.
+Corollary 3.2 Let Λ = {λ1, λ2, . . . , λn}, λi ̸= 0, i = 2, . . . , n, s1(Λ) > 0, be
+a realizable left half-plane list of complex numbers. If there is a decomposition
+of Λ as in Corollary 3.1, with all lists Γk being the spectrum of a diagonal-
+izable nonnegative companion matrix Ak, with a positive column, then Λ is
+universally realizable.
+Proof. It is enough to prove the result for two lists Γk of the decomposition
+of Λ. Let Γk−1 and Γk, k = 2, . . . , t, be the spectrum, respectively, of matrices
+Ak−1 and Ak, which are diagonalizable nonnegative companion with a posi-
+tive column (the last one). Then Ak−1 and Ak are irreducible. In particular,
+Ak has a positive eigenvector s and, since AT
+k is also irreducible, Ak has also
+a positive left eigenvector tT with tTs = 1. Now, let
+Ak−1 =
+� A1,k−1
+a
+bT
+s1(Γk−1)
+�
+.
+6
+
+Since the last column of Ak−1 is positive, the vector a is also positive and
+atT is a positive submatrix. Therefore, the glue of Ak−1 with Ak,
+Ck =
+� A1,k−1
+atT
+sbT
+Ak
+�
+,
+is a diagonalizable nonnegative matrix with its last column being positive.
+Note that Ck is also irreducible. Then Ck has, besides, a positive eigenvector,
+and from Lemma 3.1 Ck is similar to a matrix with constant row sums and
+with its last column being positive. Thus, Ck is similar to a diagonalizable
+positive matrix. Then, ˇSmigoc’s glue applied to all matrices Ak gives an n-
+by-n diagonalizable irreducible nonnegative matrix A with a positive column
+and spectrum Λ. Therefore, A is similar to a diagonalizable positive matrix
+with spectrum Λ and from the extension in [1] Λ is UR.
+Observe that if λi ̸= 0, i = 2, . . . , n; s1(Λ) > 0; b2(Ak) > 0 in Corollary
+3.2, then we can guarantee the existence of an n-by-n diagonalizable nonneg-
+ative irreducible matrix A with spectrum Λ and a positive column. Thus,
+this is enough to show the universal realizability of Λ.
+Example 3.1 Consider the list
+Λ
+=
+{23, −2, −2, −1 ± 5i, −1 ± 5i, −1 ± 5i, −2 ± 7i, −2 ± 7i}, with
+Γ1
+=
+{23, −1 ± 5i}, Γ2 = {21, −2, −1 ± 5i, −2 ± 7i},
+Γ3
+=
+{13, −2, −1 ± 5i, −2 ± 7i}.
+The diagonalizable companion matrices
+A1
+=
+
+
+0
+0
+598
+1
+0
+20
+0
+1
+21
+
+ , A2 =
+
+
+0
+0
+0
+0
+0
+57 876
+1
+0
+0
+0
+0
+35 002
+0
+1
+0
+0
+0
+6266
+0
+0
+1
+0
+0
+1695
+0
+0
+0
+1
+0
+69
+0
+0
+0
+0
+1
+13
+
+
+,
+A3
+=
+
+
+0
+0
+0
+0
+0
+35 828
+1
+0
+0
+0
+0
+20 618
+0
+1
+0
+0
+0
+3194
+0
+0
+1
+0
+0
+903
+0
+0
+0
+1
+0
+5
+0
+0
+0
+0
+1
+5
+
+
+7
+
+realize lists Γ1, Γ2 and Γ3, respectively.
+ˇSmigoc’s glue technique applied to
+matrices A1, A2 and A3 gives a 13-by-13 diagonalizable irreducible nonnega-
+tive matrix with a positive column and spectrum Λ. Therefore, from Lemma
+3.1 and [1], Λ is UR.
+4
+The effect of adding a negative real number
+to a not UR list
+In this section we show how to add a negative real number −c to a list of
+complex numbers
+Λ = {λ, −a ± bi, . . . , −a ± bi}, λ, a, b > 0, with s1(Λ) > 0,
+which is not UR or we do not know whether it is, makes
+Λc = {λ, −c, −a ± bi, . . . , −a ± bi
+�
+��
+�
+(n−2) complex numbers
+}
+UR.
+For instance, the list Λ1 = {6, −1 ± 3i, −1 ± 3i} is realizable, but
+we do not know whether it is UR, while Λ2 = {17, −3 ± 9i, −3 ± 9i} is not
+realizable. However, both lists become UR if we add an appropriate negative
+real number −c to each of them.
+We start this section with a lemma which gives a formula to compute the
+coefficient b2 in (2), Lemma 2.1, for lists Λc
+Lemma 4.1 Let
+Λc = {λ, −c, −a ± bi, . . . , −a ± bi
+�
+��
+�
+(n−2) complex numbers
+}
+be a realizable left half-plane lists of complex numbers and let Λc = Λ1 ∪ Λ2 ∪
+· · ·∪Λt be a decomposition of Λc, −c ∈ Λt, with auxiliary lists Γk with realizing
+companion matrices Ak, k = 1, 2, . . . , t, as in Corollary 3.1, associated with a
+desired JCF allowed by Λc. Then the entry in position (n − 1, n) of a matrix
+Ak, k = 1, 2, . . . , t, is
+b2 = p(2aλ − 2a2n + (4k − 2p + 1)a2 − b2) + c(λ − (n − 2)a),
+(3)
+8
+
+where (k −1) is the number of pairs −a±bi of the last list Γt of the diagonal-
+izable decomposition of Λc, plus the number of pairs −a ± bi of each previous
+list Γk, k = 1, . . . , t − 1, of the decomposition, and p is the number of pairs
+−a ± bi of the corresponding list Γk. Moreover, b2 increases if k increases.
+Proof. It is well known that b2 =
+�
+1≤j1<j2≤n
+λj1λj2, with λji ∈ Γk, from which
+b2 in (3) is obtained. Moreover it is clear that b2 increases when k increases.
+Example 4.1 Consider
+Λc = {77
+4 , −3, −2 ± 5i, . . . , −2 ± 5i
+�
+��
+�
+8 complex numbers
+}.
+The last diagonalizable list from the diagonalizable decomposition of Λc is
+Γ4 : (x − 29
+4 )(x + 3)(x + 2 − 5i)(x + 2 + 5i)
+with realizing matrix
+A4
+=
+
+
+0
+0
+0
+2523
+4
+1
+0
+0
+841
+4
+0
+1
+0
+39
+4
+0
+0
+1
+1
+4
+
+ −→ b2(A4) = 39
+4
+b2
+=
+p(2aλ − 2a2n + (4k − 2p + 1)a2 − b2) + c(λ − (n − 2)a)
+b2(A4)
+=
+(477
+4 − 80 + (8 − 2 + 1)4 − 25) + 3(77
+4 − 16) = 39
+4 .
+Suppose we want to obtain a nonnegative matrix with JCF
+J = diag{J1(77
+4 ), J1(−3), J2(−2 + 5i), (J2(−2 − 5i)}.
+Then,
+Γ′
+1
+=
+{77
+4 , −2 ± 5i, −2 ± 5i}
+Γ′
+2
+=
+{45
+4 , −3, −2 ± 5i, −2 ± 5i}.
+9
+
+If A′
+1, A′
+2 are companion realizing matrices for Γ′
+1 and Γ′
+2, respectively, then
+from Lemma 4.1, b2(A′
+2) = 103
+4 , b2(A′
+1) = 80 guarantee that A′
+1 and A′
+2 are
+nonnegative. Next, the glue of A′
+1 with A′
+2 gives a nonnegative matrix with
+JCF J.
+Theorem 4.1 Let Λ = {λ, −a ± bi, . . . , −a ± bi}, fixed λ, a, b > 0, be a list
+of complex numbers with s1(Λ) > 0. If
+(2n − 11)a2 + b2
+2a
+≤ λ,
+(4)
+and there is a real number c > 0 such that
+2a(na − λ) + b2 − 7a2
+λ − (n − 2)a
+≤ c ≤ λ − (n − 2)a,
+(5)
+then
+Λc = {λ, −c, −a ± bi, . . . , −a ± bi
+�
+��
+�
+(n−2) complex numbers
+}
+becomes universally realizable.
+Proof. Consider the decomposition Λc = Λ1 ∪ Λ2 ∪ · · · ∪ Λ n−2
+2 , with
+Λ1
+=
+{λ, −a ± bi},
+Λk
+=
+{−a ± bi}, k = 2, . . . ,n − 4
+2
+,
+Λ n−2
+2
+=
+{−c, −a ± bi}.
+We take the auxiliary sub-lists
+Γ1
+=
+Λ1 = {λ, −a ± bi}
+Γ2
+=
+{λ − 2a, −a ± bi}
+Γ3
+=
+{λ − 4a, −a ± bi}
+...
+Γ n−4
+2
+=
+{λ − (n − 6)a, −a ± bi},
+Γ n−2
+2
+=
+{λ − (n − 4)a, −c, −a ± bi},
+10
+
+where Γ n−4
+2
+and Γ n−2
+2
+are the spectrum of the diagonalizable companion ma-
+trices
+A n−4
+2
+=
+
+
+0
+0
+(a2 + b2)(λ − (n − 6)a)
+1
+0
+2aλ − a2(2n − 11) − b2
+0
+1
+λ − (n − 4)a
+
+
+and
+A n−2
+2
+=
+
+
+0
+0
+0
+(a2 + b2)(λ − (n − 4)a)c
+1
+0
+0
+(a2 + b2)(λ − (n − 4)a) + (7a2 − b2 + 2aλ − 2a2n)c
+0
+1
+0
+(λ − (n − 2)a)c + (7a2 − b2 + 2aλ − 2a2n)
+0
+0
+1
+λ − (n − 2)a − c
+
+ ,
+respectively. Observe that sub-lists Γ n−6
+2 , . . . , Γ2, Γ1 have the same pair of
+complex numbers that the list Γ n−4
+2 , but with a bigger Perron eigenvalue.
+Then, if Γ n−4
+2
+is diagonalizably companion realizable, Γ n−6
+2 , . . . , Γ2, Γ1 also
+are. Thus, from Lemma 2.1 we only need to consider the entries in position
+(2, 3) in A n−4
+2
+and in position (3, 4) in A n−2
+2 . From (4) and (5) these entries
+are nonnegative and therefore A n−4
+2
+and A n−2
+2
+are diagonalizable companion
+realizing matrices. Thus, after applying n−4
+2
+times ˇSmigoc’s glue to the ma-
+trices A1, . . . , A n−2
+2 , we obtain an n-by-n diagonalizable nonnegative matrix
+A with spectrum Λc. Thus Λc is DR.
+To obtain an n-by-n nonnegative matrix A with spectrum Λc and a non-
+diagonal JCF J, we take Λc = Λ1 ∪ · · · ∪ Λt with auxiliary lists Γk being the
+spectrum of a companion matrix Ak with JCF as a sub-matrix of J. Next we
+need to prove that all Ak are nonnegative. To do that, we compute b2(At)
+from the formula in (3), where At (with Γt containing −c) is the last diago-
+nalizable matrix in the diagonalizable decomposition of Λc. From (4) and (5)
+b2(At) ≥ 0. From Lemma 4.1 all b2(Ak), k = 1, . . . , t − 1, are nonnegative.
+Therefore the glue of matrices Ak gives an n-by-n nonnegative matrix A with
+the desired JCF J.
+Example 4.2 i) Λ = {6, −1 ± 3i, −1 ± 3i} is realizable by the companion
+matrix
+C =
+
+
+0
+0
+0
+0
+600
+1
+0
+0
+0
+140
+0
+1
+0
+0
+104
+0
+0
+1
+0
+0
+0
+0
+0
+1
+2
+
+
+,
+11
+
+with a non-diagonal JCF. We do not know whether Λ has a diagonalizable
+realization. Then, consider the list
+Λc = {6, ���c, −1 ± 3i, −1 ± 3i}.
+Condition (4) is satisfied and from (5) we have 1 ≤ c ≤ 2. Then for c = 2,
+we have that
+Γ1 = {6, −1 ± 3i},
+Γ2 = {4, −2, −1 ± 3i}
+are the spectrum of diagonalizable nonnegative companion matrices
+A1 =
+
+
+0
+0
+60
+1
+0
+2
+0
+1
+4
+
+ , and A2 =
+
+
+0
+0
+0
+80
+1
+0
+0
+36
+0
+1
+0
+2
+0
+0
+1
+0
+
+ ,
+respectively. Then, from ˇSmigoc’s glue we obtain a diagonalizable nonnegative
+matrix with spectrum Λc. It is clear that, from the characteristic polynomial
+associated to Λc, Λc has also a companion realization A3,
+A3 =
+
+
+0
+0
+0
+0
+0
+1200
+1
+0
+0
+0
+0
+880
+0
+1
+0
+0
+0
+348
+0
+0
+1
+0
+0
+104
+0
+0
+0
+1
+0
+4
+0
+0
+0
+0
+1
+0
+
+
+,
+with a JCF with blocks of maximum size. Note that the formula in (3) gives
+(k = 2, p = 1, t = 2) b2(A2) = 2, while (k = 3, p = 2) gives b2(A3) = 4.
+Therefore Λc is UR. Observe that if 1 ≤ c ≤ 2, then
+Λc = {6, −c, −1 ± 3i, −1 ± 3i}
+is also UR.
+ii) Consider the list Λ = {17, −3±9i, −3±9i}. Since s1(Λ) = 5 and s2(Λ) =
+1, Λ is not realizable. From condition (5), 24
+5 ≤ c ≤ 5. Then for c = 5,
+Λc = {17, −5, −3 ± 9i, −3 ± 9i}
+12
+
+is UR. In fact,
+Γ1 = {17, −3 ± 9i} and Γ2 = {11, −5, −3 ± 9i}
+are the spectrum of diagonalizable nonnegative companion matrices, which
+from ˇSmigoc’s glue give rise to a diagonalizable nonnegative matrix with
+spectrum Λc. From the characteristic polynomial associated to Λc we obtain
+a nonnegative companion matrix with spectrum Λc and non-diagonal JCF.
+Therefore, Λc is UR.
+Observe that in Theorem 4.1, in spite that s1(Λ) > 0, if s1(Λ) is small
+enough, there are lists Λc, which are not UR or we cannot to prove they are
+from our procedure. However, from Theorem 4.1 we may compute a Perron
+eigenvalue λ, which guarantees that for a family of lists Λc, with c > 0 and
+n ≥ 6, Λc will be UR. Then, the following result characterizes a family of
+left half-plane lists, which are UR.
+Corollary 4.1 The left half-plane lists of the family
+Λc = { 1
+2a((2n − 7)a2 + b2), −c, −a ± bi, . . . , −a ± bi
+�
+��
+�
+(n−2) complex numbers
+},
+with 0 <
+√
+3a < b, 0 < c ≤ b2−3a2
+2a
+, are universally realizable..
+Proof. It is clear that for λ =
+1
+2a ((2n − 7)a2 + b2) , conditions (4) and (5)
+in Theorem 4.1 are satisfied. Moreover, from 0 <
+√
+3a < b, λ − (n − 2)a =
+b2−3a2
+2a
+> 0.
+Then, from Corollary 4.1 some left half-plane lists that are UR are:
+i) Λc
+=
+{2n − 3
+2
+a, −c, −a ± 2ai, . . . , −a ± 2ai
+�
+��
+�
+(n−2) complex numbers
+}, with 0 < c ≤ a
+2
+ii) Λc
+=
+{(n + 1)a, −c, −a ± 3ai, . . . , −a ± 3ai
+�
+��
+�
+(n−2) complex numbers
+}, with 0 < c ≤ 3a
+.
+iii) Λc
+=
+{2n + 9
+2
+a, −c, −a ± 4ai, . . . , −a ± 4ai
+�
+��
+�
+(n−2) complex numbers
+}, with 0 < c ≤ 13
+2 a
+iv) Λc
+=
+{8n − 3
+8
+a, −c, −a ± 5
+2ai, . . . , −a ± 5
+2ai
+�
+��
+�
+(n−2) complex numbers
+}, with 0 < c ≤ 13
+8 a,
+13
+
+and so on.
+Observe that in Corollary 4.1, if c is strictly less than its upper bound,
+then Λc, as we have seen, can be realized by a diagonalizable matrix with its
+last column being positive. Then, from the extension in [1], Λc is UR.
+5
+The merge of spectra
+Let Γ1 = {λ1, λ2, . . . , λn} and Γ2 = {µ1, µ2, . . . , µm} be lists of complex
+numbers. In [5] the authors define the concept of the merge of the spectra Γ1
+with Γ2 as
+Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm},
+and prove that if Γ1 and Γ2 are diagonalizably ODP realizable, then the
+merge Γ1 with Γ2, is also diagonalizably ODP realizable, and therefore from
+the extension in [4], Γ is UR. Here we set a similar result as follows:
+Theorem 5.1 Let Γ1 = {λ1, λ2, . . . , λn}, λ1 > |λi| , i = 2, . . . , n, be the
+spectrum of a diagonalizable nonnegative n-by-n matrix A ∈ CSλ1 with its last
+column being positive. Let Γ2 = {µ1, µ2, . . . , µm}, µ1 > |µi| , i = 2, . . . , m, be
+the spectrum of a diagonalizable nonnegative m-by-m matrix B ∈ CSµ1 with
+its last column being positive. Then
+Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm}
+is universally realizable..
+Proof. Let A ∈ CSλ1 be a diagonalizable nonnegative matrix with spec-
+trum Γ1 and with its last column being positive. Then A is similar to a
+diagonalizable positive matrix A′. If α1, . . . , αn are the diagonal entires of A′,
+then
+A1 = A′ + e[0, 0, . . . , µ1] =
+� A′
+11
+a
+bT
+αn + µ1
+�
+∈ CSλ1+µ1
+is diagonalizable positive with spectrum {λ1 + µ1, λ2, . . . , λn} and diagonal
+entries α1, α2, . . . , αn + µ1. Let B ∈ CSµ1 be a diagonalizable nonnegative
+matrix with spectrum Γ2 and with its last column being positive. Then B is
+similar to a diagonalizable positive matrix B′ and
+B1 = B′ + e[αn, 0, . . . , 0]
+14
+
+is diagonalizable positive with spectrum {µ1 + αn, µ2, . . . , µm}. Now, by ap-
+plying the ˇSmigoc’s glue to matrices A1 and B1, we obtain a diagonalizable
+positive matrix C with spectrum Γ. Hence, Γ is UR
+Theorem 5.1 is useful to decide, in many cases, about the universal real-
+izability of left half-plane list of complex numbers, as for instance:
+Example 5.1 Is the list
+Γ = {30, −1, −5, −1 ± 3i, −1 ± 3i, −1 ± 3i, −3 ± 9i, −3 ± 9i} UR?
+Observe that from the results in Section 4,
+Γ1
+=
+{21, −5, −3 ± 9i, −3 ± 9i}.
+Γ2
+=
+{9, −1, −1 ± 3i, −1 ± 3i, −1 ± 3i}
+are the spectrum of a diagonalizably nonnegative matrix with constant row
+sums and a positive column (the last one). Then, they are similar to diag-
+onalizable positive matrices and from Theorem 5.1, the merge Γ is also the
+spectrum of a diagonalizable positive matrix. Therefore, Γ is UR.
+References
+[1] M. Collao, M. Salas, R. L. Soto, Spectra universally realizable by doubly
+stochastic matrices, Special Matrices 6 (2018) 301-309.
+[2] R. C. Diaz, R. L. Soto, Nonnegative inverse elementary divisors problem
+in the left half-plane, Linear Multilinear Algebra 64 (2016) 258-268.
+[3] C. R. Johnson, Row stochastic matrices similar to doubly stochastic
+matrices, Linear Multilinear Algebra 10 (1981) 113-130.
+[4] C. R. Johnson, A. I. Julio, R. L. Soto, Nonnegative realizability with
+Jordan structure, Linear Algebra Appl. 587 (2020) 302-313.
+[5] C. R. Johnson, A. I. Julio, R. L. Soto, Indices of diagonalizable and
+universal realizability of spectra, submitted.
+[6] T. J. Laffey, H. ˇSmigoc, Nonnegative realization of spectra having neg-
+ative real parts, Linear Algebra Appl. 416 (2006) 148-159.
+15
+
+[7] H. Minc, Inverse elementary divisor problem for nonnegative matrices,
+Proc. of the Amer. Math. Society 83 (1981) 665-669.
+[8] H. Minc, Inverse elementary divisor problem for doubly stochastic ma-
+trices, Linear Multilinear Algebra 11 (1982) 121-131.
+[9] H. ˇSmigoc, The inverse eigenvalue problem for nonnegative matrices,
+Linear Algebra Appl. 393 (2004) 365-374.
+[10] R. L. Soto, R. C. D´ıaz, H. Nina, M. Salas, Nonnegative matrices with
+prescribed spectrum and elementary divisors, Linear Algebra Appl. 439
+(2013) 3591-3604.
+16
+

5NFIT4oBgHgl3EQf7itm/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf,len=531
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='11398v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='SP]  26 Jan 2023  ˇSmigoc’s glue for universal realizability in the  left half-plane∗  Jaime H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Alfaro, Ricardo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Soto†  Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Matem´aticas, Universidad Cat´olica del Norte, Casilla 1280  Antofagasta, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Abstract  A list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} of complex numbers is said to be real-  izable if it is the spectrum of a nonnegative matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Λ is said to be  universally realizable (UR) if it is realizable for each possible Jordan  canonical form allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In this paper, using companion matrices  and applying a procedure by ˇSmigoc, is provides a sufficient condi-  tion for the universal realizability of left half-plane spectra, that is,  Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} with λ1 > 0, Re λi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' It is also shown  how the effect of adding a negative real number to a not UR left half-  plane list of complex numbers, makes the new list UR, and a family  of left half-plane lists that are UR is characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  AMS classification:  15A18, 15A20, 15A29  Key words: Nonnegative matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' companion matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Universal realizabil-  ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' ˇSmigoc’s glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  1  Introduction  A list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} of complex numbers is said to be realizable if it  is the spectrum of an n-by-n nonnegative matrix A, and A is said to be a  realizing matrix for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The problem of the realizability of spectra is called the  ∗Supported by Universidad Cat´olica del Norte-VRIDT 036-2020, N´ucleo 6 UCN  VRIDT 083-2020, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  †E-mail addresses: rsoto@ucn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='cl (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Soto), jaime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='alfaro@ucn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='cl (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Alfaro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  1  nonnegative inverse eigenvalue problem (NIEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' From the Perron-Frobenius  Theorem we know that if Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} is the spectrum of an n-  by-n nonnegative matrix A, then the leading eigenvalue of A equals to the  spectral radius of A, ρ(A) =: max  1≤i≤n |λi| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' This eigenvalue is called the Perron  eigenvalue, and we shall assume in this paper, that ρ(A) = λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  A matrix is said to have constant row sums, if each one of its rows sums  up to the same constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The set of all matrices with constant row sums  equal to α, is denoted by CSα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, any matrix A ∈ CSα has the eigenvector  eT = [1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , 1], corresponding to the eigenvalue α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The real matrices with  constant row sums are important because it is known that the problem of  finding a nonnegative matrix with spectrum Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn}, is equivalent  to the problem of finding a nonnegative matrix in CSλ1 with spectrum Λ (see  [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' We denote by ek, the n-dimensional vector, with 1 in the kth position  and zeros elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' If Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn}, then sk(Λ) =  n  �  i=1  λk  i , k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  A list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} of complex numbers, is said to be diagonaliz-  ably realizable (DR), if there is a diagonalizable realizing matrix for Λ The  list Λ is said to be universally realizable (UR), if it is realizable for each possi-  ble Jordan canonical form (JCF) allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The problem of the universal  realizability of spectra, is called the universal realizability problem (URP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  The URP contains the NIEP, and both problems are equivalent if the given  numbers λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In terms of n, both problems remain  unsolved for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' It is clear that if Λ is UR, then Λ must be DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The  first known results on the URP are due to Minc [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In terms of the URP,  Minc [7] showed that if a list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} of complex numbers is the  spectrum of a diagonalizable positive matrix, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The positivity  condition is necessary for Minc’s proof, and the question set by Minc himself,  whether the result holds for nonnegative realizations was open for almost 40  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Recently, two extensions of Minc’s result have been obtained in [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  In [1], Collao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' showed that a nonnegative matrix A ∈ CSλ1, with a pos-  itive column, is similar to a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Note that if A is nonnegative  with a positive row and AT has a positive eigenvector, then AT is also similar  to a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Besides, if Λ is diagonalizably realizable by a matrix  A ∈ CSλ1 having a positive column, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In [4], Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' in-  troduced the concept of ODP matrices, that is, nonnegative matrices with all  positive off-diagonal entries (zero diagonal entries are permitted) and proved  2  that if Λ is diagonalizably ODP realizable, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Note that both  extensions contain, as a particular case, Minc’s result in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Both extensions  allow us to significantly increase the set of spectra that can be proved to be  UR, as for instance, certain spectra Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} with s1(Λ) = 0, which  is not possible from Minc’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In particular, we shall use the extension  in [1] to generate some of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 In [1], Section 2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, there is an  error in assuming that if A is nonnegative with a positive row, then AT, which  has a positive column, is similar to a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The reason is that we  cannot guarantee that AT has a positive eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Regarding non-positive universal realizations, we mention that in [10,  2] the authors proved, respectively, that lists of complex numbers Λ =  {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn}, of Suleimanova type, that is,  λ1 > 0, Re λi ≤ 0, |Re λi| ≥ |Im λi| , i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n,  or of ˇSmigoc type, that is,  λ1 > 0, Re λi ≤ 0,  √  3 |Re λi| ≥ |Im λi| , i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n,  (1)  are UR if and only if they are realizable if and only if  n  �  i=1  λi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Outline of the paper: The paper is organized as follows: In Section 2, we  present the mathematical tools that will be used to generate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In  Section 3, we study the URP for a left half-plane list and we give a sufficient  condition for it to be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In Section 4, we discuss the effect of adding a  negative real number −c to a left half-plane list Λ = {λ1, −a±bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a±bi},  which is not UR (or even not realizable), or we do not know whether it is,  and we show how Λ ∪ {−c} becomes UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' We also characterize a family of  left half-plane lists that are UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In Section 5, we show that the merge of two  lists diagonalizably realizable Γ1 ∈ CSλ1 and Γ2 ∈ CSµ1 is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Examples  are shown to illustrate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  2  Preliminaries  Throughout this paper we use the following results: The first one, by ˇSmigoc  [9], gives a procedure that we call ˇSmigoc’s glue technique, to obtain from two  3  matrices A and B of size n-by-n and m-by-m, respectively, a new (n+m−1)-  by-(n + m − 1) matrix C, preserving in certain way, the corresponding JCFs  of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' The second one, by Laffey and ˇSmigoc [6] solves the NIEP for  lists of complex numbers on the left half-plane, that is, lists with λ1 > 0,  Re λi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Moreover, we also use Lemma 5 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 [9] Suppose B is an m-by-m matrix with a JCF that contains  at least one 1-by-1 Jordan block corresponding to the eigenvalue c:  J(B) =  � c  0  0  I(B)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Let t and s, respectively, be the left and the right eigenvectors of B associated  with the 1-by-1 Jordan block in the above canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Furthermore, we  normalize vectors t and s so that t  Ts = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Let J(A) be a JCF for the n-by-n  matrix  A =  �  A1  a  bT  c  �  ,  where A1 is an (n − 1)-by-(n − 1) matrix and a and b are vectors in C  n-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Then the matrix  C =  �  A1  at  T  sb  T  B  �  has JCF  J(C) =  �  J(A)  0  0  I(B)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='2 [6] Let Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} be a list of complex numbers with  λ1 ≥ |λi| and Re λi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then Λ is realizable if and only if  s1 = s1(Λ) ≥ 0,  s2 = s2(Λ) ≥ 0,  s2  1 ≤ ns2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 [6] Let t be a nonnegative real number and let λ2, λ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn be  complex numbers with real parts less than or equal to zero, such that the list  {λ2, λ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} is closed under complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Set ρ = 2t−λ2−· · ·−λn  and  f(x) = (x − ρ)  n  �  j=2  (x − λj) = xn − 2txn−1 + b2xn−2 + · · · + bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  (2)  Then b2 ≤ 0 implies bj ≤ 0 for j = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  4  3  Companion matrices and the ˇSmigoc’s glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  We say that a list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} of complex numbers is on the left half-  plane if λ1 > 0, Re λi ≤ 0, i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In this section we give a sufficient  condition for a left half-plane list of complex numbers to be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Of course,  it is our interest to consider lists of complex numbers containing elements  out of realizability region of lists of ˇSmigoc type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Our strategy consists in to  decompose the given list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} into sub-lists  Λk = {λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λkpk}, λ11 = λ1, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t,  with auxiliary lists  Γ1  =  Λ1  Γk  =  {s1(Γk−1), λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λkpk},  k = 2, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t,  each one of them being the spectrum of a nonnegative companion matrix  Ak, in such a way that it be possible to apply ˇSmigoc’s glue technique to  the matrices Ak, to obtain an n-by-n nonnegative matrix with spectrum Λ  for each possible JCF allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In the case s1(Λ) > 0, with λi ̸= 0,  i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n, we may choose, if they exist, sub-lists Γk being the spectrum  of a diagonalizable nonnegative companion matrix with a positive column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Then, after ˇSmigoc’s glue, we obtain a diagonalizable nonnegative n-by-n  matrix A with spectrum Λ and a positive column, which is similar to a  diagonalizable positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Thus, from the extension in [1], Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Next we have the following corollary from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Let Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} be a realizable left half-plane list of  complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Suppose that for each JCF J allowed by Λ, there exists a  decomposition of Λ as  Λ  =  Λ1 ∪ Λ2 ∪ · · · ∪ Λt, where  Λk  =  {λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λkpk}, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t, λ11 = λ1,  with auxiliary lists  ��1  =  Λ1,  Γk  =  {s1(Γk−1), λk1, λk2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λkpk}, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t,  being the spectrum of a nonnegative companion matrix Ak with JCF J(Ak)  as a sub-matrix of J, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Then Λ is universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Since each matrix Ak, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t, is nonnegative companion  with JCF J(Ak) being a submatrix of J, then, from ˇSmigoc’s glue applied to  matrices Ak, we obtain an n-by-n nonnegative matrix with spectrum Λ and  JCF J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' As J is any JCF allowed by Λ, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  The following result is well known and useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Let A be a diagonalizable irreducible nonnegative matrix with  spectrum Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} and a positive row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then A is similar  to a diagonalizable nonnegative matrix B ∈ CSλ1, with a positive row or  column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' If A is irreducible nonnegative, it has a positive eigenvector xT =  [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then if D = dig{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , xn}, the matrix  B = D−1AD =  �xj  xi  ai,j  �  ∈ CSλ1  is nonnegative with a positive row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Suppose all lists Γk in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, can be taken as the spectrum of a  diagonalizable nonnegative companion matrix Ak with a positive column (the  last one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, since the glue of matrices Ak gives an n-by-n diagonalizable  irreducible nonnegative matrix A with a positive column and spectrum Λ, A  is similar to a diagonalizable positive matrix with spectrum Λ and therefore  Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' This is what the next result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='2 Let Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn}, λi ̸= 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n, s1(Λ) > 0, be  a realizable left half-plane list of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' If there is a decomposition  of Λ as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, with all lists Γk being the spectrum of a diagonal-  izable nonnegative companion matrix Ak, with a positive column, then Λ is  universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' It is enough to prove the result for two lists Γk of the decomposition  of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Let Γk−1 and Γk, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t, be the spectrum, respectively, of matrices  Ak−1 and Ak, which are diagonalizable nonnegative companion with a posi-  tive column (the last one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then Ak−1 and Ak are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In particular,  Ak has a positive eigenvector s and, since AT  k is also irreducible, Ak has also  a positive left eigenvector tT with tTs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Now, let  Ak−1 =  � A1,k−1  a  bT  s1(Γk−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  6  Since the last column of Ak−1 is positive, the vector a is also positive and  atT is a positive submatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Therefore, the glue of Ak−1 with Ak,  Ck =  � A1,k−1  atT  sbT  Ak  �  ,  is a diagonalizable nonnegative matrix with its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Note that Ck is also irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then Ck has, besides, a positive eigenvector,  and from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Ck is similar to a matrix with constant row sums and  with its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Thus, Ck is similar to a diagonalizable  positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, ˇSmigoc’s glue applied to all matrices Ak gives an n-  by-n diagonalizable irreducible nonnegative matrix A with a positive column  and spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Therefore, A is similar to a diagonalizable positive matrix  with spectrum Λ and from the extension in [1] Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Observe that if λi ̸= 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' s1(Λ) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' b2(Ak) > 0 in Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='2, then we can guarantee the existence of an n-by-n diagonalizable nonneg-  ative irreducible matrix A with spectrum Λ and a positive column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Thus,  this is enough to show the universal realizability of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Consider the list  Λ  =  {23, −2, −2, −1 ± 5i, −1 ± 5i, −1 ± 5i, −2 ± 7i, −2 ± 7i}, with  Γ1  =  {23, −1 ± 5i}, Γ2 = {21, −2, −1 ± 5i, −2 ± 7i},  Γ3  =  {13, −2, −1 ± 5i, −2 ± 7i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  The diagonalizable companion matrices  A1  =  \uf8ee  \uf8f0  0  0  598  1  0  20  0  1  21  \uf8f9  \uf8fb , A2 =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  0  0  0  57 876  1  0  0  0  0  35 002  0  1  0  0  0  6266  0  0  1  0  0  1695  0  0  0  1  0  69  0  0  0  0  1  13  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ,  A3  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  0  0  0  35 828  1  0  0  0  0  20 618  0  1  0  0  0  3194  0  0  1  0  0  903  0  0  0  1  0  5  0  0  0  0  1  5  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  7  realize lists Γ1, Γ2 and Γ3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  ˇSmigoc’s glue technique applied to  matrices A1, A2 and A3 gives a 13-by-13 diagonalizable irreducible nonnega-  tive matrix with a positive column and spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Therefore, from Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 and [1], Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  4  The effect of adding a negative real number  to a not UR list  In this section we show how to add a negative real number −c to a list of  complex numbers  Λ = {λ, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± bi}, λ, a, b > 0, with s1(Λ) > 0,  which is not UR or we do not know whether it is, makes  Λc = {λ, −c, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± bi  �  ��  �  (n−2) complex numbers  }  UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  For instance, the list Λ1 = {6, −1 ± 3i, −1 ± 3i} is realizable, but  we do not know whether it is UR, while Λ2 = {17, −3 ± 9i, −3 ± 9i} is not  realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' However, both lists become UR if we add an appropriate negative  real number −c to each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  We start this section with a lemma which gives a formula to compute the  coefficient b2 in (2), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, for lists Λc  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Let  Λc = {λ, −c, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± bi  �  ��  �  (n−2) complex numbers  }  be a realizable left half-plane lists of complex numbers and let Λc = Λ1 ∪ Λ2 ∪  · ·∪Λt be a decomposition of Λc, −c ∈ Λt, with auxiliary lists Γk with realizing  companion matrices Ak, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t, as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, associated with a  desired JCF allowed by Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then the entry in position (n − 1, n) of a matrix  Ak, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t, is  b2 = p(2aλ − 2a2n + (4k − 2p + 1)a2 − b2) + c(λ − (n − 2)a),  (3)  8  where (k −1) is the number of pairs −a±bi of the last list Γt of the diagonal-  izable decomposition of Λc, plus the number of pairs −a ± bi of each previous  list Γk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t − 1, of the decomposition, and p is the number of pairs  −a ± bi of the corresponding list Γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Moreover, b2 increases if k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' It is well known that b2 =  �  1≤j1<j2≤n  λj1λj2, with λji ∈ Γk, from which  b2 in (3) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Moreover it is clear that b2 increases when k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Consider  Λc = {77  4 , −3, −2 ± 5i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −2 ± 5i  �  ��  �  8 complex numbers  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  The last diagonalizable list from the diagonalizable decomposition of Λc is  Γ4 : (x − 29  4 )(x + 3)(x + 2 − 5i)(x + 2 + 5i)  with realizing matrix  A4  =  \uf8ee  \uf8ef\uf8ef\uf8f0  0  0  0  2523  4  1  0  0  841  4  0  1  0  39  4  0  0  1  1  4  \uf8f9  \uf8fa\uf8fa\uf8fb −→ b2(A4) = 39  4  b2  =  p(2aλ − 2a2n + (4k − 2p + 1)a2 − b2) + c(λ − (n − 2)a)  b2(A4)  =  (477  4 − 80 + (8 − 2 + 1)4 − 25) + 3(77  4 − 16) = 39  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Suppose we want to obtain a nonnegative matrix with JCF  J = diag{J1(77  4 ), J1(−3), J2(−2 + 5i), (J2(−2 − 5i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Then,  Γ′  1  =  {77  4 , −2 ± 5i, −2 ± 5i}  Γ′  2  =  {45  4 , −3, −2 ± 5i, −2 ± 5i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  9  If A′  1, A′  2 are companion realizing matrices for Γ′  1 and Γ′  2, respectively, then  from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, b2(A′  2) = 103  4 , b2(A′  1) = 80 guarantee that A′  1 and A′  2 are  nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Next, the glue of A′  1 with A′  2 gives a nonnegative matrix with  JCF J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Let Λ = {λ, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± bi}, fixed λ, a, b > 0, be a list  of complex numbers with s1(Λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' If  (2n − 11)a2 + b2  2a  ≤ λ,  (4)  and there is a real number c > 0 such that  2a(na − λ) + b2 − 7a2  λ − (n − 2)a  ≤ c ≤ λ − (n − 2)a,  (5)  then  Λc = {λ, −c, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± bi  �  ��  �  (n−2) complex numbers  }  becomes universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Consider the decomposition Λc = Λ1 ∪ Λ2 ∪ · · · ∪ Λ n−2  2 , with  Λ1  =  {λ, −a ± bi},  Λk  =  {−a ± bi}, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' ,n − 4  2  ,  Λ n−2  2  =  {−c, −a ± bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  We take the auxiliary sub-lists  Γ1  =  Λ1 = {λ, −a ± bi}  Γ2  =  {λ − 2a, −a ± bi}  Γ3  =  {λ − 4a, −a ± bi}  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Γ n−4  2  =  {λ − (n − 6)a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' −a ± bi},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Γ n−2  2  =  {λ − (n − 4)a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' −c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' −a ± bi},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  10  where Γ n−4  2  and Γ n−2  2  are the spectrum of the diagonalizable companion ma-  trices  A n−4  2  =  \uf8ee  \uf8f0  0  0  (a2 + b2)(λ − (n − 6)a)  1  0  2aλ − a2(2n − 11) − b2  0  1  λ − (n − 4)a  \uf8f9  \uf8fb  and  A n−2  2  =  \uf8ee  \uf8ef\uf8ef\uf8f0  0  0  0  (a2 + b2)(λ − (n − 4)a)c  1  0  0  (a2 + b2)(λ − (n − 4)a) + (7a2 − b2 + 2aλ − 2a2n)c  0  1  0  (λ − (n − 2)a)c + (7a2 − b2 + 2aλ − 2a2n)  0  0  1  λ − (n − 2)a − c  \uf8f9  \uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Observe that sub-lists Γ n−6  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , Γ2, Γ1 have the same pair of  complex numbers that the list Γ n−4  2 , but with a bigger Perron eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Then, if Γ n−4  2  is diagonalizably companion realizable, Γ n−6  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , Γ2, Γ1 also  are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Thus, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 we only need to consider the entries in position  (2, 3) in A n−4  2  and in position (3, 4) in A n−2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' From (4) and (5) these entries  are nonnegative and therefore A n−4  2  and A n−2  2  are diagonalizable companion  realizing matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Thus, after applying n−4  2  times ˇSmigoc’s glue to the ma-  trices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , A n−2  2 , we obtain an n-by-n diagonalizable nonnegative matrix  A with spectrum Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Thus Λc is DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  To obtain an n-by-n nonnegative matrix A with spectrum Λc and a non-  diagonal JCF J, we take Λc = Λ1 ∪ · · · ∪ Λt with auxiliary lists Γk being the  spectrum of a companion matrix Ak with JCF as a sub-matrix of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Next we  need to prove that all Ak are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' To do that, we compute b2(At)  from the formula in (3), where At (with Γt containing −c) is the last diago-  nalizable matrix in the diagonalizable decomposition of Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' From (4) and (5)  b2(At) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 all b2(Ak), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , t − 1, are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Therefore the glue of matrices Ak gives an n-by-n nonnegative matrix A with  the desired JCF J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='2 i) Λ = {6, −1 ± 3i, −1 ± 3i} is realizable by the companion  matrix  C =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  0  0  600  1  0  0  0  140  0  1  0  0  104  0  0  1  0  0  0  0  0  1  2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ,  11  with a non-diagonal JCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' We do not know whether Λ has a diagonalizable  realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, consider the list  Λc = {6, −c, −1 ± 3i, −1 ± 3i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Condition (4) is satisfied and from (5) we have 1 ≤ c ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then for c = 2,  we have that  Γ1 = {6, −1 ± 3i},  Γ2 = {4, −2, −1 ± 3i}  are the spectrum of diagonalizable nonnegative companion matrices  A1 =  \uf8ee  \uf8f0  0  0  60  1  0  2  0  1  4  \uf8f9  \uf8fb , and A2 =  \uf8ee  \uf8ef\uf8ef\uf8f0  0  0  0  80  1  0  0  36  0  1  0  2  0  0  1  0  \uf8f9  \uf8fa\uf8fa\uf8fb ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, from ˇSmigoc’s glue we obtain a diagonalizable nonnegative  matrix with spectrum Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' It is clear that, from the characteristic polynomial  associated to Λc, Λc has also a companion realization A3,  A3 =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  0  0  0  1200  1  0  0  0  0  880  0  1  0  0  0  348  0  0  1  0  0  104  0  0  0  1  0  4  0  0  0  0  1  0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ,  with a JCF with blocks of maximum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Note that the formula in (3) gives  (k = 2, p = 1, t = 2) b2(A2) = 2, while (k = 3, p = 2) gives b2(A3) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Therefore Λc is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Observe that if 1 ≤ c ≤ 2, then  Λc = {6, −c, −1 ± 3i, −1 ± 3i}  is also UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  ii) Consider the list Λ = {17, ��3±9i, −3±9i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Since s1(Λ) = 5 and s2(Λ) =  1, Λ is not realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' From condition (5), 24  5 ≤ c ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then for c = 5,  Λc = {17, −5, −3 ± 9i, −3 ± 9i}  12  is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In fact,  Γ1 = {17, −3 ± 9i} and Γ2 = {11, −5, −3 ± 9i}  are the spectrum of diagonalizable nonnegative companion matrices, which  from ˇSmigoc’s glue give rise to a diagonalizable nonnegative matrix with  spectrum Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' From the characteristic polynomial associated to Λc we obtain  a nonnegative companion matrix with spectrum Λc and non-diagonal JCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Therefore, Λc is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Observe that in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, in spite that s1(Λ) > 0, if s1(Λ) is small  enough, there are lists Λc, which are not UR or we cannot to prove they are  from our procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' However, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 we may compute a Perron  eigenvalue λ, which guarantees that for a family of lists Λc, with c > 0 and  n ≥ 6, Λc will be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, the following result characterizes a family of  left half-plane lists, which are UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 The left half-plane lists of the family  Λc = { 1  2a((2n − 7)a2 + b2), −c, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± bi  �  ��  �  (n−2) complex numbers  },  with 0 <  √  3a < b, 0 < c ≤ b2−3a2  2a  , are universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='.  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' It is clear that for λ =  1  2a ((2n − 7)a2 + b2) , conditions (4) and (5)  in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Moreover, from 0 <  √  3a < b, λ − (n − 2)a =  b2−3a2  2a  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Then, from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 some left half-plane lists that are UR are:  i) Λc  =  {2n − 3  2  a, −c, −a ± 2ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± 2ai  �  ��  �  (n−2) complex numbers  }, with 0 < c ≤ a  2  ii) Λc  =  {(n + 1)a, −c, −a ± 3ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± 3ai  �  ��  �  (n−2) complex numbers  }, with 0 < c ≤ 3a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  iii) Λc  =  {2n + 9  2  a, −c, −a ± 4ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± 4ai  �  ��  �  (n−2) complex numbers  }, with 0 < c ≤ 13  2 a  iv) Λc  =  {8n − 3  8  a, −c, −a ± 5  2ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , −a ± 5  2ai  �  ��  �  (n−2) complex numbers  }, with 0 < c ≤ 13  8 a,  13  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Observe that in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, if c is strictly less than its upper bound,  then Λc, as we have seen, can be realized by a diagonalizable matrix with its  last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, from the extension in [1], Λc is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  5  The merge of spectra  Let Γ1 = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} and Γ2 = {µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , µm} be lists of complex  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' In [5] the authors define the concept of the merge of the spectra Γ1  with Γ2 as  Γ = {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , µm},  and prove that if Γ1 and Γ2 are diagonalizably ODP realizable, then the  merge Γ1 with Γ2, is also diagonalizably ODP realizable, and therefore from  the extension in [4], Γ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Here we set a similar result as follows:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Let Γ1 = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn}, λ1 > |λi| , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , n, be the  spectrum of a diagonalizable nonnegative n-by-n matrix A ∈ CSλ1 with its last  column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Let Γ2 = {µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , µm}, µ1 > |µi| , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , m, be  the spectrum of a diagonalizable nonnegative m-by-m matrix B ∈ CSµ1 with  its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then  Γ = {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , µm}  is universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='.  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Let A ∈ CSλ1 be a diagonalizable nonnegative matrix with spec-  trum Γ1 and with its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then A is similar to a  diagonalizable positive matrix A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' If α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , αn are the diagonal entires of A′,  then  A1 = A′ + e[0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , µ1] =  � A′  11  a  bT  αn + µ1  �  ∈ CSλ1+µ1  is diagonalizable positive with spectrum {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , λn} and diagonal  entries α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , αn + µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Let B ∈ CSµ1 be a diagonalizable nonnegative  matrix with spectrum Γ2 and with its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then B is  similar to a diagonalizable positive matrix B′ and  B1 = B′ + e[αn, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , 0]  14  is diagonalizable positive with spectrum {µ1 + αn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' , µm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Now, by ap-  plying the ˇSmigoc’s glue to matrices A1 and B1, we obtain a diagonalizable  positive matrix C with spectrum Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Hence, Γ is UR  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 is useful to decide, in many cases, about the universal real-  izability of left half-plane list of complex numbers, as for instance:  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1 Is the list  Γ = {30, −1, −5, −1 ± 3i, −1 ± 3i, −1 ± 3i, −3 ± 9i, −3 ± 9i} UR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Observe that from the results in Section 4,  Γ1  =  {21, −5, −3 ± 9i, −3 ± 9i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='  Γ2  =  {9, −1, −1 ± 3i, −1 ± 3i, −1 ± 3i}  are the spectrum of a diagonalizably nonnegative matrix with constant row  sums and a positive column (the last one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Then, they are similar to diag-  onalizable positive matrices and from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content='1, the merge Γ is also the  spectrum of a diagonalizable positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
+page_content=' Therefore, Γ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'}
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+1
+Vocabulary-informed Zero-shot and Open-set
+Learning
+Yanwei Fu, Xiaomei Wang, Hanze Dong, Yu-Gang Jiang, Meng Wang, Xiangyang Xue, Leonid Sigal
+Abstract—Despite significant progress in object categorization, in recent years, a number of important challenges remain; mainly, the
+ability to learn from limited labeled data and to recognize object classes within large, potentially open, set of labels. Zero-shot learning
+is one way of addressing these challenges, but it has only been shown to work with limited sized class vocabularies and typically
+requires separation between supervised and unsupervised classes, allowing former to inform the latter but not vice versa. We propose
+the notion of vocabulary-informed learning to alleviate the above mentioned challenges and address problems of supervised, zero-
+shot, generalized zero-shot and open set recognition using a unified framework. Specifically, we propose a weighted maximum margin
+framework for semantic manifold-based recognition that incorporates distance constraints from (both supervised and unsupervised)
+vocabulary atoms. Distance constraints ensure that labeled samples are projected closer to their correct prototypes, in the embedding
+space, than to others. We illustrate that resulting model shows improvements in supervised, zero-shot, generalized zero-shot, and large
+open set recognition, with up to 310K class vocabulary on Animal with Attributes and ImageNet datasets.
+Index Terms—Vocabulary-informed learning, Generalized zero-shot learning, Open-set recognition, Zero-shot learning.
+!
+1
+INTRODUCTION
+Object recognition, more specifically object categorization, has
+seen unprecedented advances in recent years with development
+of convolutional neural networks (CNNs) [41]. However, most
+successful recognition models, to date, are formulated as su-
+pervised learning problems, in many cases requiring hundreds,
+if not thousands, labeled instances to learn a given concept
+class [15]. This exuberant need for large labeled instances
+has limited recognition models to domains with hundreds to
+thousands of classes. Humans, on the other hand, are able
+to distinguish beyond 30, 000 basic level categories [8]. Even
+more impressive is the fact that humans can learn from few
+examples, by effectively leveraging information from other
+object category classes, and even recognize objects without
+ever seeing them (e.g., by reading about them on the Internet).
+This ability has spawned the research in few-shot and zero-
+shot learning.
+Zero-shot learning (ZSL) has now been widely studied
+in a variety of research areas including neural decoding of
+fMRI images [54], character recognition [44], face verification
+[42], object recognition [43], and video understanding [27],
+[82]. Typically, zero-shot learning approaches aim to recog-
+nize instances from the unseen or unknown testing target
+• Yanwei Fu and Hanze Dong are with the School of Data Science, Fudan
+University, Shanghai. Email: {yanweifu,hzdong15}@fudan.edu.cn.
+• Xiaomei Wang, Yu-Gang Jiang and Xiangyang Xue are with the School of
+Computer Science, Shanghai Key Lab of Intelligent Information Processing,
+Fudan University. Email: {17110240025,ygj,xyxue}@fudan.edu.cn.
+• Yu-Gang Jiang is the corresponding author. Yanwei Fu is also with
+AITRICS.
+• Meng
+Wang
+is
+with
+the
+School
+of
+Computer
+and
+Information
+Science,
+Hefei
+University
+of
+Technology,
+Hefei,
+China.
+Email:
+eric.wangmeng@gmail.com.
+• Leonid Sigal is with the Department of Computer Science, University of
+British Columbia, BC, Canada. Email: lsigal@cs.ubc.ca.
+categories by transferring information through intermediate-
+level semantic representations, from known observed source
+(or auxiliary) categories for which many labeled instances
+exist. In other words, supervised classes/instances, are used
+as context for recognition of classes that contain no visual
+instances at training time, but that can be put in some
+correspondence with supervised classes/instances. Therefore,
+a general experimental setting of ZSL is that the classes in
+target and source (auxiliary) dataset are disjoint. Typically, the
+learning is done on the source dataset and then information is
+transferred to the target dataset, with performance measured
+on the latter.
+This setting has a few important drawbacks: (1) it assumes
+that target classes cannot be mis-classified as source classes
+and vice versa; this greatly and unrealistically simplifies the
+problem; (2) the target label set is often relatively small,
+between ten [43] and several thousand unknown labels [24],
+compared to at least 30, 000 entry level categories that humans
+can distinguish; (3) large amounts of data in the source
+(auxiliary) classes are required, which is problematic as it has
+been shown that most object classes have very few instances
+(long-tailed distribution of objects in the world [72]); and
+(4) the vast open set vocabulary and corresponding semantic
+knowledge, defined as part of ZSL [54], is not leveraged in
+any way to inform the learning or source class recognition.
+A few works recently looked at resolving (1) through
+class-incremental learning [66], [68] or generalized zero-shot
+learning (G-ZSL) [11], [54] which are designed to distinguish
+between seen (source) and unseen (target) classes at the testing
+time and apply an appropriate model – supervised for the
+former and ZSL for the latter. However, (2)–(4) remain largely
+unresolved. In particular, while (2) and (3) are artifacts of the
+ZSL setting, (4) is more fundamental; e.g., a recent study [34]
+argues that concepts, in our own brains, are represented in
+the form of a continuous semantic space mapped smoothly
+arXiv:2301.00998v1  [cs.CV]  3 Jan 2023
+
+2
+Fig. 1.
+Illustration of the semantic embeddings learned (left) using support vector regression (SVR) and (right)
+using the proposed vocabulary-informed learning (Deep WMM-Voc) approach. In both cases, t-SNE visualization is
+used to illustrate samples from 4 source/auxiliary classes (denoted by ×) and 2 target/zero-shot classed (denoted by
+◦) from the ImageNet dataset. Decision boundaries, illustrated by dashed lines, are drawn by hand for visualization.
+The large margin constraints, both among the source/target classes and the external vocabulary atoms, are denoted
+by arrows and words on the right. Note that the WMM-Voc approach on the right leads to a better embedding with
+more compact and separated classes (e.g., see truck and car or unicycle and tricycle).
+across the cortical surface. For example, consider learning
+about a car by looking at image instances in Figure 1. Not
+knowing that other motor vehicles exist in the world, one may
+be tempted to call anything that has 4-wheels a car. As a
+result, the zero-shot class truck may have a large overlap with
+the car class (see Figure 1 (left)). However, imagine knowing
+that there also exist many other motor vehicles (trucks, mini-
+vans, etc). Even without having visually seen such objects,
+the very basic knowledge that they exist in the world and are
+closely related to a car should, in principle, alter the criterion
+for recognizing instance as a car (making the recognition
+criterion stricter in this case). Encoding this in our vocabulary-
+informed learning model results in better separation among
+classes (see Figure 1 (right)).
+To tackle the limitations of ZSL and towards the goal
+of generic recognition, we propose the idea of vocabulary-
+informed learning. Specifically, assuming we have few labeled
+training instances and a large, potentially open set, vocab-
+ulary/semantic dictionary (along with textual sources from
+which statistical semantic relations among vocabulary atoms
+can be learned), the task of vocabulary-informed learning is to
+learn a unified model that utilizes this semantic dictionary to
+help train better classifiers for observed (source) classes and
+unobserved (target) classes in supervised, zero-shot, general-
+ized zero-shot, and open set image recognition settings.
+In particular, we formulate Weighted Maximum Margin
+Vocabulary-informed Embedding (WMM-Voc), which learns
+a joint embedding for visual features and semantic words.
+In this formulation, two maximum margin sets of constraints
+are simultaneously optimized. The first set ensures that la-
+beled training visual instances, belonging to a particular class,
+project close to semantic word vector prototype corresponding
+to the class name in the embedding space. The second set
+ensures that these instances are closer to the correct class
+word vector prototype than to any of the incorrect ones in
+the embedding space; including those that may not contain
+training data (i.e., zero-shot). The constraints in the first
+set further take into the account the distribution of training
+samples for each class, and nearby classes, to dynamically
+set appropriate margins. In other words, for some classes the
+distance, between the projected training sample and the word
+vector prototype, is explicitly penalized more (or less) than for
+others. This weighting is derived using extreme values theory.
+Contributions: Our main contribution is to propose a
+novel paradigm for potentially open set image recognition:
+vocabulary-informed learning (Voc), which is capable of uti-
+lizing vocabulary over unsupervised items, during training, to
+improve recognition. We extend the model initially proposed
+by us in a conference paper [29] to include class-specific
+weighting in the data term, as well as the ability to run the
+models as an end-to-end network. Particularly, classification is
+done through the nearest-neighbor distance to class prototypes
+in the semantic embedding space. Semantic embedding is
+learned subject to constraints ensuring that labeled images
+project into semantic space such that they end up closer to
+the correct class prototypes than to incorrect ones (whether
+those prototypes are part of the source or target classes). We
+show that word embedding (word2vec) can be used effectively
+
+mini van
+X××o×0
+roller
+0
+skate
+%
+00
+XOX
+8.
+x8
+X
+%
+X
+X
+to
+8
+8
+8
+8
+to
+00
+88
+skate
+8
+board
+8
+8
+xo
+ helicopter
+motorcycle
+dirigible
+motor
+Q
+scooter3
+to initialize the semantic space. Experimentally, we illustrate
+that through this paradigm: we can achieve very competitive
+supervised (on source classes), ZSL (on target classes) and
+G-ZSL performance, as well as open set image recognition
+performance with a large number of unobserved vocabulary
+entities (up to 300, 000); effective learning with few samples
+is also illustrated. Critically, our models can be directly utilized
+in G-ZSL scenario and still has much better results than the
+baselines.
+2
+RELATED WORK
+Our model belongs to a class of transfer learning approaches
+[55], also sometimes called meta-learning [79] or learning to
+learn [70]. The key idea of transfer learning is to transfer
+the knowledge from previously learned categories to recognize
+new categories with no training examples (zero-shot learning
+[43], [59]), few examples (one-shot learning [19], [71]) or
+from vast open set vocabulary [29]. The process of knowledge
+transfer can be done by sharing features [4], [5], [22], [33],
+[73], [81], semantic attributes [43], [58], [60], or contextual
+information [74].
+Visual-semantic embeddings have been widely used for
+transfer learning. Such models embed visual features into a
+semantic space by learning projections of different forms.
+Examples include WSABIE [80], ALE [2], SJE [3], DeViSE
+[24], SVR [18], [43], kernel embedding [33] and Siamese
+networks [37].
+2.1
+Open-set Recognition
+The term “open set recognition” was initially defined in [65],
+[66] and formalized in [6], [7], [63] which mainly aim at
+identifying whether an image belongs to a seen or unseen
+classes. The problem is also known as class-incremental
+learning. However, none of these methods can further identify
+classes for unseen instances. The exceptions are [24], [53]
+which augment zero-shot (unseen) class labels with source
+(seen) labels in some of their experimental settings. Similarly,
+we define the open set image recognition as the problems of
+recognizing the class name of an image from a potentially
+very large open set vocabulary (including, but not limited
+to source and target labels). Note that methods like [65],
+[66] are orthogonal but potentially useful here – it is still
+worth identifying seen or unseen instances to be recognized
+with different label sets. Conceptually similar, but different
+in formulation and task, open-vocabulary object retrieval [32]
+focused on retrieving objects using natural language open-
+vocabulary queries.
+2.2
+One-shot Learning
+While most of machine learning-based object recognition
+algorithms require a large amount of training data, one-shot
+learning [20] aims to learn object classifiers from one, or very
+few examples. To compensate for the lack of training instances
+and enable one-shot learning, knowledge must be transferred
+from other sources, for example, by sharing features [5],
+semantic attributes [27], [43], [58], [60], or contextual infor-
+mation [74]. However, none of the previous work had used
+the open set vocabulary to help learn the object classifiers.
+2.3
+Zero-shot Learning
+Zero-shot Learning (ZSL) aims to recognize novel classes with
+no training instance by transferring knowledge from source
+classes. ZSL was first explored with use of attribute-based
+semantic representations [18], [26], [27], [28], [42], [56]. This
+required pre-defined attribute vector prototypes for each class,
+which is costly to obtain for a large-scale dataset. Recently,
+semantic word vectors were proposed as a way to embed
+any class name without human annotation effort; they can
+therefore serve as an alternative semantic representation [3],
+[24], [31], [53] for ZSL. Semantic word vectors are learned
+from large-scale text corpus by language models, such as
+word2vec [52] or GloVec [57]. However, most of the previous
+work only use word vectors as semantic representations in ZSL
+setting, but have neither (1) utilized semantic word vectors
+explicitly for learning better classifiers; nor (2) for extending
+ZSL setting towards open set image recognition. A notable
+exception is [53] which aims to recognize 21K zero-shot
+classes given a modest vocabulary of 1K source classes; we
+explore vocabularies that are up to an order of the magnitude
+larger – 310K.
+Generalized zero-shot recognition (G-ZSL) [11] relaxed the
+problem setup of conventional zero-shot learning by consider-
+ing the training classes in the recognition step. Chao et al.
+[11] investigated the G-ZSL task and found that it is less
+effective to directly extend the existing zero-shot learning
+algorithms to deal with G-ZSL setting. Recently, Xian et
+al. [54] systematically compared the evaluation settings for
+ZSL and G-ZSL. Comparing against existing ZSL models,
+which are inferior in the G-ZSL scenario, we show that our
+vocabulary-informed frameworks can be directly utilized for
+G-ZSL and achieve very competitive performance.
+2.4
+Visual-semantic Embedding
+Mapping between visual features and semantic entities has
+been explored in three ways: (1) directly learning the embed-
+ding by regressing from visual features to the semantic space
+using Support Vector Regressors (SVR) [18], [43] or neural
+network [68]; (2) projecting visual features and semantic
+entities into a common new space, such as SJE [3], WSABIE
+[80], ALE [2], DeViSE [24], and CCA [25], [28]; (3) learning
+the embeddings by regressing from the semantic space to
+visual features, including [1], [10], [38], [49].
+In contrast to other embedding methods, our model trains a
+better visual-semantic embedding from only few training in-
+stances with the help of a large amount of open set vocabulary
+items (using a maximum margin strategy). Our formulation
+is inspired by the unified semantic embedding model of
+[35], however, unlike [35], our formulation is built on word
+vector representation, contains a data term, and incorporates
+constraints to unlabeled vocabulary prototypes.
+3
+VOCABULARY-INFORMED LEARNING
+3.1
+Problem setup
+Assume a labeled source dataset Ds = {xi, zi}Ns
+i=1 of Ns
+samples, where xi ∈ Rp is the image feature representation
+
+4
+of image i and zi ∈ Ws is a class label taken from a set
+of English words or phrases W; consequently, |Ws| is the
+number of source classes. Further, suppose another set of class
+labels for target classes Wt, also taken from W, such that
+Ws ∩Wt = ∅, for which no labeled samples are available. We
+note that potentially |Wt| >> |Ws|.
+Given a new test image feature vector x∗ the goal is then to
+learn a function z∗ = f(x∗), using all available information,
+that predicts a class label z∗. Note that the form of the problem
+changes drastically depending on the label set assumed for z∗:
+• Supervised learning: z∗ ∈ Ws;
+• Zero-shot learning: z∗ ∈ Wt ;
+• Generalized zero-shot learning: z∗ ∈ {Ws, Wt};
+• Open set recognition: z∗ ∈ W.
+Note that open set recognition is similar to generalized zero-
+shot learning, however, in open set setting additional distractor
+classes that do not exist in either source or target datasets are
+present. We posit that a single unified f(x∗) can be learned for
+all cases. We formalize the definition of vocabulary-informed
+learning (Voc) as follows:
+Definition 3.1. Vocabulary-informed Learning (Voc):
+is a
+learning setting that makes use of complete vocabulary data
+(W) during training. Unlike a more traditional ZSL that typi-
+cally makes use of the vocabulary (e.g., semantic embedding)
+at test time, Voc utilizes exactly the same data during training.
+Notably, Voc requires no additional annotations or semantic
+knowledge; it simply shifts the burden from testing to training,
+leveraging the vocabulary to learn a better model.
+The vocabulary W can be represented by semantic embed-
+ding space learned by word2vec [52] or GloVec [57] on large-
+scale corpus; each vocabulary entity w ∈ W is represented as
+a distributed semantic vector u ∈ Rd. Semantics of embedding
+space help with knowledge transfer among classes, and allow
+ZSL, G-ZSL and open set image recognition. Note that such
+semantic embedding spaces are equivalent to the “semantic
+knowledge base” for ZSL defined in [54] and hence make it
+appropriate to use Vocabulary-informed Learning in ZSL.
+3.2
+Learning Embedding and Recognition
+Assuming we can learn a mapping g : Rp → Rd, from image
+features to this semantic space, recognition can be carried out
+using simple nearest neighbor distance, e.g., f(x∗) = car if
+g(x∗) is closer to ucar than to any other word vector; uj in
+this context can be interpreted as the prototype of the class
+j. Essentially, the attribute or semantic word vector of the
+class name can be taken as the class prototype [30]. The
+core question is then how to learn the mapping g(x) and
+what form of inference is optimal in the semantic space.
+For learning we propose the discriminative maximum margin
+criterion that ensures that labeled samples xi project closer
+to their corresponding class prototypes uzi than to any other
+prototype ui in the open set vocabulary i ∈ W \ zi.
+Learning Embedding: To learn the function f(x), one needs
+to establish the correspondence between visual feature space
+and semantic space. Particularly, in the training step, each
+image sample xi is regressed towards its corresponding class
+prototype uzi by minimizing
+W = arg min
+W
+Ns
+�
+i=1
+L (xi, uzi) + λ ∥ W ∥2
+F
+(1)
+where L (xi, uzi) = ∥g (xi) − uzi∥2
+2 ; and g : Rp → Rd is
+the mapping from image features to semantic space; ∥ · ∥F
+indicates the Frobenius Norm. If g (x) = W T x is a linear
+mapping, we have the closed form solution for Eq. (1). The
+loss function in Eq. (1) can be interperted as a variant of SVR
+embedding. However, this is too limiting. To learn the linear
+embedding matrix W, we introduce and discuss two sets of
+methods in Section 3.3 and Section 3.4.
+Recognition: The recognition step can be formulated using
+the nearest neighbor classifier. Given a testing instance x⋆,
+z⋆ = arg min
+i
+��W T x⋆ − ui
+��2
+2 .
+(2)
+Eq. (2) measures the distance between predicted vector and the
+class prototypes in the semantic space. In terms of different
+label set, we can do supervise, zero-shot, generalized zero-shot
+or open set recognition without modifications.
+In particular, we explore a simple variant of Eq. (2) to
+classify the testing instance x⋆,
+z∗ = arg min
+i
+∥ W T x∗ − ω (ui) ∥2
+2,
+(3)
+where the Nearest Neighbor (NN) classifier measures distance
+between the predicted semantic vectors and a function of pro-
+totypes in the semantic space, e.g., ω (ui) = ui is equivalent
+to Eq (2). In practice, we employ semantic vector prototype
+averaging to define ω (·). For example, sometimes, there might
+be more than one positive prototype, such as pig, pigs and hog.
+In such the circumstance, choosing the most likely prototype
+and using NN may not be sensible, hance we introduce the
+averaging strategy to consider more prototypes for robustness.
+Note that this strategy is known as Rocchio algorithm in infor-
+mation retrieval. Rocchio algorithm is a method for relevance
+feedback that uses more relevant instances to update the query
+for better recall and possibly precision in the vector space
+(Chap 14 in [51]). It was first suggested for use in ZSL in [27];
+more sophisticated algorithms [25], [58] are also possible.
+3.3
+Maximum Margin Voc Embedding (MM-Voc)
+The maximum margin vocabulary-informed embedding learns
+the mapping g(x) : Rp → Rd, from low-level features x to the
+semantic word space by utilizing maximum margin strategy.
+Specifically, consider g(x) = W T x, where1 W ⊆ Rp×d.
+Ideally we want to estimate W such that uzi = W T xi for all
+labeled instances in Ds. Note that we would obviously want
+this to hold for instances belonging to unobserved classes as
+well, but we cannot enforce this explicitly in the optimization
+as we have no labeled samples for them.
+Data Term: The easiest way to enforce the above objective
+is to minimize Euclidian distance between sample projections
+1. Generalizing to a kernel version is straightforward, see [76].
+
+5
+and appropriate prototypes in the embedding space,
+D (xi, uzi) =
+��W T xi − uzi
+��2
+2 .
+(4)
+Where we need to minimize this term with respect to each
+instance (xi, uzi), where zi is the class label of xi in Ds. Such
+embedding is also known, in the literature, as data embedding
+[35] or compatibility function [3].
+To make the embedding more comparable to support vector
+regression (SVR), we employ the maximal margin strategy –
+ϵ−insensitive smooth SVR (ϵ−SSVR) [46] in Eq. (1). That is,
+L (xi, uzi) = Lϵ (xi, uzi) + λ ∥ W ∥2
+F
+(5)
+where Lϵ (xi, uzi) = 1T | ξ |2
+ϵ; λ is regularization coefficient.
+(|ξ|ϵ)j = max
+�
+0,
+���W T
+⋆jxi − (uzi)j
+��� − wzi · ϵ
+�
+(6)
+|ξ|ϵ ∈ Rd; ()j indicates the j-th value of corresponding vector;
+W⋆j is the j-th column of W, and wzi is the scaling weight
+derived from the density of class zi and it’s neighboring
+classes. In our conference version [29], equal weight wzi is
+used for all classes. Here we notice that it is beneficial to
+use the density/coverage of each labeled training class as the
+constraint in learning the projection from visual feature space
+to semantic space. We introduce a specific weighting strategy
+to compute wzi in Section 3.4.
+The conventional ϵ−SVR is formulated as a constrained
+minimization problem, i.e., convex quadratic programming
+problem, while ϵ−SSVR employs quadratic smoothing [89] to
+make Eq. (5) differentiable everywhere, and thus ϵ−SSVR can
+be solved as an unconstrained minimization problem directly2.
+Triplet Term: Data term above only ensures that labelled
+samples project close to their correct prototypes. However,
+since it is doing so for many samples and over a number
+of classes, it is unlikely that all the data constraints can be
+satisfied exactly. Specifically, consider the following case, if
+uzi is in the part of the semantic space where no other entities
+live (i.e., distance from uzi to any other prototype in the em-
+bedding space is large), then projecting xi further away from
+uzi is asymptomatic, i.e., will not result in misclassification.
+However, if the uzi is close to other prototypes then minor
+error in regression may result in misclassification.
+To embed this intuition into our learning, we enforce more
+discriminative constraints in the learned semantic embedding
+space. Specifically, the distance of D (xi, uzi) should not only
+be as small as possible, but should also be smaller than the
+distance D (xi, ua), ∀a ̸= zi. Formally, we define the triplet
+term
+MV (xi, uzi) = 1
+2
+AV
+�
+a=1
+�
+C + 1
+2D (xi, uzi) − 1
+2D (xi, ua)
+�2
++
+,
+(7)
+where a ∈ Wt (or more precisely a ∈ W \ Ws) is selected
+from the open vocabulary; C is the margin gap constant. Here,
+[·]2
++ indicates the quadratically smooth hinge loss [89] which
+2. In practice, our tentative experiments shows that the Eq. (4) and Eq. (5)
+will lead to the similar results, on average; but formulation in Eq. (5) is more
+stable and has lower variance.
+is convex and has the gradient at every point. To speedup
+computation, we use the closest AV target prototypes to each
+source/auxiliary prototype uzi in the semantic space. We also
+define similar constraints for the source prototype pairs:
+MS (xi, uzi) = 1
+2
+BS
+�
+b=1
+�
+C + 1
+2D (xi, uzi) − 1
+2D (xi, ub)
+�2
++
+(8)
+where b ∈ Ws is selected from source/auxiliary dataset
+vocabulary. This term enforces that D (xi, uzi) should be
+smaller than the distance D (xi, ub), ∀b ̸= zi. To facilitate the
+computation, we similarly use closest BS prototypes that are
+closest to each prototype uzi in the source classes. Note that,
+the Crammer and Singer loss [13], [75] is the upper bound of
+Eq. (7) and (8) which we use to tolerate slight variants of uzi
+(e.g., the prototypes of ’pigs’ Vs. ’pig’).
+To sum up, the complete triplet maximum margin term is:
+M (xi, uzi) = MV (xi, uzi) + MS (xi, uzi) .
+(9)
+We note that the form of rank hinge loss in Eq. (7) and (8) is
+similar to DeViSE [24], but DeViSE only considers loss with
+respect to source/auxiliary data and prototypes.
+Maximum Margin Vocabulary-informed Embedding: The
+complete combined objective can now be written as:
+W = argmin
+W
+nT
+�
+i=1
+(αLϵ (xi, uzi) +
+(1 − α)M (xi, uzi)) + λ ∥ W ∥2
+F , (10)
+where α ∈ [0, 1] is the coefficient that controls contribution
+of the two terms. One practical advantage is that the objective
+function in Eq. (10) is an unconstrained minimization problem
+which is differentiable and can be solved with L-BFGS. W is
+initialized with all zeros and converges in 10 − 20 iterations.
+3.4
+Weighted
+Maximum
+Margin
+Voc
+Embedding
+(WMM-Voc)
+We note that there is no previous method that directly estimates
+the density of source training classes in the semantic space.
+However, doing so may lead to several benefits. First, the num-
+ber of training instances in source classes may be unbalanced.
+In such a case, an estimate of the density of samples in a
+training class can be utilized as a constraint in learning the
+embedding characterized by Eq. (6). Second, in the semantic
+space, the instances from the classes whose data samples
+span a large radius [62] may reside in the neighborhood
+of many other classes or open vocabulary. This can happen
+when the embedding is not well learned. We can interpret this
+phenomenon as hubness [45], [67]3. Adding a penalty based
+on the density of each training class may be helpful in better
+learning the embedding and alleviating the hubness problem.
+This subsection introduces a strategy for estimating the
+density of each known class in the semantic space (i.e.,
+wzi in Eq. (6)). Generally, we know the prototype of each
+known and novel class in the semantic space. To estimate
+3. However, the causes for hubness are still under investigation [16], [67].
+
+6
+Fig. 2. Illustration of margin distribution of prototypes in
+the semantic space.
+the density/coverage of a known class, one needs to look at
+pairwise distance between a prototype and the nearest negative
+instance and the furthest positive instance. This intuition leads
+us to introduce the concept of margin distribution.
+Margin Distribution: The concept of margin is fundamen-
+tal to maximum margin classifiers (e.g., SVMs) in machine
+learning. The margin enables an intuitive interpretation of such
+classifiers in searching for the maximum margin separator in a
+Reproducing Kernel Hilbert Space. Previous margin classifiers
+[92] aim to maximize a single margin across all training
+instances. In contrast, some recent studies [17], [62], [64], [90]
+suggest that the knowledge of margin distribution of instances,
+rather than a single margin across all instances, is crucial for
+improving the generalization performance of a classifier.
+The “instance margin” is defined as the distance between
+one instance and the separating hyperplane. Formally, for one
+instance i in the semantic space g (xi) and sufficiently many4
+samples g (xj) (zi ̸= zj) drawn from well behaved class
+distributions5. We define the distance dij = ∥g (xi) − g (xj)∥.
+For instance i, we can obtain a set of distances Di
+=
+{dij, zj ̸= zi} with the minimal values ¯di⋆ = minDi. As
+shown in [62], the distribution for the minimal values of the
+margin distance is characterized by a Weibull distribution.
+Based on this finding, we can express the probability of g (x)
+being included in the boundary estimated by g (xi):
+ψ (g (x) ; g (xi)) = exp
+�
+−
+�∥g (x) − g (xi)∥
+λi
+�κi�
+,
+(11)
+where κi and λi are Weibull shape and scale parameters
+obtained by fitting Di using Maximum Likelihood Estimate
+(MLE), which is summarized6 in Alg. 1. Equation (11) quan-
+titatively describes the margin of one specific class, probabilis-
+4. In our experiments, we use all available training instances here.
+5. The well behaved indicates that the moments of the distribution should
+be well-defined. For example, Cauchy distribution is not well-behaved [39].
+6. codes released in https://github.com/xiaomeiyy/WMM-Voc.
+tically, in our semantic embedding space. Note that Eq. (11)
+requires ψ (·) to be non-degenerate margin distribution, which
+is essentially guaranteed by Extreme Value Theorem [40].
+Algorithm 1 EVT estimator by Weibull distribution.
+Input: Extreme values x1, · · · , xn
+Output: Estimated parameters ˆκ, ˆλ
+If n == 1:
+ˆκ = ∞, ˆλ = x1.
+Else:
+1. Sort x1, · · · , xn to get x[1] ≥ · · · ≥ x[n]
+(where x[i] is the re-ordered value).
+2. Maximum likelihood estimator for κ:
+n � �
+xκ
+[i] log x[i] − xκ
+[n] log x[n]
+�
+� �
+xκ
+[i] − xκ
+[n]
+�
+=
+�
+log x[i]
+(12)
+3. Solve Eq. (12), and numerically estimate ˆκ.
+(e.g., using fzero function in MATLAB)
+4. Compute ˆλ =
+�� �
+xˆκ
+[i] − xˆκ
+[n]
+�
+/n
+�1/ˆκ
+.
+Margin Distribution of Prototypes: Consider a class zi
+which in the embedding space is represented by a prototype
+uzi. In accordance with above formalism, we can also assume
+sufficiently many samples g (xj) drawn from other (zi ̸= zj)
+well behaved class distributions. We can also consider the
+prototypes of vast open vocabulary uzj (zi ̸= zj, zj ∈ Wt).
+Under these assumptions, we can obtain a set of distances
+Duzi = {
+��uzi − gzj
+�� , zj ̸= zi, gzj ∈
+�
+g (xj) , uzj
+�
+} for the
+prototype uzi. As a result, the distribution for the minimal
+values of the margin distance for uzi is given by a Weibull
+distribution. The probability that gzi is included in the bound-
+ary estimated by uzi is given by
+ψ (gzi; uzi) = exp
+�
+−
+�
+∥gzi − uzi∥
+λuzi
+�κuzi �
+.
+(13)
+The above equation models the distribution of minimum value;
+thus it can be used to estimate the boundary density (or more
+specifically, the boundary distribution) of class zi.
+We set significant level to 0.05 to approximately esti-
+mate the minimal value ¯duzi⋆. As illustrated in Figure 2, if
+ψ (gzi; uzi) < 0.05, we will assume gzi does not belong to
+the prototype uzi; otherwise, gzi is included in the boundary
+estimated by uzi. In term of the significant level of 0.05,
+we can further denote the minimal values as ¯d(0.05)
+uzi⋆ , i.e.,
+exp
+�
+−
+� ¯d(0.05)
+uzi ⋆
+λuzi
+�κuzi �
+= 0.05. Thus we have
+¯d(0.05)
+uzi⋆ = λuzi · log1/κuzi
+� 1
+0.05
+�
+(14)
+Coverage Distribution of Prototypes: Now, for class zi
+consider the nearest instance from another class g (xj) where
+zi ̸= zj; with sufficient many instances g (xk) from class zi,
+we have pairwise unique ("within class") distance:
+cuzik = ∥g (xk) − uzi∥.
+(15)
+
+Positiveinstances
+Negativeinstances
+★ Prototypes
+id7
+We consider outliers those instances g (xk) that have larger
+distance to uzi than the nearest instance g (xj) (zi ̸= zj)
+of another class. To remove the outliers we hence consider
+Cuzi =
+�
+cuzik|cuzik ≤ minzj̸=zk ∥g (xj) − uzi∥
+�
+. As illus-
+trated in Figure 2, we only consider positive instances within
+the orange circle and all other instances with larger distance
+are removed. Then the distribution of the largest distance
+¯cuzi⋆ = max Cuzi will follow a reversed Weibull distribution.
+This allows us to get the probability distribution to describe
+positive instances,
+φ (g (xk) ; uzi) = 1 − exp
+�
+�
+�−
+�
+∥g (xk) − uzi∥
+λ′uzi
+�κ
+′
+uzi
+�
+�
+�
+(16)
+where κ
+′
+i and λ
+′
+i are reverse Weibull shape and scale pa-
+rameters individually obtained from fitting the largest Cuzi ,
+¯cuzi⋆ is the distance between instance and prototype, φ is the
+probability that the instance is in the class.
+Similar to the margin distribution, we can estimate the
+coverage by setting the significant level to 0.05. As shown in
+Figure 2, we establish two boundaries to estimate the scale of
+each class probabilistically. If φ (g (xk) ; uzi) ⩾ 0.05, g (xk)
+is included in the coverage distribution uzi. The maximum
+values ¯c(0.05)
+uzi⋆
+can be computed as φ (g (xk) ; uzi) = 0.05. It
+results in,
+¯c(0.05)
+uzi⋆ = λ′
+uzi · log
+1/κ
+′
+uzi
+�
+1
+1 − 0.05
+�
+.
+(17)
+By combining the terms computed in Eq. (14) and (17), we
+can obtain the weight wzi for class zi in Eq. (6),
+wzi ∝
+�
+¯d(0.05)
+uzi⋆ + ¯c(0.05)
+uzi⋆
+�
+(18)
+As explained in Algorithm 1, we set ˆκ = ∞, ˆλ = x1 in
+one-shot setting. In few-shot learning setting, we can estimate
+ˆκ and ˆλ directly. In addition, such an initialization of weights
+(ˆκ and ˆλ) intrinsically helps learn the embedding weight W.
+The learning process of parameters: The process could
+be interpreted as a form of block coordinate descent where
+we estimate the embedding/mapping; then density within that
+embedding and so on. In practice, the weights wzi are initially
+randomized. But they do not play an important role at the
+beginning of the optimization, since
+���W T
+⋆jxi − (uzi)j
+��� is very
+large in the first few iterations. In other words, the optimization
+is initially dominated by the data term and maximum margin
+terms play little role. However, once we can get a relative
+good mapping (i.e., smaller
+���W T
+⋆jxi − (uzi)j
+���) after several
+training iterations, the weight wzi starts becoming significant.
+By virtue of such an optimization, the weighted version can
+achieve better performance than the previous non-weighted
+version in our conference paper [29].
+Deep Weighted Maximum Margin Voc Embedding (Deep
+WMM-Voc). In practice, we extend WMM-Voc to include
+a deep network for feature extraction. Rather than extracting
+low-level features using an off-the-shelf pre-trained model in
+Eq. (10), we use an integrated deep network to extract xi
+from the raw images. As a result, the loss function in Eq. (10)
+is also used to optimize the parameters of the deep network.
+In particular, we fix the convolutional layers of corresponding
+network and fine-tune the last fully connected layer in our task.
+The network was trained using stochastic gradient descendent.
+4
+EXPERIMENTS
+4.1
+Experimental setup
+We conduct our experiments on Animals with Attributes
+(AwA) dataset, and ImageNet 2012/2010 dataset.
+AwA dataset: AwA consists of 50 classes of animals (30, 475
+images in total). In [43] standard split into 40 source/auxiliary
+classes (|Ws| = 40) and 10 target/test classes (|Wt| = 10)
+is introduced. We follow this split for supervised and zero-
+shot learning. We use ResNet101 features (downloaded from
+[54]) on AwA to make the results more easily comparable to
+state-of-the-art.
+ImageNet 2012/2010 dataset: ImageNet is a large-scale
+dataset. We use 1000 (|Ws| = 1000) classes of ILSVRC 2012
+as the source/auxiliary classes and 360 (|Wt| = 360) classes
+of ILSVRC 2010 that are not used in ILSVRC 2012 as target
+data. We use pre-trained VGG-19 model [12] to extract deep
+features for ImageNet.
+Recognition tasks: We consider several different settings in
+a variety of experiments. We first divide the two datasets into
+source and target splits. On source classes, we can validate
+whether our framework can be used to solve one-shot and
+supervised recognition. By using both the source and target
+classes, transfer learning based settings can be evaluated.
+1) SUPERVISED recognition: learning is on source classes;
+test instances come from the same classes with Ws as
+recognition vocabulary. In particular, under this setting,
+we also validate the one- and few-shot recognition sce-
+narios, i.e., classes have one or few training examples.
+2) ZERO-SHOT recognition: In ZSL, learning is on the
+source classes with Ws vocabulary; test instances come
+from target dataset with Wt as recognition vocabulary.
+3) GENERAL-ZERO-SHOT
+recognition:
+G-ZSL
+uses
+source classes to learn, with test instances coming from
+either target Wt or original Ws recognition vocabulary.
+4) OPEN-SET recognition: Again source classes are used
+for learning, but the entire open vocabulary with |W| ≈
+310K atoms is used at test time. In practice, test images
+come from both source and target splits (similar to
+G-ZSL); however, unlike G-ZSL there are additional
+distractor classes. In other words, chance performance
+for open-set recognition is much lower than for G-ZSL.
+We test both our Voc variants – MM-Voc and WMM-Voc.
+Additionally, we also validate the Deep WMM-Voc by fine-
+tuning the WMM-Voc on VGG-19 architecture and optimizing
+the weights with respect to the loss in Eq. (10).
+Competitors: We compare to a variety of the models in the
+literature, including:
+1) SVM: SVM classifier trained directly on the training
+instances of source data, without the use of semantic
+embedding. This is the standard (SUPERVISED/ONE-
+SHOT) learning setting and the learned classifier can only
+
+8
+predict the labels within the source classes. Hence, SVM
+is inapplicable in ZSL, G-ZSL, and open-set recognition
+settings.
+2) SVR-Map: SVR is used to learn W and the recognition
+is done, similar to our method, in the resulting semantic
+manifold. This corresponds to only optimizing Eq. (5).
+3) Deep-SVR: This is a variant SVR, which further allows
+fine-tuning of the underlying neural network generating
+the features. In this case, W is expressed as the last
+linear layer and the entire network is fine-tuned with
+respect to the loss encoding only the data term (Eq. (5)).
+4) SAE: SAE is a semantic encoder-decoder paradigm that
+projects visual features into a semantic space and then
+reconstructs the original visual feature representation
+[38]. The SAE has two variants in learning the embed-
+ding space, i.e., semantic space to feature space (S→F),
+and feature space to semantic space (F→S). By default,
+the best result of these two variants are reported.
+5) ESZSL: ESZSL first learns the mapping between visual
+features and attributes, then models the relationship
+between attributes and classes [61].
+6) DeVise, ConSE, AMP: To compare with state-of-the-art
+large-scale zero-shot learning approaches we implement
+DeViSE [24] and ConSE [53]7. ConSE uses a multi-
+class logistic regression classifier for predicting class
+probabilities of source instances; and the parameter
+T (number of top-T nearest embeddings for a given
+instance) was selected from {1, 10, 100, 1000} that gives
+the best results. ConSE method in supervised setting
+works the same as SVR. We use the AMP code provided
+on the author webpage [31].
+Metrics: Classification accuracies are reported as the eval-
+uation metrics on most of tasks. In our conference version
+[29], we further introduce an evaluation setting for OPEN-
+SET tasks where we do not assume that test data comes from
+either source/auxiliary domain or target domain. Thus we split
+the two cases (i.e., SUPERVISED-like, and ZERO-SHOT-like
+settings), to mimic SUPERVISED and ZERO-SHOT scenarios
+for easier analysis. Particularly, in G-ZSL task, this newly
+introduced evaluation setting is corresponding to the evaluation
+metrics defined in [85]: (1) S → T: Test instances from seen
+classes, the prediction candidates include both seen and unseen
+classes; (2) U → T: Test instances from unseen classes, the
+prediction candidates include both seen and unseen classes.
+(3) The harmonic mean is used as the main evaluation metric
+to further combine the results of both S → T and U → T:
+H = 2·(Acc(U → T) × Acc(S → T))
+(Acc(U → T) + Acc(S → T)).
+(19)
+Setting of Parameters: For the recognition tasks, we learn
+classifiers by using various number of training instances.
+We compare relevant baselines with results of our method
+variants: MM-Voc, WMM-Voc, Deep WMM-Voc. Each setting
+is repeated/tested 10 times. The averaged results are reported
+to reduce the variance. For each setting, our Voc methods are
+trained by a single model to be capable of solving the tasks
+7. Codes for [24] and [53] are not publicly available.
+of supervised, zero-shot, G-ZSL and open-set recognition.
+Specifically,
+1) In Deep WMM-Voc, we fix λ to 0.01 and α = 0.6 with
+the learning rate initially set to 1e−5 and is reduced by
+1
+2 every 10 epochs. AV and BS are set to 5 in order to
+balance performance and computational cost of pairwise
+constraints.
+2) To solve Eq. (10) at a scale, one can use Stochastic
+Gradient Descent (SGD) which makes great progress ini-
+tially, but often is slow when approaching convergence.
+In contrast, the L-BFGS method mentioned above can
+achieve steady convergence at the cost of computing the
+full objective and gradient at each iteration. L-BFGS
+can usually achieve better results than SGD with good
+initialization, however, is computationally expensive. To
+leverage benefits of both of these methods, we utilize a
+hybrid method to solve Eq. (10) in large-scale datasets:
+the solver is initialized with few instances to approx-
+imate the gradients using SGD first; then gradually
+more instances are used and switch to L-BFGS is made
+with iterations. This solver is motivated by Friedlander
+et al. [23], who theoretically analyzed and proved the
+convergence for the hybrid optimization methods. In
+practice, we use L-BFGS and the Hybrid algorithms for
+AwA and ImageNet respectively. The hybrid algorithm
+can save between 20 ∼ 50% training time as compared
+with L-BFGS.
+Open set vocabulary. We use Google word2vec to learn
+the open set vocabulary set from a large text corpus of
+around 7 billion words: UMBC WebBase (3 billion words),
+the latest Wikipedia articles (3 billion words) and other web
+documents (1 billion words). Some rare (low frequency)
+words and high frequency stopping words were pruned in the
+vocabulary set: we remove words with the frequency < 300
+or > 10 million times. The result is a vocabulary of around
+310K words/phrases with openness ≈ 1, which is defined as
+openness = 1 −
+�
+(2 × |Ws|) / (|W|) [66].
+4.2
+Experimental results on AwA dataset
+4.2.1
+Learning Classifiers from Few Source Training
+Instances
+We are particularly interested in learning of classifiers from
+few source training instances. This is inclined to mimic human
+performance of learning from few examples and illustrate
+ability of our model to learn with little data8. We show
+that, our vocabulary-informed learning is able to improve the
+recognition accuracy on all settings.
+By only using 200 training instances, we report the results
+on standard supervised (on source classes), zero-shot (on
+target classes), and generalized zero-shot recognition (both
+on source and target classes) as shown in Table 2. Note that
+for ZSL and G-ZSL, our settings is a more realistic and yet
+8. As for feature representations, the ResNet100 features from [54] are
+trained from ImageNet 2012 dataset, which potentially have some overlapped
+classes with AwA dataset.
+
+9
+Methods
+S. Sp
+Features
+Acc.
+WMM-Voc
+W
+CNNresnet101
+90.79
+WMM-Voc: closed
+W
+CNNresnet101
+84.51
+Deep WMM-Voc
+W
+CNNresnet101
+90.65
+Deep WMM-Voc: closed
+W
+CNNresnet101
+83.85
+SAE
+W
+CNNresnet101
+71.42
+ESZSL
+W
+CNNresnet101
+74.17
+Deep-SVR
+W
+CNNresnet101
+67.22
+Akata et al. [3]
+A+W
+CNNGoogleNet
+73.90
+TMV-BLP [25]
+A+W
+CNNOverFeat
+69.90
+AMP (SR+SE) [31]
+A+W
+CNNOverFeat
+66.00
+PST [58]
+A+W
+CNNOverFeat
+54.10
+Latem [83]
+A+W
+CNNresnet101
+74.80
+SJE [3]
+A+W
+CNNresnet101
+76.70
+DeViSE [24]
+W
+CNNresnet101
+72.90
+ConSE [53]
+W
+CNNresnet101
+63.60
+CMT [68]
+W
+CNNresnet101
+58.90
+SSE [91]
+W
+CNNresnet101
+54.50
+SSE [91]
+W
+CNNVGG19
+57.49
+TASTE [87]
+W
+CNNVGG19
+89.40
+KLDA+KRR [48]
+W
+CNNGoogleNet
+79.30
+CLN+KRR [48]
+W
+CNNVGG19
+81.00
+UVDS [50]
+W
+CNNVGG19
+62.88
+DEM [10]
+W
+CNNInception-V2
+86.70
+DS [60]
+W/A
+CNNOverFeat
+52.70
+SYNC [9]
+W/A
+CNNresnet101
+72.20
+Relation Net [69]
+A
+CNNInception-V2
+84.50
+ESZSL [61]
+A
+CNNresnet101
+74.70
+UVDS [50]
+A
+CNNGoogleNet
+80.28
+GFZSL [78]
+A
+CNNVGG19
+80.50
+DEM [10]
+A
+CNNInception-V2
+78.80
+SE-GZSL [77]
+A
+CNNVGG19
+69.50
+cycle-CLSWGAN [21]
+A
+CNNresnet101
+66.30
+f-CLSWGAN [84]
+A
+CNNresnet101
+68.20
+PTMCA [47]
+A
+CNNresnet101
+66.20
+Jayaraman et al. [36]
+A
+low-level
+48.70
+DAP [43]
+A
+CNNVGG19
+57.50
+DAP [43]
+A
+CNNresnet101
+57.10
+DAP [43]
+A
+CNNOverFeat
+53.20
+ALE [2]
+A
+CNNresnet101
+78.60
+Yu et al. [86]
+A
+low-level
+48.30
+IAP [43]
+A
+CNNOverFeat
+44.50
+HEX [14]
+A
+CNNDECAF
+44.20
+AHLE [2]
+A
+low-level
+43.50
+TABLE 1
+Zero-shot comparison on AwA. We compare the
+state-of-the-art ZSL results using different semantic
+spaces (S. Sp) including word vector (W) and attribute
+(A). 1000 dimension word2vec dictionary is used for our
+model. (Chance-level =10%). Different types of features
+are used by different methods. WMM-Voc: closed and
+Deep WMM-Voc: closed are the two variants of our
+model obtained by learning the vocabulary-informed
+constraints only from known classes (i.e., closed set),
+similar to our conference version [29].
+ZSL results by 100-d word vector
+40
+200
+800
+1600
+12139
+24295
+Training Instance Number
+0
+20
+40
+60
+80
+100
+Top1 Accuracy(%)
+WMM-Voc
+Deep WMM-Voc
+Deep-SVR
+SAE(F->S)
+SAE(S->F)
+ESZSL
+ZSL results by 1000-d word vector
+40
+200
+800
+1600
+12139
+24295
+Training Instance Number
+0
+20
+40
+60
+80
+100
+Top1 Accuracy(%)
+WMM-Voc
+Deep WMM-Voc
+Deep-SVR
+SAE(F->S)
+SAE(S->F)
+ESZSL
+Fig. 3. The ZSL results on AwA by different settings.
+more challenging than those in previous methods [38], [61],
+since the source classes have few training instances. We also
+compare using 100/1000-dimensional word2vec representation
+(i.e., d = 100/1000). Both the Top-1 and Top-5 classification
+accuracy is reported. Note that the key novelty of our WMM-
+Voc comes from directly estimating the density of source train-
+ing classes. Such an approach would be helpful in alleviating
+the hubness problem and should lead to better performance
+in zero-shot learning. As shown in Table 2, the improvement
+from MM-Voc to WMM-Voc and then further to Deep WMM-
+Voc validate this point.
+We highlight the following observations: (1) Deep WMM-
+Voc achieves the best zero-shot learning accuracy compared
+with the other state-of-art methods. It is 18.45% and 21.02%
+higher than SAE and ESZSL respectively on Top-1 accuracy.
+Our WMM-Voc can still beat the state-of-the-art SAE and
+ESZSL by outperforming 17.67% and 20.24% individually on
+Top-1 accuracy. (2) In supervised learning task, the ESZSL
+and Deep WMM-Voc have almost the same performance, if we
+consider the variances in sampling the 200 training instances.
+Our WMM-Voc is slightly better than these two methods.
+(3) In G-ZSL setting, our two models get significantly better
+performance compared with the other competitors. Notably,
+the Top-1 accuracy of SAE and ESZSL is 0. While Deep
+WMM-Voc and WMM-Voc both have higher accuracy. This
+shows the effectiveness of our two models. (4) As expected,
+the deep models that fine-tune features along with classifiers
+(Deep-SVR and Deep WMM-Voc) are better than counterparts
+with pre-extracted representations (SVR-Map, WMM-Voc).
+4.2.2
+Results on different training/testing splits
+We conduct experiments using different number of training
+instances and compare results on tasks of supervised, zero-
+shot and generalized zero-shot learning. On each split, we use
+both 100 and 1000 dimensional word vectors. We use 12,156
+testing instances from source classes in supervised, G-ZSL
+and open-set setting as well as 6,180 testing instances from
+
+10
+Dimension
+SVR-Map
+Deep-SVR
+SAE
+ESZSL
+MM-Voc
+WMM-Voc
+Deep WMM-Voc
+Supervised
+100-dim
+51.4/-
+71.59/91.98
+70.22/92.60
+74.86/94.85
+58.01/87.88
+75.57/94.31
+76.23/94.85
+1000-dim
+57.1/-
+76.32/95.22
+75.32/94.17
+75.08/94.27
+59.1/77.73
+79.44/96.01
+76.55/96.22
+Zero-shot
+100-dim
+52.1/-
+53.12/84.24
+67.96/95.08
+73.69/95.83
+61.10/96.02
+82.78/98.92
+84.87/98.87
+1000-dim
+58.0/-
+64.29/88.71
+71.42/97.18
+74.17/97.12
+83.84/96.74
+89.09/99.21
+88.07/99.40
+G-ZSL
+100-dim
+-
+5.65/54.45
+2.15/52.7
+2.88/68.37
+19.74/85.79
+28.92/88.01
+33.04/89.11
+1000-dim
+-
+0/39.84
+0/35.91
+0/33.09
+8.54/59.79
+27.98/90.47
+34.77/90.76
+TABLE 2
+Classification accuracy (Top-1 / Top-5) on AwA dataset for SUPERVISED, GENERAL ZERO-SHOT and ZERO-SHOT settings for
+100-dim and 1000-dim word2vec representation (200 instances).
+target classes in zero-shot, G-ZSL and open-set setting. All the
+competitors are using the same types of features – ResNet101.
+Supervised learning: The results are compared in Figure 4.
+As shown in the figure, we observe that our method shows
+significant improvements over the competitors in few-shot
+setting; however, as the number of instance increasing, the
+visual semantic mapping, g(x), can be well learned, and the
+effects of additional vocabulary-informed constraints, become
+less pronounced.
+Zero-shot learning: The ZSL results are compared in Figure
+3. On all the settings, our two Voc methods – Deep WMM-
+Voc and WMM-Voc outperforms all the other baselines. This
+validates the importance of information learning from the
+open vocabulary. Further, we compare our results with the
+state-of-the-art ZSL results on AwA dataset in Table 1. Our
+two models achieve 90.79% and 90.65% accuracy, which
+is markedly higher than all previous methods. This is par-
+ticularly impressive, if we take into account the fact that
+we use only a semantic space and no additional attribute
+representations (unlike many other competitor methods). We
+argue that much of our success and improvement comes from
+a more discriminative information obtained using the open
+set vocabulary and corresponding large margin constraints,
+rather than from the features. Varying the number of training
+instances may slightly affect accuracy of methods reported
+in Table 1. Therefore we report the best results of each
+competitor and our own method at different number of train-
+ing instances {200, 800, 1600, 121319, 24295}.All competing
+methods in Figure 3 use the same features.
+General zero-shot learning:. The general zero-shot learning
+results are compared in Table 3. We consider the accuracies of
+both U → T (ZERO-SHOT-like) and S → T (SUPERVISED-like).
+In term of the harmonic mean H, our methods have signifi-
+cantly better performance in the general zero-shot setting. This
+again shows that our framework can have better generalization
+by learning from the open vocabulary. On the other hand, in the
+terms of Area Under Seen-Unseen accuracy Curve (AUSUC),
+the performance of Deep-SVR is very weak and the scores
+of ESZSL and SAE are lower than our method. Overall, the
+results of AUSUC still support the superiority of our methods
+on G-ZSL tasks. Notably, since the source domain only have
+24295 instances (including training and testing images), we
+are unable to obtain the results of SUPERVISED-like setting
+(S → T), H and AUSUC with all source instances.
+Supervised results by 100-d word vector
+200
+800
+1600
+12139
+Training Instance Number
+0
+20
+40
+60
+80
+100
+Top1 Accuracy(%)
+WMM-Voc
+Deep WMM-Voc
+Deep-SVR
+SAE(F->S)
+SAE(S->F)
+ESZSL
+Supervised results by 1000-d word vector
+200
+800
+1600
+12139
+Training Instance Number
+0
+20
+40
+60
+80
+100
+Top1 Accuracy(%)
+WMM-Voc
+Deep WMM-Voc
+Deep-SVR
+SAE(F->S)
+SAE(S->F)
+ESZSL
+Fig. 4. Supervised learning results of AwA datasets.
+4.2.3
+Large-scale open set recognition
+We also compare the results on OPEN-SET310K setting with
+the large vocabulary of approximately 310K entities; as such
+the chance performance is much lower. We use 100-dim word
+vector representations as the semantic space. While our OPEN-
+SET variants do not assume that test data comes from either
+source/auxiliary domain or target domain, we split the two
+cases to mimic SUPERVISED and ZERO-SHOT scenarios for
+easier analysis. The results are shown in Figure 5.
+On SUPERVISED-like setting, Figure 5 (left), our Deep
+WMM-Voc and WMM-Voc have better performance than the
+other baselines. The better results are largely due to the better
+embedding matrix W learned by enforcing maximum margins
+between training class name and open set vocabulary on source
+training data. This validates the effectiveness of proposed
+framework. In particular, we find that (1) The “deep” version
+always has better performance than their corresponding “non-
+deep” counterparts. For example, the Deep-SVR and Deep
+WMM-Voc achieve higher open-set recognition accuracy than
+SVR-Map and WMM-Voc. (2) The WMM-Voc has better
+performance than MM-Voc; this shows that the weighting
+strategy introduced in Section 3.4 can indeed help better learn
+
+11
+TABLE 3
+The G-ZSL results (100-dim/1000-dim) of AwA dataset. We compare the results by varying the number of training
+instances (No.of Tr. Ins.) of each class. Where H is defined in Eq. (19) without calibrated stacking, AUSUC means
+the Area Under Seen-Unseen accuracy Curve with calibrated stacking and ’-’ represents the unavailable results.
+Metrics
+ESZSL
+SAE
+Deep-SVR
+WMM-Voc
+Deep WMM-Voc
+200
+U → T
+2.88/0
+2.15/0
+5.65/0
+28.92/27.98
+33.04/34.77
+S → T
+75.76/76.08
+70.13/75.32
+71.22/76.32
+70.20/74.20
+71.16/69.48
+H
+5.55/0
+4.17/0
+10.47/0
+40.96/40.64
+45.13/46.35
+AUSUC
+0.4231/0.4344
+0.3885/0.4556
+0.3048/0.3939
+0.4840/0.5190
+0.5028/0.4776
+800
+U → T
+0.19/0
+0.78/0
+5.34/0.02
+25.57/25.68
+27.59/27.77
+S → T
+81.14/83.95
+78.02/83.41
+78.92/81.46
+74.23/77.33
+75.53/77.19
+H
+0.38/0
+1.54/0
+10.00/0.04
+38.04/38.56
+40.42/40.85
+AUSUC
+0.4409/0.4710
+0.3870/0.4483
+0.3452/0.4400
+0.4764/0.5387
+0.4953/0.5353
+1600
+U → T
+0.71/0
+0.87/0
+4.69/0
+24.63/27.22
+33.66/32.86
+S → T
+85.62/86.24
+81.08/85.48
+83.30/86.02
+74.99/77.67
+78.96/78.64
+H
+1.41/0
+1.72/0
+8.88/0
+37.08/40.31
+47.20/46.35
+AUSUC
+0.4507/0.5139
+0.4190/0.4740
+0.3776/0.4780
+0.5016/0.5572
+0.5554/0.5733
+12139
+U → T
+0.37/0
+0.44/0
+5.19/0
+27.80/30.53
+32.23/28.19
+S → T
+89.98/91.16
+88.18/91.26
+85.37/85.64
+77.36/78.34
+80.64/78.32
+H
+0.74/0
+0.88/0
+9.79/0
+40.90/43.94
+46.05/41.46
+AUSUC
+0.5096/0.5294
+0.4493/0.5120
+0.3353/0.4397
+0.5144/0.5319
+0.5525/0.5394
+24295
+U → T
+0.83/0
+0.37/0
+5.39/0
+27.15/29.42
+35.65/31.78
+S → T
+-
+-
+-
+-
+-
+H
+-
+-
+-
+-
+-
+AUSUC
+-
+-
+-
+-
+-
+the embedding from visual to semantic space.
+On ZERO SHOT-like setting, our method still has a notable
+advantage over that of SVR-Map, Deep-SVR methods on Top-
+k (k > 3) accuracy, again thanks to the better embedding
+W learned by Eq. (10). However, we notice that our top-
+1 accuracy on ZERO SHOT-like setting is lower than Deep
+SVR method. We find that our method tends to label some
+instances from target data with their nearest classes from
+within source label set. For example, “humpback whale” from
+testing data is more likely to be labeled as “blue whale”.
+However, when considering Top-k (k > 3) accuracy, our
+method still has advantages over baselines. It suggests that
+the semantic embeddings may be suffering from the problem
+that density of source classes is more concentrated than that
+of target classes. To show the effectiveness of WMM-Voc, as
+opposed to MM-Voc, we employ the False Positive Rate as the
+metric, rfp = Ne/Nun, where Ne means the number of testing
+unseen instances predicted as seen ones and Nun defines
+the number of testing unseeen instances. Experiments are
+conducted on AwA dataset with all training instances, and 100-
+dim word vector prototypes. The false positive rates are 0.16,
+0.10, 0.12, 0.05 and 0.06 by using SVR, Deep-SVR, MM-Voc,
+WMM-Voc and Deep WMM-Voc, respectively. They further
+validates that WMM-Voc outperforms MM-Voc.
+4.3
+Experimental results on ImageNet dataset
+We validate our methods on large-scale ImageNet 2012/2010
+dataset; the 1000-dimensional word2vec representation is used
+here since this dataset has larger number of classes than AwA.
+Testing Classes
+AwA dataset
+Aux.
+Targ.
+Total
+Vocab
+OPEN-SET310K
+(left)
+(right)
+40 / 10
+310K
+0
+5
+10
+15
+20
+Hit@k
+50
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Accuracy (%)
+SUPERVISED-like(Aux)
+0
+5
+10
+15
+20
+Hit@k
+0
+10
+20
+30
+40
+50
+60
+70
+Accuracy (%)
+ZERO SHOT-like(Tag)
+SVR-Map
+MM-Voc
+Deep-SVR
+Deep WMM-Voc
+WMM-Voc
+Fig. 5.
+Openset results of AwA datasets. We use
+1600 training instances equally sampled from all source
+classes to train the model.
+The instances of testing classes are equally sampled; making
+experiment less sensitive to the problem of unbalanced data.
+To be specific, 50×1, 000 testing instances from source classes
+are used in supervised, G-ZSL and open-set setting as well as
+
+12
+TABLE 4
+The classification accuracy (Top-1 / Top-5) of ImageNet 2012/2010 dataset on ZERO-SHOT and SUPERVISED settings using
+3000 source training instances.
+Settings
+SVR-Map
+Deep-SVR
+ESZSL
+SAE
+MM-Voc
+WMM-Voc
+Deep WMM-Voc
+Supervised
+25.6/–
+31.26/50.51
+38.26/64.38
+32.95/54.44
+37.1/62.35
+35.95/62.77
+38.92/65.35
+Zero-shot
+4.1/–
+5.29/13.32
+5.86/13.71
+5.11/12.62
+8.90/14.90
+8.50/20.73
+9.26/21.99
+360 × 100 testing instances from target classes are used in
+zero-shot, G-ZSL and open-set setting. The VGG-19 features
+of ImageNet pre-trained network are utilized as the input of
+all algorithms to make a fair comparison. We employ the
+Deep-SVR, SAE, ESZSL as baselines under the SUPERVISED,
+ZERO-SHOT and GENERAL ZERO-SHOT settings respectively.
+4.3.1
+Pseudo-few-shot Source Training instances
+The standard few-shot learning assumes disjoint instance set
+on source and target domains, as discussed in Sec. 4.3.2.
+As an ablation study, we would like to simulate a few-shot-
+like learning task on source domain by slightly violating
+the standard few-shot learning assumption. We name this
+setting “Pseudo-few-shot learning”: only few source training
+instances are used here and the feature extractor – VGG-
+19 model is pre-trained on ILSVRC 2012 dataset [12]. The
+“Pseudo-” here indicates that large amount of instances are
+used to train the feature extractor, but not used in training
+classifiers. Thus the experiments in this section can be served
+as an additional ablation study to reveal the insights of our
+model in addressing the few-shot-like task on source domain.
+Particularly, we conduct the experiments of using few-shot
+source training instance, i.e., 3,000 training instances used
+here. The results are listed in Table 4. We introduce this
+setting to particularly focus on learning from few training
+samples per class, in order to mimic human capability and
+performance in learning from few examples. We show that
+our vocabulary-informed learning framework enables learning
+with little data. In particular, we highlight that the Top-5
+performance of WMM-Voc is much higher (>5%) than that
+of MM-Voc, despite the slightly worse performance on Top-
+1 accuracy. Note that the degradation of Top-1 results on
+ImageNet is also understandable. Note, WMM-Voc is only
+fitting the 3000 training instances on ImageNet dataset, and
+the features of these training instances may not be fine-
+tuned/optimized for the newly introduced penalty term of
+WMM-Voc. Once the features of training instances are fine-
+tuned by the deep version; we can show that the Deep WMM-
+Voc can improve from MM-Voc and WMM-Voc.
+Critically, with different settings in Table 4, our vocabulary-
+informed learning can beat the other baselines under all
+settings. We highlight several findings:
+(1) The supervised performance of our methods stands
+out from the state-of-art. Specifically, our Deep WMM-Voc
+achieves the highest supervised recognition accuracy, with
+ESZSL following closely. SVR-Map appears to be the worst.
+(2) On Zero-shot learning task, our proposed Deep WMM-
+Voc gets 9.26% Top-1 and 21.99% Top-5 accuracy. It outper-
+forms all the other baselines. Comparing with our previous
+Supervised Learning results of ImageNet
+3k
+5k
+10k
+20k
+50k
+Training Instance Number
+0
+20
+40
+60
+Top1 Accuracy(%)
+WMM-Voc
+Deep WMM-Voc
+Deep-SVR
+SAE(F->S)
+SAE(S->F)
+ESZSl
+Zero-shot Learning results of ImageNet
+3k
+5k
+10k
+20k
+50k
+Training Instance Number
+0
+2
+4
+6
+8
+10
+Top1 Accuracy(%)
+WMM-Voc
+Deep WMM-Voc
+Deep-SVR
+SAE(F->S)
+SAE(S->F)
+ESZSL
+Fig. 6. The supervised and zero-shot learning results on
+ImageNet 2012/2010 dataset.
+MM-Voc result in [29], our result is 0.36% higher than MM-
+Voc. This improvement is statistically significant due to the
+few number of training instances and large number of testing
+instances. Additionally, the Top-1 result of WMM-Voc is
+8.5% which is also comparable to that of MM-Voc, and far
+higher than those of SVR, Deep-SVR, ESZSL and SAE. This
+validates the effectiveness of learning from open vocabulary
+proposed in our two variants.
+(3) In G-ZSL setting, we observe that both Deep WMM-
+Voc and WMM-Voc outperform all the other baselines. The
+full set of experiments on G-ZSL under different settings are
+reported in Table 5.
+4.3.2
+Few-shot Target Training instances
+We further introduce few-shot learning experiments on target
+instances to validate the performance of our methods. The
+experiments are conducted on ImageNet dataset. In total, there
+are 360 target classes from ImageNet 2010 data split with
+100 instances per class; the feature extractor – VGG-19 is
+trained on the 1000 classes from ImageNet 2012. The 1 or
+3 training instances are sampled from each target class. The
+other instances of the target classes are utilized as the test
+set. This is the few-shot learning setting, which is consistent
+with general definition [20]. We compare to SVM, KNN,
+Deep SVR, and SAE. The results are shown in Table 6. We
+can see that our method (WMM-Voc) can beat all the other
+competitors. Particularly, we have an obvious advantage in 1-
+shot target setting. Our Deep variant (Deep WMM-Voc) has
+
+13
+TABLE 5
+The G-ZSL results (1000-dim) of ImageNet datasets. We
+compare the results of using different number of training
+instances (No.) D-SVR, W-V and D-W-V indicate Deep
+SVR, WWM-Voc, and Deep WWM-Voc, respectively.
+No.
+Metrics
+ESZSL
+SAE
+D-SVR
+W-V
+D-W-V
+3000
+U → T
+0.46
+0.24
+0.20
+2.02
+1.93
+S → T
+38.07
+32.86
+31.06
+32.40
+36.61
+H
+0.91
+0.48
+0.40
+3.80
+3.67
+10000
+U → T
+0.38
+0.18
+0.18
+2.01
+1.99
+S → T
+49.65
+46.23
+33.54
+32.87
+43.53
+H
+0.75
+0.36
+0.36
+3.79
+3.81
+50000
+U → T
+0.37
+0.19
+0.20
+2.11
+2.15
+S → T
+57.43
+54.55
+36.75
+33.16
+47.28
+H
+0.74
+0.38
+0.40
+3.97
+4.11
+TABLE 6
+Results of few-shot target training instances on
+ImageNet dataset.
+Method
+1-instance
+3-instance
+SVM
+2.65
+9.81
+KNN
+5.23
+13.3
+Deep SVR
+14.01
+25.00
+SAE
+14.93
+26.42
+WMM-Voc
+17.26
+26.59
+Deep WMM-Voc
+17.95
+30.44
+better performance both in 1- and 3-shot setting. This shows
+the efficacy of proposed methods in few-shot learning task.
+4.3.3
+Results on different training/testing splits
+We further validate our findings on ImageNet 2012/2010
+dataset. In general, our framework has advantages over the
+baselines since open vocabulary helps inform the learning
+process when few training instances or limited training data is
+available. The results are compared in Figure 6.
+Supervised learning: As shown in Figure 6, we compare the
+supervised results by increasing the training instances from
+3,000 to 50,000. With 3,000 training instances, the results of
+Deep WMM-Voc are better than all the other baselines with
+the help of learning from free vocabulary. We further evaluate
+our models with larger number of training instances (> 3
+per class). We observe that for standard supervised learning
+setting, the improvements achieved using vocabulary-informed
+learning tend to somewhat diminish as the number of training
+instances substantially grows. With large number of training
+instances, the mapping between low-level image features and
+semantic words, g(x), becomes better behaved and effect of
+additional constraints, due to the open-vocabulary, becomes
+less pronounced.
+Zero-shot Learning: We further validate the results on zero-
+shot learning setting. Figure 6 shows that our models can beat
+all other baselines. Our Deep WMM-Voc always performs
+the best with the source training instances increased from
+TABLE 7
+ImageNet comparison to state-of-the-art on ZSL: We
+compare the results of using 3, 000/all training instances
+for all methods; T-1 (top 1) and T-5 (top 5) classification
+in (%) is reported. The VGG-19 features are used for all
+methods.
+Methods
+S. Sp
+T-1
+T-5
+Deep WMM-Voc
+W
+9.26/10.29
+21.99/23.12
+WMM-Voc
+W
+8.5/8.76
+20.30/21.36
+MM-Voc
+W
+8.9/9.5
+14.9/16.8
+SAE
+W
+5.11/9.32
+12.26/21.04
+ESZSL
+W
+5.86/8.3
+13.71/18.2
+Deep-SVR
+W
+5.29/5.7
+13.32/14.12
+Embed [88]
+W
+–/11.00
+–/25.70
+ConSE [53]
+W
+5.5/7.8
+13.1/15.5
+DeViSE [24]
+W
+3.7/5.2
+11.8/12.8
+AMP [31]
+W
+3.5/6.1
+10.5/13.1
+Chance
+–
+2.78e-3
+–
+3,000 to 50,000. The WMM-Voc always has the second
+best performance; especially when only few source training
+instances are available, i.e., 3,000 and 5,000 training instances.
+Our Deep WMM-Voc and WMM-Voc demonstrate significant
+improvements over the competitors in ZSL task. The good
+performance of Deep WMM-Voc and WMM-Voc is largely
+due to our vocabulary-informed learning framework which can
+leverage the discriminative information from open vocabulary
+and max-margin constraints, helping to improve performance.
+General Zero-shot Learning: In G-ZSL, our methods still
+have the best performance compared to the baselines, as seen
+from Table 5. The Top-1 results of WMM-Voc and Deep
+WMM-Voc are beyond 2%; in contrast, the performance of
+other state-of-art methods are lower than 0.5%.
+Varying training set size: In Figure 6 we also evaluate our
+model with the larger number of training instances (> 3 per
+class) in all settings. The results are inline with prior findings.
+The state-of-the-art on ZSL: We compare our results to sev-
+eral state-of-the-art large-scale zero-shot recognition models.
+Our results are better than those of ConSE, DeViSE, Deep-
+SVR, SAE, ESZSL and AMP on both T-1 and T-5 metrics
+with a very significant margin. Poor results of DeViSE with
+3, 000 training instances are largely due to the inefficient
+learning of visual-semantic embedding matrix. AMP algorithm
+also relies on the embedding matrix from DeViSE, which
+explains similar poor performance of AMP with 3, 000 training
+instances. Table 7 shows that our Deep WMM-Voc obtains
+good performance with (all) 50,000 training instances. Top-5
+accuracy of our methods are beyond 20%. This again validates
+that our proposed methods can have the advantages of learning
+from limited available training instances by leveraging the
+discriminative information from open vocabulary. Embed [1]
+has slightly better ZSL performance compared to our models.
+However, unlike the other works that directly use word vector
+representations of class names, [1] require additional textual
+descriptions of each class to learn better class prototypes.
+Open-set recognition: The open set image recognition results
+
+14
+Testing Classes
+ImageNet Data
+Aux.
+Tag.
+Total
+Vocab
+OPEN-SET310K
+(left)
+(right)
+1000 / 360
+310K
+0
+5
+10
+15
+20
+Hit@k
+5
+10
+15
+20
+25
+30
+Accuracy (%)
+SUPERVISED-like(Aux)
+0
+5
+10
+15
+20
+Hit@k
+0
+1
+2
+3
+4
+5
+6
+Accuracy (%)
+ZERO SHOT-like(Tag)
+MM-Voc
+Deep-SVR
+Deep WMM-Voc
+WMM-Voc
+Fig. 7.
+Open set recognition results on ImageNet
+2012/2010 dataset: Openness=0.9839. Chance=3.2e −
+4%. We use the synsets of each class— a set of synony-
+mous (word or prhase) terms as the ground truth names
+for each instance. We use the model trained with 50,000
+instances sampled equally from source classes.
+are shown in Figure 7. On SUPERVISED-like settings, we
+notice the MM-Voc and WMM-Voc have similar open set
+recognition accuracy. Since this dataset is very large, linear
+mapping g(x) may not have enough capacity to model the
+embedding mapping from visual space to semantic space. Thus
+adding constraints on source training classes in WMM-Voc
+may slightly hinder the learning such an embedding. That
+explains why the results of WMM-Voc are slightly inferior
+to MM-Voc. Deep WMM-Voc has the best performance, due
+to its ability to fine-tune low-level feature representation while
+learning the embedding. On the ZERO-SHOT-like setting, our
+WMM-Voc and Deep WMM-Voc have the best performance.
+Qualitative visualization: We illustrate the embedding space
+learned
+by
+our
+Deep
+WMM-Voc
+model
+for
+the
+Ima-
+geNet2012/2010 dataset in Figure 1. In particular, we have
+4 source/auxiliary and 2 target/zero-shot classes in this figure.
+The better separation among classes is largely attributed to
+open-set max-margin constraints introduced in our vocabulary-
+informed learning model. We further visualize the semantic
+space in Figure 8. Critically, we list seven target classes
+on AwA dataset, as well as their surrounding neighbor-
+hood open vocabulary. For example, “orcas” is very near to
+“killer_whale”. While “orcas” are semantically different from
+“killer_whale”, the difference is much smaller if we compare
+the “orcas” with the other classes, such as “spider monkey”,
+“grizzly_bear” and so on. Hence the “orcas” can be used
+to help learn the class of “killer_whale” in our vocabulary-
+informed learning framework.
+killer_whale
+chihuahua_dog
+dalmatian_dog
+buffalo
+grizzly_bear
+collie
+spider_monkey
+Word prototype:
+golden_retriever
+collies
+fox_terrier
+sheepdog
+shetland_pony
+jack_russell_terrier
+cattle_dog
+wellard
+drover
+collie
+killer_whale
+tilikum
+orcas
+brancheau
+orca
+seaworld
+bottlenose_dolphin
+sea_lion
+whale
+seaworld_orlando
+spider_monkey
+capybara
+giant_anteater
+marmoset
+tamarin
+macaw
+howler_monkeys
+squirrel_monkeys
+macaque
+macaws
+grizzly_bear
+grizzly
+polar_bear
+elk
+grizzly_bears
+coyote
+caribou
+mountain_lion
+mountain_goats
+bighorn_sheep
+buffalo
+minneapolis
+detroit
+chicago
+kansas_city
+pittsburgh
+erie
+duluth
+minnesota
+grand_rapids
+chihuahua_dog
+ciudad_ju_rez
+sonora
+sinaloa
+coahuila
+durango
+jalisco
+michoac_n
+zacatecas
+nuevo_leon
+dalmatian_dog
+istria
+ragusa
+dalmatian_coast
+kor_ula
+gradisca
+sardinian
+zadar
+croatian
+istrian
+Fig. 8.
+Visualization of the semantic space: We
+show the t-SNE visualization of the semantic space. The
+words in boxes are the mapping of training image in the
+semantic space, and close neighbors are shown. The
+neighborhoods extend the single training data to a space
+semantically meaningful.
+5
+CONCLUSION AND FUTURE WORK
+This paper introduces the learning paradigm of vocabulary-
+informed learning, by utilizing open set semantic vocabulary
+to help train better classifiers for observed and unobserved
+classes in supervised learning, ZSL, G-ZSL, and open set
+image recognition settings. We formulate vocabulary-informed
+learning in the maximum margin frameworks. Extensive ex-
+perimental results illustrate the efficacy of such learning
+paradigm. Strikingly, it achieves competitive performance with
+very few training instances and is relatively robust to a large
+open set vocabulary of up to 310, 000 class labels.
+ACKNOWLEDGMENTS
+This work was supported in part by NSFC Project (61702108,
+61622204), STCSM Project (16JC1420400), Eastern Scholar
+(TP2017006), Shanghai Municipal Science and Technol-
+ogy Major Project (2017SHZDZX01, 2018SHZDZX01) and
+ZJLab.
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+2014. 3.4
+Yanwei Fu received the Ph.D. degree from
+Queen Mary University of London in 2014, and
+the M.Eng. degree from the Department of Com-
+puter Science and Technology, Nanjing Univer-
+sity, China, in 2011. He held a post-doctoral po-
+sition at Disney Research, Pittsburgh, PA, USA,
+from 2015 to 2016. He is currently a tenure-track
+Professor with Fudan University. His research
+interests are image and video understanding,
+and life-long learning.
+
+17
+Xiaomei Wang is a PhD student in the School of
+Computer Science of Fudan University. She re-
+ceived the Master degree of communication and
+information system from Shanghai University in
+2016 and the Bachelor degree of electronic infor-
+mation engineering from Shandong University of
+Technology in 2012. Her reaseach interests in-
+clude zero-shot/few-shot learning, image/video
+captioning and visual question answering.
+Hanze Dong is an undergraduate student ma-
+joring in mathematics (data science track) at
+the School of Data Science, Fudan University.
+He works in Shanghai Key Lab of Intelligent
+Information Processing under the supervision
+of Professor Yanwei Fu. His current research
+interests include both machine learning theory
+and its applications.
+Yu-Gang Jiang is Professor of Computer Sci-
+ence at Fudan University and Director of Fudan-
+Jilian Joint Research Center on Intelligent Video
+Technology, Shanghai, China. He is interested
+in all aspects of extracting high-level informa-
+tion from big video data, such as video event
+recognition, object/scene recognition and large-
+scale visual search. His work has led to many
+awards, including the inaugural ACM China Ris-
+ing Star Award, the 2015 ACM SIGMM Rising
+Star Award, and the research award for out-
+standing young researchers from NSF China. He is currently an as-
+sociate editor of ACM TOMM, Machine Vision and Applications (MVA)
+and Neurocomputing. He holds a PhD in Computer Science from City
+University of Hong Kong and spent three years working at Columbia
+University before joining Fudan in 2011.
+Meng Wang is a professor at the Hefei Univer-
+sity of Technology, China. He received his B.E.
+degree and Ph.D. degree in the Special Class
+for the Gifted Young and the Department of
+Electronic Engineering and Information Science
+from the University of Science and Technology of
+China (USTC), Hefei, China, in 2003 and 2008,
+respectively. His current research interests in-
+clude multimedia content analysis, computer vi-
+sion, and pattern recognition. He has authored
+more than 200 book chapters, journal and con-
+ference papers in these areas. He is the recipient of the ACM SIGMM
+Rising Star Award 2014. He is an associate editor of IEEE Transactions
+on Knowledge and Data Engineering (IEEE TKDE), IEEE Transactions
+on Circuits and Systems for Video Technology (IEEE TCSVT), IEEE
+Transactions on Multimedia (IEEE TMM), and IEEE Transactions on
+Neural Networks and Learning Systems (IEEE TNNLS).
+Xiangyang Xue received the BS, MS, and PhD
+degrees in communication engineering from Xi-
+dian University, Xi’an, China, in 1989, 1992, and
+1995, respectively. He is currently a professor of
+computer science with Fudan University, Shang-
+hai, China. His research interests include com-
+puter vision, multimedia information processing
+and machine learning.
+Leonid Sigal is an Associate Professor in the
+Department of Computer Science at the Univer-
+sity of British Columbia and a Faculty Member
+of the Vector Institute for Artificial Intelligence.
+He is a recipient of Canada CIFAR AI Chair
+and NSERC Canada Research Chair (CRC) in
+Computer Vision and Machine Learning. Prior
+to this he was a Senior Research Scientist at
+Disney Research. He completed his Ph.D. at
+Brown University in 2008; received his M.A.
+from Boston University in 1999, and M.Sc. from
+Brown University in 2003. Leonid’s research interests lie in the areas
+of computer vision, machine learning, and computer graphics. Leonid’s
+research emphasis is on machine learning and statistical approaches
+for visual recognition, reasoning, understanding and analytics. He has
+published more than 70 papers in venues and journals in these fields
+(including TPAMI, IJCV, CVPR, ICCV and NeurIPS).
+

6NE4T4oBgHgl3EQf1g37/content/tmp_files/2301.05292v1.pdf.txt

@@ -0,0 +1,659 @@
+A Novel Framework for Handling Sparse Data in Traffic Forecast
+Nikolaos Zygouras
+Huawei Amsterdam Research Center
+Netherlands
+nikolas.zygouras@huawei.com
+Dimitrios Gunopulos
+National and Kapodistrian University of Athens
+Greece
+dg@di.uoa.gr
+ABSTRACT
+The ever increasing amount of GPS-equipped vehicles provides in
+real-time valuable traffic information for the roads traversed by
+the moving vehicles. In this way, a set of sparse and time evolving
+traffic reports is generated for each road. These time series are a
+valuable asset in order to forecast the future traffic condition. In
+this paper we present a deep learning framework that encodes the
+sparse recent traffic information and forecasts the future traffic con-
+dition. Our framework consists of a recurrent part and a decoder.
+The recurrent part employs an attention mechanism that encodes
+the traffic reports that are available at a particular time window.
+The decoder is responsible to forecast the future traffic condition.
+CCS CONCEPTS
+• Information systems → Data stream mining; Location based
+services; Geographic information systems.
+KEYWORDS
+travel time estimation, traffic forecasting, deep learning, trans-
+former, GPS trajectories, mining mobility data
+ACM Reference Format:
+Nikolaos Zygouras and Dimitrios Gunopulos. 2022. A Novel Framework
+for Handling Sparse Data in Traffic Forecast. In The 30th International
+Conference on Advances in Geographic Information Systems (SIGSPATIAL
+’22), November 1–4, 2022, Seattle, WA, USA. ACM, New York, NY, USA, 4 pages.
+https://doi.org/10.1145/3557915.3560968
+1
+INTRODUCTION
+In recent years, the wide usage of mobile devices and the corre-
+sponding collection of vast amounts of spatiotemporal data have
+resulted in the development of various novel Location Based Ser-
+vices (LBS). The LBS are software services that integrate geographic
+information providing appropriate services and information to the
+users [7]. Traffic forecasting and travel time estimation are un-
+doubtedly two of the widely used LBS and a lot of recent research
+work has been conducted towards improving their performance.
+The importance of such services is indicated by the fact that the
+vast majority of drivers consults several times a day services that
+Part of this work was done while N. Zygouras was at the National and Kapodistrian
+University of Athens, Greece.
+Permission to make digital or hard copies of part or all of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for third-party components of this work must be honored.
+For all other uses, contact the owner/author(s).
+SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA
+© 2022 Copyright held by the owner/author(s).
+ACM ISBN 978-1-4503-9529-8/22/11.
+https://doi.org/10.1145/3557915.3560968
+Porto
+Airport
+Estádio do 
+Dragão
+Timestamp now
+?
+?
+Historical Data
+Predictions
+Travel
+Times
+...
+...
+?
+r1
+r2
+Porto
+Airport
+Estádio 
+do 
+Dragão
+r|P |
+q
+Figure 1: The travel time estimation problem for a given
+query path 𝑃𝑞 (blue line) and time of departure 𝑡𝑞 in the city
+of Porto, that starts at 10:00 from the airport of Porto and
+ends at the Estádio do Dragão, the entire path is decomposed
+by a set of |𝑃𝑞| road segments 𝑟1 → 𝑟2 → · · · → 𝑟 |𝑃 | and for
+each road segment we have a time series of travel time re-
+ports, received by the available probe vehicles.
+perform travel time estimation in order to appropriately choose the
+fastest route to follow.
+Motivated by this, in this paper we propose a novel path based
+travel time estimation technique that considers the available traffic
+reports that have been received by the set of the available probe
+vehicles. Each probe vehicle moves in the road network and reports
+the time that was required to traverse each individual road segment.
+In this way, for each road segment of the road network a time series
+of the reported travel times are generated, illustrated at the right
+part of Figure 1. Our technique receives a query path along with
+a time of departure and estimates the time of arrival considering
+the current traffic condition of the road network. Our problem is
+illustrated in Figure 1. A query path 𝑃𝑞 and a time of departure 𝑡𝑞
+are received as input and the task is to estimate the time that is
+required to traverse the whole path 𝑃𝑞 if the driver departs at 𝑡𝑞.
+We propose a novel deep learning framework which is comprised
+of a recurrent part and a decoder. The recurrent part encodes the
+sparse traffic reports that are available at each time window using
+an attention mechanism and an embedding representations for
+each road segment. The decoder is responsible to forecast the traffic
+condition of the next time window.
+2
+RELATED WORK
+In DeepGTT the travel time distribution for any route is learnt by
+conditioning on the real-time traffic [5]. Initially, an embedding is
+arXiv:2301.05292v1  [cs.LG]  12 Jan 2023
+
+12:30Aveleda
+Casteloda
+Maia
+Lavra
+Gondim
+EN542
+EN13
+Mioue
+Silva Escura
+Vila Novada
+Barca
+EN105-2
+Telha
+Moreira
+A28
+SaoPedro
+Fins
+160m
+NTo7Maig-Este,Vermoim
+PortoViaNorte)
+A3Porto.Braqo
+Alfen
+Vermoim
+Nogueira
+ZonoIndustrial
+A41
+Maiat
+deifena
+A.41
+OLIPORI
+A3
+Perafita
+A41 /Moia /(AE) Broga/A42 Felgueiras
+EN14
+Milheiros
+Santa Cruz
+Gueifaes
+Ermesinde
+doBispo
+Refinaria
+Balio
+deMatosinhos
+EN15-1
+VILPL
+dstoias
+Aguas Santas
+A4
+Guifoes
+Sao Mamede
+deInfesta
+Pedroucos
+Baguimdo
+Senhora da
+Monte
+Hora
+34m
+Matosinhos
+EN12
+RioTinto
+Paranhos
+EM612
+EN12
+Pargue
+Aldoar
+daCidode
+Ranalde
+EN15
+Bogvista
+Fanzeres
+Nevogilde
+tecedc
+For
+Fan
+Cedofeita
+Areias/W12Circunva/acao
+Lordelo.do
+Campanttransavia
+F-GZHUSUPERBOCKSIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA
+Zygouras et al.
+estimated for each link considering its characteristics, then a non-
+linear factorization model generates the speed and finally an atten-
+tion mechanism is used to generate the observed travel time. Also,
+in HETETA [3] the road map is translated into a multi-relational
+network, considering the traffic behavior patterns. Temporal and
+graph convolutions are used in order to learn spatiotemporal het-
+erogeneous information, considering recent, daily and weekly traf-
+fic. CompactETA [2] provides an accurate ETA estimation with
+low latency. Graph attention network was employed in order to
+encode spatial and temporal dependencies of the weighted road
+network and the sequential information of the route is encoded
+with positional encoding. A multi-layer perceptron was used for
+online inference. The authors in [6] proposed a multitask represen-
+tation learning model which predicts the travel time of an origin-
+destination pair extracting a representation that preserves trips
+properties and road network structure. ConSTGAT [1] proposed
+a spatiotemporal graph neural network exploiting the spatial and
+temporal information with a 3D-attention mechanism and a model
+with convolutions over local windows in order to capture route’s
+contextual information. STGNN-TTE [4] adopted a spatial–temporal
+module to capture the real-time traffic condition and a transformer
+layer to estimate the links’ travel time and the total routes’ travel
+time synchronously.
+3
+OUR APPROACH
+3.1
+Problem Definition
+Road Network is represented as a directed graph 𝐺(𝑉, 𝐸), where
+the nodes 𝑉 represent the junctions and the edges 𝐸 represent the
+|𝐸| roads segments. A road segment 𝑟 ∈ 𝐸 is the part of the road net-
+work between two consecutive junctions without any intermediate
+junction between them.
+Trip 𝑇 is a time ordered sequence of |𝑇 | points 𝑝1 → · · · → 𝑝 |𝑇 |;
+each point 𝑝 contains the geospatial coordinates of the moving
+object along with the corresponding timestamp 𝜏 that the vehicle
+was at this particular location 𝑝 = (𝑙𝑜𝑛,𝑙𝑎𝑡,𝜏).
+Map-matched Trip 𝑇𝐺 is a sequence of |𝑇𝐺 | consecutive points
+𝑝′
+1 → · · · → 𝑝′
+|𝑇𝐺 | that comes from map matching trip 𝑇 on the
+road network 𝐺. Each point 𝑝′ corresponds to a road segment that
+was traversed by𝑇. Each point 𝑝′ of the map matched trip contains
+a triplet (𝑟,𝑡𝑡,𝜏); 𝑟 is the traversed road segment, 𝑡𝑡 is the travel
+time of the road segment 𝑟 and is computed assuming that the
+vehicle moved with the same speed in the road network between
+two consecutive GPS points and 𝜏 is the timestamp that the travel
+time is reported to the system.
+Travel time reports 𝐷 is the collection of travel times for the
+road segments as they are extracted by the trips of all the available
+probe vehicles that traverse the road network. Each travel time
+report (𝑟,𝑡𝑡,𝑡,𝑇𝑖𝑑) contains the information of the map-matched
+trips enriched by the id of the trip 𝑇𝑖𝑑.
+Path 𝑃 is a sequence of |𝑃| consecutive road segments 𝑟1 → · · · →
+𝑟 |𝑃 |, where 𝑟𝑖 is the 𝑖th road segment of 𝑃.
+Below we define formally the traffic forecasting problem.
+Traffic Forecasting: Given the available travel times of the last
+𝐿 time windows T𝑡−𝐿+1:𝑡 a traffic forecasting model forecasts the
+travel times of the next 𝐻 time windows T𝑡+1:𝑡+𝐻, where the vector
+T𝑡 contains the travel times of the 𝐸 road segments at time 𝑡. The
+Road 
+Network
+Trajectories
+Travel Times 
+Reports
+D
+Travel Times
+ZScores 
+Aggregated 
+Travel Times  
+M 
+Time 
+Window 
+length
+Roads 
+Embeddings
+Matrix 
+Factorization
+Map 
+Matching
+Extracting 
+Roads Segs 
+Statistics
+Figure 2: Data preparation.
+input matrix T𝑡−𝐿+1:𝑡 ∈ R|𝐸 |×𝐿 has missing values for the roads
+that were not traversed by any vehicle at a given time window. The
+forecasted matrix T𝑡+1:𝑡+𝐻 ∈ R|𝐸 |×𝐻 contains forecasts for all the
+road segments 𝐸 for the next 𝐻 time windows.
+3.2
+Data Preparation
+The first step of the proposed framework is to preprocess the raw
+data and prepare them appropriately in order to feed them to the
+neural network. The overview of the data preparation approach is
+illustrated in Figure 2 and described below.
+Map Matching. Firstly, we map-match the available trips matching
+them to the road network 𝐺. Each trip 𝑇 is transformed into a map-
+matched trip 𝑇𝐺. This procedure generates the set of the available
+travel time reports 𝐷. This step is common to both the historical
+data that are used to train our model and the streaming traffic data
+that will be used to make forecasts in real time.
+Modeling the periodicity of traffic. In order to model the peri-
+odicity of traffic we estimate from the historical travel time reports
+the average travel time 𝑎𝑣𝑔_𝑡𝑡𝑖,ℎ𝑜𝑢𝑟 for each road segment 𝑟𝑖 ∈ 𝐸
+and for different hours of day ℎ𝑜𝑢𝑟 ∈ [1 . . . 24]. Then, we subtract
+from each travel time the historical average travel time for that road
+segment at the given hour. In this way, we force the deep learning
+framework to model, for each different road segment, the deviation
+from the average travel time for the different hours of the day.
+Standardizing Travel Times. Since road segments have different
+lengths and speed limits we selected to standardize the travel time
+reports, considering the average behaviour of each different road
+segment. More specifically, for each road segment 𝑟𝑖 we compute
+the historical average travel time 𝜇𝑖 and standard deviation of travel
+times 𝜎𝑖 and we use these values in order to standardize the travel
+times per road segment. For instance, if 𝑡𝑡5 is a travel time that is re-
+ported for the road segment 𝑟5 then the corresponding Z-Score will
+be 𝑡𝑡5−𝜇5
+𝜎5
+. In the rest of the paper we assume that travel times are
+the Z-Scores of travel times with subtracted the average historical
+travel time for the different hours of the day.
+Aggregating travel times. The historical travel time reports 𝐷
+are grouped together generating a sparse matrix 𝑀 ∈ R|𝐸 |×𝑊 . The
+rows of 𝑀 correspond to the |𝐸| road segments of the road network
+𝐺 and the columns correspond to the𝑊 time windows. In this work
+we use time windows of 15 minutes. If more than one travel time
+reports are available for a particular road segment 𝑟𝑖 at the same
+time window 𝑤𝑗 then 𝑀𝑖𝑗 contains the average travel time of the
+available travel times.
+
+A Novel Framework for Handling Sparse Data in Traffic Forecast
+SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA
+Roads
+ Embeddings
+Travel 
+Times
+Scaled Dot-Product 
+Attention
+Ki
+Qi
+Vi
+headi
+MatMul
+MatMul
+i=1...h
+Concatenate
+Concatenate
+Linear
+Linear
+Linear
+Linear
+Linear
+h
+h
+Figure 3: Multi-Head Scaled Dot-Product
+Attention
+Roads
+ Embeddings
+Travel 
+Times
+Roads
+ Embeddings
+Travel 
+Times
+Attention Mechanism
++
+Norm.
++
++
+Linear
+x2
+Roads
+ Embeddings
+Travel 
+Times
+Attention Mechanism
++
+Norm.
++
++
+Conv1D
+x2
+Norm.
+V
+K
+Q
+Encoder 
+Output
+N
+Figure 4: Encoder Block.
+V
+Roads
+ Embeddings
+Travel 
+Times
+Attention Mechanism
++
+Norm.
++
++
+Linear
+x2
+Norm.
+V
+K
+Q
+Attention Mechanism
++
+Norm.
++
+K
+Q
+Roads
+ Embeddings
+Travel Times
+Encoder 
+Output
+Figure 5: Decoder Block.
+Extracting Road Segments Embeddings. An embedding repre-
+sentation 𝐸𝑖 is detected for each road segment 𝑟𝑖 considering its
+historical travel time reports. Here, we follow the process intro-
+duced by [9]. We perform matrix factorization in the sparse matrix
+𝑀, learning a matrix P ∈ R|𝐸 |×𝑑 contains a 𝑑-dimensional embed-
+ding representation of the available road segments
+Feeding the Model. The deep learning model that is described
+in Section 3.4 receives as input two vectors that contain: (i) the
+aggregated travel times that are available for a given time window
+and (ii) the corresponding road segments. For instance, consider
+a road network 𝐺 that is comprised of |𝐸| = 5 road segments [𝑟𝑖],
+𝑖 ∈ [1, . . . , 5]. If for a particular time window only the travel times
+𝑡𝑡2 and 𝑡𝑡5 of road segments 𝑟2 and 𝑟5 respectively are available,
+then the inputs to the deep learning model will be the following: the
+vector of the travel times T = [𝑡𝑡2,𝑡𝑡5, ∅, ∅, ∅] ∈ R|𝐸 | and the vector
+of the road segments ids R = [𝑟2,𝑟5, ∅, ∅, ∅] ∈ Z|𝐸 |. Then, inside the
+deep learning model the ids of the road segments are transmitted
+to an embedding layer. This layer transforms the vector R into a
+matrix of road segments embeddings E = [𝐸2, 𝐸5, ∅, ∅, ∅] ∈ R|𝐸 |×𝑑.
+The embedding representation of the road segments is trainable
+and initialized with matrix P, computed using matrix factorization
+as it was described above.
+3.3
+Attention Mechanism
+Here, we extend the "Scaled Dot-Product Attention" that was intro-
+duced in [8]. The proposed attention mechanism encodes a varying
+number of travel time reports received at a particular time window.
+We consider as input here the following: (i) the embeddings’ matrix
+of the road segments that have been traversed by the probe vehi-
+cles at a particular time window along with (ii) the vector of the
+corresponding reported travel times. The overview of the proposed
+attention mechanism is illustrated in Figure 3.
+Initially, the Query 𝑄𝑖, Key 𝐾𝑖 and Value 𝑉𝑖 matrices are com-
+puted using the embeddings E of the available road segments, com-
+puted earlier. Therefore, three parameter matrices 𝑊 𝑄
+𝑖
+∈ R𝑑×𝑑,
+𝑊 𝐾
+𝑖
+∈ R𝑑×𝑑 and 𝑊 𝑉
+𝑖
+∈ R𝑑×𝑑 are trained using the training in-
+stances and are used to compute the matrices 𝑄𝑖 = E𝑊 𝑄
+𝑖 , 𝐾𝑖 =
+E𝑊 𝐾
+𝑖
+and 𝑉𝑖 = E𝑊 𝑉
+𝑖 . The index 𝑖 ∈ [1, . . . ,ℎ] of the different
+parameter matrices stands for the ℎ parallel attention layers.
+The next step is to compute the attention scores using the 𝑄𝑖
+and 𝐾𝑖 matrices. The scores indicate the focus that will be placed at
+the travel times of other road segments, that have been reported at
+the same time window. Multiple attention heads ℎ𝑒𝑎𝑑𝑖 ∈ R|𝐸 |×|𝐸 |
+are computed in parallel according to eq. 1 indicating the attention
+at each particular road segment.
+ℎ𝑒𝑎𝑑𝑖 = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(
+𝑄𝑖𝐾𝑇
+𝑖
+√
+𝑑
+), 𝑖 ∈ [1, . . . ,ℎ]
+(1)
+The road segments’ embeddings and travel times are then up-
+dated considering the computed attention heads. More specifi-
+cally we train the parameter matrices 𝑊 𝑂1 ∈ Rℎ𝑑×𝑑 and 𝑊 𝑂2 ∈
+Rℎ|𝐸 |×|𝐸 | that are multiplied with the concatenated values 𝑉𝑖 and
+travel times T respectively.
+E′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1𝑉1, . . . ,ℎ𝑒𝑎𝑑ℎ𝑉ℎ)𝑊 𝑂1
+(2)
+T ′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1T, . . . ,ℎ𝑒𝑎𝑑ℎT)𝑊 𝑂2
+(3)
+3.4
+Traffic Transformer’s Architecture
+Here we describe our model’s architecture, which is based on the
+original implementation of Transformer model described in [8].
+3.4.1
+Encoder. The encoder considers all travel time reports that
+are available at a given time window, encodining the traffic con-
+dition of that time window. It is comprised by a set of 𝑁 identical
+blocks. The first block receives as input the roads segments embed-
+dings and the travel times that are available at a given time window,
+following the data preparation procedure described in Section 3.2.
+The rest encoder blocks receive as input the output of the previous
+block. Figure 4 illustrates the overview of the encoder block.
+
+SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA
+Zygouras et al.
+Encoder
+Roads
+ Embeddings
+Travel 
+Times
+Encoder
+Block
+Roads
+ Ids
+Travel 
+Times
+Decoder
+Block
+N
+Encoder
+Encoder
+Block
+Roads
+ Ids
+Travel 
+Times
+Decoder
+Block
+N
+...
+...
+Decoder
+Decoder
+Block
+Query
+Roads Ids
+∅
+N
+∅
+∅
+t  - L - 1
+t
+t + 1
+...
+...
+1st Recurrent Cell
+Lth Recurrent Cell
+Embedding
+Embedding
+Embedding
+Figure 6: Overview of our model.
+Each block first transmits the matrix of the available roads em-
+beddings E and the corresponding vector of travel times T at the
+attention mechanism. The attention mechanism produces the ma-
+trix E′ and the vector T ′. Then, residual connections are employed
+at the output of the attention mechanism, normalizing the sum of
+the received roads segments embeddings E with the output of the
+attention mechanism E′. The output is transmitted to two dense
+layers followed by another residual connection. For the travel times
+the output of each encoder block is the sum of the received travel
+times T and the output of the attention mechanism T ′.
+3.4.2
+Decoder. The decoder (Figure 5) is responsible to forecast
+the travel times of the next time window, considering the encoder’s
+output. The decoder consists of a set of 𝑁 blocks, similarly to the
+encoder. Each block receives as input the output of the encoder and
+the output of the previous block. In the training phase the first block
+receives as input (i) the embeddings of the road segments that are
+available in the target time window and (ii) a vector of zeros. For the
+testing phase the first block receives as input (i) the embeddings
+of all the road segments 𝐸 and (ii) a vector of zeros. Recall that
+we are working with the Z-Scores of travel times aggregated per
+road segment. Consequently, the vector of zeros corresponds to the
+average travel for each road segment.
+Each block of the decoder contains two attention mechanisms.
+Firstly, the embeddings of the queried road segments and the travel
+times are transmitted to the first attention mechanism, followed
+by a residual connection. Then a second attention mechanism is
+employed, receiving as input the embedding matrix E′
+1 that resulted
+from the first attention mechanism along with the embeddings and
+the travel times that come from the output of the encoder. The main
+difference here is that the matrices 𝑉𝑖 and 𝐾𝑖 are computed from
+the output of the encoder and that the considered travel times come
+from the encoder. Then, the embedding output E′
+2 of the second
+attention mechanism is followed again by a residual connection.
+This is followed by two dense layers and a second residual connec-
+tion. Finally, the travel times that result from each block is the sum
+of the original travel times T that were received as input along
+with travel times that result from the first and the second attention
+mechanism T ′
+1 and T ′
+2 respectively.
+3.4.3
+Recurrent Neural Network. The final module of our proposed
+model is a recurrent model that considers the sequence of the last
+𝐿 time windows. Each cell of the recurrent network encapsulates
+an encoder (consisting of 𝑁 encoder blocks) along with a single
+decoder block. Here the decoder block is responsible to aggregate
+the information that has been encoded from the previous time win-
+dow with the information that has been encoded from the current
+time window. Figure 6 illustrates this recurrent architecture. The
+encoder and the decoder blocks of the different recurrent cells share
+the same weights among the 𝐿 different time windows.
+The output of the last recurrent cell is used by the decoder model
+in order to make forecasts. The decoder model consists of 𝑁 decoder
+blocks that are different from each other and from the decoder
+block that lies inside the recurrent cells. The output of the last
+decoder block contains the predicted travel times of the queried
+road segments for the next time window. This will be the Z-Scores
+of the travel times for the road segments that were queried at the
+first decoder block.
+4
+CONCLUSION
+In this paper we presented a novel deep learning framework that
+considers the current traffic condition of the road network and is
+used to forecast the traffic condition. Our framework can efficiently
+encode the travel time reports that are available at a particular
+time window via an attention mechanism that considers only the
+available travel times reports and the corresponding embeddings
+of the road segments.
+ACKNOWLEDGMENTS
+This research has been financed by the European Union through the
+H2020 LAMBDA Project (No. 734242), the EU ICT-48 2020 project
+TAILOR (No. 952215) and the Horizon Europe AUTOFAIR Project
+(No. 101070568).
+REFERENCES
+[1]
+Xiaomin Fang, Jizhou Huang, Fan Wang, Lingke Zeng, Haijin Liang, and Haifeng
+Wang. 2020. Constgat: contextual spatial-temporal graph attention network for
+travel time estimation at baidu maps. In Proceedings of the 26th ACM SIGKDD
+International Conference on Knowledge Discovery & Data Mining, 2697–2705.
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+Kun Fu, Fanlin Meng, Jieping Ye, and Zheng Wang. 2020. Compacteta: a fast
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+Huiting Hong, Yucheng Lin, Xiaoqing Yang, Zang Li, Kung Fu, Zheng Wang,
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+Guangyin Jin, Min Wang, Jinlei Zhang, Hengyu Sha, and Jincai Huang. 2022.
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+Xiucheng Li, Gao Cong, Aixin Sun, and Yun Cheng. 2019. Learning travel time
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+Jochen Schiller and Agnès Voisard. 2004. Location-based services. Elsevier.
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+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones,
+Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you
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+Nikolaos Zygouras, Nikolaos Panagiotou, Yang Li, Dimitrios Gunopulos, and
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+tional Conference on Advances in Geographic Information Systems, 99–108.
+

6NE4T4oBgHgl3EQf1g37/content/tmp_files/load_file.txt

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+page_content='A Novel Framework for Handling Sparse Data in Traffic Forecast  Nikolaos Zygouras  Huawei Amsterdam Research Center  Netherlands  nikolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='zygouras@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='com  Dimitrios Gunopulos  National and Kapodistrian University of Athens  Greece  dg@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='gr  ABSTRACT  The ever increasing amount of GPS-equipped vehicles provides in  real-time valuable traffic information for the roads traversed by  the moving vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In this way, a set of sparse and time evolving  traffic reports is generated for each road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' These time series are a  valuable asset in order to forecast the future traffic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In  this paper we present a deep learning framework that encodes the  sparse recent traffic information and forecasts the future traffic con-  dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Our framework consists of a recurrent part and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The recurrent part employs an attention mechanism that encodes  the traffic reports that are available at a particular time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The decoder is responsible to forecast the future traffic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  CCS CONCEPTS  Information systems → Data stream mining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Location based  services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Geographic information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  KEYWORDS  travel time estimation, traffic forecasting, deep learning, trans-  former, GPS trajectories, mining mobility data  ACM Reference Format:  Nikolaos Zygouras and Dimitrios Gunopulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' A Novel Framework  for Handling Sparse Data in Traffic Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In The 30th International  Conference on Advances in Geographic Information Systems (SIGSPATIAL  ’22), November 1–4, 2022, Seattle, WA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' ACM, New York, NY, USA, 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='1145/3557915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='3560968  1  INTRODUCTION  In recent years, the wide usage of mobile devices and the corre-  sponding collection of vast amounts of spatiotemporal data have  resulted in the development of various novel Location Based Ser-  vices (LBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The LBS are software services that integrate geographic  information providing appropriate services and information to the  users [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Traffic forecasting and travel time estimation are un-  doubtedly two of the widely used LBS and a lot of recent research  work has been conducted towards improving their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The importance of such services is indicated by the fact that the  vast majority of drivers consults several times a day services that  Part of this work was done while N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Zygouras was at the National and Kapodistrian  University of Athens, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Permission to make digital or hard copies of part or all of this work for personal or  classroom use is granted without fee provided that copies are not made or distributed  for profit or commercial advantage and that copies bear this notice and the full citation  on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA  © 2022 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  ACM ISBN 978-1-4503-9529-8/22/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='1145/3557915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='3560968  Porto  Airport  Estádio do  Dragão  Timestamp now  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Historical Data  Predictions  Travel  Times  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  r1  r2  Porto  Airport  Estádio  do  Dragão  r|P |  q  Figure 1: The travel time estimation problem for a given  query path 𝑃𝑞 (blue line) and time of departure 𝑡𝑞 in the city  of Porto, that starts at 10:00 from the airport of Porto and  ends at the Estádio do Dragão, the entire path is decomposed  by a set of |𝑃𝑞| road segments 𝑟1 → 𝑟2 → · · · → 𝑟 |𝑃 | and for  each road segment we have a time series of travel time re-  ports, received by the available probe vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  perform travel time estimation in order to appropriately choose the  fastest route to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Motivated by this, in this paper we propose a novel path based  travel time estimation technique that considers the available traffic  reports that have been received by the set of the available probe  vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each probe vehicle moves in the road network and reports  the time that was required to traverse each individual road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  In this way, for each road segment of the road network a time series  of the reported travel times are generated, illustrated at the right  part of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Our technique receives a query path along with  a time of departure and estimates the time of arrival considering  the current traffic condition of the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Our problem is  illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' A query path 𝑃𝑞 and a time of departure 𝑡𝑞  are received as input and the task is to estimate the time that is  required to traverse the whole path 𝑃𝑞 if the driver departs at 𝑡𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  We propose a novel deep learning framework which is comprised  of a recurrent part and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The recurrent part encodes the  sparse traffic reports that are available at each time window using  an attention mechanism and an embedding representations for  each road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The decoder is responsible to forecast the traffic  condition of the next time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  2  RELATED WORK  In DeepGTT the travel time distribution for any route is learnt by  conditioning on the real-time traffic [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Initially, an embedding is  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='05292v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='LG]  12 Jan 2023  12:30Aveleda  Casteloda  Maia  Lavra  Gondim  EN542  EN13  Mioue  Silva Escura  Vila Novada  Barca  EN105-2  Telha  Moreira  A28  SaoPedro  Fins  160m  NTo7Maig-Este,Vermoim  PortoViaNorte)  A3Porto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='Braqo  Alfen  Vermoim  Nogueira  ZonoIndustrial  A41  Maiat  deifena  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='41  OLIPORI  A3  Perafita  A41 /Moia /(AE) Broga/A42 Felgueiras  EN14  Milheiros  Santa Cruz  Gueifaes  Ermesinde  doBispo  Refinaria  Balio  deMatosinhos  EN15-1  VILPL  dstoias  Aguas Santas  A4  Guifoes  Sao Mamede  deInfesta  Pedroucos  Baguimdo  Senhora da  Monte  Hora  34m  Matosinhos  EN12  RioTinto  Paranhos  EM612  EN12  Pargue  Aldoar  daCidode  Ranalde  EN15  Bogvista  Fanzeres  Nevogilde  tecedc  For  Fan  Cedofeita  Areias/W12Circunva/acao  Lordelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='do  Campanttransavia  F-GZHUSUPERBOCKSIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA  Zygouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  estimated for each link considering its characteristics, then a non-  linear factorization model generates the speed and finally an atten-  tion mechanism is used to generate the observed travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Also,  in HETETA [3] the road map is translated into a multi-relational  network, considering the traffic behavior patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Temporal and  graph convolutions are used in order to learn spatiotemporal het-  erogeneous information, considering recent, daily and weekly traf-  fic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' CompactETA [2] provides an accurate ETA estimation with  low latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Graph attention network was employed in order to  encode spatial and temporal dependencies of the weighted road  network and the sequential information of the route is encoded  with positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' A multi-layer perceptron was used for  online inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The authors in [6] proposed a multitask represen-  tation learning model which predicts the travel time of an origin-  destination pair extracting a representation that preserves trips  properties and road network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' ConSTGAT [1] proposed  a spatiotemporal graph neural network exploiting the spatial and  temporal information with a 3D-attention mechanism and a model  with convolutions over local windows in order to capture route’s  contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' STGNN-TTE [4] adopted a spatial–temporal  module to capture the real-time traffic condition and a transformer  layer to estimate the links’ travel time and the total routes’ travel  time synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  3  OUR APPROACH  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='1  Problem Definition  Road Network is represented as a directed graph 𝐺(𝑉, 𝐸), where  the nodes 𝑉 represent the junctions and the edges 𝐸 represent the  |𝐸| roads segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' A road segment 𝑟 ∈ 𝐸 is the part of the road net-  work between two consecutive junctions without any intermediate  junction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Trip 𝑇 is a time ordered sequence of |𝑇 | points 𝑝1 → · · · → 𝑝 |𝑇 |;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  each point 𝑝 contains the geospatial coordinates of the moving  object along with the corresponding timestamp 𝜏 that the vehicle  was at this particular location 𝑝 = (𝑙𝑜𝑛,𝑙𝑎𝑡,𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Map-matched Trip 𝑇𝐺 is a sequence of |𝑇𝐺 | consecutive points  𝑝′  1 → · · · → 𝑝′  |𝑇𝐺 | that comes from map matching trip 𝑇 on the  road network 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each point 𝑝′ corresponds to a road segment that  was traversed by𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each point 𝑝′ of the map matched trip contains  a triplet (𝑟,𝑡𝑡,𝜏);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 𝑟 is the traversed road segment, 𝑡𝑡 is the travel  time of the road segment 𝑟 and is computed assuming that the  vehicle moved with the same speed in the road network between  two consecutive GPS points and 𝜏 is the timestamp that the travel  time is reported to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Travel time reports 𝐷 is the collection of travel times for the  road segments as they are extracted by the trips of all the available  probe vehicles that traverse the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each travel time  report (𝑟,𝑡𝑡,𝑡,𝑇𝑖𝑑) contains the information of the map-matched  trips enriched by the id of the trip 𝑇𝑖𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Path 𝑃 is a sequence of |𝑃| consecutive road segments 𝑟1 → · · · →  𝑟 |𝑃 |, where 𝑟𝑖 is the 𝑖th road segment of 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Below we define formally the traffic forecasting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Traffic Forecasting: Given the available travel times of the last  𝐿 time windows T𝑡−𝐿+1:𝑡 a traffic forecasting model forecasts the  travel times of the next 𝐻 time windows T𝑡+1:𝑡+𝐻, where the vector  T𝑡 contains the travel times of the 𝐸 road segments at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The  Road  Network  Trajectories  Travel Times  Reports  D  Travel Times  ZScores  Aggregated  Travel Times  M  Time  Window  length  Roads  Embeddings  Matrix  Factorization  Map  Matching  Extracting  Roads Segs  Statistics  Figure 2: Data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  input matrix T𝑡−𝐿+1:𝑡 ∈ R|𝐸 |×𝐿 has missing values for the roads  that were not traversed by any vehicle at a given time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The  forecasted matrix T𝑡+1:𝑡+𝐻 ∈ R|𝐸 |×𝐻 contains forecasts for all the  road segments 𝐸 for the next 𝐻 time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='2  Data Preparation  The first step of the proposed framework is to preprocess the raw  data and prepare them appropriately in order to feed them to the  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The overview of the data preparation approach is  illustrated in Figure 2 and described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Map Matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Firstly, we map-match the available trips matching  them to the road network 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each trip 𝑇 is transformed into a map-  matched trip 𝑇𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' This procedure generates the set of the available  travel time reports 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' This step is common to both the historical  data that are used to train our model and the streaming traffic data  that will be used to make forecasts in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Modeling the periodicity of traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In order to model the peri-  odicity of traffic we estimate from the historical travel time reports  the average travel time 𝑎𝑣𝑔_𝑡𝑡𝑖,ℎ𝑜𝑢𝑟 for each road segment 𝑟𝑖 ∈ 𝐸  and for different hours of day ℎ𝑜𝑢𝑟 ∈ [1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Then, we subtract  from each travel time the historical average travel time for that road  segment at the given hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In this way, we force the deep learning  framework to model, for each different road segment, the deviation  from the average travel time for the different hours of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Standardizing Travel Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Since road segments have different  lengths and speed limits we selected to standardize the travel time  reports, considering the average behaviour of each different road  segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' More specifically, for each road segment 𝑟𝑖 we compute  the historical average travel time 𝜇𝑖 and standard deviation of travel  times 𝜎𝑖 and we use these values in order to standardize the travel  times per road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' For instance, if 𝑡𝑡5 is a travel time that is re-  ported for the road segment 𝑟5 then the corresponding Z-Score will  be 𝑡𝑡5−𝜇5  𝜎5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In the rest of the paper we assume that travel times are  the Z-Scores of travel times with subtracted the average historical  travel time for the different hours of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Aggregating travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The historical travel time reports 𝐷  are grouped together generating a sparse matrix 𝑀 ∈ R|𝐸 |×𝑊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The  rows of 𝑀 correspond to the |𝐸| road segments of the road network  𝐺 and the columns correspond to the𝑊 time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In this work  we use time windows of 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' If more than one travel time  reports are available for a particular road segment 𝑟𝑖 at the same  time window 𝑤𝑗 then 𝑀𝑖𝑗 contains the average travel time of the  available travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  A Novel Framework for Handling Sparse Data in Traffic Forecast  SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA  Roads  Embeddings  Travel  Times  Scaled Dot-Product  Attention  Ki  Qi  Vi  headi  MatMul  MatMul  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='h  Concatenate  Concatenate  Linear  Linear  Linear  Linear  Linear  h  h  Figure 3: Multi-Head Scaled Dot-Product  Attention  Roads  Embeddings  Travel  Times  Roads  Embeddings  Travel  Times  Attention Mechanism  +  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  +  +  Linear  x2  Roads  Embeddings  Travel  Times  Attention Mechanism  +  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  +  +  Conv1D  x2  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  V  K  Q  Encoder  Output  N  Figure 4: Encoder Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  V  Roads  Embeddings  Travel  Times  Attention Mechanism  +  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  +  +  Linear  x2  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  V  K  Q  Attention Mechanism  +  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  +  K  Q  Roads  Embeddings  Travel Times  Encoder  Output  Figure 5: Decoder Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Extracting Road Segments Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' An embedding repre-  sentation 𝐸𝑖 is detected for each road segment 𝑟𝑖 considering its  historical travel time reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Here, we follow the process intro-  duced by [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' We perform matrix factorization in the sparse matrix  𝑀, learning a matrix P ∈ R|𝐸 |×𝑑 contains a 𝑑-dimensional embed-  ding representation of the available road segments  Feeding the Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The deep learning model that is described  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='4 receives as input two vectors that contain: (i) the  aggregated travel times that are available for a given time window  and (ii) the corresponding road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' For instance, consider  a road network 𝐺 that is comprised of |𝐸| = 5 road segments [𝑟𝑖],  𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' , 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' If for a particular time window only the travel times  𝑡𝑡2 and 𝑡𝑡5 of road segments 𝑟2 and 𝑟5 respectively are available,  then the inputs to the deep learning model will be the following: the  vector of the travel times T = [𝑡𝑡2,𝑡𝑡5, ∅, ∅, ∅] ∈ R|𝐸 | and the vector  of the road segments ids R = [𝑟2,𝑟5, ∅, ∅, ∅] ∈ Z|𝐸 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Then, inside the  deep learning model the ids of the road segments are transmitted  to an embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' This layer transforms the vector R into a  matrix of road segments embeddings E = [𝐸2, 𝐸5, ∅, ∅, ∅] ∈ R|𝐸 |×𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The embedding representation of the road segments is trainable  and initialized with matrix P, computed using matrix factorization  as it was described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='3  Attention Mechanism  Here, we extend the "Scaled Dot-Product Attention" that was intro-  duced in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The proposed attention mechanism encodes a varying  number of travel time reports received at a particular time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  We consider as input here the following: (i) the embeddings’ matrix  of the road segments that have been traversed by the probe vehi-  cles at a particular time window along with (ii) the vector of the  corresponding reported travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The overview of the proposed  attention mechanism is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Initially, the Query 𝑄𝑖, Key 𝐾𝑖 and Value 𝑉𝑖 matrices are com-  puted using the embeddings E of the available road segments, com-  puted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Therefore, three parameter matrices 𝑊 𝑄  𝑖  ∈ R𝑑×𝑑,  𝑊 𝐾  𝑖  ∈ R𝑑×𝑑 and 𝑊 𝑉  𝑖  ∈ R𝑑×𝑑 are trained using the training in-  stances and are used to compute the matrices 𝑄𝑖 = E𝑊 𝑄  𝑖 , 𝐾𝑖 =  E𝑊 𝐾  𝑖  and 𝑉𝑖 = E𝑊 𝑉  𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The index 𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' ,ℎ] of the different  parameter matrices stands for the ℎ parallel attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The next step is to compute the attention scores using the 𝑄𝑖  and 𝐾𝑖 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The scores indicate the focus that will be placed at  the travel times of other road segments, that have been reported at  the same time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Multiple attention heads ℎ𝑒𝑎𝑑𝑖 ∈ R|𝐸 |×|𝐸 |  are computed in parallel according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 1 indicating the attention  at each particular road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  ℎ𝑒𝑎𝑑𝑖 = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(  𝑄𝑖𝐾𝑇  𝑖  √  𝑑  ), 𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' ,ℎ]  (1)  The road segments’ embeddings and travel times are then up-  dated considering the computed attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' More specifi-  cally we train the parameter matrices 𝑊 𝑂1 ∈ Rℎ𝑑×𝑑 and 𝑊 𝑂2 ∈  Rℎ|𝐸 |×|𝐸 | that are multiplied with the concatenated values 𝑉𝑖 and  travel times T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  E′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1𝑉1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' ,ℎ𝑒𝑎𝑑ℎ𝑉ℎ)𝑊 𝑂1  (2)  T ′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' ,ℎ𝑒𝑎𝑑ℎT)𝑊 𝑂2  (3)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='4  Traffic Transformer’s Architecture  Here we describe our model’s architecture, which is based on the  original implementation of Transformer model described in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='1  Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The encoder considers all travel time reports that  are available at a given time window, encodining the traffic con-  dition of that time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' It is comprised by a set of 𝑁 identical  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The first block receives as input the roads segments embed-  dings and the travel times that are available at a given time window,  following the data preparation procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The rest encoder blocks receive as input the output of the previous  block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Figure 4 illustrates the overview of the encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA  Zygouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Encoder  Roads  Embeddings  Travel  Times  Encoder  Block  Roads  Ids  Travel  Times  Decoder  Block  N  Encoder  Encoder  Block  Roads  Ids  Travel  Times  Decoder  Block  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Decoder  Decoder  Block  Query  Roads Ids  ∅  N  ∅  ∅  t  - L - 1  t  t + 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  1st Recurrent Cell  Lth Recurrent Cell  Embedding  Embedding  Embedding  Figure 6: Overview of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Each block first transmits the matrix of the available roads em-  beddings E and the corresponding vector of travel times T at the  attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The attention mechanism produces the ma-  trix E′ and the vector T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Then, residual connections are employed  at the output of the attention mechanism, normalizing the sum of  the received roads segments embeddings E with the output of the  attention mechanism E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The output is transmitted to two dense  layers followed by another residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' For the travel times  the output of each encoder block is the sum of the received travel  times T and the output of the attention mechanism T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='2  Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The decoder (Figure 5) is responsible to forecast  the travel times of the next time window, considering the encoder’s  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The decoder consists of a set of 𝑁 blocks, similarly to the  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each block receives as input the output of the encoder and  the output of the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' In the training phase the first block  receives as input (i) the embeddings of the road segments that are  available in the target time window and (ii) a vector of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' For the  testing phase the first block receives as input (i) the embeddings  of all the road segments 𝐸 and (ii) a vector of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Recall that  we are working with the Z-Scores of travel times aggregated per  road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Consequently, the vector of zeros corresponds to the  average travel for each road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Each block of the decoder contains two attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  Firstly, the embeddings of the queried road segments and the travel  times are transmitted to the first attention mechanism, followed  by a residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Then a second attention mechanism is  employed, receiving as input the embedding matrix E′  1 that resulted  from the first attention mechanism along with the embeddings and  the travel times that come from the output of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The main  difference here is that the matrices 𝑉𝑖 and 𝐾𝑖 are computed from  the output of the encoder and that the considered travel times come  from the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Then, the embedding output E′  2 of the second  attention mechanism is followed again by a residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  This is followed by two dense layers and a second residual connec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Finally, the travel times that result from each block is the sum  of the original travel times T that were received as input along  with travel times that result from the first and the second attention  mechanism T ′  1 and T ′  2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='3  Recurrent Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The final module of our proposed  model is a recurrent model that considers the sequence of the last  𝐿 time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Each cell of the recurrent network encapsulates  an encoder (consisting of 𝑁 encoder blocks) along with a single  decoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Here the decoder block is responsible to aggregate  the information that has been encoded from the previous time win-  dow with the information that has been encoded from the current  time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Figure 6 illustrates this recurrent architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The  encoder and the decoder blocks of the different recurrent cells share  the same weights among the 𝐿 different time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  The output of the last recurrent cell is used by the decoder model  in order to make forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The decoder model consists of 𝑁 decoder  blocks that are different from each other and from the decoder  block that lies inside the recurrent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' The output of the last  decoder block contains the predicted travel times of the queried  road segments for the next time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' This will be the Z-Scores  of the travel times for the road segments that were queried at the  first decoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  4  CONCLUSION  In this paper we presented a novel deep learning framework that  considers the current traffic condition of the road network and is  used to forecast the traffic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' Our framework can efficiently  encode the travel time reports that are available at a particular  time window via an attention mechanism that considers only the  available travel times reports and the corresponding embeddings  of the road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This research has been financed by the European Union through the  H2020 LAMBDA Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 734242), the EU ICT-48 2020 project  TAILOR (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 952215) and the Horizon Europe AUTOFAIR Project  (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
+page_content=' 101070568).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'}
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+arXiv:2301.02634v1  [math.LO]  6 Jan 2023
+ON DISJOINT STATIONARY SEQUENCES
+MAXWELL LEVINE
+Abstract. We answer a question of Krueger by obtaining disjoint stationary
+sequences on successive cardinals. The main idea is an alternative presentation
+of a mixed support iteration, using it even more explicitly as a variant of
+Mitchell forcing. We also use a Mahlo cardinal to obtain a model in which
+ℵ2 /∈ I[ℵ2] and there is no disjoint stationary sequence on ℵ2, answering a
+question of Gilton.
+1. Introduction and Background
+In order to develop a more vivid picture of the infinite cardinals, set theorists
+study a variety of objects that can potentially exist on these cardinals. The objects
+of interest for this paper are called disjoint stationary sequences. These were intro-
+duced by Krueger to answer a question of Abraham and Shelah about forcing clubs
+through stationary sets. Beginning in joint work with Friedman, Krueger wrote a
+series of papers in this area, connecting a wide range of concepts and answering
+seemingly unrelated questions of Foreman and Todorˇcevi´c [4, 11, 12, 13, 14, 15].
+Generally, the new arguments hinged on the behavior of two-step iterations of the
+form Add(τ) ∗ P.
+In order to extend the application of these arguments as widely as possible,
+Krueger developed the notion of mixed support forcing [12, 15]. These forcings
+are to some extent an analog of the forcing that Mitchell used to obtain the tree
+property at double successors of regular cardinals. Their most notable feature is the
+appearance of quotients insofar as the forcings took the form M ≃ ¯M ∗ Add(τ) ∗ Q
+where ¯M is a partial mixed support iteration. The appearance of Add(τ) after
+the initial component, together with the preservation properties of the quotient Q,
+allowed Krueger’s new arguments to go through various complicated constructions.
+Mixed support iterations have found several applications since [5], particularly in
+regard to guessing models [16].
+The main idea in this paper is to use a version of Mitchell forcing to accomplish
+the task of a mixed support iteration. Specifically, this version of Mitchell forcing
+takes the form M ≃ ¯M ∗ Add(τ) ∗ Q.1 The trick used to obtain this structural
+property is reminiscent of the one usd by Cummings et al. in “The Eightfold Way”
+to demonstrate that subtle variations in the definitions of Mitchell forcing—up to
+merely shifting a L´evy collapse by a single coordinate—can substantially alter the
+properties of the forcing extension. The benefit of the forcing used here is that it
+comes with a projection analysis of the sort that Abraham used for Mitchell forcing
+[1]. Both the forcing itself and its quotients are projections of products of the form
+1The extent to which all variations of these forcings are equivalent or not is left as a loose
+end. Here we only deal with the case where the two-step iteration Add(τ) ∗ P takes the form
+Add(τ) ∗
+˙
+Col(µ, δ).
+1
+
+2
+MAXWELL LEVINE
+A× T where A has a good chain condition and T has a good closure property. This
+allows us to obtain preservation properties conveniently, without having to delve
+into too many technical details. Abraham in fact used this projection analysis to
+extend Mitchell’s result to successive cardinals. This is exactly what we do here
+for disjoint stationary sequences, answering the first component of a question of
+Krueger [15, Question 12.8]:
+Theorem 1. Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a
+forcing extension in which there are disjoint stationary sequences on ℵ2 and ℵ3.
+We lay out the basic definition and concepts in the following subsections and
+then develop the proof in Section 2. We also achieve one of Krueger’s separations
+for successive cardinals, which answers a component of another one of his questions
+[15, Question 12.9]:
+Theorem 2. Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forc-
+ing extension in which for µ ∈ {ℵ1, ℵ2}, there are stationarily many N ∈ [H(µ+)]µ
+that are internally stationary but not internally club.
+The last main result is motivated by work of Gilton and Krueger, who answered
+a question from “The Eightfold Way” by obtaining stationary reflection for subsets
+of ℵ2 ∩ cof(ω) together with failure of approachability at ℵ2 (i.e. ℵ2 /∈ I[ℵ2]) using
+disjoint stationary sequences [5]. This result used the fact that the existence of a
+disjoint stationary sequence implies failure of approachability. Gilton asked for the
+exact consistency strength of the failure of approachability at ℵ2 together with the
+nonexistence of a disjoint stationary set on ℵ2 [7, Question 9.0.15]. (He pointed
+out that Cox found this separation using PFA [2].) It is known that the failure
+of approachability requires the consistency strength of a Mahlo cardinal, and in
+Section 3 we show that a Mahlo cardinal is sufficient for the separation:
+Theorem 3. Suppose that λ is Mahlo in V . Then there is a forcing extension in
+which ℵ2 /∈ I[ℵ2] and there is no disjoint stationary sequence on ℵ2.
+Disjoint stationary sequences are known to be interpretable in terms of canonical
+structure (see Fact 6 below), and the main idea for Theorem 3 is a simple master
+condition argument that exploits this connection.
+We note that all three of these theorems can be generalized to arbitrarily high
+cardinals.
+1.1. Basic Definitions. We assume familiarity with the basics of forcing and large
+cardinals. We use the following conventions: If P is a forcing poset, then p ≤ q
+for p, q ∈ P means that p is stronger than q. We say that P is κ-closed if for all
+≤P-decreasing sequences ⟨pξ : ξ < τ⟩ with τ < κ, there is a lower bound p, i.e.
+p ≤ pξ for all ξ < τ. We say that P has the κ-chain condition of all antichains
+A ⊆ P have cardinality strictly less than κ.
+Now we give our main definitions:
+Definition 4. Given a regular cardinal µ, a disjoint stationary sequence on µ+ is
+a sequence ⟨Sα : α ∈ S⟩ such that:
+• S ⊆ µ+ ∩ cof(µ) is stationary,
+• Sα is a stationary subset of Pµ(α) for all α ∈ S,
+• Sα ∩ Sβ = ∅ if α ̸= β.
+
+ON DISJOINT STATIONARY SEQUENCES
+3
+We write DSS(µ+) to say that there is a disjoint stationary sequence on µ+.
+Definition 5. Given a stationary N ∈ [H(Θ)]κ,2 we say:
+• N is internally unbounded if ∀x ∈ Pκ(N), ∃M ∈ N, x ⊆ M,
+• N is internally stationary if Pκ(N) ∩ N is stationary in Pκ(N),
+• N is internally club if Pκ(N) ∩ N is club in Pκ(N),
+• N is internally approachable if there is an increasing and continuous con-
+tinuous chain ⟨Mξ : ξ < κ⟩ such that |Mξ| < κ and ⟨Mη : η < ξ⟩ ∈ Mξ+1
+for all ξ < κ such that N = �
+ξ<κ Mξ.
+Although disjoint stationary sequences may seem unrelated the separation of
+variants of internal approachability, there are deep connections here, for example:
+Fact 6 (Krueger, [15]). If µ is regular and 2µ = µ+, then DSS(µ+) is equivalent to
+the existence of a stationary set U ⊆ [H(µ+)]µ such that every N ∈ U is internally
+internally unbounded but not internally club.
+1.2. Projections and Preservation Lemmas. Technically speaking, our main
+goal is to show that certain forcing quotients behave nicely.
+We will make an
+effort to demonstrate the preservation properties of these quotients directly. These
+quotients will be defined in terms of projections:
+Definition 7. If P1 and P2 are posets, a projection is an onto map π : P1 → P2
+such that:
+• p ≤ q implies that π(p) ≤ π(q),
+• if r ≤ π(p), then there is some q ≤ p such that π(q) ≤ r.
+A projection is trivial if π(p) = π(q) implies that p and q are compatible.
+Trivial projections are basically ismorphisms:
+Fact 8. If π : P1 → P2 is a trivial projection, then P1 ≃ P2.
+For our purposes, we are interested in the preservation of stationary sets. The
+chain condition gives us preservation fairly straightforwardly. The following fact is
+implicit in parts of the literature, and a version of it can be found in this paper in
+the form of Proposition 26.
+Fact 9. If P has the µ-chain condition and S ⊂ Pµ(X) is stationary, then P forces
+that S is stationary in P V
+µ (X).
+However, we must place demands on our stationary sets in order for them to be
+preserved by closed forcings.
+Definition 10. A stationary set S ⊂ Pµ(H(Θ)) is internally approachable of length
+τ if for all N ∈ S with N ≺ H(Θ), there is a continuous chain of elementary
+submodels ⟨Mi : i < τ⟩ such that N = �
+i<τ Mi and for all i < τ, ⟨Mi : i < j⟩ ∈
+Mj+1. In this case we write S ⊆ IA(τ).
+Fact 11. If S ⊂ Pµ(H(Θ)) ∩ IA(τ) is an internally approachable stationary set,
+τ < µ, and P is µ-closed, then P forces that S is stationary in Pµ(H(Θ)V ).
+2See Jech for details on stationary sets [10].
+
+4
+MAXWELL LEVINE
+1.3. Costationarity of the Ground Model. The notion of ground model co-
+stationarity is a key ingredient in arguments pertaining to disjoint stationary se-
+quences. It will specifically give us the disjointness, since we will be picking sta-
+tionary sets that are not added by initial segments of these forcings.
+Gitik obtained the classical result:
+Fact 12 (Gitik [8]). If V ⊂ W are models of ZFC with the same ordinals, W \ V
+contains a real, and κ is a regular cardinal in W such that (κ+)W ≤ λ, then
+P W
+κ (λ) \ V is stationary.
+Because we will need Fact 11, we will actually use Krueger’s refinement of Gitik’s
+theorem:
+Fact 13 (Krueger [15]). Suppose V ⊂ W are models of ZFC with the same ordinals,
+W \ V contains a real, µ is a regular cardinal in W, and X ∈ V is such that
+(µ+)W ⊆ X, and that in W, Θ is a regular cardinal such that X ⊂ H(Θ). Then in
+W the set {N ∈ Pµ(H(Θ)) ∩ IA(ω) : N ∩ X /∈ V } is stationary.
+2. The New Mitchell Forcing
+2.1. Defining the Forcing. In this subsection we will illustrate the basic idea of
+this paper by using our new take on Mitchell forcing to prove a known result:
+Theorem 14 (Krueger [15]). If λ is a Mahlo cardinal and µ < λ are regular
+cardinals, there is a forcing extension in which 2ω = µ+ = λ and there is a disjoint
+stationary sequence on λ.
+Specifically, we will define a forcing M+(τ, µ, λ) such that the model W in
+Theorem 14 can be realized as an extension by M+(ω, µ, λ).
+For standard technical reasons, we define a poset ismorphic to Add(τ, λ):
+Definition 15. Given a regular τ and a set of ordinals Y , we let Add∗(τ, Y ) be
+the poset consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1}
+where | dom p| < τ. We let p ≤Add∗(τ,Y ) q if and only if p ⊇ q.
+Note: In later subsections we will conflate Add(τ, λ) and Add∗(τ, λ) to simplify
+notation.
+Definition 16. Let λ be inaccessible and let τ < µ < λ be regular cardinals such
+that τ <τ = τ. We define a forcing M+(τ, µ, λ) that consists of pairs (p, q) such
+that:
+(1) p ∈ Add∗(τ, λ),
+(2) q is a function such that:
+(a) dom q ⊂ {δ < λ : δ is inaccessible},
+(b) | dom q| < µ,
+(c) ∀δ ∈ dom(q), p↾((δ + 1) × τ) ⊩Add∗(τ,δ+1) “q(δ) ∈
+˙
+Col(µ, δ)”.
+We let (p, q) ≤ (p′, q′) if and only if:
+(i) p ≤Add∗(τ,λ) p′,
+(ii) dom q ⊇ dom q′,
+(iii) for all δ ∈ dom q′, p↾((δ + 1) × τ) ⊩Add∗(τ,δ+1) “q(δ) ≤ ˙
+Col(µ,δ) q′(δ)”
+First we go through the more routine properties that one would expect of this
+forcing.
+
+ON DISJOINT STATIONARY SEQUENCES
+5
+Proposition 17. M+(τ, µ, λ) is τ-closed and λ-Knaster.
+Proof. Closure uses the facts that Add∗(τ, λ) is τ-closed and ⊩Add(τ,δ+1) “ ˙
+Col(µ, δ)
+is µ-closed” for all δ. Knasterness uses a standard application of the Delta System
+Lemma.
+□
+Crucially, we get a nice termspace:
+Definition 18. Let T = T(M+(τ, µ, λ)) be the poset consisting of conditions q
+such that:
+(1) dom q ⊂ λ ∩ {δ < λ : δ is inaccessible},
+(2) | dom q| < µ,
+(3) ∀δ ∈ dom q, ⊩Add∗(τ,δ+1) q(δ) ∈
+˙
+Col(µ, δ)”.
+Most importantly, we let q ≤ q′ if and only if:
+(i) dom q ⊇ dom q′,
+(ii) for all δ ∈ dom q, ⊩Add∗(τ,δ+1) “q(δ) ≤ q′(δ)”.
+Proposition 19. There is a projection Add∗(τ, λ)×T(M+(τ, µ, λ)) ։ M+(τ, µ, λ).
+Proof. We let π be the projection with the definition π(p, q) = (p, q). This is au-
+tomatically order-preserving because the ordering ≤Add∗(τ,λ)×T is coarser than the
+ordering ≤M+(τ,µ,λ). For obtaining the density condition, suppose (r, s) ≤M+(τ,µ,λ)
+(p0, q0). We want to find some (p1, q1) such that (p1, q1) ≤Add∗(τ,λ)×T (p0, q0) and
+(p1, q1) ≤M+(τ,µ,λ) (r, s). To do this, we first let p1 = r, and then we define q1
+with dom q1 = dom r such that at each coordinate δ ∈ dom q1, we use standard
+arguments on names to show that we can get both p0 ↾ ((δ + 1) × τ) ⊩Add∗(τ,λ)
+“q1(δ) ≤ s(δ)” as well as 1Add∗(τ,λ) ⊩Add∗(τ,λ) “q1(δ) ≤ q0(δ)”.
+□
+Proposition 20. T is µ-closed.
+Proof. This as an application of the Mixing Principle. Given a ≤T-decreasing se-
+quence ⟨qi : i < τ⟩ with τ < µ we let d = �
+i<τ dom qi. Then we define a lower
+bound ¯q with domain d such that for all δ ∈ d, q(δ) is a canonically-defined name
+for a lower bound of the qi(δ)’s (where i is large enough that δ ∈ dom qi).
+□
+Then we get the standard consequences of the termspace analysis:
+Proposition 21. The following are true in any extension by M+(τ, µ, λ):
+(1) V -cardinals up to and including µ are cardinals.
+(2) For all α < λ, |α| = µ.
+(3) λ = µ+.
+(4) 2τ = λ.
+Proof. (1) follows from the projection analysis and the fact that T is µ-closed and
+Add∗(τ, λ) is τ +-cc, and from τ-closure of M+(τ, µ, λ). (2) follows from the fact
+that for all inaccessible δ < λ, M+(τ, µ, λ) projects onto Col(µ, δ). (3) follows from
+(1) and (2) plus λ-Knasterness. (4) follows from the fact that M+(µ, λ) projects
+onto Add∗(τ, λ), so it forces that 2τ ≥ λ. Since the poset has size λ, it also forces
+that 2τ ≤ λ.
+□
+The following lemma is the crux of the new idea.
+
+6
+MAXWELL LEVINE
+Lemma 22. If δ0 < λ is inaccessible, then there is a forcing equivalence
+M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ) ∗ Q
+where M+(τ, µ, δ0) ∗ Add(τ) forces that Q is a projection of a product of a µ-closed
+forcing and a τ +-cc forcing.
+Proof. More precisely, we will show that there is a forcing equivalence M+(τ, µ, λ) ≃
+M+(τ, µ, δ0) ∗ Add(τ) ∗ (P × R) where the following hold in the extension by
+M+(τ, µ, δ0) ∗ Add(τ):
+• R is a projection of a product of a µ-closed forcing and Add∗(τ, λ), and
+• V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P is µ-closed”.
+The statement of the lemma can then be obtained by merging P with the closed
+component of the product that projects onto R.
+First we describe P and R. To do this, we fix some notation. Given Y ⊆ λ, we let
+πY
+Add denote the projection (p, q) → p↾(Y × τ) from M+(τ, µ, λ) onto Add∗(τ, Y ).
+For any poset P, we employ the convention that Γ(P) denotes a canonical name for
+a P-generic. If X ⊂ P, then we use the notation ↑ X := {q ∈ P : ∃p ∈ X, p ≤ q}.
+We will let
+P := Col(µ, δ0)V [(↑(πδ0
+Add”Γ(M+(τ,µ,δ0))))×Γ(Add(τ))]
+if we are working in an extension by M+(τ, µ, δ0) ∗ Add(τ). (In other words, the
+poset P will be the version of Col(µ, δ0) as interpreted in the extension of V by
+Add∗(τ, δ + 1) where the initial coordinates come from M+(τ, µ, δ0) and the last
+coordinate comes from the additional copy of Add(τ).)
+Still working in an extension by M+(τ, µ, δ0) ∗ Add(τ), the poset R consists of
+pairs (p, q) such that the following hold:
+(1) p ∈ Add∗(τ, (δ0, λ)),
+(2) q is a function such that
+(a) dom q ⊂ {δ ∈ (δ0, λ) : δ is inaccessible},
+(b) | dom q| < µ,
+(c) ∀δ ∈ dom(q), p↾((δ0, (δ + 1)) × τ) ⊩Add∗(τ,(δ0,δ+1)) “q(δ) ∈
+˙
+Col(µ, δ)”.
+The ordering is the one analogous to that of M+(τ, µ, λ). An easy adaptation of
+the arguments for the projection analysis for M+(τ, µ, λ) will then give a projection
+analysis for R.
+The rest of the proof of the lemma consists of verifying the more substantial
+claims.
+Claim 23. M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ, 1) ∗ (P × R).
+Proof. We identify M+(τ, µ, δ0) ∗ Add(τ, 1) ∗ (P × R) with the dense subset of con-
+ditions ((r, s), t, u, (r, ˙s′)) such that ˙s′ is forced to have a specific domain in V . The
+fact that this subset is dense follows from the fact that M+(τ, µ, λ) ∗ Add(τ) has
+the µ-covering property.
+We will argue that there is a trivial projection defined by
+π : (p, q) �→ ((p↾(δ0 × τ), q↾δ0)
+�
+��
+�
+M+(µ,δ0)
+, p↾({δ0} × τ)
+�
+��
+�
+Add(τ)
+, q∗(δ0)
+� �� �
+P
+, (¯p, ¯q)
+� �� �
+R
+)
+such that
+• ¯p := p↾((δ0, λ) × τ);
+
+ON DISJOINT STATIONARY SEQUENCES
+7
+• q∗(δ0) is obtained by changing q(δ0) from an Add∗(τ, δ0 + 1)-name to an
+Add(τ) as interpreted in the extension by the relevant generic, namely
+(↑ (πδ0
+Add”Γ(M+(τ, µ, δ0))));
+• ¯q has domain (δ0, λ), and for each δ ∈ (δ0, λ), ¯q(δ) has changes analogous
+to the changes made to q∗(δ0).
+It is clear that π is order-preserving. We also want to show that if
+((r, s), t, u, (r′, ˙s′)) ≤M+(τ,µ,δ0)∗Add(τ)∗(P×R) π(p0, q0)
+then there is some (p1, q1) ≤M+(µ,λ) (p0, q0) such that π(p1, q1) ≤ ((r, s), t, u, (r′, s′)).
+This can be done by taking:
+• p1 = r ∪ ˜t ∪ r′ where ˜t writes t as as a partial function {δ} × τ → {0, 1},
+• q1 = s ∪ ˜u ∪ ˜s′ where ˜u reinterprets u as a Add∗(δ0 + 1)-name and for each
+δ ∈ dom( ˙s′), ˜s′ reinterprets ˙s′(δ) as a Add∗(δ + 1)-name.
+Last, we argue that π(p0, q0) = π(p1, q1) implies that (p0, q0) and (p1, q1) are
+compatible. Suppose that (p0, q0) and (p1, q1) are incompatible. If p0 and p1 are
+incompatible as elements of Add∗(τ, λ), then one of pi ↾ (δ0 × τ), pi ↾ ({δ0} × τ),
+and pi ↾((δ0, λ) × τ) must be distinct for i = 0 and i = 1. Otherwise, there is some
+p′ ≤ p0, p1 and some δ ∈ dom q0∩dom q1 inaccessible such that p′ ⊩ “q0(δ) ⊥ q1(δ)”,
+which implies that q0(δ) ̸= q1(δ). Therefore, one of qi ↾ δ0, qi(δ0), or qi ↾ (δ0, λ) is
+distinct for i ∈ {0, 1}.
+□
+Claim 24. V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P is µ-closed”.
+Proof. In fact, our argument will also show that V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P =
+Col(µ, δ0)”. We fix some arbitrary generics:
+• G is M+(τ, µ, δ0)-generic over V ,
+• r is Add(τ)-generic over V [G],
+• H is the Add∗(τ, δ0)-generic induced from G by πδ0
+Add,
+• K is the generic for the quotient of M+(τ, µ, δ0) by Add∗(τ, δ0), i.e. the
+generic such that V [H][K] = V [G],
+• T is the generic for the termspace forcing T(M+(τ, µ, δ0)), so that V [G] ⊂
+V [T ][H].
+It is enough to argue that V [G][r] |= “P is µ-closed” knowing that V [H][r] |= “P
+is µ-closed”. Because adjoining G does not change the definition of Add(τ), and
+because K is defined in terms of the subsets of τ adjoined by the filter H, we have
+V [G][r] = V [H][K][r] = V [H][r][K]. Therefore, it is enough to show that K does
+not add <µ-sequences over V [H][r], so that V [H][r]’s version of Col(µ, δ0) remains
+µ-closed in V [G][r]. We have
+V [H][r] ⊂ V [H][r][K] = V [H][K][r] = V [G][r] ⊂ V [T ][H][r] = V [H][r][T ],
+and Easton’s Lemma implies that T does not add new <µ-sequences over V [H][r],
+so therefore K does not add new <µ-sequences over V [H][r] since it is an interme-
+diate factor of the extension.
+□
+This completes the proof of the lemma.
+□
+Now we have an application for the case where τ = ω.
+Proposition 25. If λ is Mahlo then V [M+(ω, µ, λ)] |= DSS(λ).
+This basically repeats Krueger’s argument for [15, Theorem 9.1].
+
+8
+MAXWELL LEVINE
+Proof. Let G be M+(ω, µ, λ)-generic over V . The set of V -inaccessibles in λ will
+form the stationary set S ⊂ µ+ ∩ cof(µ) carrying the disjoint stationary sequence
+in the extension by M+(ω, µ, λ). For every such δ ∈ S, let ¯G be the generic on
+M+(ω, µ, δ) induced by G and let r be the Add(ω)-generic induced by G via π{δ}
+Add.
+We use Fact 13 to obtain a stationary set S∗
+δ ⊂ Pµ(H(δ))V [ ¯
+G][r] such that for all
+N ∈ S∗
+δ, N ∩ δ /∈ V [ ¯G] and such that S∗
+δ is also internally approachable by a ω-
+sequence. Therefore we can apply Lemma 22 with Fact 11 and then Fact 9 to find
+that S∗
+δ is stationary in V [G]. We then let Sδ = {N ∩ δ : N ∈ S∗
+δ}, and we see that
+⟨Sδ : δ ∈ S⟩ is a disjoint stationary sequence.
+□
+2.2. Proving the Main Theorems. Now we will apply the new version of Mitchell
+forcing to answer Krueger’s questions. Theorem 1 follows quickly:
+Proof of Theorem 1. Begin with a ground model V in which λ1 < λ2 and the λ’s
+are Mahlo.
+Let M1 = M+(ω, ℵ1, λ1).
+(Any λ1-sized forcing that turns λ1 into
+ℵ2 and adds a disjoint stationary sequence on ℵ2 would work, so we could also
+use a more standard mixed support iteration.) Then let ˙M2 be an M1-name for
+M+(ω, λ1, λ2). We argue that if G1 is M1-generic over V and G2 is ˙M2[G1]-generic
+over V [G1], then V [G1][G2] |= “DSS(λ1)∧DSS(λ2)”. We get DSS(λ2) from the fact
+that λ2 remains Mahlo in V [G1] together with Proposition 25, so we only need to
+argue that the disjoint stationary sequence ⃗S := ⟨Sα : α ∈ S⟩ ∈ V [G1] remains a
+disjoint stationary sequence in V [G1][G2].
+Working in V [G1], preservation of ⃗S follows from the projection analysis: Let H1
+and H2 be chosen so that H1 is T := T(M2)-generic over V [G1], H2 is Add(ω, λ2)V [G1]-
+generic over V [G1][H1], and V [G1][G2] ⊆ V [G1][H1][H2]. Since T is λ1-closed, it
+preserves stationarity of S and the Sα’s, and Add(ω, λ2)V [G1] still has the countable
+chain condition in V [G1][H1]. It follows that the stationarity of S is preserved in
+V [G1][H1][H2], as well as the stationarity of the Sα’s (by Fact 9). Therefore ⃗S is a
+disjoint stationary sequence on λ1 in V [G1][G2].
+□
+It will take a bit more work to show that Theorem 2 holds in the same model
+given for Theorem 1. Note that we cannot just apply Fact 6 because 2ω = ℵ3 in
+the model for Theorem 1, plus it is consistent that there can be a stationary set
+which is internally unbounded but not internally stationary [13].
+We will give some facts on preservation of the distinction between stationary
+sets that are internally stationary but not internally club:
+Proposition 26. Suppose P is ν-closed and S ⊆ Pδ(X) is a stationary set such
+that |X|<δ ≤ ν and δ ≤ ν. Then ⊩P “S is stationary in Pδ(X)”.
+Proof. Let
+˙C be a P-name for a club in Pδ(X).
+Let ⃗x = ⟨xξ : ξ ≤ ¯ν⟩ be an
+enumeration of Pδ(X) (where ¯ν ≤ ν). We construct a sequence ⃗z = ⟨zξ : ξ ≤ ¯ν⟩ ⊆
+Pδ(X) and a ≤P-descending sequence ⟨pξ : ξ ≤ ¯ν⟩ such that for all ξ, pξ ⊩ “xξ ⊆
+zξ ∈ ˙C”. Let D be the set of unions �
+i<¯δ zξi for all increasing chains ⟨zξi : i < ¯δ⟩ ⊂
+⃗z (where ¯δ < δ). Since D is a club in Pδ(X) defined in V , there is some w ∈ D ∩ S.
+Let ⟨zξi : i < ¯δ⟩ be an ⊆-increasing chain with ¯δ < δ such that �
+i<¯δ zξi = w and
+let ξ∗ < ¯ν be such that ξ∗ > supi<¯δ ξi. Then pξ∗ ⊩ “w ∈ ˙C ∩ S”.
+□
+Proposition 27. Let P1 have the δ-chain condition, let P2 be ν-closed, and let X
+be a set such that |X|δ ≤ ν with δ+ ≤ ν. If S ⊆ [X]δ is stationary and internally
+
+ON DISJOINT STATIONARY SEQUENCES
+9
+stationary but not internally club, then P1 × P2 forces that S is stationary and
+internally stationary but not internally club.
+Proof. First, S remains stationary in the extension by P2 by Proposition 26, and it
+remains stationary in the further extension by P1 by the fact that P1 still has the
+δ-chain condition together with Fact 9. If N ∈ S, then N ∩ Pδ(N) is stationary, so
+its stationarity is preserved by the same reasoning, using the fact that we still have
+the appropriate chain condition. The fact that N is not internally club is preserved
+in the extension by P2 because of ν-closure and the fact that δ ≤ ν, and then it
+is preserved in the further extension by P1 because the proof of Fact 9 shows that
+added clubs contain ground model clubs.
+□
+We use a concept from Harrington and Shelah to handle Mahlo cardinals:
+Definition 28. [9] Let N be a model of some fragment of ZFC. We say that M ≺ N
+is rich if the following hold:
+(1) λ ∈ M;
+(2) ¯λ := M ∩ λ ∈ λ;
+(3) ¯λ is an inaccessible cardinal in N;
+(4) The size of M is ¯λ;
+(5) M is closed under < ¯λ-sequences and ¯λ < λ.
+Lemma 29. If λ is Mahlo, then M+(ω, µ, λ) forces that there are stationarily many
+Z ∈ [µ+]µ which are internally stationary but not internally club.
+This follows Krueger’s proof of [15, Theorem 10.1], making necessary changes
+for Mahlo cardinals, and including enough details to show that we can get the
+necessary preservation of stationarity simply from the projection analysis. We do
+not need guessing functions (which are used in Krueger’s argument) because we are
+only obtaining one instance of separation per large cardinal.
+Proof of Lemma 29. Denote M := M+(ω, µ, λ) and let ˙C be an M-name for a club
+in ([H(µ+)]µ)V [M]. We want to find an M-name ˙Z for an element of ([H(µ+)]µ)V [M]∩
+˙C that is internally stationary but not internally club. Let ˙F be an M-name for
+a function (H(µ+)V [M])<ω → H(µ+)V [M] with the property that all of its closure
+points are in ˙C. Let Θ be as large as needed for the following discussion and let N
+be the structure (H(Θ), ∈, <Θ, M, ˙F, λ, µ) where <Θ is a well-ordering of H(Θ).
+Since λ is Mahlo, we can find some M ≺ N with µ ⊂ M that is a rich submodel of
+cardinality ¯λ. Now set G to be M-generic over V . Note that H(λ)V [G] = H(λ)[G]
+because M has the λ-chain condition and M ⊂ H(λ). We will argue that Z :=
+M[G] ∩ H(λ)[G] is what we are looking for.
+Claim 30. Z ∈ C := ˙C[G].
+Proof. We have ¯λ ≤ |Z| ≤ |M| ≤ ¯λ and ¯λ has cardinality µ in N[G], so Z ∈
+[H(λ)V [G]]µ. If a1, . . . , an ∈ Z, there are M-names ˙b1, . . . , ˙bn ∈ M ∩ H(λ) such
+that ai = ˙bi[G] for all 1 ≤ i ≤ n.
+By elementarity, M contains the <Θ-least
+maximal antichain A ⊂ M of conditions deciding ˙F(˙b1, . . . , ˙bn). Since |A| < λ,
+|A| ∈ M∩λ = ¯λ, so it will follow that A ⊂ M. Therefore if p ∈ G∩A, then p ∈ M in
+particular, so p ⊩ ˙F(˙b1, . . . , ˙bn) = ˙b∗ for some ˙b∗ ∈ M∩H(λ) where we automatically
+get ˙b∗ ∈ H(¯λ), and therefore F(a1, . . . , an) = a∗ := ˙b∗[G] ∈ M[G] ∩ H(λ)[G] = Z
+(where of course F := ˙F[G]).
+□
+
+10
+MAXWELL LEVINE
+For the rest of the proof let ¯G := πM(G) where πM is the Mostowski collapse
+relative to M. Since πM(M) = M+(ω, µ, ¯λ), there is an extension πM : M[G] ∼=
+πM(M)[ ¯G]. We also denote h := πM(H(λ)[G] ∩ M[G]). Note that h<¯λ ⊂ h by the
+facts that M is rich and πM(M) has the ¯λ-chain condition.
+Claim 31. Z is internally stationary.
+Proof. First, we argue that S := Pµ(h)N[ ¯
+G] is stationary in N[G]. By Lemma 22,
+the quotient M/ ¯G is a projection of a forcing of the form A1 ∗ ( ˙T × A2) where A1
+has the countable chain condition, ˙T is an A1-name for a µ-closed forcing, and A2
+also has the countable chain condition. Let K1, KT , and K2 be respective generics
+such that V [G] ⊆ V [ ¯G][K1][KT ][K2]. Working in N[ ¯G], note that S′ ∩ IA(ω) is
+stationary, and therefore has its stationarity preserved in V [ ¯G][K1] by Fact 9.
+We must also show that the stationarity of S′ will be preserved by countably
+closed forcings over N[ ¯G][K1]. Suppose ⟨Mn : n < ω⟩ witnesses internal approach-
+ability of some N ∈ S′ in V [ ¯G] with respect to the structure H(λ+)V [ ¯
+G], and let
+Mω := �
+n<ω Mn. Then we can see that ⟨Mn[K1] : n < ω⟩ is a chain of elementary
+submodels of H(λ)[ ¯G][K1] = H(λ)V [ ¯
+G][K1]. We also have Mn[K1] ∩ V [ ¯G] = M and
+Mω[K1] ∩ V [ ¯G] = Mω ∈ S′ with Mω[K1] ≺ H(λ)V [ ¯
+G][K1]. If we choose the Mn’s to
+be elementary substructures of H(λ+)V [ ¯
+G](∈, <∗, ˙C, . . .) where <∗ is a well-ordering
+and ˙C is a A1 ∗ ˙T-name for a club, then an argument almost exactly like the one
+showing that internal approachability is preserved (i.e. the proof of Fact 11) will
+show that S′ is stationary in N[ ¯G][K1][KT ].
+Then the extension of N[ ¯G][K1][KT][K2] over N[ ¯G][K1][KT ] preserves the sta-
+tionarity of S′ by another application of Fact 9, so we get stationarity in N[G].
+Now that we have established preservation of stationarity of S′, we can finish
+the argument. Since |h| = µ in N[G], we can write h = �
+i<µ xi where ⟨xi : i < µ⟩
+is a continuous and ⊂-increasing chain of elements of Pµ(h). The chain is a club
+in h, so there is a stationary X ⊆ µ such that {xi : i ∈ X} ⊆ T . For all i < µ,
+the fact that |xi| < µ implies that xi ∈ h, and so xi = πM(yi) for some yi ∈ Z.
+Therefore ⟨yi : i < µ⟩ is ⊂-increasing and continuous with union Z, and in particular
+⟨yi : i ∈ X⟩ is stationary in Z.
+□
+Claim 32. Z is not internally club.
+Proof. Suppose for contradiction that Z is internally club and hence that there is
+a ⊂-increasing and continuous chain ⟨Zi : i < µ⟩ ∈ N[G] with |Zi| < µ for all i < µ
+and �
+i<µ Zi = Z. So for all i < µ, Zi ⊂ Z, and so ⟨πM[Zi] : i < µ⟩ is an ⊂-
+increasing and continuous chain with union h. If we let Wi := πM[Zi] for all i < µ,
+then the fact that |Wi| < µ implies that Wi = πM(Zi). Therefore ⟨Wi : i < µ⟩ is a
+continuous and ⊂-increasing chain of sets in Pµ(h) with union h.
+Next we define a set U ∈ N[ ¯G][r] (where r is the generic induced by G from
+π{¯λ}
+Add) as
+{A ∈ Pµ(H(χ)) ∩ IA(ω) : A �� h /∈ N[ ¯G]}.
+We have a real in N[ ¯G][r] \ N[ ¯G] and (µ+)N[ ¯
+G][r] = λ ⊂ H(λ). Hence we apply
+Fact 13 to see that U is stationary in N[ ¯G][r], and it remains stationary in N[G] by
+the preservation properties of the quotient (i.e. Lemma 22 combined with Fact 11
+and Fact 9). Therefore in N[G], {A ∩ h : A ∈ U} is stationary in Pµ(h). Since
+⟨Wi : i < µ⟩ is club in h, there is some i < µ such that Wi = A ∩ h for some A ∈ U.
+
+ON DISJOINT STATIONARY SEQUENCES
+11
+But by definition, A ∩ h /∈ N[ ¯G], and subsets of Wi of cardinality < ¯λ are in N[ ¯G],
+so this is a contradiction.
+□
+This completes the proof of the lemma.
+□
+Proof of Theorem 2. Let M1 be any λ1-sized forcing that turns λ1 into ℵ2 and adds
+stationarily many N ∈ [H(ℵ2)]ℵ1 that are internally stationary but not internally
+club. Let ˙M2 be an M1-name for M+(ω, λ1, λ2), let G1 be M1-generic over V , and
+let G2 be ˙M2[G1]-generic over V [G1]. Then we can see that the theorem holds
+in V [G1][G2]: the distinction between internally stationary and internally club on
+[H(ℵ2)]ℵ1 is preserved in V [G1][G2] by Proposition 27, and we get a distinction
+between internally stationary and internally club for [H(ℵ3)]ℵ2 by Lemma 29.
+□
+3. A Club Forcing and a Guessing Sequence
+3.1. A review of the tools. The main idea of the proof of Theorem 3 is to force
+a club through the complement of a canonical stationary set, which is described as
+follows:
+Fact 33 (Krueger,[15]). Suppose µ is an uncountable regular cardinal and µ<µ ≤
+µ+. Let x = ⟨xα : α < µ+⟩ enumerate [µ+]<µ and let
+S(x) := {α ∈ µ+ ∩ cof(µ) : Pµ(α) \ ⟨xβ : β < α⟩ is stationary}.
+Then DSS(µ+) holds if and only if S(x) is stationary.
+The natural thing to do is to define the following:
+Definition 34. Let µ be an uncountable regular cardinal such that µ<µ = µ+
+and let x and S(x) be defined as in Fact 33. Then let P(x) be the set of closed
+bounded subsets p of µ+ such that p ∩ S(x) = ∅. We let p′ ≤ p if and only if
+p′ ∩ (max p + 1) = p.
+We will also crucially need a characterization of diamonds. This following ap-
+pears in joint work with Gilton and Stejskalov´a [6].
+Fact 35. The following are equivalent:
+(1) λ is Mahlo and ♦λ(Reg) (where of course Reg = {τ < λ : τ regular}) holds.
+(2) There is a function ℓ : λ → Vλ such that for every transitive structure N
+satisfying a rich fragment of ZFC that is closed under λ+-sequences in V ,
+the following holds: For every A ∈ N with A ∈ H(λ+) and any a ⊂ H with
+|a| < λ, there is a rich M ≺ N with a ∪ {ℓ} ⊂ M such that ℓ(¯λ) = πM(A)
+(where ¯λ = M ∩ λ and πM is the Mostowski collapse).3
+We can always use such an ℓ assuming the consistency of a Mahlo cardinal: If λ
+is Mahlo in a model V , then it is Mahlo in G¨odel’s class L where ♦λ(S) holds for
+all regular λ and stationary S ⊂ λ.
+We use a poset that appears in Gilton’s thesis [7] and is discussed in the same
+paper with the guessing sequence [6]. We denote this poset MG
+ℓ (κ, λ) and black-box
+its basic properties:
+Fact 36. [3, 7] The following hold for MG
+ℓ (κ, λ):
+3The original is stated with a different quantification—for all rich structures, there exists a
+function, not the other way around. However, the proof works with the quantification used here.
+
+12
+MAXWELL LEVINE
+• MG
+ℓ (κ, λ) has the λ-chain condition;
+• MG
+ℓ (κ, λ) is κ-closed;
+• If ℓ(δ) = P for some κ+-closed forcing, then we have the forcing equivalence:
+MG
+ℓ (κ, λ) ≃ MG
+ℓ (κ, δ) ∗ (P × Add(κ, δ⊕)) ∗ Nδ⊕
+where:
+– α⊕ takes the least inaccessible larger than α, and
+– Nδ⊕ is a projection of a product of a square-κ+-cc and a κ+-closed
+forcing.
+3.2. The proof. Now we prove Theorem 3. Fix κ and λ as in the statement of
+the theorem and let µ = κ+. We can assume that ♦λ(Reg) holds, so let ℓ witness
+Fact 35 and let M = MG
+ℓ (κ, λ). We have V [M] |= µ<µ ≤ µ+, so we fix an M-name
+˙x of [µ+]<µ in V [M] as well as a sequence of names ⟨ ˙xα : α < µ+⟩ that canonically
+represent the elements listed by ˙x. Then let ˙P be an M-name for P( ˙x). Let G be
+M-generic over V and let H be P := ˙P[G]-generic over V [G]. Then the model in
+which the theorem is realized is V [G][H].
+Note: If M ≺ N is rich and πM is the Mostowski collapse relative to M, we will
+typically denote πM(a) as ¯a.
+The following lemma is the crux of the proof:
+Lemma 37. Let M ≺ N be a rich model chosen to witness Fact 35 in the sense
+of having the properties that M ∩ λ = ¯λ and ℓ(¯λ) = πM( ˙P(˙x)). Suppose ¯G0 ∗ ¯H0 is
+¯M ∗ ¯P-generic over V .
+Then there is a G0 ∗ H0 which is M ∗ P( ˙x)-generic over V and a rich M ≺ N
+such that:
+(1) if j : ¯M → M ⊂ N is the inverse of the Mostowski collapse, then there is a
+lift j : ¯M[ ¯G0][ ¯H0] → N[G0][H0];
+(2) ¯M[ ¯G0][ ¯H0]<¯λ ⊆ ¯M[ ¯G0];
+(3) N[G0] is an extension of N[ ¯G0][ ¯H0] by Add(κ, (M ∩ λ)⊕) ∗ N(M∩λ)⊕.
+Proof. We will lift the elementary embedding j : ¯M → N to j : ¯M[ ¯G0][ ¯H0] →
+N[G0][H0]. We therefore fix the notation ¯λ = M ∩ λ, and we have an ¯M-generic
+¯G0, so we let P = ˙P( ˙x)[G0].
+To perform the lift, we need to show that we can absorb the generic ¯H0. We
+can see that N[ ¯G0] |= “πM(P) is ¯λ-closed”, which follows from the fact that ¯M has
+the ¯λ-chain condition. By the guessing property of ℓ we have a forcing equivalence
+M/G0 ≃ (¯P × Add(κ, ¯λ⊕)) ∗ ˙N¯λ⊕, giving us (3).
+The first stage of the lift j : ¯M[ ¯G0] → N[G0] works by choosing a generic G′ over
+M/ ¯G0 such that G′ projects to ¯H0. Then we let G0 = ¯G0 × G′ and we see that
+j” ¯G0 ⊆ G0.
+To lift the embedding further, we use a master condition argument. Specifically,
+we want to show that ∪ ¯H0 ∪ {¯λ} is a condition in P. This follows because ¯λ /∈ S(x)
+(as evaluated in N[G0]) because ¯M[ ¯G0]<¯λ ⊂ ¯M[ ¯G0] and therefore Pµ(¯λ) \ ⟨xβ : β <
+¯λ⟩ will be empty, so of course it will be nonstationary. Hence we choose H0 to be a
+generic containing ∪ ¯H0 ∪ {¯λ} . It then follows that ¯M[ ¯G0][ ¯H0]<¯λ ⊆ ¯M[ ¯G0], giving
+us (2).
+□
+Proposition 38. ˙P[G] is λ-distributive over V [G].
+
+ON DISJOINT STATIONARY SEQUENCES
+13
+Proof. Suppose there were (m, ˙p) ∈ M ∗ ˙P forcing that some ˙f collapses λ over
+V . Then a suitably-chosen N := (H(Θ), ∈, <Θ, M ∗ ˙P, (m, ˙p), ˙f, . . .) would contain
+the <Θ-least such example, and so we can find a rich M ≺ N witnessing Fact 35
+with (m, ˙p) ∈ M and such that ℓ(¯λ) = πM( ˙P). Then (2) from Lemma 37 obtains a
+contradiction.
+□
+Proposition 39. V [G][H] |= ¬DSS(µ+).
+Proof. Since P is λ-distributive over V [G], x remains an enumeration of [µ+]<µ in
+V [M][P]. Moreover, P forces that S(x) is nonstationary in V [M][P], so we can apply
+Fact 33.
+□
+Proposition 40. V [G][H] |= ¬AP(µ+).
+Proof. This is exactly as in Lemma 5.9 [6], where we imitate the argument of “The
+Eightfold Way” and use property (3) of the lift, except that here P stands for a
+P(x) rather than the iteration Pα used in [6]. The main point is that if we are
+using an embedding j : ¯M[ ¯G][ ¯H] → N[G][H], then the extension by G ∗ H over
+the extension by ¯G ∗ ¯H has the correct branch preservation properties (as given by
+the distributivity of ˙P[G] and the closure and square-chain condition of the posets
+projecting onto N¯λ⊕).
+□
+Now we are finished with the proof of Theorem 3.
+4. Further directions
+We propose some other considerations along the lines of the question: Why did
+we have to do more work to get Theorem 2 after obtaining Theorem 1? Or rather,
+is the assumption 2µ = µ+ necessary for Fact 6?
+Question 1. Is it consistent for µ regular that exactly one of DSS(µ+) and “inter-
+nally club and internally unbounded are distinct for [H(µ+)]µ” holds?
+On a similar note, the assumption that 2µ = |H(µ+)| is also used in a folklore
+result that assuming 2µ = µ+, the distinction between internally unbounded and
+internally approachable for [µ+]µ requires a Mahlo cardinal.
+Question 2. What is the exact equiconsistency strength of the separation of inter-
+nally approachable and internally unbounded for [H(µ+)]µ for regular µ?
+References
+[1] Uri Abraham. Aronszajn trees on ℵ2 and ℵ3. Ann. Pure Appl. Logic, 24(3):213–230, 1983.
+[2] Sean D. Cox. Forcing axioms, approachability, and stationary set reflection. J. Symb. Log.,
+86(2):499–530, 2021.
+[3] James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot, and Dima Sinapova.
+The eightfold way. Journal of Symbolic Logic, 83(1):349–371, 2018.
+[4] Sy-David Friedman and John Krueger. Thin stationary sets and disjoint club sequences.
+Trans. Amer. Math. Soc., 359(5):2407–2420, 2007.
+[5] Thomas Gilton and John Krueger. A note on the eightfold way. Proc. Amer. Math. Soc.,
+148(3):1283–1293, 2020.
+[6] Thomas Gilton, Maxwell Levine, and ˇS´arka Stejskalov´a. Trees and stationary reflection at
+double successors of regular cardinals. Journal of Symbolic Logic. To appear.
+[7] Thomas Daniells Gilton. On the Infinitary Combinatorics of Small Cardinals and the Car-
+dinality of the Continuum. ProQuest LLC, Ann Arbor, MI, 2019. Thesis (Ph.D.)–University
+of California, Los Angeles.
+
+14
+MAXWELL LEVINE
+[8] Moti Gitik. Nonsplitting subset of Pκ(κ+). J. Symbolic Logic, 50(4):881–894 (1986), 1985.
+[9] Leo Harrington and Saharon Shelah. Some exact equiconsistency results in set theory. Notre
+Dame Journal of Formal Logic, 26(2):178–188, 1985.
+[10] Thomas Jech. Set Theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, the
+third millennium, revised and expanded edition, 2003.
+[11] John Krueger. Internally club and approachable. Adv. Math., 213(2):734–740, 2007.
+[12] John Krueger. A general Mitchell style iteration. MLQ Math. Log. Q., 54(6):641–651, 2008.
+[13] John Krueger. Internal approachability and re��ection. J. Math. Log., 8(1):23–39, 2008.
+[14] John Krueger. Internally club and approachable for larger structures. Fund. Math.,
+201(2):115–129, 2008.
+[15] John Krueger. Some applications of mixed support iterations. Ann. Pure Appl. Logic, 158(1-
+2):40–57, 2009.
+[16] Matteo Viale. Guessing models and generalized Laver diamond. Ann. Pure Appl. Logic,
+163(11):1660–1678, 2012.
+

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+1
+MissBeamNet: Learning Missing Doppler Velocity
+Log Beam Measurements
+Mor Yona and Itzik Klein
+Abstract—One of the primary means of sea exploration is
+autonomous underwater vehicles (AUVs). To perform these tasks,
+AUVs must navigate the rough challenging sea environment.
+AUVs usually employ an inertial navigation system (INS), aided
+by a Doppler velocity log (DVL), to provide the required
+navigation accuracy. The DVL transmits four acoustic beams to
+the seafloor, and by measuring changes in the frequency of the
+returning beams, the DVL can estimate the AUV velocity vector.
+However, in practical scenarios, not all the beams are successfully
+reflected. When only three beams are available, the accuracy of
+the velocity vector is degraded. When fewer than three beams
+are reflected, the DVL cannot estimate the AUV velocity vector.
+This paper presents a data-driven approach, MissBeamNet, to
+regress the missing beams in partial DVL beam measurement
+cases. To that end, a deep neural network (DNN) model is
+designed to process the available beams along with past DVL
+measurements to regress the missing beams. The AUV velocity
+vector is estimated using the available measured and regressed
+beams. To validate the proposed approach, sea experiments were
+made with the ”Snapir” AUV, resulting in an 11 hours dataset of
+DVL measurements. Our results show that the proposed system
+can accurately estimate velocity vectors in situations of missing
+beam measurements. Our dataset and codebase implementing
+the described framework is available at our GitHub repository.
+Index Terms—Autonomous Underwater Vehicles, Navigation,
+Doppler Velocity Log, Deep-Learning
+I. INTRODUCTION
+The demand for autonomous underwater vehicles (AUV)
+is significantly growing [1], [2], [3], [4]. AUVs are used in
+a variety of applications, such as seafloor exploration and
+mapping [5], pipeline inspection [6], [7], and underwater mine
+detection [8]. An accurate navigation system is necessary
+for the AUV to navigate challenging sea conditions and
+successfully perform the required tasks. From a navigational
+perspective, the commonly used global navigation satellite
+system (GNSS) is unavailable underwater. Furthermore, un-
+derwater currents and the ever-changing landscape make it
+difficult to use simultaneous localization and mapping (SLAM)
+[9]. Consequently, most AUVs employ an inertial navigation
+system (INS) aided by a Doppler velocity log (DVL). The INS
+provides a complete navigation solution comprising position,
+velocity, and orientation using three-axis accelerometers and
+three-axis gyroscopes. However, due to inertial measurement
+errors, the pure inertial solution will drift over time [10].
+The DVL provides an accurate estimate of the AUV velocity
+vector, which is used to aid the INS and obtain an accurate
+navigation solution. The fusion between INS and DVL is well
+addressed in the literature under normal DVL operating condi-
+tions. For example, a rotational dynamic model was shown to
+improve the INS/DVL fusion performance [11]. Furthermore,
+an adaptive Kalman filter aimed at finding the optimal window
+length for each measurement has been suggested [12]. In
+order to improve the extended Kalman filter, an innovative
+unscented Kalman filter was developed for AUV navigation
+[13]. Recently, a dedicated neural network was proposed to
+cope with current estimation during INS/DVL fusion [14].
+The DVL emits four acoustic beams to the seafloor and mea-
+sures the changes in the reflected beams’ frequency. Using the
+frequency shift, the beam’s velocity is calculated. The AUV
+velocity vector can be estimated when at least three beams
+are reflected back. In real-life scenarios, however, beams may
+not reflect back to the DVL for several reasons, such as if the
+AUV passes over a deep trench in one of the directions, an
+underwater sand wave changes the seafloor surface, or when
+the AUV operates in extreme roll and pitch angles. In such
+scenarios, the DVL cannot estimate the AUV velocity vector,
+and the INS/DVL loosely coupled approach cannot be applied.
+Since the tightly coupled approach uses any of the available
+beams, it can be implemented for the fusion process. Yet, for
+practical considerations, the loosely coupled method is usually
+implemented [15], [16]. To cope with situations of partial
+beam measurement, a model-based extended loosely coupled
+approach was suggested [17].
+The use of data-driven approaches in navigation and their
+benefits over model-based approaches were recently summa-
+rized in [18]. A novel method of improving the accuracy of
+the estimated DVL velocity in underwater navigation using a
+neural network structure was suggested [19]. Furthermore, a
+deep learning network that utilizes attitude and heading data
+in order to improve navigation accuracy and fault tolerance
+was developed [20].
+This paper presents, a learning framework, MissBeamNet, to
+regress the missing DVL beams and enable AUV velocity vec-
+tor estimation. To that end, we leveraged our initial research
+to regress only a single beam [21]. The contributions of this
+research are:
+‚ A modular framework capable of regressing one, two, or
+three missing beams.
+‚ A robust long short-term memory network architecture
+able to accurately regress the missing beams.
+‚ Inclusion of depth measurements to improve beam re-
+gression accuracy.
+‚ A GitHub repository containing our code and dataset as a
+benchmark dataset and solution and to encourage further
+research in the field.
+Here, we provide a thorough analysis of the missing beam sce-
+narios. In addition, we compare our results to two model-based
+approaches: 1) an average of the missing beam to estimate the
+current one (baseline) and 2) the virtual beam approach [17].
+arXiv:2301.11597v1  [cs.RO]  27 Jan 2023
+
+2
+All analyses were made on a dataset consisting of 11 hours
+of DVL recordings made by the Snapir AUV [23] during its
+mission in the Mediterranean Sea. We further demonstrate
+the superiority of MissBeamNet over current model-based
+approaches and its ability to estimate the AUV velocity vector
+in situations of missing DVL beam measurements.
+The remainder of the paper is organized as follows: Section II
+describes the AUV sensors and the model-based partial DVL
+approaches. Section III presents our MissBeamNet framework,
+while Section IV gives our sea experiment results. Finally, our
+conclusions are presented in Section V.
+II. PROBLEM FORMULATION
+This section briefly describes the AUV sensors used in this
+research and presents the baseline model-based approaches to
+coping with missing beam measurements.
+A. AUV Sensors
+1) DVL: The DVL transmits four acoustic beams to the
+seafloor, which reach the seafloor and bounce back to the DVL
+transducers. The DVL measures the change in frequency in
+each direction. Based on [24], the relative velocity of each
+beam is calculated by:
+Vrel “ pFD ` bF,D ` nF,Dq1000 ¨ Cp1 ` SFcq
+2fs
+(1)
+where FD is the Doppler frequency shift, bF,D and nF,D are
+the bias and noise of the Doppler frequency shift, respectively,
+SFc is the scale factor error, C is the speed of sound, and fs
+is the transmitted acoustic frequency. The DVL transducers
+send acoustic beams in four directions. The standard DVL
+configuration is the ”Janus Doppler configuration”.In this
+configuration, the transducers are in an ”X” shape, and the
+direction of each beam is described by the following equation:
+bi “
+»
+—–
+cos ˜ψi sin ˜θ
+sin ˜ψi sin ˜θ
+cos ˜θ
+fi
+ffifl
+(2)
+where ˜θ is the (fixed) pitch angle and ˜ψi is the yaw angle
+defined for each beam i as:
+˜ψi “ pi ´ 1q90 deg `45 deg, i “ 1, 2, 3, 4.
+(3)
+The estimated DVL velocity in the platform frame is:
+˜vp
+t{p “ pAT Aq´1AT y
+(4)
+where ˜vp
+t{p is the velocity vector, A is the direction matrix
+defined as:
+A “
+»
+———–
+bT
+1
+bT
+2
+bT
+3
+bT
+4
+fi
+ffiffiffifl
+(5)
+and y is the measured beams vector
+y “
+“
+˜y1
+˜y2
+˜y3
+˜y4
+‰T .
+(6)
+2) Pressure sensor: A pressure sensor measures the pres-
+sure of a fluid or gas. In underwater navigation, a pressure
+sensor can be used to measure the water pressure at different
+depths, which can be used to determine the depth of the sub-
+merged vehicle. The underlying physical equation to estimate
+the AUV depth is [26]:
+p “ ρ ¨ g ¨ h ` ρp0
+(7)
+where p [Kpa] is the measured pressure, p0 is the pressure
+in
+the
+atmosphere
+equalling
+101.3[Kpa],
+ρ
+is
+water
+density[kg{m3],
+g
+is
+the
+gravity
+magnitude,
+assumed
+here constant and equal to 9.81 [m{s2], and h [m] is the
+depth of the AUV.
+B. Model-based approaches for missing beams
+1) Average: An average in a time window refers to the
+average value of a measurement over a specific period of time
+(the ’time window’). This can be useful for smoothing out
+noisy or erratic measurements, and reducing the effects of
+random errors. In the context of measurement synthesizing the
+average is a standard method that uses the average between the
+measurements in the previous time window, to assume the cur-
+rent measurement. For a time window with N measurements,
+the average is :
+AV pxq “ 1
+N
+N
+ÿ
+k“1
+xk
+(8)
+The size of the time window is chosen based on the charac-
+teristics of the sensor and system. For example, a small time
+window may be used for measurements that change rapidly,
+while a larger time window may be more suitable for relatively
+stable measurements.
+2) Virtual Beam: The last velocity vector measurement can
+be utilized to predict the current velocity vector [17]. This
+method replaces the missing DVL beam measurement with
+the previously available measurement. For example, if beam
+#1 is absent, solving (4) with the known velocity vector at
+k ´ 1 gives an estimate of its velocity:
+y1,k « bT
+1 rˆvx,k´1
+ˆvy,k´1
+ˆvz,k´1s
+(9)
+where k is the time index and ˆvj,k´1 is the estimated velocity
+component from the previous step for j “ x, y, z. This
+approximated beam velocity is then used, together with the
+measured beams, in (4) to predict the current velocity vector.
+III. MISSBEAMNET FRAMEWORK
+We propose a deep learning framework, MissBeamNet, as a
+mechanism to handle missing DVL beam measurements (1,2,
+or 3 beams) and allow the estimation of the AUV velocity
+vector. The MissBeamNet framework utilizes n past DVL
+beam measurements and the currently available beams as input
+to an end-to-end neural network, which regresses the missing
+beams. Then, the regressed and currently available measured
+beams are plugged into the model-based least squares (LS)
+estimator to estimate the AUV velocity vector. Figure 1
+
+3
+describes our MissBeamNet framework.
+Our proposed MissBeamNet can cope with the following
+scenarios:
+‚ If three beams are available, MissBeamNet will regress
+one missing beam.
+‚ If two beams are available, MissBeamNet will regress
+two missing beams.
+‚ If one beam is available, MissBeamNet will regress three
+missing beams.
+Note, that MissBeamNet was not designed to handle complete
+DVL outages, as it requires at least one available beam. For
+total outages, other solutions exist [27], [28]. We consider two
+Fig. 1.
+MissBeamNet framework utilizing past DVL beam measurements to
+regress the missing beams.
+types of neural networks as our baseline network architectures.
+The first is based on a one-dimension convolution neural
+network (CNN), while the other is based on long-short-term
+memory (LSTM) cells. Both networks have been proven to
+work with time-series data, such as those considered in our
+scenario.
+A. Baseline Network Architectures
+1) Convolutional Neural Network: In CNN layers, there is
+a sparse interaction between the input and output, as appose to
+fully connected layers, where all the input parameters directly
+interact with the output. The convolution operator is a linear
+operator that involves multiplying an input with a kernel
+containing learned parameters. The kernel slides through the
+input, and the result is the sum of all the multiplications:
+yt “
+pÿ
+k“1
+xt`kwk
+(10)
+where t is the timestamp, p is the kernel length, w is the
+learned kernel parameter, and x,y are the input and output, re-
+spectively. The fact that CNN shares parameters by passing the
+same kernels through all the input makes CNN architectures
+very popular in situations with large inputs. Figure 2 describes
+our baseline CNN architecture, including network parameters,
+for a scenario of two missing beams. The network is a multi-
+head network where past DVL measurements are the input to
+the first head, and current DVL measurements are the input to
+the second head. The same structure and parameters are used
+when one or three beams are missing. The selected activation
+function between the layers is Relu and the stride and padding
+are set to one.
+2) Long Short-Term Memory Network: LSTM is an ad-
+vanced version of a recurrent neural network (RNN) and solves
+its shortcomings. RNNs are capable of handling temporal
+data by using information from prior inputs. However, if
+the sequence is long, the RNN may face a problem known
+as vanishing/exploding gradients [29]. For example, when a
+Fig. 2.
+Baseline CNN architecture with an example of two missing beams.
+gradient is small, it may continue to decrease until the model
+is no longer learning. The LSTM addresses these problems
+using three types of gates: The forget gate, the input gate, and
+the output gate.
+The role of the forget gate is to forget unwanted information
+from the previous output and current input:
+ft “ σpxtU f ` ht´1W f ` bfq
+(11)
+where xt is the input, ht´1 is the output of the previous LSTM
+cell, W f and bf are the weights and biases of the forget gate,
+respectively. In (11) sigmoid function is employed to bring the
+parameter it wants to forget closer to zero. The output of the
+forget gate is then multiplied by the previous cell state. The
+role of the input gate is to update the cell state Ct´1, by first
+calculating the input gate it:
+it “ σpxtU i ` ht´1W i ` biq
+(12)
+where U i and wi are the gate weights and bi is the bias.
+Second, calculating the estimated cell state ˜Ct:
+˜Ct “ tanhpxtU g ` ht´1W g ` bgq
+(13)
+where U g and wg are the gate weights and bg is the bias. The
+results of (11),(12), and (13) are used for the current cell state
+calculations:
+Ct “ ft ¨ Ct´1 ` it ¨ ˜Ct
+(14)
+As the name implies, the output gate ot determines which
+parameters are important as the output and next hidden state.
+ot “ σpxtU o ` ht´1W o ` boq
+(15)
+where U t and wt are the gate weights and bt is the bias. The
+output gate results are then multiplied by a tanh layer of the
+cell state to calculate the current output and hidden state
+ht “ ot ¨ tanhpCtq
+(16)
+Figure 3 describes our LSTM baseline network structure.
+Previous beam measurements are used as input to the LSTM
+layers. After the LSTM features extraction, the features are
+concatenated with available beam measurements into a fully
+connected layer, which performs the final process resulting in
+the output of the regressed missing beams. Note that, like our
+baseline CNN network, this is a multi-head network where past
+DVL measurements are inputs to the first head, and current
+DVL measurements are inputs to the second head. Figure 4
+describes the LSTM architecture parameters in the scenario of
+two missing beams. The activation function between the layers
+is Relu. The same structure and parameters are also used when
+one or three beams are missing.
+
+Current beams
+E-
+LSEstimator
+★Velocityvector
+Past n DVL beam measurements,
+Neural network
+Regressed beamsInput:
+1DCNN1
+1DCNN2
+1DCNN3
+1DCNN 4
+1DCNN5
+1DCNN6
+1DCNN7
+4X6
+16@2X1
+32@3X1
+64@3X1
+64@3X1
+128@3X1
+128@3X1
+256@3X1
+Flatten layer
+Fully connected 1
+Dropoutlayer
+Fully connected 2
+1280
+probability = 0.3
+640
+Fully connected 3
+Output:
+2 missing
+10
+beams
+Current beams
+24
+Fig. 3.
+Baseline LSTM structure.
+Fig. 4.
+Baseline LSTM architecture with an example of two missing beams.
+B. Training Process
+The training process of deep neural networks requires defin-
+ing a loss function. The common loss functions for regression
+problems are mean absolute error (MAE) or mean squared
+error (MSE). In this paper, we use MSE loss defined by:
+MSE “ 1
+n
+n
+ÿ
+i“1
+pyactual ´ ypredictedq2
+(17)
+where n is the number of samples, yactual is the target, and
+ypredicted is the model output. Generally, the MSE loss func-
+tion will try to adjust the model to better handle outliers than
+MAE due to the MSE squared error. However, in our scenarios,
+an AUV operates in varying sea conditions, therefore we adopt
+the MSE loss. During training, the loss function is calculated
+after each forward propagation in order to use the method of
+gradient descent and set the DNN initial weight and biases on
+values that will provide the desired result. Forward propagation
+is the process of the data going through all the layers of
+the architecture, like (10) for CNN and (11)-(16) for LSTM
+networks. After completing the forward propagation process,
+the back propagation process updates the weights and biases
+of all the layers with a gradient descent principle
+θ “ θ ´ η∇θJpθq
+(18)
+where θ is the vector of weights and biases, Jpθq is the loss
+function with the DNN weights and biases set to θ, ∇θ is the
+gradient operator, and η is the learning rate.
+The learning rate is a crucial hyperparameter, which dictates
+how fast the weights and biases change after each training
+batch. If the selected learning rate is too low, it might converge
+in a local minimum, and if it is too high, the model might not
+converge at a minimum. Our selected optimizer for all tested
+architectures is an adaptive moment estimation (ADAM) [30].
+IV. ANALYSIS RESULTS
+A. Dataset Description
+To examine the proposed approach, data from sea experi-
+ments were employed. All experiments were conducted in the
+Mediterranean Sea by the ”Snapir” (ECA A18D), a 5.5[m]
+long AUV capable of reaching 3000[m] depth. It is equipped
+with the Teledyne RDI Work Horse navigator DVL[31], which
+has a four-beams Janus convex configuration with a sample
+rate of 1[Hz]. To train the deep neural network, first, all invalid
+Fig. 5.
+The ”Snapir” being pulled out of the water after a successful mission.
+DVL data was removed (some of the invalid readings occurred
+when actual beams were missing). Then, the data was divided
+into routes that the AUV performed. Approximately 60% of
+the missions were used as the training dataset and the rest as
+the test dataset. The training dataset comprised 23,243 samples
+corresponding to 387 minutes of recording, and the test dataset
+had 276 minutes of recording (16,618 samples).
+Figure 6 shows an experiment with challenging dynamics
+which is part of the test dataset.
+Fig. 6.
+Experiment example from the test dataset.
+The total of 663 minutes of recordings consists of two
+parts: 300 minutes from our initial data collection cam-
+paign [22] and 363 minutes in the current campaign.
+The complete dataset is publicly available at our GitHub
+https://github.com/ansfl/MissBeamNet.
+
+output
+fully connected 2
+fully connected 1
+OOOOOOO
+flatten layer
+co
+LSTM
+LSTM
+LSTM
+LSTM
+LSTM
+LSTM
+ho
+cell
+cell
+cell
+cell
+cell
+cell
+Beams
+Beams t
+Beams t
+Beams t.
+Beams t.
+Beams t,
+Beam 1
+current
+beams
+Beam 2
+5
+Beam 3
+5
+Beam 4Input:
+LSTM
+Fully connected 1
+4X6
+hidden state = 500
+3000
+Fully connected 1
+2 missing
+Output
+6
+beams
+Current beams:
+20
+9-
+10
+15
+-20
+-25
+30
+35
+0
+1000
+2000
+3000
+4000
+5000
+Time [s]5
+B. Performance Metric
+Performance metrics compare different models/methods and
+choose the one with the best performance. Throughout the
+research, we used the performance metric of root mean squared
+error (RMSE), which is widely used to evaluate models on
+regression tasks. RMSE is calculated by taking the root of the
+average of squared differences between the predicted values
+and the target values
+RMSE “
+cřn
+i“1pyactual ´ ypredictedq2
+n
+(19)
+The RMSE results are in the same units as the original data,
+making it easy to interpret.
+C. Baseline Architectures Comparison
+To compare our two baseline architectures described in
+Section
+III-B, we consider a scenario with two missing
+beams, namely, beams #1 and #2, and assume six past beam
+measurements are used. In addition to these two baseline
+architectures, we examine the possibility of using only
+past beam measurements instead of the baseline multi-head
+approach. These two architectures are denoted as CNN A
+and LSTM A. The results of the test dataset in terms of
+RMSE are presented in Figure 7. The results shows that
+Fig. 7.
+RMSE results for network architecture comparison.
+using the baseline architectures (multi-head) obtained better
+performance than working with all the inputs in a single
+head. In addition, the performance of the baseline LSTM
+showed an improvement of 27 % over the baseline CNN.
+D. Number of Past Beam Measurement Influence
+The number of past measurements utilized by the network
+is defined as the window-size length. The length of the
+optimal window size is crucial for model performance. The
+window size regularizes the model performance between the
+long and short movement patterns. If the selected window
+size is too short, the model might miss the pattern of the
+AUV movement, and if it is too long, the model might not
+react well enough to a movement that just started. Figure 8
+shows the RMSE of the baseline LSTM model with different
+window-size lengths (between 3-10) when beams #1 and
+#2 are being regressed. The results suggest that the optimal
+window-size length on our dataset is six measurements.
+Fig. 8.
+RMSE as a function of the window size for the baseline LSTM
+network.
+E. Additional Input Information
+To improve the model performance even further, additional
+inputs are considered.
+1) Depth Sensor: The last depth sensor reading.
+2) AUV Velocity Vector: Domain knowledge is used to
+transform the raw data (in this case, the beams) into
+meaningful features using feature engineering. Feature
+engineering is prevalent in classical machine learning
+methods, but less in deep neural networks. The assump-
+tion when using a neural network is thet model will
+learn the essential relations between features indepen-
+dently. The beams and the velocity vector are related,
+as the latter is estimated using the former. That is,
+the model is not receiving new information. Yet, in
+the proposed LSTM-based model, there are only two
+fully connected layers, and therefore feature engineering
+may help achieve better accuracy or shorten the network
+convergence time.
+Figure 9 describes the performance of each input with our
+baseline LSTM architecture, including additional inputs of 1)
+depth, 2) velocity vector, and 3) depth and velocity vector.
+The tested case is when beams #1 and #2 are missing, and
+beams #3 and #4 are inserted as a two-phase input to our
+baseline LSTM network. All of the additional inputs improved
+Fig. 9.
+Velocity RMSE as a function of different input selection for our
+baseline LSTM network.
+the performance of the baseline LSTM, and the best approach
+was obtained using all three input types - beam measurements
+(baseline), depth sensors, and the velocity vector. In this
+instance, there was a 16% improvement compared to the
+LSTM baseline.
+F. Missing Beams Analysis
+There are 14 combinations of missing beams: four combina-
+tions of one missing beam, six of two missing beams, and four
+of three missing beams. In the proposed approach a different
+network needs to be trained for each of those combinations.
+Training for all networks used the same hyper-parameters:
+
+0.14
+0.12 
+0.10
+MSE
+0.06
+0.04
+0.02
+0.00
+LSTMA
+Baseline LSTM
+CNN-A
+Baseline CNN
+Architecture0.100
+0.095
+[m/s]
+0.090
+E
+0.085
+0.080
+0.075
+3
+4
+5
+6
+7
+8
+9
+10
+# of past measurements0.0750
+0.0725
+0.0700
+兰0.0675
+S
+MSE
+20.0650
+0.0625
+0.0600
+0.0575
+Baseline LSTM
+Baseline LSTM +
+Baseline LSTM +
+Baseline LSTM +
+altitude
+velocity vector
+altitude + velocity vector
+Architecture6
+MSE loss function with a learning rate of 0.00005, batch size
+of 1 sequence, and 150 epochs. In the following sections, we
+present the performance of our MissBeamNet approach com-
+pared to the average (baseline) and virtual beam approaches.
+For this analysis, we employed our baseline LSTM network
+described in Section 3.1.2. Based on the results of Section
+4.4, we use six past DVL beam measurements and, based on
+Section 4.5, both the depth sensor reading and velocity vector
+were are added as additional inputs. The results were obtained
+on the testing dataset.
+1) One missing beam: When one beam is missing, the
+least squared approach (4) can be used to obtain the estimated
+AUV velocity vector. Table I presents the results of estimating
+the missing beam, the speed error obtained when using the
+estimated fourth beam together with the measured three, and
+the improvement of our MissBeamNet approach over the two
+model-based approaches.
+Both model-based and MissBeamNet methods were supe-
+rior tp the three beams solution, indicating that regressing the
+fourth beam is critical to improving the AUV speed estima-
+tion accuracy. Specifically, MissBeamNet, improved the speed
+accuracy by over 90%. In addition, MissBeamNet performed
+significantly better than the model-based approaches, with a
+40%-68% improvement. Taking the mean of performance of
+all four scenarios MissBeamNet improved the model-based
+approaches by over 49.8 %.
+2) Two Missing Beams: When considering two missing
+beams, six different combinations exist. In such scenarios,
+the AUV velocity cannot be estimated. Following the same
+procedure as the previous one missing beam scenarios, Table
+II presents the results of two missing beams. The results show
+a significant difference between the speed error in each com-
+bination, even in the model-based approaches, emphasizing
+the problem’s complexity. Yet, in all cases, MissBeamNet
+was more accurate than the model-based approaches, with a
+minimum improvement of 20% that reached almost 50%. The
+average improvement over the baseline model-based approach
+was 28.7% compared to 49.8% when only one beam was miss-
+ing. This is attributed to the model receiving less information
+from two current beams compared to three when only one is
+missing.
+3) Three Missing Beams: Table III presents the results
+for the four scenarios in which three beams are missing. As
+expected, the speed error when three beams are missing is
+higher than in the two or one missing beams scenarios. Yet,
+MissBeamNet use improved results by at least 21% over the
+model-based approaches. For three missing beams, the average
+improvement was 24% compared to 28.7% when two beams
+were missing, only 4.7% less, indicating that even with only
+one beam at hand, the AUV velocity can be estimated.
+4) Hyperparameter Tuning: One of the main challenges in
+deep learning research is to find the best combination of hy-
+perparameters for the proposed architecture. Each architecture
+has several parameters that can influence model performance,
+including the number of layers, the number of parameters in
+each layer, the type of cost function, the learning rate, and
+batch size. To demonstrate the potential of hyperparameter
+tuning, we evaluated three different hyperparameters. The
+first was the learning rate, which affects how much each
+batch changes the weights and biases III-B. The second hyper
+parameter was the number of hidden parameters in the LSTM
+layer ht III-A2, and the third hyperparameter is the number
+of parameters in the LSTM output. To test the importance of
+hyperparameter tuning, each parameter was set with a few
+available options, and a seed was set (equal initialization
+in each run). We focused on a one missing beam scenario,
+which has four options - missing beam #1, #2, #3, or #4.
+For each case, 15 randomly selected combinations of the
+three hyperparameters were examined. Table IV presents the
+potential of hyperparameter tuning. It is important to note that
+out of the 15 tested hyperparameter combinations, only a few
+were better than the results before tuning. Yet, they were able
+to improve the missing beam estimation and, consequently,
+reduce the speed error and increase the rate of improvement
+compared to the two model-based approaches.
+V. CONCLUSIONS
+Here, we presented MissBeamNet, a deep learning-based
+framework developed to compensate for partial DVL measure-
+ment scenarios (1, 2, or 3 missing beams). To that end, an
+LSTM-based dedicated DNN was derived. We demonstrated
+that the best input to the network is past DVL measurements,
+past depth sensor measurements, previous velocity vectors,
+and the currently available measured beams. Once the missing
+beams are regressed, they are combined with the available
+beams and plugged into the classical model-based approach
+to estimate the AUV velocity vector.
+To evaluate MissBeamNet, sea experiments with the Uni-
+versity of Haifa’s ”Snapir” AUV were conducted. The data
+included several trajectories collected for different purposes
+and under various sea conditions. We provide a thorough
+analysis of all 14 missing beam combinations and explore
+several means to enhance our baseline architecture. The results
+show that MissBeamNet allows estimating the missing DVL
+beams and, consequently, the AUV velocity vector. Addi-
+tionally, MissBeamNet significantly improves the accuracy of
+the velocity vector in all examined scenarios compared to
+the model-based approaches. The improvement of all three
+missing beam combinations was above 20 % over the model-
+based approaches. For two missing beams, performance was
+generally better compared to three missing beams since the
+model uses one additional measured beam. Finally, we show
+that hyperparameters-tuned models improve the accuracy of
+MissBeamNet by more than 40%.
+REFERENCES
+1 Q. Luo, Y. Shao, J. Li, X. Yan and C. Liu, A multi-AUV cooperative
+navigation method based on the augmented adaptive embedded cuba-
+ture Kalman filter algorithm., Neural Comput and Applic vol. 34, pp.
+18975–18992 2022.
+2 M. Mohammadi, M.M. Arefi, N. Vafamand and O. Kaynak Control of
+an AUV with completely unknown dynamics and multi-asymmetric input
+constraints via off-policy reinforcement learning, Neural Comput and
+Applic vol. 34, pp. 5255–5265 2022.
+3 RR.B. Wynn, V.A.I. Huvenne, T.P. Le Bas, B.J. Murton, D.P. Connelly,
+B.J. Bett, H.A. Ruhl, K.J. Morris, J. Peakall, D.R. Parsons, E.J. Sumner,
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+(AUVs): their past, present and future contributions to the advancement
+of marine geoscience Marine Geology., vol. 352, pp. 451-468. 2014.
+
+7
+TABLE I
+ONE MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET
+Case
+Approach
+Beam 1
+Beam 2
+Beam 3
+Beam 4
+Avg.
+results
+Missing beam [m/s]
+Average (baseline)
+0.110
+0.101
+0.101
+0.111
+0.106
+Virtual beam
+0.139
+0.109
+0.110
+0.129
+0.121
+MissBeamNet (ours)
+0.065
+0.048
+0.034
+0.067
+0.053
+Speed error [m/s]
+Average (baseline)
+0.066
+0.061
+0.061
+0.067
+0.064
+Virtual beam
+0.079
+0.065
+0.066
+0.077
+0.072
+Three beams
+0.450
+0.437
+0.438
+0.449
+0.443
+MissBeamNet (ours)
+0.039
+0.029
+0.021
+0.040
+0.032
+MissBeamNet improvement %
+Average (baseline)
+40.9
+52.4
+65.6
+40.3
+49.8
+Virtual beam
+50.6
+55.3
+68.1
+48.0
+55.5
+Three beams
+91.3
+93.3
+95.2
+91.1
+92.7
+TABLE II
+TWO MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET
+Case
+Approach
+Beam
+1,2
+Beam
+1,3
+Beam
+1,4
+Beam
+2,3
+Beam
+2,4
+Beam
+3,4
+Avg.
+result
+Missing beams [m/s]
+Average (baseline)
+0.106
+0.106
+0.111
+0.101
+0.107
+0.106
+0.106
+Virtual beam
+0.121
+0.121
+0.131
+0.110
+0.119
+0.120
+0.120
+MissBeamNet (ours)
+0.062
+0.052
+0.085
+0.076
+0.057
+0.066
+0.066
+Speed error [m/s]
+Average (baseline)
+0.092
+0.077
+0.096
+0.088
+0.079
+0.092
+0.087
+Virtual beam
+0.106
+0.069
+0.114
+0.096
+0.065
+0.105
+0.092
+MissBeamNet (ours)
+0.055
+0.057
+0.075
+0.066
+0.061
+0.058
+0.062
+MissBeamNet improvement %
+Average (baseline)
+40.2
+25.9
+21.9
+25.0
+22.8
+36.9
+28.7
+Virtual beam
+48.1
+17.4
+34.2
+31.25
+6.15
+44.7
+30.3
+TABLE III
+THREE MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET
+Case
+Approach
+Beam
+1,2,3
+Beam
+2,3,4
+Beam
+1,2,4
+Beam
+1,3,4
+Avg.
+results
+Missing beams [m/s]
+Average (baseline)
+0.104
+0.108
+0.107
+0.105
+0.106
+Virtual beam
+0.118
+0.124
+0.124
+0.116
+0.120
+MissBeamNet (ours)
+0.071
+0.073
+0.077
+0.071
+0.073
+Speed error [m/s]
+Average (baseline)
+0.102
+0.108
+0.106
+0.103
+0.105
+Virtual beam
+0.102
+0.109
+0.111
+0.099
+0.105
+MissBeamNet (ours)
+0.077
+0.081
+0.083
+0.078
+0.079
+MissBeamNet improvement %
+Average (baseline)
+24.5
+25.0
+21.7
+24.3
+23.9
+Virtual beam
+24.5
+25.7
+25.2
+21.2
+24.1
+TABLE IV
+HYPERPARAMETERS TUNING
+Case
+Approach
+Beam 1
+Beam 2
+Beam 3
+Beam 4
+Missing beam [m/s]
+Before tuning
+0.065
+0.048
+0.034
+0.067
+After tuning
+0.011
+0.017
+0.02
+0.012
+Speed error [m/s]
+Before tuning
+0.039
+0.029
+0.021
+0.040
+After tuning
+0.007
+0.010
+0.012
+0.007
+Learning rate
+Before tuning
+5e-05
+5e-05
+5e-05
+5e-05
+After tuning
+1e-04
+1e-04
+5e-05
+1e-05
+hidden LSTM
+parameters ht
+Before tuning
+500
+500
+500
+500
+After tuning
+250
+750
+750
+100
+LSTM output
+parameters
+Before tuning
+7
+7
+7
+7
+After tuning
+7
+5
+7
+5
+MissBeamNet Tuning
+improvement %
+Average (baseline)
+89.4
+83.6
+80.3
+89.5
+Virtual beam
+91.1
+84.6
+81.8
+90.9
+Three beams
+98.4
+97.7
+97.2
+98.4
+Before tuning
+82.2
+67.5
+42.8
+82.5
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+23 The Hatter department of marine technologies, ocean instruments:
+https://www.marinetech.haifa.ac.il/ocean-instruments
+24 Teledyne RD Instruments, Adcp Coordinate Transformation Formulas and
+Calculation.
+25 P. A. Miller, J. A. Farrell, Y. Zhao and V. Djapic, Autonomous Underwater
+Vehicle Navigation, IEEE Journal of Oceanic Engineering, vol. 35, no. 3,
+pp. 663-678, 2010.
+26 C.T. Crowe, D.F. Elger, and J.A. Roberson, Engineering Fluid Mechanics,
+Boston: Cengage Learning, pp.21-27, 2016.
+27 I. Klein and Y. Lipman, Continuous INS/DVL Fusion in Situations of DVL
+Outages 2020 IEEE/OES Autonomous Underwater Vehicles Symposium
+(AUV), pp. 1-6,2020.
+28 I. Klein, Y. Gutnik and Y. Lipman, Estimating DVL Velocity in Complete
+Beam Measurement Outage Scenarios, IEEE Sensors Journal, vol. 22, no.
+21, pp. 20730-20737, 2022,
+29 R. Pascanu, T. Mikolov and Y. Bengio, On the difficulty of training re-
+current neural networks Proceedings of the 30th International Conference
+on Machine Learning, pp.1310-1318, 2013.
+30 P. Diederik and B. Jimmy Adam: A Method for Stochastic Optimization,
+3rd International Conference for Learning Representations, 2015.
+31 Teledyne marine manual for the Teledyne RDI Work Horse navigator
+DVL.
+

ANE3T4oBgHgl3EQfsQvt/content/tmp_files/2301.04667v1.pdf.txt

@@ -0,0 +1,1522 @@
+MNRAS 000, 1–13 (2023)
+Preprint 13 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Unravelling the mass spectrum of destroyed dwarf galaxies with the
+metallicity distribution function
+Alis J. Deason1,2★, Sergey E. Koposov3,4,5†, Azadeh Fattahi1, Robert J. J. Grand6,7
+1 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
+2 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
+3Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
+4 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
+5 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
+6Instituto de Astrofisica de Canarias, Calle Via Lactea s/n, E-38205 La Laguna, Tenerife, Spain
+7Departamento de Astrofisica, Universidad de La Laguna, Av. del Astrofisico Francisco Sanchez s/n, E-38206, La Laguna, Tenerife, Spain
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Accreted stellar populations are comprised of the remnants of destroyed galaxies, and often dominate the ‘stellar haloes’ of
+galaxies such as the Milky Way (MW). This ensemble of external contributors is a key indicator of the past assembly history
+of a galaxy. We introduce a novel statistical method that uses the unbinned metallicity distribution function (MDF) of a stellar
+population to estimate the mass spectrum of its progenitors. Our model makes use of the well-known mass-metallicity relation of
+galaxies and assumes Gaussian MDF distributions for individual progenitors: the overall MDF is thus a mixture of MDFs from
+smaller galaxies. We apply the method to the stellar halo of the MW, as well as the classical MW satellite galaxies. The stellar
+components of the satellite galaxies have relatively small sample sizes, but we do not find any evidence for accreted populations
+with 𝐿 > 𝐿host/100. We find that the MW stellar halo has 𝑁 ∼ 1 − 3 massive progenitors (𝐿 ≳ 108𝐿 ⊙) within 10 kpc, and likely
+several hundred progenitors in total. We also test our method on simulations of MW-mass haloes, and find that our method is
+able to recover the true accreted population within a factor of two. Future datasets will provide MDFs with orders of magnitude
+more stars, and this method could be a powerful technique to quantify the accreted populations down to the ultra-faint dwarf
+mass-scale for both the MW and its satellites.
+Key words: Galaxies: dwarf – Galaxy: halo – Local Group – galaxies: luminosity function
+1 INTRODUCTION
+Dark matter haloes of all shapes and sizes grow by accumulating
+lower mass constituents (or subhaloes). The galaxies at the centres
+of these haloes grow via ongoing star formation, but can also form
+diffuse ‘stellar haloes’ from the stellar material deposited by the
+accretion of subhaloes (if they contain stars). Depending on the mass-
+scale, this accreted stellar material can amount to significant (e.g.
+clusters, ∼ 20 − 30%) or minuscule (e.g. dwarfs, ∼ 0 − 5%) fractions
+of the overall stellar mass of the central galaxy (Purcell et al. 2007).
+Despite having a relatively low stellar mass and surface brightness,
+stellar haloes retain a record of the lower mass systems that have been
+digested by haloes over time, and quantifying and understanding this
+accreted relic has been a major research focus in astronomy for several
+decades (see e.g. Helmi 2008; Belokurov 2013).
+The most-studied stellar halo is, unsurprisingly, that of our own
+Milky Way (MW) galaxy. However, despite significant progress in
+recent years, we still only have a qualitative view of the mass spectrum
+of dwarf galaxies that have been consumed by the MW. Most notably,
+it has become clear since the game-changing Gaia mission (Gaia
+★ E-mail: alis.j.deason@durham.ac.uk (AD)
+† E-mail: sergey.koposov@ed.ac.uk (SK)
+Collaboration et al. 2016), that the inner stellar halo (within ∼ 20
+kpc) is dominated by one ancient, massive accretion event, dubbed
+the Gaia-Enceladus-Sausage (GES, Belokurov et al. 2018; Helmi
+et al. 2018). There is also some evidence that an additional massive
+structure resides in the very central regions of the galaxy (within ∼ 4
+kpc), and was accreted even earlier than the GES (Kruijssen et al.
+2019; Horta et al. 2021a). However, it is debated whether or not this
+is really an accreted structure, or rather in-situ MW material (see
+e.g. Myeong et al. 2022; Rix et al. 2022). These massive progenitors,
+join the already discovered streams and substructures, such as the
+Sagittarius and Orphan streams (e.g. Newberg et al. 2003; Majewski
+et al. 2004; Belokurov et al. 2007b), and the Virgo (Jurić et al. 2008)
+and Hercules-Aquila (Belokurov et al. 2007a) clouds (although the
+latter structures may be related to the GES, see e.g. Simion et al.
+2019; Chandra et al. 2022), and more stellar structures in the halo are
+continuously being discovered (e.g. Naidu et al. 2020). The overall
+inventory of the Galactic stellar halo is evolving, but the picture
+is far from complete, and we have no quantitative ‘mass-spectrum’
+of destroyed dwarfs akin to the surviving satellite dwarf luminosity
+function (Koposov et al. 2008; Tollerud et al. 2008; Drlica-Wagner
+et al. 2020; Nadler et al. 2020), which is a pillar of the field.
+Many of the halo structures that have been discovered in the
+MW are identified in phase-space and/or action-angle space. This,
+© 2023 The Authors
+arXiv:2301.04667v1  [astro-ph.GA]  11 Jan 2023
+
+2
+Deason, Koposov et al.
+of course, is where an astrometric mission such as Gaia has en-
+abled a deeper understanding of the phase-space structure of the halo
+by providing 6D measurements (at least for the inner halo). How-
+ever, even with perfect 6D data, robustly identifying distinct halo
+substructures is challenging. Indeed, massive progenitors can have
+several ‘clumps’ in dynamical spaces which cannot be unambigu-
+ously disentangled (e.g. Callingham et al. 2022) and when the stellar
+material is fully phase-mixed it becomes more difficult to identify
+from the background (e.g. Johnston et al. 2008). Furthermore, even
+in the space of conserved quantities the clumps may not stay com-
+pact due to perturbations from massive systems such as the LMC
+(Koposov et al. 2022b). This is where chemical information can be
+crucial, as galaxies of different mass (and star formation history) can
+have distinct chemical signatures (e.g. Venn et al. 2004; Tolstoy et al.
+2009). Most notable, is the well-known mass-metallicity relation for
+galaxies, which extends down to the dwarf mass-scales (e.g. Skillman
+et al. 1989; Kirby et al. 2011).
+More massive galaxies are, on average, more metal-rich, and the
+relation between mass and metallicity exists over several orders of
+magnitude in mass (e.g. Tremonti et al. 2004; Kirby et al. 2013).
+This relation can, to first order, be explained by the larger gravi-
+tational wells of more massive galaxies, which are able to retain
+metals (Dekel & Silk 1986). Lower mass galaxies lack the gravity to
+resist the expulsion of metals due to feedback mechanisms. On the
+dwarf mass-scale, not only does the average metallicity vary with
+mass, but the width of the metallicity distribution function (MDF)
+also varies, with the lowest mass dwarfs having a wider spread of
+metallicities (e.g. Kirby et al. 2011). The combined MDF of a popu-
+lation of accreted dwarf galaxies, such as a stellar halo, is therefore
+the superposition of several individual MDFs. Thus, in principle,
+metallicity measurements alone contain a unique record of the mass
+spectrum of accreted dwarfs. Indeed, the disentangling of a MDF
+into its individual components is the main focus of this work. Fi-
+nally, it is worth noting that previous work on the MDFs of dwarf
+galaxies has focused on surviving dwarfs, which, depending on the
+largely unknown redshift evolution of the mass-metallicity relation,
+may or may not be relevant for the destroyed dwarfs that make-up
+stellar haloes (see e.g. Fattahi et al. 2020; Naidu et al. 2022).
+In this work, we consider Galactic-sized stellar haloes as well as
+the (potential) stellar haloes of dwarf galaxies. In principle, dwarf
+galaxies themselves can cannibalise lower-mass dwarfs, and form
+what we classically think of as a ‘stellar halo’. However, unlike larger
+mass-scales where the merging dark matter clumps all contain stars,
+at lower mass scales (below ∼ 109𝑀⊙ in halo mass) dark matter sub-
+haloes may not have any stars at all (e.g. Benitez-Llambay & Frenk
+2020). A recent study by Deason et al. (2022) showed that the very
+existence of a stellar halo around a dwarf galaxy can have important
+implications for both small-scale galaxy formation and the nature of
+dark matter. For example, the mass-threshold for galaxy formation,
+which is largely determined by the epoch of reionization, can have
+a major effect on the stellar haloes of dwarf galaxies: for models
+with a high mass threshold for galaxy formation (≳ 109𝑀⊙) dwarf
+galaxies should not have stellar haloes at all! Thus, the detection
+or non-detection of lower mass accretion events surrounding dwarf
+galaxies, particularly at the ultra-faint mass scale (𝑀star ≲ 105𝑀⊙),
+is of utmost importance.
+In order to study the MDFs of accreted populations, we need
+large, ideally unbiased, spectroscopic samples with metallicity mea-
+surements. For both the Galactic halo, and dwarf satellite galaxies in
+the MW, extensive samples are hard to come by, but there has been
+significant progress in recent years (e.g. Kirby et al. 2011; Zhao et al.
+2012; Kunder et al. 2017; Majewski et al. 2017; Walker et al. 2007;
+Conroy et al. 2019; Taibi et al. 2022). Moreover, and importantly,
+we are entering a new era of spectroscopic surveys in the MW, with
+several projects such as DESI, WEAVE, and 4MOST on the horizon
+(Cooper et al. 2022; Dalton et al. 2012; de Jong et al. 2019). Thus,
+with these new surveys in mind, we develop a new modeling method
+to extract the mass spectrum of accreted components from a sample
+of [Fe/H] measurements and apply this to current datasets.
+The paper is organised as follows. In Section 2 we outline our
+methodology and introduce the statistical model. This is a fairly tech-
+nical section that some readers may want to skip over! The method is
+applied to spectroscopic samples of classical dwarf satellite galaxies,
+and Galactic halo data in Section 3. We test the method on state-of-
+the-art cosmological simulations of MW-mass galaxies in Section
+4, and discuss caveats and future prospects in Section 5. Finally, we
+summarise our main findings in Section 6.
+2 MDF MODELING
+In this Section, we present the methodology that allows us to take
+samples with measured [Fe/H], and some estimate of the total lu-
+minosity of the system, and use them to provide constraints on the
+number of discrete stellar systems of different luminosities that can
+contribute to a given galaxy.
+This next Section is fairly technical, so a less statistically-minded
+reader may want to skip it and continue with Section 3. The Python
+code implementing the inference method presented in this section is
+released on GitHub1.
+2.1 General statistical model
+We construct a generative model that allows us to represent the
+metallicity distribution function (MDF) as a mixture of MDFs from
+smaller galaxies. Throughout this work, we will assume that the MDF
+of each smaller galaxy can be represented by a Gaussian.
+The generic model, where the sample of stars for the MDF is
+coming from several galaxies, can be described with these model
+parameters:
+• Number of galaxies N
+• 𝐿𝑖 individual galaxy luminosities (where 1 < 𝑖 < 𝑁)
+• 𝜇𝑖 mean galaxy metallicities
+• 𝜎𝑖 widths of MDF of individual galaxies.
+We can then assume that the number of stars in the sample scales
+linearly with galaxy luminosity. This assumption is accurate for stel-
+lar populations of similar ages. For that assumption to hold, our
+sample must not be biased towards one galaxy or another (e.g. if
+our sample comes from a small volume that has an unrepresentative
+subsample of certain galaxies). If the proportionality holds, one can
+write the MDF as
+𝑃(𝑧|𝑁, {𝐿𝑖}, {𝜇𝑖}, {𝜎𝑖}) =
+1
+� 𝐿𝑖
+𝑖=𝑁
+∑︁
+𝑖=1
+𝐿𝑖N (𝑧|𝜇𝑖, 𝜎𝑖)
+(1)
+Here, for clarity, we use 𝑧 as a short-hand notation of [Fe/H]. Given
+our expectation that galaxy luminosities and metallicities are corre-
+lated (Tremonti et al. 2004; Kirby et al. 2011), we can assume that
+galaxies follow a mass metallicity relation (or luminosity metallicity
+relation)
+𝜇𝑖 ∼ N (𝐴 + 𝐵 log 𝐿𝑖|S)
+(2)
+1 https://github.com/segasai/mdf_modeling_paper
+MNRAS 000, 1–13 (2023)
+
+Destroyed dwarfs with the MDF
+3
+where 𝐴 and 𝐵 are constants i.e. taken from the mass metallicity
+relation presented in Kirby et al. (2011) and Simon (2019). S is a
+constant representing a scatter in the relation (found to be 0.15 dex
+by Simon 2019, for MW satellites).
+The individual widths 𝜎𝑖 of MDFs differ from galaxy to galaxy
+but have been approximated to be slowly dependent on the galaxy
+luminosity 𝜎 = 𝐶 + 𝐷 log 𝐿 (see Simon 2019). If we specify the
+constants 𝐴, 𝐵, 𝐶, 𝐷, and S we have a model for the distribution
+of metallicities, and this model has an integer parameter 𝑁 and
+2𝑁 floating point parameters for luminosities and metallicities of 𝑁
+individual galaxies.
+While this model for the MDF is valid and can be applied to real
+data, it has the problem of having a variable number of parameters
+and therefore is difficult to sample in practice (i.e. Green 1995).
+Therefore, it would be beneficial to reformulate the model in a way
+that makes the number of parameters fixed.
+The first modification we can do is to group galaxies in 𝑀 lumi-
+nosity bins, so that rather than represent their luminosities by discrete
+parameters we represent the number of galaxies in certain luminosity
+bins. Now we define:
+• ˆ𝐿 𝑗 are the grid of galaxy luminosities 1 ≤ 𝑗 ≤ 𝑀
+• 𝑁 𝑗 are the numbers of galaxies with luminosities ˆ𝐿 𝑗.
+• 𝜇 𝑗,𝑘 are mean metallicities of k-th galaxy with luminosity ˆ𝐿 𝑗.
+1 < 𝑘 < 𝑁 𝑗
+Where due to mass metallicity relation
+𝜇 𝑗,𝑘 ∼ N (𝐴 + 𝐵 log ˆ𝐿 𝑗 |S)
+or
+𝜇 𝑗,𝑘 = 𝐴 + 𝐵 log ˆ𝐿 𝑗 + S𝜖 𝑗,𝑘
+where 𝜖 𝑗,𝑘 ∼ N (0, 1). Here S could either be a constant or a deter-
+ministic function of ˆ𝐿 𝑗
+The MDF model is now
+𝑃 �𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘}� =
+1
+� 𝑁 𝑗 𝐿 𝑗
+𝑗=𝑀
+∑︁
+𝑗=1
+ˆ𝐿 𝑗
+������
+𝑘=𝑁𝑗
+∑︁
+𝑘=1
+N (𝑧|𝜇 𝑗,𝑘, 𝜎𝑗,𝑘)
+������
+.
+The likelihood of the data consisting of (for simplicity) a single
+star with metallicity z would be exactly 𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘}). The only
+problem with this formulation is that this likelihood still depends
+on a variable number of parameters 𝜖 𝑗,𝑘 so one would prefer to
+marginalise over these.
+𝑃(𝑧|{𝑁 𝑗}) =
+∫
+𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘})N ({𝜖 𝑗,𝑘}|0, 1)𝑑𝜖 𝑗,𝑘
+While this marginalisation is difficult, and may be impossible to
+do analytically, one can simply perform a Monte-Carlo integration
+over 𝑄 samples from a normal distribution, where 𝜖 𝑗,𝑘,𝑞 are the q-th
+sample 1 ≤ 𝑞 ≤ 𝑄 from N (0, 1)
+𝑃(𝑧|{𝑁 𝑗}) ≈ 1
+𝑄
+𝑞=𝑄
+∑︁
+𝑞=1
+𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘,𝑞})
+Finally, instead of directly doing the summation we can simply
+treat this as likelihood with integer parameter 𝑞
+𝑃(𝑧|{𝑁 𝑗}, 𝑞) = 𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘,𝑞})
+(3)
+where 𝑞 is a nuisance seed parameter that we marginalise over
+−4.0
+−3.5
+−3.0
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+[Fe/H]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+dP
+d[Fe/H]
+MV =-4.0
+MV =-4.0
+MV =-10.0
+MV =-10.0
+MV =-10.0,
+20 x {-4.0}
+Figure 1. The simulated MDFs for a few systems of different luminosities.
+The black lines show the expected MDFs in our model for a system with
+𝑀𝑉 = −4, with solid and dashed curves showing the MDFs when using a
+different random seed that controls the offset of the galaxy with respect to
+the mass metallicity relation. Red curves similarly show the MDF of a single
+𝑀𝑉 = −10 galaxy with different random seeds. The green curve shows the
+MDF for a synthetic galaxy that consists of stars coming from one galaxy
+with 𝑀𝑉 = −10 and 20 galaxies with 𝑀𝑉 = −4.
+under the uniform prior 𝑈(1, 𝑄). Here, we assume that 𝜖 𝑗,𝑘,𝑞 are
+coming from a pseudo-random number generator that is seeded by
+𝑞 and provides normally distributed samples. We then will need to
+sample the posterior over {𝑁 𝑗} and 𝑞, which gives the model with
+𝑀 + 1 parameters. Armed with Eqn 3 that specifies the likelihood
+function for the metallicity distribution, the only missing ingredient
+for the model are the priors.
+We assume that occupation numbers {𝑁𝑖} (i.e. numbers of galaxies
+in luminosity bins) have a prior distribution of ⌊10𝑥⌋ where 𝑥 ∼
+𝑈(−1, 4). This is essentially the log uniform of integers distribution
+with 20% prior volume at 𝑁𝑖 = 0 and 20% for 1 ≤ 𝑁𝑖 ≤ 10, and
+20 ≤ 𝑁𝑖 ≤ 100 etc.
+Finally, we complement the model with the constraint on the total
+luminosity of the system. Specifically, we require that the combined
+luminosity of multiple galaxies must match certain known total lu-
+minosity log 𝐿tot with some uncertainty 𝜎𝐿. This provides a term for
+the log of the posterior.
+log(
+∑︁
+𝑁𝑖 ˆ𝐿𝑖) ∼ 𝑁(log 𝐿tot|𝜎𝐿)
+A final remark that despite the introduction of the formalism based
+on binned number of galaxies, we have found the model is more stable
+when at least one contributor to the MDF (likely the one being the
+most massive main progenitor) is represented directly (rather than
+in a bin) by the satellite luminosity 𝐿main, metallicity 𝑧main and that
+also obeys the mass-metallicity relation.
+To illustrate our modeling approach, in Figure 1 we show the
+expected [Fe/H] distributions given our model. Specifically, solid
+black and red curves show possible MDFs for a single galaxy of
+𝑀𝑉 = −4 and 𝑀𝑉 = −10, respectively. Dashed lines of same colours
+show the MDFs when different random seeds are used. The green
+curve shows a distribution that we might expect if we observe stars
+coming from a single 𝑀𝑉 = −10 galaxy and 20 𝑀𝑉 = −4 systems.
+This shows a prominent tail towards low metallicities, and this is
+exactly what allows us to probe the number of possible mergers with
+low luminosity systems.
+MNRAS 000, 1–13 (2023)
+
+4
+Deason, Koposov et al.
+2.2 Sampling
+In the previous section, we have introduced the likelihood function
+for the metallicity distribution that is conditional on the number of
+different dwarf galaxies 𝑁 𝑗 on a grid of luminosities. The model
+also has an integer seed parameter 𝑞. It is not trivial to sample inte-
+ger parameters, especially if we expect multiple modes. To perform
+the sampling we decide to use the dynamic nested sampling as im-
+plemented in the dynesty package (Speagle 2020; Koposov et al.
+2022a). As nested sampling is technically invalid if the likelihood
+surface has plateaus (Fowlie et al. 2020), we add a a small level of
+deterministic noise with standard deviation of 0.01 to the likelihoods,
+which should not affect the inference.
+3 APPLICATIONS
+We now apply the method described above to observational data.
+Here, we focus on the classical MW satellites (§3.1) and the MW
+stellar halo (§3.2).
+3.1 Classical dwarf satellite galaxies
+We start from the homogeneous sample of dwarf galaxy members
+presented in Kirby et al. (2011) as provided in the Strasbourg astro-
+nomical Data Center (CDS). As mentioned in the previous section,
+the key assumption that we rely on for our method is that the abun-
+dances that we model are random samples from the system. This is
+likely not technically correct for the data at hand since the stellar
+samples in dwarfs tend to be biased towards the centres of systems
+(see e.g. Walker & Peñarrubia 2011), and may have slight metallic-
+ity biases caused by the colour-magnitude selection of spectroscopic
+targets. We will, however, proceed ignoring these issues.
+We take the sample of stars from Kirby et al. (2011) and only
+consider stars with small metallicity uncertainty 𝜎[Fe/H] < 0.2. This
+catalogue has measurements of 10 MW satellites with more than
+10 stars: Canes Venatici I, Draco, Fornax, Hercules, Leo I, Leo II,
+Sculptor, Sextans, Ursa Minor and Ursa Major I. We then proceed to
+model each of the dwarfs with the machinery presented in Section 2.
+We take the luminosities of each system from McConnachie (2012)
+(using an updated catalogue from January 2021) and adopt an 𝑀𝑉
+uncertainty for each system of 0.1 mag. For each system, we use
+the luminosity bins that are 1 magnitude wide from 𝑀𝑉 = 0 to the
+luminosity of the dwarf itself minus 2.5 magnitudes.
+The posterior samples on the number of possible dwarf galaxies
+that contributed to the systems’ MDF are shown in Figures 2 and
+3. We show measurements for 8 out of 10 systems spanning the
+luminosity range from 𝑀𝑉 ∼ −5 for Ursa Major I to 𝑀𝑉 ∼ −13
+for Fornax. The panels are ordered by system luminosity. The total
+number of stars varies from 𝑁 = 15 for Ursa Major I to 𝑁 = 789
+for Leo I. Figure 2 shows the constraints on the differential number
+of systems that have contributed to the dwarfs’ MDF, while Figure 3
+shows constraints on the cumulative counts of the number of systems
+brighter than a certain value. The blue/orange bands show the 16/84
+and 1/99 percentiles, and the black line shows the median of the
+posterior. The green bands show the constraints if we do not use
+metallicities at all. This is essentially a prior and corresponds to the
+case where the only constraint comes from ensuring the combination
+of galaxies matches the total luminosity of the system. Note that,
+because we include all the stars in the galaxy, we expect to measure
+𝑁merged = 1 at around the total luminosity of the dwarf galaxy (shown
+with the solid red line). Although technically this is an ‘in-situ’ rather
+than an accreted component, what we are actually constraining are
+the contributors to the MDF, regardless of their origin.
+We now look at the posteriors in more detail. First, we focus on
+clear cases where the data is particularly constraining. These are the
+cases of Fornax, Leo I, Leo II, and Draco, where the spectroscopic
+samples have hundreds of members. We see that their differential
+posterior distributions (Figure 2) have a peak with a value of one
+next to the system luminosity (highlighted in red) and show the value
+consistent with zero for 𝑀𝑉 ,host ≲ 𝑀𝑉 ≲ 𝑀𝑉 ,host+5. Thus, the data
+suggests that these systems did not experience a merger with a dwarf
+that is larger than 1% of the system luminosity. This is also seen in the
+cumulative plots, where we see the implied number 𝑁merged(< 𝑀𝑉 )
+is flat and equal to one in the range 𝑀𝑉 ,host ≲ 𝑀𝑉 ≲ 𝑀𝑉 ,host + 5.
+Looking at the implications for the number of faint contributors to
+the MDF for Fornax, Leo I, Leo II, and Draco systems, we can see
+that our constraints on 𝑁merged shoot up and become significantly
+broader. The differential counts are essentially unconstrained. For
+example, for the Fornax MDF contributors at 𝑀𝑉 = 0 (top left panel
+of Figure 2) the 1-𝜎 confidence interval is 0 < 𝑁merged < 100 as the
+data allows many faint dwarfs before the observed MDF is affected
+significantly. The behavior of the cumulative counts for the faint
+MDF contributors is somewhat misleading as it rises at faint 𝑀𝑉
+purely because we are summing over bins with non-negative values.
+Fainter dwarf galaxies like CVnI or UMa have a smaller number of
+spectroscopic observations. In Figure 2 we see that the posteriors on
+the number of MDF contributors start to rise next to 𝑀𝑉 = 𝑀𝑉 ,host,
+which indicates that we cannot even rule out that the galaxy is a
+product of a merger of two systems with similar luminosities. The
+constraints on the cumulative number of mergers for fainter dwarfs
+do not show a flat 𝑁merged = 1 part next to the luminosity of the
+system and instead rises to faint luminosities. We also see that for
+faint systems, the posteriors basically look very close to priors.
+3.2 Galactic stellar halo
+We next apply our method to the Galactic stellar halo. It has been
+realised for some time that the stellar halo of the MW comprises
+an assortment of destroyed dwarf debris, and thus the metallicity
+distribution of these halo stars retains a memory of their dwarf galaxy
+progenitors.
+Large, homogeneous samples of halo stars with metallicity mea-
+surements are hard to come by, and this is a significant limitation of
+our current study. At present, we build a sample of halo stars based
+on several spectroscopic surveys and use the latest Gaia data (Gaia
+Collaboration et al. 2021), to help select a clean halo sample. We
+begin by cross-matching stars with spectroscopic data from SDSS
+(Abolfathi et al. 2018), RAVE (Kunder et al. 2017), LAMOST (Zhao
+et al. 2012), APOGEE (Majewski et al. 2017), and GALAH (Buder
+et al. 2021) with Gaia DR3. This results in 𝑁 = 656, 0819 stars. To
+estimate distances to the stars we use the Bailer-Jones et al. (2021)
+photogeometric distances computed from Gaia EDR3. We only con-
+sider stars with reasonable parallax 𝜎𝜛/𝜛 < 0.5, and restrict our
+sample to 𝑟 < 10 kpc and |𝑧| > 1 kpc. Finally, to avoid disk con-
+tamination, we apply a cut on the rotational velocity of the stars. We
+impose a fairly strict cut to remove the majority of thick disk and/or
+splash stars (Belokurov et al. 2020), and only include those with ret-
+rograde orbits 𝑣𝜙 < −50 km/s. The resulting spatial (top-panel) and
+metallicity distribution (bottom-panel) of the stars are shown in Fig.
+4. In the bottom panel, we also show the MDF for the stars without the
+𝑣𝜙 cut in grey. Our restriction to retrograde orbits is fairly stringent
+but, as can be seen in the figure, it is effective at removing disk stars,
+which have prograde orbits and are generally more metal-rich. We
+MNRAS 000, 1–13 (2023)
+
+Destroyed dwarfs with the MDF
+5
+−10
+−5
+100
+101
+102
+Nmerged
+For
+−10
+−5
+MV
+LeoI
+−10
+−5
+MV
+LeoII
+−10
+−5
+MV
+Sex
+−10
+−5
+MV
+100
+101
+102
+Nmerged
+UMi
+−10
+−5
+MV
+Dra
+−10
+−5
+MV
+CVnI
+−10
+−5
+MV
+UMaI
+Figure 2. The inferred contributions from systems to the MDF of different dwarf galaxies from our analysis. In each panel, the black curve shows the median
+number of galaxies of a given luminosity that could have contributed to the MDF. The blue and orange bands show the 16/84 and 1/99 percentiles, respectively.
+The green band shows the sampling of the prior with only the constraint on total luminosity of the system. The vertical red line on each panel shows the
+luminosity of each system. Note that the logarithmic y-axis is cut-off at 𝑁merged = 10−1, so median values at this level are consistent with zero.
+−10
+−5
+100
+101
+102
+Nmerged(< MV )
+For
+−10
+−5
+MV
+LeoI
+−10
+−5
+MV
+LeoII
+−10
+−5
+MV
+Sex
+−10
+−5
+MV
+100
+101
+102
+Nmerged(< MV )
+UMi
+−10
+−5
+MV
+Dra
+−10
+−5
+MV
+CVnI
+−10
+−5
+MV
+UMaI
+Figure 3. The inferred contributions to the MDF from our analysis. This is similar to Figure 2 but shows the cumulative numbers. Each panel shows a different
+dwarf galaxy. In each panel, the black curve shows the median number of galaxies of a given luminosity or brighter that could have contributed to the MDF.
+The blue and orange bands show the 16/84 and 1/99 percentiles, respectively. The green band shows the sampling of the prior with only the constraint on total
+luminosity of the system. The vertical red line on each panel shows the luminosity of each system.
+apply our modelling procedure to stars with −4 < [Fe/H] < −0.82,
+and 𝜎([Fe/H]) < 0.2, which results in a sample of 𝑁 = 21, 813 stars.
+2 Note that this metallicity cut is made in both the data and model, so there
+is no metallicity bias introduced with our selection.
+Our sample is comprised of 5 different spectroscopic surveys, with
+varying selection functions. Here, we aim to maximise the number of
+halo stars with metallicity measurements by combining these surveys
+but note that, ideally, a more homogeneous sample would be used.
+For now, we continue on, under the assumption that there are no
+MNRAS 000, 1–13 (2023)
+
+6
+Deason, Koposov et al.
+0
+2
+4
+6
+8
+10
+R [kpc]
+-10
+-5
+0
+5
+10
+|z| [kpc]
+0
+2
+4
+6
+8
+10
+-10
+-5
+0
+5
+10
+-4
+-3
+-2
+-1
+0
+1
+[Fe/H]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+dN
+All
+vφ < -50 km/s
+log(g) < 3.5
+log(g) > 3.5
+Figure 4. Top panel: The spatial distribution in the 𝑧 vs. 𝑅 plane of our MW
+halo sample. Bottom panel: The metallicity distribution of the sample. The
+grey line indicates the MDF without a cut in 𝑣𝜙, which leads to the inclusion of
+(metal-rich) disk stars. The dashed red line indicates the median metallicity
+of our halo sample ([Fe/H] = −1.5). We also show the MDFs of our halo
+sample split by log(𝑔) with the blue and purple dotted lines, respectively.
+The gray line-filled region indicates the metal-rich regime ([Fe/H] > −0.8)
+that is excluded in our modelling.
+significant metallicity biases in this combined sample. However, we
+stress that future work with upcoming spectroscopic surveys such as
+DESI (Cooper et al. 2022) and WEAVE (Dalton et al. 2012) will be
+much better suited for this type of analysis.
+In our analysis, we adopt a total halo luminosity of 𝑀𝑉
+=
+−17.7 ± 0.5. This is consistent with recent measurements which
+suggest 𝐿 ∼ 1 × 109��⊙, and also allows a range around this value
+encompassing the majority of observational constraints and their un-
+certainties (Deason et al. 2019; Mackereth & Bovy 2020; Horta et al.
+2021b). The top panels of Fig. 5 show the resulting number of de-
+stroyed dwarfs in the MW halo as a function of 𝑀𝑉 . We consider
+dwarfs with 0 > 𝑀𝑉 > −22, using 22 bins with 1 mag bin size. Note
+that the size of our sample means that we are unlikely constraining
+dwarfs with 𝑀𝑉 ≳ −10, which will not be represented by a large
+enough number of stars (see e.g. Section 5.2). In blue, we show the
+results when the Kirby et al. (2011) mass-metallicity relation is used,
+which is appropriate for surviving dwarf galaxies in the MW. In re-
+cent work, Naidu et al. (2022) (see also Fattahi et al. 2020) argue
+that destroyed dwarfs may not lie on this relation, and a relation with
+∼ 0.3 dex offset to lower metallicities is more appropriate. We show
+the results with this offset applied in orange.
+Our model predicts several hundred (𝑁 ∼ 400) destroyed dwarfs
+with 𝑀𝑉 ≲ −10. However, the different mass-metallicity relations
+(relevant for either ‘surviving’ or ‘destroyed’ dwarfs) predict different
+distributions of progenitor masses, particularly at larger masses. For
+example, when using the Kirby et al. (2011) mass-metallicity relation
+applicable for surviving dwarf galaxies, we estimate 𝑁 = 1 massive
+dwarf progenitor with 𝐿 ∼ 108.5𝐿⊙, but this rises to 𝑁 = 3 when the
+relation more relevant to destroyed dwarf galaxies is used instead.
+This seems to be at odds with our adopted total halo luminosity of 𝐿 ∼
+1 × 109𝐿⊙. Indeed, by summing the predicted numbers of destroyed
+dwarfs we find that the total luminosity when the Kirby et al. (2011)
+relation is used is 1.1+0.2
+−0.2 × 109𝐿⊙, but this rises to 3.4+7.2
+−2.3 × 109𝐿⊙
+when an 0.3 dex offset is applied to the mass-metallicity relation.
+Clearly, in this latter case, the bias in metallicity has pushed the
+progenitor masses higher, and, because we have allowed a fairly
+flexible total luminosity, resulted in a high halo luminosity. However,
+it is still consistent with the input luminosity within 1 − 𝜎.
+In the bottom panel of Fig. 5 we show the results when the total
+halo luminosity is fixed to 𝑀𝑉 = −17.7 (technically, an uncertainty
+of 0.01 dex is adopted). Here, the ‘fiducial’ result using the Kirby
+et al. (2011) mass-metallicity relation is only slightly changed. For
+example, the most massive progenitor is shifted to a slightly lower
+luminosity (by ∼ 1 dex in 𝑀𝑉 ), and the total number of dwarfs with
+𝑀𝑉 < −10 is reduced (𝑁 ∼ 300). In general, the changes are within
+the predicted uncertainties. When an 0.3 dex metallicity offset is
+applied to the mass-metallicity relation, fixing the halo luminosity
+has a greater effect. This is unsurprising given that allowing for a
+more flexible halo luminosity favours a higher value than the fiducial
+1×109𝐿⊙. In this case, the most massive progenitor has 𝐿 ∼ 108.1𝐿⊙
+(compared to 𝑁 ∼ 3 with 𝐿 ∼ 108.5𝐿⊙ when the total luminosity
+is more flexible). The number of low mass dwarfs is also reduced,
+with 𝑁 ∼ 110 with 𝑀𝑉 < −10. This exercise emphasizes how
+important the assumed total halo luminosity, as well as the adopted
+mass-metallicity relation are for this type of analysis.
+We also show the surviving dwarf satellite luminosity function for
+comparison in Fig. 5. Here, we show the observed (solid purple) and
+completeness-corrected (dashed purple) cumulative number counts
+given by Drlica-Wagner et al. (2020). The numbers of low luminosity
+satellite systems are much lower than the predicted number of de-
+stroyed dwarfs. This is perhaps unsurprising given that our estimates
+are likely overestimated at low luminosities, owing both to sample
+size and our assumption of Gaussian MDFs (see Section 5.1). Inter-
+estingly, the (cumulative) number counts are similar at intermediate
+luminosities (−16 ≲ 𝑀𝑉 ≲ −12) but destroyed dwarfs as massive
+as the LMC are not favoured unless the adopted mass-metallicity
+relation is adjusted from the fiducial 𝑧 = 0 form.
+Fattahi et al. (2020) show using the Auriga simulation suite that
+the number of destroyed dwarfs in MW-mass haloes is larger than
+MNRAS 000, 1–13 (2023)
+
+Destroyed dwarfs with the MDF
+7
+20
+15
+10
+5
+0
+MV
+10
+1
+100
+101
+102
+103
+N(merged)
+MW halo: MV =
+17.7 ± 0.5
+20
+15
+10
+5
+0
+MV
+10
+1
+100
+101
+102
+103
+N(merged < MV)
+Satellite dwarf LF
+Mass-Metallicity relation:
+Kirby+2011
+Kirby+2011 
+0.3 dex
+20
+15
+10
+5
+0
+MV
+10
+1
+100
+101
+102
+103
+N(merged)
+MW halo: MV =
+17.7 (fixed)
+20
+15
+10
+5
+0
+MV
+10
+1
+100
+101
+102
+103
+N(merged < MV)
+Satellite dwarf LF
+Mass-Metallicity relation:
+Kirby+2011
+Kirby+2011 
+0.3 dex
+Figure 5. The estimated differential (left) and cumulative (right) number of destroyed dwarfs in the MW halo. The dark(light) shaded regions show the
+16-84(1-99) percentiles, and the solid lines are the medians. The dashed black line indicates the assumed total stellar halo luminosity (𝑀𝑉 = −17.7). In the top
+panels, the total luminosity has a flexible uncertainty of ±0.5 dex, whereas in the bottom panel the total luminosity is kept fixed. The results in blue are for when
+the 𝑧 = 0 mass-metallicity relation for dwarfs is assumed (Kirby et al. 2011). In orange, we show the results when an −0.3 dex offset is applied to the relation,
+which has been postulated to be more applicable to destroyed dwarfs (Naidu et al. 2022). For comparison, we show the surviving dwarf satellite luminosity
+function in purple. The dashed line indicates the completeness-corrected LF derived by Drlica-Wagner et al. (2020).
+the number of surviving satellites, at least down to 𝑀𝑉 ∼ −8. This is
+in agreement with our results, however, our estimated total number
+of destroyed dwarfs is far higher than these models (by a factor of
+∼ 3 − 10, see also Fig. 7). This could be a genuine tension with
+the models, but it is worth stressing that our number estimates at
+low luminosities are likely biased high, and the numbers could be
+reduced if we had larger sample sizes and/or the metal-poor tails of
+higher mass systems are taken into account (see Section 5.1).
+Finally, given the heterogeneous nature of our sample of halo stars,
+we consider how different cuts in surface gravity affect the results.
+Namely, dwarf stars and giants can have different metallicity biases,
+and probe different volumes in magnitude-limited surveys. The MDF
+of our halo sample split by log(𝑔) was shown in Fig. 4. Here, we
+MNRAS 000, 1–13 (2023)
+
+8
+Deason, Koposov et al.
+22
+20
+18
+16
+14
+12
+10
+8
+6
+4
+2
+0
+MV
+10
+1
+100
+101
+102
+103
+N(merged < MV)
+MV =
+17.7 (fixed)
+All
+log(g) > 3.5
+log(g) < 3.5
+Figure 6. The estimated cumulative number of destroyed dwarfs in the MW
+halo. Same as Fig. 5, but split into two bins with low (log(g) < 3.5) and high
+(log(g) > 3.5) surface gravity stars. The thick gray line indicates the overall
+sample. The stellar halo luminosity is fixed (𝑀𝑉 = −17.7).
+can see there are slight differences for low and high log(𝑔), and now
+we consider how our inferred number counts of destroyed dwarfs
+are affected. The cumulative number of destroyed dwarfs is shown
+in Fig. 6 with two different bins of log(𝑔), appropriate for dwarf
+stars (log(𝑔) > 3.5) and giants (log(𝑔) < 3.5). It is worth bearing in
+mind that our overall sample is dominated by the high surface gravity
+dwarf stars (approximately ∼ 2/3 have log(𝑔) > 3.5). Note that here
+we only use the Kirby et al. (2011) mass-metallicity relation, and
+the total halo luminosity is fixed. Encouragingly, the total number of
+progenitors (for 𝑀𝑉 ≲ −10) is very similar for the two bins of log(𝑔).
+However, massive progenitors (𝐿 ≳ 108𝐿⊙) are only favoured in the
+high log(𝑔) sample. This is likely because the MDF is biased towards
+lower metallicities for the giant star sample (see Fig. 4). Moreover,
+the giant and dwarfs are probing slightly different volumes, with
+the high surface gravity dwarfs more concentrated around the solar
+neighbourhood. This exercise highlights the difficulty of using a
+‘hodge-podge’ of halo stars for our analysis, and it will clearly be
+preferable for future work to have a more homogeneous sample,
+where the selection function is clearly defined.
+4 AURIGA SIMULATIONS
+Our modeling procedure makes various assumptions and simplifica-
+tions. For example, it assumes each progenitor galaxy is sampled in a
+representative way, and that their MDFs are adequately described by a
+Gaussian distribution. In reality, this may not be the case, particularly
+for volume-limited Galactic-sized stellar haloes. To this end, we test
+our model on simulated MW stellar haloes, which are representative
+of ‘realistic’ accreted populations. We apply our modeling procedure
+to halo stars in the Auriga simulations (Grand et al. 2017); these cos-
+mological hydrodynamical simulations are a suite of 𝑁 ∼ 30 high
+resolution (𝑚 𝑝 ∼ 5×104𝑀⊙) MW-mass (1−2×1012𝑀⊙) haloes. In
+this work, we make use of the 𝑁 = 28 haloes studied in Fattahi et al.
+(2019), which omits two haloes currently undergoing major mergers.
+We only consider accreted halo stars, which are identified in Fattahi
+et al. (2019) as those that formed in subhaloes other than the main
+progenitor galaxy.
+For each halo, we construct a sample of halo star particles within
+𝑟 < 20 kpc. This is chosen to roughly mimic the volume limit of
+current observations, and ensure large enough sample sizes. The in-
+put into the model is the [Fe/H] values of the stellar particles. Of
+course, in the simulations, we also know the progenitor galaxy of
+each star particle, and can thus test the estimated mass spectrum of
+accreted dwarfs from our modeling procedure. The final ingredient
+we need to define is the mass-metallicity relation for the Auriga sim-
+ulations. Grand et al. (2021) show that the mass-metallicity relation
+for dwarf galaxies in Auriga is in good agreement with low mass
+dwarfs (𝑀star ∼ 106𝑀⊙), but is too metal-rich by ∼ 0.5 dex for more
+massive dwarfs (see Figure 13 in Grand et al. 2021). We use all the
+destroyed dwarf progenitors across the 𝑁 = 28 Auriga haloes to cali-
+brate this relation3. However, we do exclude dwarfs that are accreted
+recently (less than 5 Gyr ago) as these can have significantly different
+metallicities due to ongoing star formation. The debris from these
+events is still included in the analysis, but our calibration is only based
+on the relatively old dwarf galaxies. Note that we only consider dwarf
+progenitors with 𝑀𝑉 > −7, which corresponds to a stellar mass of
+𝑀star > 105𝑀⊙ or 𝑁 > 2 star particles. We use the ‘peak’ stellar
+mass of each dwarf, which corresponds to the maximum stellar mass
+the progenitor has reached. Note that we get similar results if the
+stellar mass at infall is used instead. The resulting mass-metallicity4
+relation for Auriga is: [Fe/H] = −1.69 + 0.39 × (log10𝐿 − 6). To es-
+timate the scatter around this mean relation, we calculate the scatter
+for each individual halo, and use the median value across all haloes.
+This results in a scatter around the mean [Fe/H] relation of 0.3 dex.
+Finally, we consider the spread in [Fe/H] for individual dwarfs. Un-
+like the observations, we find no strong evidence for a variation with
+dwarf mass, so instead adopt a constant dispersion of 0.4 dex of
+the MDF for all dwarfs. Armed with the mass-metallicity relation
+appropriate for Auriga, we can now test our modeling procedure on
+these cosmological haloes.
+When applying our method to the Auriga haloes, we assume the
+total luminosity of the halo is known. This of course results in addi-
+tional uncertainty in the real observations, but we particularly want to
+investigate the systematic influences present in the cosmological sim-
+ulations. We consider accreted dwarfs in the range −7 > 𝑀𝑉 > −22,
+and estimate the number of dwarfs in 15 bins with bin size of 1 mag.
+Fig. 7 shows the resulting cumulative number of destroyed dwarfs
+in the Auriga haloes. Each panel shows a different halo, and our
+estimated numbers are shown with the solid black lines (median),
+and blue/orange shaded regions (16-84/1-99 percentiles). The points
+with error bars are the true values, with Poisson noise adopted for
+the uncertainties in each 𝑀𝑉 bin. Note that the ‘true’ values include
+all dwarfs that have deposited any material within 20 kpc of the
+host halo. Thus, there can be cases where only a small fraction of
+a destroyed dwarf is included in the sample (see below). The green
+values in Fig. 7 are for all progenitors, while the purple are only
+those accreted earlier than 5 Gyr ago. In many cases, there is little
+difference between the green and purple values, because most dwarfs
+are accreted at earlier times. However, we highlight the most recently
+accreted dwarfs because these are likely not fully phase-mixed, and
+can significantly deviate from the mass-metallicity relation for (old)
+dwarf galaxies in Auriga (see above). In reality, we find that these
+recently accreted dwarfs only cause a significant effect if the progen-
+itors are relatively massive (e.g. Halo 25).
+3 To clarify, all destroyed dwarfs are used, not just those that have debris
+within 20 kpc of the host halo
+4 Note that we assume a stellar mass-to-light ratio of (𝑀/𝐿) = 2 to convert
+stellar mass to luminosity.
+MNRAS 000, 1–13 (2023)
+
+Destroyed dwarfs with the MDF
+9
+1
+10
+100
+N(merged < MV)
+halo_2
+halo_3
+halo_21
+halo_23
+halo_1
+halo_22
+halo_26
+1
+10
+100
+N(merged < MV)
+halo_4
+halo_5
+halo_27
+halo_19
+halo_25
+halo_7
+halo_6
+1
+10
+100
+N(merged < MV)
+halo_24
+halo_30
+halo_18
+halo_15
+halo_29
+halo_28
+halo_14
+10
+15
+20
+MV
+1
+10
+100
+N(merged < MV)
+halo_16
+10
+15
+20
+MV
+halo_8
+10
+15
+20
+MV
+halo_9
+10
+15
+20
+MV
+halo_17
+10
+15
+20
+MV
+halo_13
+10
+15
+20
+MV
+halo_12
+10
+15
+20
+MV
+halo_10
+Figure 7. The estimated cumulative number of destroyed dwarfs for the 𝑁 = 28 Auriga haloes. The solid black line shows the median, and the shaded
+blue(orange) regions the 16-84(1-99) percentiles. The red dashed line indicates the assumed total luminosity of the halo. For each halo, accreted star particles are
+selected within 𝑟 < 20 kpc. The points with (Poisson) error bars indicate the ‘truth’, with all progenitors shown in green, and only those accreted > 5 Gyr ago
+in purple. The latter are shown because recently accreted dwarfs are likely (i) not fully phase-mixed, and (ii) can significantly deviate from the mass-metallicity
+relation for (old) dwarf galaxies in Auriga.
+We discuss these results more quantitatively below, but first cast
+a qualitative eye on Fig. 7. In general, our estimates agree well with
+the true mass spectrum of accreted dwarfs. However, in some cases,
+there can be notable differences. We find that the most significant
+deviations are due to the following: (1) relatively massive progenitors
+that lie off the mass-metallicity relation (e.g. Halo 2, 6) and/or (2)
+progenitors with a low fraction of their material within the given
+radial range (e.g. Halo 15, 27). These systematics, and sometimes the
+combination of both, are most likely to cause our method to fail. On
+the other hand, there are a significant number of haloes for which we
+recover the mass spectrum very well, which is encouraging given the
+complexity of these hydrodynamic simulations, and the cosmological
+nature of their assembly histories.
+In Fig. 8 we give a more quantitative summary of our tests of
+the Auriga haloes. Here, for each halo (identified in the x-axis) we
+show the fraction of 𝑀𝑉 bins that have number estimates that agree
+within the 16-84, 5-95, and 1-99 percentile confidence limits. The
+median recovery fractions across all haloes are 0.61, 0.77, and 0.83,
+respectively. These fractions are below the expected fractions for a
+‘perfect’ procedure, but this is unsurprising given the various sys-
+tematic influences present in the simulations, such as deviations from
+the adopted mass-metallicity relation and the presence of stellar de-
+bris that does not fully occupy the available phase-space. These, of
+course, are realistic effects that could be present in the observational
+data.
+In Fig. 9 we explore the halo-to-halo scatter more closely. In the
+left-hand panel, we show the difference between the estimated and
+1
+2
+3
+4
+5
+6
+7
+8
+9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
+Halo ID
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction
+16-84
+5-95
+1-99
+Figure 8. Quantifying the test with Auriga haloes. For each halo, we show
+the fraction of 𝑀𝑉 bins (1 mag wide) that have estimated numbers that
+agree within the 16 − 84 (red-filled circles), 5 − 95 (blue-filled squares), and
+1−99 (green-filled diamonds) percentage confidence limits, respectively. The
+median values are shown with the horizontal coloured lines.
+true cumulative numbers of destroyed dwarfs as a function of 𝑀𝑉 .
+The black line shows the median of the 𝑁 = 28 haloes, and the
+blue and orange shaded regions show the 16-84 and 1-99 percentiles,
+respectively. The deviation from the true numbers is fairly symmet-
+MNRAS 000, 1–13 (2023)
+
+10
+Deason, Koposov et al.
+22
+20
+18
+16
+14
+12
+10
+8
+MV
+40
+20
+0
+20
+40
+N(<MV)Est - N(<MV)True
+100
+101
+102
+N(<MV)True
+100
+101
+102
+N(<MV)Est
+MV < -17.5
+MV < -14.5
+MV < -11.5
+MV < -8.5
+Figure 9. Comparing the estimated and true numbers of destroyed progenitors in the (𝑁 = 28) Auriga haloes. Left panel: We show the difference between the
+estimated and true (cumulative) numbers as a function of 𝑀𝑉 . The solid black line shows the median, and the shaded blue(orange) regions the 16-84(1-99)
+percentiles. Right panel: The estimated versus the true number in different ranges of 𝑀𝑉 . Error bars show the Poisson errors in the true numbers and the 16-84
+percentiles in the estimated numbers.
+rical and only starts to shift from zero for very low-mass progenitors.
+It is worth noting that there is a trend toward overestimating the
+number of accreted dwarfs at lower masses. This could be a real
+effect, caused by e.g the assumption of Gaussian MDFs, however,
+this low-mass regime may also be affected by resolution limitations
+in the simulations, as the MDFs of these dwarfs are only represented
+by a handful of star particles. In the right-hand panel, we show the
+estimated vs. true cumulative number of progenitors in four differ-
+ent magnitude ranges. Here, we can see that there is considerable
+scatter around the 1-to-1 line, but the spread is fairly symmetrical.
+Finally, to quantify these findings we compute the typical accuracy
+of our 𝑁(< 𝑀𝑉 ) estimates (averaged over all 𝑀𝑉 bins); we find that
+𝑁(< 𝑀𝑉 )est/𝑁(< 𝑀𝑉 )true = 0.9+0.6
+−0.4. Thus, we estimate that our
+method is able to recover the true 𝑁(< 𝑀𝑉 ) within 50% for most
+𝑀𝑉 bins.
+5 DISCUSSION
+5.1 Caveats and potential improvements
+Probably the most significant caveat in our modeling approach is the
+assumption of Gaussian MDFs. We know that galaxies are expected
+to have metallicity distributions that are not Gaussian (Revaz et al.
+2009; Kirby et al. 2011, 2013). The details of non-Gaussianity heav-
+ily depend on the star formation history, the timescale and intensity
+of gas inflows (Lanfranchi & Matteucci 2004; Romano & Starken-
+burg 2013), and likely other processes. The non-Gaussianity is likely
+a bigger problem for more luminous systems, as they have more ex-
+tended star-formation histories compared to faint systems, which, in
+some cases, are consistent with a single burst of star formation be-
+fore reionization (Weisz et al. 2014). What is the possible systematic
+effect of neglecting the non-Gaussianity? Assuming that the non-
+Gaussianity is not caused by accreted systems, but is intrinsic, that
+would lead us to overestimate the number of accreted fainter systems.
+Thus our constraints would be upper limits on the number of accreted
+events. However, the Gaussian assumption is something that can be
+potentially fixed in our formalism. For example it could be done by
+assuming parametric MDF families from Kirby et al. (2011), where
+one would need to assume some dependence of the MDF parameters
+on galaxy luminosity.
+Another key assumption is that all of the accreted stars are com-
+ing from dwarf galaxies. However, it is likely that some fraction of
+stars (at least in the MW) are coming from disrupted globular clus-
+ters. If trends seen in more massive galaxies extend to faint dwarfs
+(Forbes et al. 2018; Huang & Koposov 2021; Eadie et al. 2022), we
+may expect that 0.1-1% of stars coming from disrupted GCs. The
+metallicity distribution of clusters is poorly understood, and since
+individual GCs have extremely narrow MDFs it is unclear if there is
+a solution to take GCs into account in our model.
+5.2 Future prospects
+The MDF modeling procedure we have outlined in this work has
+compelling potential when applied to future spectroscopic datasets.
+In particular, the availability of much larger numbers of stars with
+metallicity measurements will allow us to probe to lower dwarf mass
+scales, and potentially constrain the number of destroyed ultra-faint
+dwarfs. These latter measurements would not only inform us about
+the low mass accretion history of galaxies but could also be used to
+constrain small-scale galaxy formation and the nature of dark matter
+(Deason et al. 2022).
+Here, we use toy models to estimate the sample sizes needed to
+probe down to the ultra-faint mass scale (𝑀𝑉 ≳ −8). Note here we
+focus on the ideal case and ignore the potential caveats discussed
+in the previous sub-section and elsewhere. We generate Gaussian
+MDFs that follow the Kirby et al. (2011) mass-metallicity relation,
+with varying sample sizes. We consider two example cases, one
+similar to a classical dwarf galaxy (𝑀𝑉 = −13.5), and another akin
+to a Galactic stellar halo with one main progenitor (𝑀𝑉 = −17.5).
+For each case, we generate the central MDFs with no lower mass
+progenitors, or with an additional 𝑁 = 10−50 low luminosity systems
+(𝑀𝑉 = −7.5). The results of this exercise are shown in Figs. 10 and
+11.
+It is immediately clear that as the sample sizes increase, we can
+probe to lower mass scales. However, to probe down to the ultra-
+MNRAS 000, 1–13 (2023)
+
+Destroyed dwarfs with the MDF
+11
+0
+5
+10
+100
+101
+102
+N(merged < MV)
+N =500
+0
+5
+10
+100
+101
+102
+N =1000
+0
+5
+10
+100
+101
+102
+N =5000
+0
+5
+10
+100
+101
+102
+N =10000
+0
+5
+10
+100
+101
+102
+N(merged < MV)
+0
+5
+10
+MV
+100
+101
+102
+0
+5
+10
+MV
+100
+101
+102
+0
+5
+10
+MV
+100
+101
+102
+Figure 10. Testing the method on dwarf galaxies with toy fake data. Here, dwarfs are generated with Gaussian MDFs following the adopted 𝑧 = 0 mass-metallicity
+relation. In the top row, there are no merger events (just the MDF of the central galaxy, 𝑀𝑉 = −13.5). In the bottom row, 𝑁 = 10 low mass (𝑀𝑉 = −7.5)
+systems are included. The size of the samples generated increases with each column. The solid black line shows the median, and the shaded blue(orange) regions
+the 16-84(1-99) percentiles. The vertical red dashed line indicates the 𝑀𝑉 of the central galaxy, and the green dashed line shows the 𝑀𝑉 of the accreted system
+(if included). The true number of lower-mass systems is shown with the solid horizontal green line.
+0
+5
+10
+15
+20
+100
+101
+102
+N(merged < MV)
+N =10000
+0
+5
+10
+15
+20
+100
+101
+102
+N =50000
+0
+5
+10
+15
+20
+100
+101
+102
+N =100000
+0
+5
+10
+15
+20
+100
+101
+102
+N =500000
+0
+5
+10
+15
+20
+MV
+100
+101
+102
+N(merged < MV)
+0
+5
+10
+15
+20
+MV
+100
+101
+102
+0
+5
+10
+15
+20
+MV
+100
+101
+102
+0
+5
+10
+15
+20
+MV
+100
+101
+102
+Figure 11. Same as Fig. 10 but for MW haloes. Here, one massive progenitor is generated (𝑀𝑉 = −17.5) with no other progenitors (top panel), and with
+𝑁 = 50 additional low mass progenitors (bottom panel).
+MNRAS 000, 1–13 (2023)
+
+12
+Deason, Koposov et al.
+faint regime requires significant sample sizes that are not currently
+available. For example, for a typical classical dwarf 𝑁 ≳ 5000 stars
+are needed to unambiguously detect low mass progenitors. On the
+other hand, for Galactic stellar haloes the sample sizes likely need to
+exceed 𝑁 ≳ 105. Although these numbers are larger than the sample
+sizes currently available, they are achievable with upcoming spectro-
+scopic surveys. Indeed, the large field-of-view and copious number
+of fibres available in the DESI, WEAVE, and 4MOST instruments,
+make them ideal tools for this task. Dedicated programs focusing
+on classical dwarf satellite galaxies could yield thousands of mem-
+ber stars with spectroscopic measurements. Furthermore, the MW
+surveys planned with these facilities are predicted to obtain measure-
+ments for 𝑁 ∼ 106 halo stars between 10 − 30 kpc (e.g. Cooper et al.
+2022). These survey data will not only provide significant numbers
+of dwarf members and halo stars with metallicity measurements, but
+will also provide more homogeneous sampling, and well-defined se-
+lection functions. This latter point is a particular downside of the
+current implementation in this work, which relies on a combination
+of data samples with ill-defined selection functions. In summary, the
+method we propose here is poised to exploit upcoming datasets to
+robustly quantify the accreted populations of stars in the MW and
+their dwarf galaxies.
+6 CONCLUSIONS
+We have introduced a new statistical method to model the MDF of
+a stellar population as an ensemble of individual components. These
+components follow the galaxy mass-metallicity relation and are as-
+sumed to be Gaussian distributed around their mean values (with a
+mass-dependent spread). We apply the method to observations of the
+MW halo and classical dwarf satellites, and we also test the procedure
+on cosmological hydrodynamical simulations of MW-mass haloes.
+Our main conclusions are summarised as follows:
+• Most samples of stars associated with MW dwarf satellites are
+too small to robustly probe lower mass accretion events. However, we
+do not find any evidence for significant mergers, and can indeed in
+some cases (e.g. Fornax, Leo I), rule out accreted components more
+massive than 𝑀𝑉 ,host + 5 (or 𝐿host/100).
+• We constructed a sample of MW halo stars within 𝑟 < 10 kpc
+using several spectroscopic surveys and Gaia data. By adopting the
+mass-metallicity relation applicable to surviving dwarf galaxies we
+find that one massive progenitor is favoured with 𝐿 ∼ 108.5𝐿⊙, and
+there are several hundred (𝑁 ∼ 400) progenitors in total down to
+𝑀𝑉 < −10.
+• We also consider a mass-metallicity relation more appropriate
+for destroyed dwarf galaxies for the MW stellar halo, as suggested by
+Naidu et al. (2022). Here, 𝑁 = 3 massive progenitors are favoured,
+but the total number of progenitors down to 𝑀𝑉 < −10 is similar
+to the fiducial case. By placing a stringent constraint on the total
+halo luminosity (𝐿tot = 109𝑀⊙), the two different mass-metallicity
+relations give more similar results for massive progenitors, but the
+total number of progenitors differs more significantly (by a factor of
+3).
+• We find that the total halo luminosity in our model, and the
+adopted mass-metallicity relation, are both important assumptions.
+The former can be constrained by other means (e.g. Deason et al.
+2019; Mackereth & Bovy 2020), and more work needs to be done to
+understand the redshift evolution of the mass-metallicity relation.
+• Our modeling procedure is applied to the hydrodynamic cos-
+mological Auriga simulations, a suite of 𝑁 ∼ 30 MW-mass haloes.
+Here, many of our assumptions (e.g. phase-mixed material, Gaus-
+sian MDFs) are unlikely to hold, so this provides a strong test for
+our method. We find that, in many cases, our procedure works well,
+and most failures come from scatter in the mass-metallicity relation
+and/or recent accretion events not fully occupying the phase-space
+we are probing. In general, we find that we can recover the true lumi-
+nosity function (𝑁(< 𝑀𝑉 )) of destroyed dwarfs to within 50% for
+most 𝑀𝑉 bins.
+• Finally, we consider how the increase in sample sizes from
+future spectroscopic surveys can allow us to probe down to the ultra-
+faint dwarf mass scale (𝑀𝑉 > −10). We find that MW stellar halo
+samples with 𝑁 ∼ 106 tracers will allow us to probe down to 𝑀𝑉 >
+−10; encouragingly, this should be feasible with upcoming surveys
+such as DESI and WEAVE. Moreover, with sample sizes exceeding
+𝑁 ∼ 5000 we should be able to probe the lower mass accretion events
+associated with classical dwarf satellites in the MW. Our ability to
+probe down to these puny stellar systems will enable us to address
+fundamental questions about galaxy formation at the lowest mass
+scales and, potentially, the nature of dark matter.
+We have shown that using only the MDF of an (accreted) stellar
+population, the mass-spectrum of its progenitors can be uncovered.
+This is encouraging for the upcoming generation of spectroscopic sur-
+veys of the MW. However, a possible extension of this work would
+be to combine the MDF modeling with phase-space data and/or ad-
+ditional chemical dimensions (see e.g. Cunningham et al. 2022).
+The addition of dynamical information could provide tighter con-
+straints on the luminosity function of destroyed dwarfs. In particular,
+where the MDF modeling is weakest, i.e. when the stellar material
+is un-mixed in phase-space, is likely where the dynamical data is the
+most informative. Moving forward, modeling in the chemodynami-
+cal space is the next logical step, and, importantly, we will have the
+data to do this. Thus, it is clear that future datasets combined with
+modeling methods such as that presented here will provide all the
+tools needed to finally quantify the accretion history of the Galaxy
+and its satellite population.
+ACKNOWLEDGEMENTS
+AD is supported by a Royal Society University Research Fellow-
+ship. AD acknowledges support from the Leverhulme Trust and the
+Science and Technology Facilities Council (STFC) [grant numbers
+ST/P000541/1, ST/T000244/1]. AF is supported by a UKRI Future
+Leaders Fellowship (grant no MR/T042362/1). RG acknowledges
+financial support from the Spanish Ministry of Science and Innova-
+tion (MICINN) through the Spanish State Research Agency, under
+the Severo Ochoa Program 2020-2023 (CEX2019-000920-S).
+This work used the DiRAC@Durham facility managed by the In-
+stitute for Computational Cosmology on behalf of the STFC DiRAC
+HPC Facility (www.dirac.ac.uk). The equipment was funded
+by BEIS capital funding via STFC capital grants ST/K00042X/1,
+ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham Univer-
+sity and STFC operations grant ST/R000832/1. DiRAC is part of the
+National e-Infrastructure.
+For the purpose of open access, the author has applied a Cre-
+ative Commons Attribution (CC BY) licence to any Author Accepted
+Manuscript version arising from this submission.
+AD thanks Ethan Nadler for providing the completeness-corrected
+estimates of the MW dwarf satellite luminosity function.
+MNRAS 000, 1–13 (2023)
+
+Destroyed dwarfs with the MDF
+13
+DATA AVAILABILITY
+The data analysed in this article can be made available upon reason-
+able request to the corresponding authors.
+The code used to perform the MDF modeling is available on
+Github5
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+Including the effect of depth-uniform ambient currents on waves in a
+non-hydrostatic wave-flow model
+Dirk P. Rijnsdorpa,∗, Arnold van Rooijenb, Ad Reniersa, Marion Tissiera, Floris de Wita,c, Marcel Zijlemaa
+aEnvironmental Fluid Mechanics section, Faculty of Civil Engineering and Geosciences, Delft University of Technology,
+Netherlands
+bOceans Graduate School & UWA Oceans Institute, The University of Western Australia, Australia
+cSvasek Hydraulics, The Netherlands
+Abstract
+Currents can affect the evolution of waves in nearshore regions through altering their wavenumber and
+amplitude. Including the effect of ambient currents (e.g., tidal and wind-driven) on waves in phase-resolving
+wave models is not straightforward as it requires appropriate boundary conditions in combination with a
+large domain size and long simulation duration. In this paper, we extended the non-hydrostatic wave-flow
+model SWASH with additional terms that account for the influence of a depth-uniform ambient current on
+the wave dynamics, in which the current field can be taken from an external source (e.g., from observations
+or a circulation model). We verified the model ability by comparing predictions to results from linear theory,
+laboratory experiments and a spectral wave model that accounts for wave interference effects. With this
+extension, the model was able to account for current-induced changes to the wave field (i.e., changes to the
+wave amplitude, length and direction) due to following and opposing currents, and two classical examples of
+sheared currents (a jet-like current and vortex ring). Furthermore, the model captured the wave dynamics
+in the presence of strong opposing currents. This includes reflections of relatively small amplitude waves at
+the theoretical blocking point, and transmission of breaking waves beyond the theoretical blocking point for
+larger wave amplitudes. The proposed model extension allows phase-resolving models to more accurately
+and efficiently simulate the wave dynamics in coastal regions with tidal and/or wind-driven flows.
+Keywords:
+wave-current interactions, non-hydrostatic, SWASH.
+1. Introduction
+Complex coastal regions such as estuaries and tidal inlets often feature the joint occurrence of surface
+gravity waves (e.g., swell and wind seas) and currents (e.g., riverine, tidal, and wind-driven flows). These
+processes typically occur at different spatial and temporal length scales.
+Currents generally experience
+∗Corresponding author.
+Email address: d.p.rijnsdorp@tudelft.nl (Dirk P. Rijnsdorp)
+Preprint submitted to Elsevier
+January 5, 2023
+arXiv:2301.01725v1  [physics.flu-dyn]  4 Jan 2023
+
+variations at hour to day timescales and over O(km) length scales. To the contrary, waves have periods of
+several seconds and length scales of O(10 − 100 m).
+Waves propagating over spatially varying currents conserve wave action (e.g., Bretherton and Garret,
+1968; Mei et al., 2005) but experience a change in their wavelength associated with the Doppler’ shift (e.g.,
+Peregrine, 1976; Holthuijsen, 2007). As a result, the wave celerity and group velocity change, resulting in
+changes in wave amplitude and wave direction (current-induced shoaling and refraction). In strong currents
+that oppose the direction of wave propagation, the group velocity cg approaches zero, resulting in significant
+increases of the wave height and wave-blocking when cg = 0 (e.g., Chawla and Kirby, 2002). Furthermore,
+waves steepen in opposing currents which may trigger wave breaking resulting in additional dissipation of
+wave energy (e.g., Chawla and Kirby, 2002). Current-induced changes in the wave shape can in turn impact
+the magnitude of wave-driven sediment transport (e.g., Roelvink and Stive, 1989; Hoefel and Elgar, 2003).
+Including for the current effects on waves is thus important when predicting sediment transport and the
+resulting morphological changes in coastal regions.
+To date, modelling of combined wave-current actions in coastal regions has generally relied on the
+coupling of phase-averaged wave models and circulation models (e.g., Lesser et al., 2004; Roelvink et al.,
+2009; Uchiyama et al., 2010; Kumar et al., 2012; Dodet et al., 2013; Olabarrieta et al., 2014) through either
+the radiation stress (e.g., Longuet-Higgins and Stewart, 1962, 1964) or vortex force formalism (e.g., Craik
+and Leibovich, 1976; McWilliams et al., 2004). Such coupled models have been successfully adopted to
+simulate the hydrodynamics in a variety of nearshore regions, ranging from sandy beaches (e.g., Orzech
+et al., 2011; Hansen et al., 2015; Luijendijk et al., 2017; Rafati et al., 2021) to tidal inlets and rivers where
+strong ambient currents can occur (e.g., Dodet et al., 2013; Chen et al., 2015; Nienhuis et al., 2016; Hopkins
+et al., 2018). However, such a coupled approach relies on spectral wave models that do not intrinsically
+account for phase-dependent (e.g., wave-interference and diffraction) and nonlinear wave processes (e.g.,
+triad interactions and wave breaking) but rely on parametrizations thereof.
+As an alternative to phase-averaged wave models, phase-resolving wave models have been developed
+to simulate the nearshore evolution of waves in the presence of ambient currents. Linear phase-resolving
+wave models based on the mild-slope equations have been shown to capture changes to the wave kinematics
+associated with the Doppler shift (e.g., Booij, 1981; Kirby and Dalrymple, 1986). This has allowed such
+models to capture the effect of prescribed ambient currents on the nearshore wave evolution (e.g., Chen
+et al., 2005; Touboul et al., 2016). Models based on the mild-slope equations generally rely on assumptions
+of linear wave theory, although they can be extended to account for higher order wave effects (e.g., Kaihatu
+and Kirby, 1995). Furthermore, they do not inherently account for wave-induced currents but require a
+coupling to a circulation model to capture such effects.
+Alternatively, weakly to fully nonlinear phase-resolving wave-flow models based on Boussinesq-type for-
+mulations (e.g., Peregrine, 1967; Madsen et al., 1991; Nwogu, 1993; Kirby, 2016) or the non-hydrostatic
+2
+
+approach (e.g., Zijlema et al., 2011; Ma et al., 2012; Wei and Jia, 2014) can be used to simulate waves and
+wave-induced currents in coastal regions (e.g., Chen et al., 1999; Feddersen et al., 2011; Rijnsdorp et al.,
+2015; Baker et al., 2021). Such models intrinsically account for phase-dependent wave effects, nonlinear
+wave interactions, and the generation of wave-induced currents (e.g., longshore currents and rip currents).
+However, directly including tidal and/or wind-driven currents in such models is not straightforward due to
+the range of spatial and temporal scales required. For example, including tidal currents in a phase-resolving
+model would typically require a significantly larger computational time to allow for spin-up of the tidal flow
+and a larger domain with appropriate boundary conditions to allow for the propagation of the tidal wave
+in and out of the domain. Due to the excessive computational costs of such a model setup, this presently
+inhibits a direct inclusion of such currents in phase-resolving wave-flow models.
+Several efforts have been made to account for the interactions between waves and a prescribed ambient
+current in nonlinear phase-resolving models based on the Boussinesq or non-hydrostatic approach. Most
+efforts focused on extending Boussinesq-type formulations to account for interactions between waves and an
+ambient current (e.g., Son and Lynett, 2014; Yang and Liu, 2020, 2022). Efforts to extend non-hydrostatic
+models have been limited to de Wit et al. (2017), who added a spatially homogeneous pressure term in the
+alongshore momentum equation of a non-hydrostatic model to simulate the nearshore wave dynamics in the
+presence of alongshore tidal flows at a sandy beach. Despite this progress on including the effect of ambient
+currents on waves in nonlinear phase-resolving wave-flow models, their application at complex coastal sites
+have not yet been able to account for the effect of spatially varying current fields from tides and/or wind on
+the wave dynamics (e.g., Risandi et al., 2020; Rijnsdorp et al., 2021; Baker et al., 2021).
+In this work, we extend the non-hydrostatic wave model SWASH (Zijlema et al., 2011) to account
+for the effect of a prescribed depth-uniform ambient current on the wave dynamics, in which the current
+field can be obtained from an external source (e.g., observations or a circulation model). By introducing a
+separation of scales and assuming vertically uniform mean flows, we derive additional terms to the governing
+equations that account for the effect of a spatially varying depth-uniform current on the waves (Section 2).
+Comparisons with linear wave theory, a spectral wave model and flume experiments show that the proposed
+model is able to account for changes in the wave height and wavelength due to an ambient currents (Section
+3-4). In Section 5-6, we conclude that the proposed extension allows non-hydrostatic models to account for
+the effect of ambient currents on waves.
+2. Numerical Methodology
+2.1. Governing equations
+The governing equations of the model are the Reynolds-Averaged Navier-Stokes (RANS) equations for
+an incompressible fluid that is bounded by the bottom d(x, y) and a free-surface ζ(x, y, t), where (x, y, z)
+3
+
+are the Cartesian coordinates and t is time,
+∂u
+∂x + ∂v
+∂y + ∂w
+∂z = 0,
+(1)
+∂u
+∂t + u∂u
+∂x + v ∂u
+∂y + w∂u
+∂z + g ∂ζ
+∂x + ∂pnh
+∂x
+= ∂τxx
+∂x + ∂τxy
+∂y
++ ∂τxz
+∂z ,
+(2)
+∂v
+∂t + u∂v
+∂x + v ∂v
+∂y + w∂v
+∂z + g ∂ζ
+∂y + ∂pnh
+∂y
+= ∂τyx
+∂x + ∂τyy
+∂y + ∂τyz
+∂z ,
+(3)
+∂w
+∂t + u∂w
+∂x + v ∂w
+∂y + w∂w
+∂z + ∂pnh
+∂z
+= ∂τzx
+∂x + ∂τzy
+∂y + ∂τzz
+∂z .
+(4)
+In this set of equations, pnh is the non-hydrostatic pressure, (u, v, w) are the velocity components in (x, y, z)
+direction, respectively, τ represents the turbulent stress (estimated using an eddy viscosity approximation).
+The kinematic boundary conditions at the bottom and the free-surface follow from the assumption that the
+vertical boundaries of the fluid are single valued functions of the horizontal coordinates,
+wz=ζ = ∂ζ
+∂t + u ∂ζ
+∂x + v ∂ζ
+∂y ,
+(5)
+wz=−d = −u∂d
+∂x − v ∂d
+∂y .
+(6)
+Integrating the local continuity equation over the water column results in a global continuity equation that
+describes the temporal evolution of the free-surface,
+∂ζ
+∂t + ∂
+∂x
+ζ
+�
+−d
+udz + ∂
+∂y
+ζ
+�
+−d
+vdz = 0.
+(7)
+Assuming a constant atmospheric pressure (equal to zero for convenience) and neglecting viscous stresses
+at the free-surface, the non-hydrostatic pressure is set to zero at the free-surface (e.g., Stelling and Zijlema,
+2003). At the bottom, the tangential stress is prescribed based on the quadratic friction law (in the case
+of a coarse vertical resolution) or the law of the wall (in the case of a fine vertical resolution). Turbulent
+stresses are modelled using the eddy-viscosity model and the k-ϵ turbulence closure model (See Rijnsdorp
+et al., 2017, for more details). Combined with boundary conditions at all horizontal edges of the physical
+domain, the above set of equations forms the basis of the SWASH model.
+2.2. Including the effect of currents on waves
+In this work, we set out to decouple the modelling of the surface waves and the currents that are
+slowly-varying with respect to the wave timescale (e.g., tidal currents and wind-driven currents). With this
+approach, we aim to account for the current effect on waves through prescribing an ambient current field
+from an other model (e.g., a circulation model) that alters the wave dynamics solved by the RANS equations.
+4
+
+To this end, we separate the horizontal flow variables and surface elevation as,
+u(x, y, z, t) = U(x, y) + u′(x, y, z, t),
+(8)
+v(x, y, z, t) = V (x, y) + v′(x, y, z, t),
+(9)
+ζ(x, y, t) = η(x, y) + ζ′(x, y, t).
+(10)
+In these equations, [...]′ denotes variables which we associate with wave-related motions and wave-induced
+currents. Capital letters (U and V ) represent vertically uniform horizontal flow velocities and η a mean
+water level, which both vary over a timescale much larger than the wave motions and are considered to be
+constant over the wave-timescale. Substituting this separation of variables into the governing equations and
+neglecting the viscous contributions and tangential stress at the bottom yields,
+∂U + u′
+∂x
++ ∂V + v′
+∂y
++ ∂w
+∂z = 0,
+(11)
+∂U + u′
+∂t
++ (U + u′)∂U + u′
+∂x
++ (V + v′)∂U + u′
+∂y
++ w∂U + u′
+∂z
++ g ∂η + ζ′
+∂x
++ ∂pnh
+∂x
+= 0,
+(12)
+∂V + v′
+∂t
++ (U + u′)∂V + v′
+∂x
++ (V + v′)∂V + v′
+∂y
++ w∂V + v′
+∂z
++ g ∂η + ζ′
+∂y
++ ∂pnh
+∂y
+= 0,
+(13)
+∂w
+∂t + (U + u′)∂w
+∂x + (V + v′)∂w
+∂y + w∂w
+∂z + ∂pnh
+∂z
+= 0,
+(14)
+∂η + ζ′
+∂t
++ ∂
+∂x
+η+ζ′
+�
+−d
+(U + u′)dz + ∂
+∂y
+η+ζ′
+�
+−d
+(V + v′)dz = 0.
+(15)
+By taking the temporal average over the wave-motion scales and integrating the horizontal momentum
+equations over the vertical we obtain the following depth-averaged mean flow equations,
+∂U
+∂x + ∂V
+∂y = 0,
+(16)
+∂U
+∂t + U ∂U
+∂x + V ∂U
+∂y + g ∂η
+∂x = −
+η
+�
+−d
+(u′ ∂u′
+∂x + v′ ∂u′
+∂y )dz,
+(17)
+∂V
+∂t + U ∂V
+∂x + V ∂V
+∂y + g ∂η
+∂y = −
+η
+�
+−d
+(u′ ∂v′
+∂x + v′ ∂v′
+∂y )dz,
+(18)
+∂η
+∂t + ∂ (d + η) U
+∂x
++ ∂ (d + η) V
+∂y
+= − ∂
+∂x
+η+ζ′
+�
+−d
+u′dz − ∂
+∂y
+η+ζ′
+�
+−d
+v′dz.
+(19)
+In these equations, we can recognise the contribution to the radiation stress gradient from the orbital
+velocities (e.g., u′ ∂u′
+∂x ) and contributions in the global continuity equation that are related to stokes drift
+(i.e., the part of the integral above the wave trough in the right-hand-side of Eq. 19). In the following we
+assume that waves do not influence the ambient currents, and neglect these contributions in the mean flow
+equations.
+5
+
+Subsequently, we derive a new set of wave equations by subtracting the mean equations (16)-(19) from
+the instantaneous equations (11)-(15),
+∂u′
+∂x + ∂v′
+∂y + ∂w
+∂z = 0,
+(20)
+∂u′
+∂t + u′ ∂u′
+∂x + v′ ∂u′
+∂y + w∂u′
+∂z + g ∂ζ′
+∂x + ∂pnh
+∂x
+= −(U ∂u′
+∂x + u′ ∂U
+∂x + V ∂u′
+∂y + v′ ∂U
+∂y ),
+(21)
+∂v′
+∂t + u′ ∂v′
+∂x + v′ ∂v′
+∂y + w∂v′
+∂z + g ∂ζ′
+∂y + ∂pnh
+∂y
+= −(U ∂v′
+∂x + u′ ∂V
+∂x + V ∂v′
+∂y + v′ ∂V
+∂y ),
+(22)
+∂w
+∂t + u′ ∂w
+∂x + v′ ∂w
+∂y + w∂w
+∂z + ∂pnh
+∂z
+= −(U ∂w
+∂x + V ∂w
+∂y ),
+(23)
+∂ζ′
+∂t + ∂
+∂x
+η+ζ′
+�
+−d
+u′dz + ∂
+∂y
+η+ζ′
+�
+−d
+v′dz = −∂ζ′U
+���x
+− ∂ζ′V
+∂y .
+(24)
+In the above set of equations, we can recognise the original set of equations (when dropping the prime
+superscripts) including several additional terms (on the right-hand-side) that account for the influence of a
+depth-uniform ambient current on the wave motions. We note that the influence of changes in the mean
+water level associated with the ambient current in the global continuity equation (i.e., the integral up to η+ζ′
+in Eq. (24)) can be straightforwardly incorporated by incorporating η in the still water depth (d = d + η).
+2.3. Numerical implementation
+In the numerical implementation of the governing set of equations, the continuous description of time
+and horizontal dimensions are replaced by discrete approximations. In SWASH, the equations are discretised
+on regular or curvilinear grid for the horizontal dimensions and a terrain-following layering system for the
+vertical coordinate. A staggered grid arrangement is used to position the flow variables on the grid. Further
+details regarding the numerical implementation of the original set of equations can be found in several
+previous papers (e.g., Stelling and Zijlema, 2003; Zijlema and Stelling, 2005; Zijlema et al., 2011), and will
+not be detailed here.
+𝑖!
+𝑖 + 1!
+𝑖"
+𝑖
+𝑖 + 1
+𝜁#, 𝑈
+𝑢#
+𝑖 − 1"
+𝑖 − 1
+Figure 1: Illustration of the arrangement of the ambient velocity U and wave-related variables [ζ, u] on the computational grid.
+6
+
+The flow velocities [U, V ] from the ambient current are positioned on the grid at the same location as
+the free-surface variable of the original set of equations ζ′ (i.e., at horizontal cell centres, see Fig. 1). Linear
+interpolation is used to define the ambient current on the SWASH grid in the case that the ambient current is
+provided on a coarser grid. The numerical implementation of the additional terms is – where possible – based
+on the existing implementation of the advective terms. The terms in the horizontal momentum equations
+are discretised using the MacCormack predictor-corrector technique (MacCormack, 1969) combined with
+flux limiters (See Zijlema et al., 2011, for more details). We use a flux limited first-order Euler scheme to
+discretise the terms in the vertical momentum equation. Finally, the terms in the global continuity equation
+are discretised using central differences and the Crank-Nicholson method.
+3. Linear properties of the model equations
+We analysed the linear properties of the model equations by deriving the numerical linear dispersion
+relationship (see Appendix
+C) to verify that the model captures the effect of currents on waves.
+The
+numerical dispersion relationship derived from the extended model equations (20)-(24) provides a polynomial
+relationship fN between the absolute wave frequency ω (in the reference frame of a stationary observer) and
+the wavenumber k for depth d and current velocity U depending on the number of layers N,
+ω = fN(k, d, U, N).
+(25)
+We compared linear wave properties based on this numerical dispersion relationship with the Doppler
+shifted dispersion relationship from linear theory (e.g., Holthuijsen, 2007),
+ω − kU = σ =
+�
+gk tanh kd,
+(26)
+in which σ is the intrinsic angular frequency (in the reference frame of an observer that is moving with the
+current). Based on this numerical and linear dispersion relationship, several wave properties can be derived.
+The relative group velocity (in a reference frame moving with the current) is given by cg,r = ∂σ
+∂k , and the
+absolute group velocity (in the reference frame of a fixed observer) is cg = cg,r + U.
+Furthermore, we also compared the numerical dispersion relationship of the extended model equations
+to the Doppler shifted numerical relationship of the original model equations,
+ω − kU = σ = fN,U=0(k, d, N),
+(27)
+where fN,U=0 is the numerical dispersion relationship in the absence of a current (Smit et al., 2014). This
+Doppler shifted numerical dispersion relationship provides the influence of a current on waves when the
+current is simulated as part of the model equations (e.g., by means of a pump system as described in
+Appendix B). Importantly, we found that all linear properties based on Eq. (25) (the numerical dispersion
+7
+
+Figure 2: Absolute relative error in the absolute wave frequency ω (panel a-c) and relative group velocity cg,r = ∂σ
+∂k (panel
+d-f) as a function of the normalized water depth kd for U = [0, −2, −4] m/s (left to right panels, as indicated by the subplot
+titles) based on the numerical dispersion relationship of the N layer system. Results are shown for N = [1, 2, 4] layers. The
+vertical dashed lines indicate where blocking occurs, with the colors indicating the number of layers of the numerical dispersion
+relationship. The vertical black line indicates where blocking occurs according to the linear dispersion relionship.
+relationship of the extended model equations) and Eq.
+(27) (the Doppler shifted numerical dispersion
+relationship) were identical. This confirms that the linear effect of current on waves can be captured by
+including additional terms in the model equations.
+In the remainder of this section, we therefore only
+compared linear wave properties based on the numerical dispersion relationship of the extended model
+equations (25) and the Doppler shifted dispersion relationship based on linear theory (26).
+Assuming that the horizontal scales are sufficiently resolved, the dispersive property of the model depends
+on the number of vertical layers (Fig. 2b). Introducing a current does not significantly affect the error in
+wave dispersion, as ∆ω under currents is comparable to the case with U = 0 m/s (compare Fig. 2a,c,d with
+Fig. 2b). Discrepancies in cg,r similarly depend on the number of layers and are not significantly affected
+by introducing a current (Fig. 2e-f). When introducing an opposing current (U < 0), no wave solution
+exists beyond a certain kd as indicated by the vertical lines in Fig. 2c-d and 2g-h. Here, waves are blocked
+as cg = 0. The kd at which blocking occurs is sensitive to the number of layers, and is in better agreement
+with linear theory when a larger number of layers is used. This is further illustrated in Fig. 3, which shows
+the current velocity at which blocking occurs (Ub) as a function of kd based on the linear and numerical
+dispersion relationship. With coarse vertical resolutions, waves are blocked on weaker opposing currents
+compared to linear theory. Increasing the number of vertical layers improves Ub, with errors in Ub < 10%
+for kd < [2, 7, 30] in the case of N = [1, 2, 4] layers, respectively. These findings show that the number of
+8
+
+U= 4m/s
+U=0m/s
+U=-2m/s
+U=-4m/s
+a)
+b)
+()
+(p
+4
+白4
+32
+0
+100
+100
+101
+10°
+100
+15
+f)
+h)
+e
+g)
+10
+4
+△5
+0
+10°
+100
+101
+100
+101
+10°101
+kd (rad)
+kd (rad)
+kd (rad)
+kd (rad)Figure 3: Panel a: Blocking current velocity Ub (panel a) as a function of kd based on the linear dispersion relationship (black
+line) and the numerical dispersion relationship for N = [1, 2, 4] (blue, red, and yellow line, respectively). Panel b: Absolute
+relative error in Ub from the numerical dispersion relationship for N = [1, 2, 4] relative to the linear dispersion relationship as
+a function of kd.
+layers controls the accuracy with which the model recovers the linear wave properties in the presence of a
+current.
+4. Test Cases
+4.1. Linear waves on opposing and following currents
+To verify the numerical implementation of the additional terms in the governing equations, we compared
+model predictions of changes in the wavelength and wave amplitude due to a gradient in the current velocity
+to linear wave theory. As illustrated by the linear properties of the equations (Sec. 3), waves that travel over
+a current gradient experience a change in their kinematics. The wavelength decreases and the amplitude
+increases for waves on an opposing current and vice-versa on a following current. In this section, we verify
+if the developed model captures these changes to the wave field for linear waves that interact with opposing
+and following currents. We considered monochromatic waves with a height of H = 0.01 m and wave periods
+T = [5, 10, 15] s in water of constant depth d = 10 m (corresponding to kd = [1.7, 0.7, 0.4] in the absence
+of a current). A range of current velocities was simulated with U ranging from -6 to 4 m/s with 0.25 m/s
+increments.
+We compared the influence of the current on the wave height and the wavelength with linear wave theory.
+The change in wavelength and group velocity follows from the linear dispersion relationship (26). The change
+in wave height follows from the conservation of wave action,
+∂
+∂x
+cgE
+σ
+= 0,
+(28)
+with the wave energy density E of a monochromatic wave (E = 1/8H2) and the absolute group velocity cg
+taken from linear theory (with cg = cg,r +U, and cg,r = ∂σ
+∂k obtained from the linear dispersion relationship).
+9
+
+0
+20
+a
+(s /u)
+-5
+△Ub
+10
+°-10
+-15
+0
+10-1
+100
+101
+100
+101
+10-1
+kd (rad)
+kd4.1.1. Model set-up
+To allow for the current effect on the waves to develop, the model setup included a transition region with
+a width of several wavelengths to gradually transition from no current to the respective maximum current
+velocity. The transition region had a width of 10L0, and the region with maximum flow had a width of 10L0
+(with L0 the wavelength in the absence of a current). These widths were found to be sufficient to allow
+for a gradual change in the wave dynamics, and provided a sufficiently large domain to determine the wave
+parameters in the presence of the current. Waves were generated at the left boundary with a wavemaker
+based on linear wave theory which was positioned 3L0 away from the transition region. A sponge layer with
+a width of 5L0 was positioned in front of the right boundary to absorb the waves and prevent any wave
+reflections. The sponge layer was positioned at a distance of 3L0 from the transition region. The model
+was set-up with two layers in the vertical. The horizontal resolution and time-step were selected based on a
+sensitivity study (Appendix A): the horizontal grid resolution was set at ∆x = L0/100 and the time-step
+was set at ∆t = T/1000 (with T the incident wave period). The surface elevation ζ was outputted at all
+computational grid points for a duration of 5 wave periods after a spin-up time that ensured statistically
+stationary results inside the numerical domain.
+We used zero-crossing analysis in the maximum current region to determine the wavelength in presence
+of a current. First, the surface elevation ζ was interpolated to a fine horizontal grid in the current region to
+allow for an accurate estimation of the wavelength independent of the grid resolution. The wavelength was
+subsequently computed from the zero-crossing analysis as the average wavelength over the current region
+and the output duration. We computed the wave height in the current region as H = 2√2m0 (with the
+zeroth order moment m0 computed as the standard deviation of the surface elevation ζ). To gain insight in
+the spatial variation of H, we computed the mean, the maximum and minimum value of H in the current
+region. Results were excluded when wave-blocking occurred in the model simulation. Wave blocking was
+recognised when the wave energy at the down-wave end of the domain (behind the current region) was < 1%
+of the incident wave energy at the numerical wavemaker.
+4.1.2. Results
+To illustrate the impact of the current on the wave field, Fig. 4 shows an example of the surface elevation
+inside the model domain for three different current velocities. For these three cases, modelled changes to
+the surface elevation in opposing and following currents qualitatively agreed with the expected changes to
+the wave field. In an opposing current, the wavelength decreased and the wave height increased (Fig. 4a).
+In contrast, the wavelength increased and the wave height decreased for a following current (Fig. 4c). In
+all three illustrative cases, the wave signal at the downwave end of the flume (x > 3000 m) was identical to
+the incident wave signal (x = 0). This confirms that wave action is conserved in these simulations.
+To verify the model results quantitatively, we compared the change in the wave height and wavelength
+10
+
+Figure 4: Snapshot of the modelled surface elevation (blue line, left axis) and ambient current velocity (red line, right axis)
+in the numerical domain for three different current velocities (U = [−3, 0, 3] m/s) for a monochromatic wave with amplitude
+a = 0.01 m and period T = 10 s). The dashed black line indicates the envelope of the wave elevation, and the title of each
+panel indicates the respective current velocity.
+11
+
+opposing current (U= -3 m/s)
+1.0
+a)
+(-) 
+ 0.5 
+(s/w) n
+H/2
+0.0 
+0
+0.5
+1.0
+3
+no current
+1.0
+b)
+0.5
+(s/w) n
+H/2
+0.0
+0
+0.5
+1.0
+following current (U= 3 m/s)
+1.0
+C)
+(-) H/2
+0.5
+(s/
+0.0 
+0
+U (m/
+0.5
+1.0
+-3
+0
+500
+1000
+1500
+2000
+2500
+3000
+x (m)Figure 5: Normalized change to the wave height H (panel a) and wavelength L (panel b) as a function of the current velocity
+U for small-amplitude monochromatic waves with T = [5, 10, 15] s. The wave height and wavelength were normalized by the
+wave parameters in absence of a current (indicated by [...]0). Converged model results of simulations (with ∆t = T/1000 and
+∆x = L/100) are indicated by colored lines (see legend) and results from linear wave theory are indicated by the thick black
+line. In the left panel, the horizontal blue line with dotted markers indicates the average change to the simulated wave height
+H in the current region and the vertical line with horizontal bars indicates the maximum and minimum simulated H inside the
+current region. The dashed vertical black lines indicate the current velocity at which wave blocking occurs according to linear
+wave theory.
+inside the current region with the results from linear wave theory (Fig. 5). For all three wave periods,
+linear wave theory predicted that the wave height and wavelength varied significantly for the considered
+range of current velocities (using Eq. 28). For opposing currents, the wave height H increased and the
+wavelength L decreased, and vice versa for following currents (as was visually observed in Fig. 4). Current
+induced changes to the wave field were larger for shorter wave periods, with wave blocking occurring for
+T = [5, 10, 15] s at U ≈ [−1.92, −3.74, −4.87] m/s (indicated by the vertical black dashed line).
+SWASH captured the changes to the wave height and wavelength for the range of ambient current
+velocities and the three wave periods (Fig. 5). This included the nonlinear dependence of H and L for
+U < 0 m/s. Furthermore, the model captured blocking of waves for opposing currents that are stronger
+than the critical flow velocity of linear wave theory (indicated by the dashed black lines in Fig. 5). For all
+three wave periods, simulations with current velocities stronger than the theoretical blocking velocity showed
+a strong decay of the wave height down-wave of the blocking point (not shown). For simulations with U close
+to but just weaker than the theoretical blocking velocity, dissipation of wave energy occurred in the model
+over the current region (visible as the difference between the vertical lines with horizontal bars in Fig. 5a,
+which indicates the maximum and minimum H in the current region). In the absence of physical mechanisms
+for dissipation, this is likely related to numerical diffusion when the waves (with shorter lengths) propagate
+in the current region. For weaker U this dissipation becomes smaller and the model results were in good
+agreement with linear theory. This numerical dissipation was found to be dependent on the horizontal grid
+12
+
+2.0
+4 -
+a)
+T=5 s
+b)7
+T=10 s
+1.5
+1
+T=15 s
+1
+H/Ho
+L/Lo
+1.0
+2
+-
+1 
+0.5
+0 -
+0.0
+-6
+-4
+-2
+0
+2
+4
+-6
+-4
+2
+2
+U (m/s)
+U (m/s)resolution and time step, with improved agreement for strong U for finer spatial and temporal resolutions
+(in accordance with the results in Appendix A).
+4.2. Sheared current fields
+In coastal regions, spatially varying current fields exist (e.g., tidal currents) that can induce wave re-
+fraction and result in focal zones that give rise to wave interference patterns (e.g., Yoon and Liu, 1989;
+Akrish et al., 2020). In this section, we verify the ability of the model to capture such wave patterns using
+two classical examples of wave-current interactions: the interactions of waves with a jet-like current and a
+vortex ring. Model results were compared with the spectral wave model SWAN (Booij et al., 1999) extended
+with a quasi-coherent formulation that accounts for wave interference due to variable topography (Smit and
+Janssen, 2013; Smit et al., 2015) and currents (Akrish et al., 2020).
+4.2.1. Model set-up
+The model set-up was based on the work of Akrish et al. (2020).
+The region of interest spanned a
+domain of 4 × 4 km. Two different simulations were considered, one with a jet-shaped and the other with
+a vortex-shaped current field, positioned along the central axis of the domain. The maximum velocities for
+the simulations were 0.38 m/s and 1.0 m/s, respectively (refer to Akrish et al., 2020, for a mathemetical
+formulation of the current fields). At the wavemaker positioned along the western boundary, a Gaussian
+shaped wave-spectrum in frequency and direction was forced with Hs = 1 m, Tp = 20 s and a standard
+deviation of 0.0015 Hz in frequency space and 1.78◦ in directional space. The waves had a mean direction
+of θ0=15◦ and 0◦ (in Cartesian coordinates) for the jet and vortex current, respectively.
+In the SWAN model, the physical domain was discretised with ∆x = ∆y = 50 m. The spectral domain
+was discretised with 45 discrete frequencies that were logarithmically spaced between 0.005 and 0.085 Hz,
+and with a directional resolution of 2◦ between -90 and 90◦.
+For the SWASH model, we extended the
+domain with a 500 m wide sponge layer at the eastern side of the domain to prevent any wave reflections.
+The domain was discretised with a resolution of ∆x = 2 m and ∆y =4 m (which resulted in ≈ 100 points
+per wavelength throughout the domain). The time step was set at ∆t = 0.05 s, equalling 300 points per
+wave period and resulting in CFL ≈ 0.6.
+4.2.2. Results
+Due to changes in wavelength induced by the current, waves were refracted by the vortex ring (Fig.
+6a-c and 6g-i). This current-induced refraction resulted in considerable variations in the significant wave
+height, with ridges of larger wave heights where waves focussed and depressions of lower wave heights where
+waves diverged (Fig.
+6a-f).
+For this current field, quasi-coherent (QC) effects needed to be taken into
+account in SWAN to resolve the constructive and de-constructive wave interference that altered the wave
+field downstream of the vortex ring (e.g., Akrish et al., 2020).
+The bulk wave heights and mean wave
+13
+
+Figure 6: Changes to the significant wave height Hm0 and mean wave direction θ due to a vortex ring current field. Panels
+a-c show a spatial overview of the significant wave height (colors) and mean wave direction (black arrows), with the red arrows
+indicating the ambient current field, for SWASH (panel a), SWAN including the Quasi-Coherent (QC) formulation (panel b)
+and default SWAN (panel c). Panels d-i show the wave height (d-f) and mean wave direction (g-i) along three alongshore
+transects predicted by SWASH (black lines), SWAN QC (orange lines) and default SWAN (blue lines).
+directions predicted by the extended SWASH model were in satisfactory agreement with the results from
+the SWAN QC model throughout the domain.
+Similarly, waves refract as they propagated into the jet-like current field, resulting in a change of
+mean wave direction (Fig.
+7g-i) and in regions with increased and decreased wave heights due to con-
+vergence/divergence of wave energy (Fig. 7a-f). Similar to the vortex ring, quasi-coherent effects need to be
+incorporated in SWAN to account for the constructive and de-constructive wave interference that altered the
+wave field, although this effect was smaller compared to the vortex ring. In general, the SWASH predictions
+were in good agreement with SWAN QC. The results of this test case, and the vortex ring, illustrate that
+SWASH including the additional terms in the model equations is able to capture the effect of current-induced
+14
+
+SWASH
+SWAN QC
+SWAN
+4000
+ 1.5
+af
+bi
+3000
+ 1.0
+(w)
+2000
+1000
+ 0.5
+0.0
+0
+1000
+2000
+3000
+0
+1000
+2000
+3000
+0
+1000
+2000
+3000
+x (m)
+x (m)
+x (m)
+x=1000 m
+x=2000 m
+x=3000 m
+1.5
+(p
+el
+1.0
+0.5
+0.0
+SWASH
+SWAN QC
+15
+(6
+15
+h)
+15
+SWAN
+(6ap) 
+0
+0
+0
+15
+15
+15
+2000
+4000
+2000
+4000
+0
+2000
+4000
+y (m)
+y (m)
+y (m)Figure 7: Changes to the significant wave height Hm0 and mean wave direction θ due to a jet-like current field. Panels a-c
+show a spatial overview of the significant wave height (colors) and mean wave direction (black arrows), with the red arrows
+indicating the ambient current field, for SWASH (panel a), SWAN including the Quasi-Coherent (QC) formulation (panel b)
+and default SWAN (panel c). Panels d-i show the wave height (d-f) and mean wave direction (g-i) along three alongshore
+transects predicted by SWASH (black lines), SWAN QC (orange lines) and default SWAN (blue lines).
+refraction on the wave propagation and the resulting spatial variability in the wave field.
+4.3. Wave blocking, reflections and breaking on opposing currents
+As a final test case, we compare model predictions with the laboratory experiment of Chawla and Kirby
+(1999, 2002) that considered wave blocking on opposing currents. The flume had a length of 30 m, a width
+of 0.6 m and still water depth of 0.5 m, with a pump system to generate a recirculating current (with a
+discharge of 0.095 m3/s) and a perforated wavemaker to generate waves on the current. A spatially varying
+current was generated by means of a false wall constricting the width of the flume, with a minimal width of
+0.36 m (see black line in Fig. 8a). Blocking of waves occurred close to the start of this narrow part of the
+flume.
+15
+
+SWASH
+SWAN QC
+SWAN
+4000
+ 1.5
+a
+h)
+3000
+ 1.0
+(w) ^
+2000
+1000
+ 0.5
+0-
+0.0
+0
+10002000 3000
+0
+1000
+2000
+3000
+0
+1000
+20003000
+x (m)
+x (m)
+x (m)
+x=1000 m
+x=2000 m
+x=3000 m
+1.5
+e)
+f)
+1.0
+0.5
+0.0
+SWASH
+30
+30
+30
+SWAN OC
+(6
+h)
+i)
+SWAN
+(deg)
+15
+15
+15
++0
++0
+0
+2000
+4000
+0
+2000
+4000
+2000
+4000
+y (m)
+y (m)
+y (m)Figure 8: Overview of the numerical setup of the Chawla and Kirby (1999) flume experiment. The top panel (a) shows the
+flume width (black line, left axis) and a snapshot of the modelled free-surface elevation for test case 1 (blue line, right axis).
+The bottom panel (b) shows the modelled (red line) and measured (black markers) current velocity (in the absence of waves).
+The experiments with monochromatic waves considered a total of 23 test conditions that included 3
+different incident wave periods (T = [1.2, 1.3, 1.4] s) for a range of wave heights (H = 0.012 − 0.14 m).
+For the low amplitude and low period waves, waves reflected with negligible transmission of wave energy
+beyond the blocking point. For increasing wave heights, waves started breaking at the blocking point of
+linear theory combined with increased transmission of wave energy beyond this theoretical blocking point.
+In this paper, we considered 4 out of the 23 test cases: the largest and smallest wave height of both the
+smallest and largest wave period (see Table 1). For case R1 and R11 waves reflected at the blocking point,
+whereas waves were breaking and wave energy was transmitted beyond the theoretical blocking point for
+case B6 and B18.
+We compared model predictions with these laboratory observations for these 4 test cases. Furthermore,
+we also computed the wave height transformation based on conservation of wave action.
+∂
+∂x
+cgE
+σ
+= 0,
+(29)
+Conservation of wave action is computed based on the linear dispersion relationship (similar to Sec. 4.1) and
+also based on the nonlinear dispersion relationship from 2nd order Stokes theory (e.g., Dean and Dalrymple,
+1991). This nonlinear dispersion relationship accounts for the effect of amplitude dispersion, which was
+found to be important for these laboratory experiments (Chawla and Kirby, 2002).
+16
+
+0.6
+(w) 
+a)
+width
+0.3 
+0.03
+(w) 2
+0.0
+0.00
+EO'O-
+0.0
+b)
+(s/w) n
+0.2
+-0.4
+0.6
+15
+10
+-5
+0
+n
+10
+x (m)H (m)
+T (s)
+Ub (m/s)
+kd (U = 0 m/s)
+kd (U = −0.32 m/s)
+kd (U = Ub m/s)
+R1
+0.012
+1.2
+-0.47
+1.53
+2.36
+5.59
+B6
+0.126
+1.2
+-0.47
+1.53
+2.36
+5.59
+R11
+0.015
+1.4
+-0.55
+1.22
+1.69
+4.14
+B18
+0.141
+1.4
+-0.55
+1.22
+1.69
+4.14
+Table 1: Experimental conditions (wave height H, wave period T, theoretical blocking velocity Ub, and normalized water depth
+kd for three current velocities) of the four test cases of the Chawla and Kirby (1999) flume experiment that were considered
+in this paper. The wave height and wave period were measured at the first wave gauge inside the flume, at a distance of 4.2
+m (case 1 and 3) and 5.2 m downstream (case 2 and 4) of the start of the narrow flume section. The blocking velocity was
+computed based on the linear dispersion relationship. The normalized water depth based on linear wave theory is provided in
+the absence of the current, for U = −0.32 m/s, and at the theoretical blocking velocity Ub.
+4.3.1. Model setup
+We used a curvilinear grid with a constant streamwise resolution but varying alongshore width and
+resolution to replicate the flume in the numerical model. Based on the sensitivity study for linear waves
+(Appendix A), the horizontal grid resolution in streamwise direction was set to ensure at least 100 points per
+wavelength (in the absence of a current). This resulted in a total of 1500 cells in the streamwise direction.
+We used 3 cells in the spanwise direction to reduce computational overhead. This implies that spanwise
+effects were not included in the modelling, such as the sidewall boundary layers that were observed in the
+flume (Chawla and Kirby, 2002). To investigate the influence of the vertical resolution, simulations were
+run with [2, 4, 20] layers. The model time step was set at a value that corresponds to CFL ≈ 0.4 resulting in
+about 250-500 points per wave period, which was found to be sufficiently fine for these test conditions. Waves
+were generated based on linear wave theory at x = −15 m (in the absence of a current), with the incident
+wave height calculated from conservation of wave action (Eq. (28)) based on the measured wave height at
+the first wave gauge (located at x ≈ −5 m). A sponge layer with a width of at least three wavelengths was
+positioned at the end of the flume to prevent wave reflections.
+We conducted two sets of simulations to replicate the four test cases. In the first set, which serves as a
+benchmark for the proposed model extension, the waves and current were modelled simultaneously through
+the original set of model equations. A re-circulating current was generated through modifying the kinematic
+boundary condition at the bottom (see Appendix B). The resulting discharge that is imposed at the bottom
+replicates a pump system through which a volume of water is pumped into the domain at one end of the
+flume and is taken out at the other end of the flume. In this manner, a current was generated inside the
+numerical flume. In all simulations with this pump system, the discharge was set at Q = 0.095 m3/s based
+on Chawla and Kirby (2002). With this model set-up, the modelled depth-averaged current field was in
+good agreement with observations taken in the flume for a reference case excluding waves (Fig. 8b). In the
+17
+
+second set of simulations, we account for the current through the additional terms in the equations that
+were derived in Section 2. The ambient current velocities were obtained from the simulation with the pump
+system without waves (Fig. 8b). In the following, we refer to the simulations with the additional terms to
+model the influence of the current on waves as an Ambient Current (AC) simulation, and we refer to the
+benchmark simulations as a Pump simulation.
+Non-hydrostatic models like SWASH inherently account for the dissipation by breaking waves but require
+high vertical resolutions to capture the onset of wave breaking correctly (e.g,. Smit et al., 2013). To capture
+the onset of breaking with coarse resolutions, Smit et al. (2013) introduced the Hydrostatic Front Approxi-
+mation (HFA) that neglects the non-hydrostatic pressure locally to trigger wave breaking (i.e., switching to
+the Non-Linear Shallow Water Equations, NSLWE). However, numerical instabilities developed in all 2-layer
+simulations with HFA. We believe this is related to the normalized water depth of the waves at breaking.
+For depth-induced wave breaking in the absence of currents (for which the HFA is normally applied), the
+normalized water depth is relatively low at breaking (kd < 1), resulting in a relatively small non-hydrostatic
+pressure contribution. In the wave-current simulations of this test case, the normalized water depth is rela-
+tively large (kd > 4 near wave blocking, see Table 1). As a results, the contribution from the non-hydrostatic
+pressure is relatively large at the location of incipient wave breaking. Excluding a relatively large contri-
+bution from the non-hydrostatic pressure likely resulted in numerical instabilities and caused the model to
+crash. As a result, the HFA approach cannot be used to improve the model predictions of the 2-layer model
+in the case of breaking waves on a strong opposing current. In the following, we therefore only show results
+for 2-layer simulations excluding HFA.
+4.3.2. Results - wave reflections
+For the wave condition with the smallest wave height and wave period (case R1, Table 1), waves reflected
+at the blocking point, resulting in a nodal pattern in the wave height H and negligible transmission of wave
+energy for x > 0 m (Fig. 9a). An energy balance based on conservation of wave action (eq. (28)) provided
+a reasonable good description of the location of wave blocking. Differences between the energy balance with
+the linear dispersion relationship and 2nd order Stokes dispersion relationship were generally small except
+near the blocking location, where the blocking location is spatially shifted by approximately 0.25 m when
+accounting for amplitude dispersion.
+In Fig. 9a, we compare both results of the simulations with an Ambient Current (AC) and of a benchmark
+simulation in which the current is included through a re-circulating pump. Both model setups (AC and
+Pump) reproduced this blocking and reflection of waves as the simulations captured the nodal pattern in
+the wave height for x < 0 m and wave energy was not transmitted beyond x = 0 m (Fig. 9a). Model
+simulations were found to be sensitive to the number of layers, and were approximately converged for 4
+layers (as illustrated by the results of the AC simulations). The nodal structure in H was stronger and
+18
+
+Figure 9: Comparisons between the measured and modelled wave height for test cases R1 and R11 of the wave-current flume
+experiment of Chawla and Kirby (1999). The black circle markers indicate the experimental observations, and the coloured
+lines indicate the model predictions (light and dark blue, 2 and 4-layer simulations with AC (Ambient Current), respectively;
+black, benchmark 4-layer simulation with pump system). The thin red lines show the results from an energy balance based
+on conservation of wave action using the linear dispersion relationship (dashed red line) and the 2nd order Stokes dispersion
+relationship (full red line).
+spatially shifted towards the wavemaker using two vertical layers with both the AC (Fig. 9a) and Pump
+setup (not shown). This indicates that reflections were stronger and blocking occurred at a weaker opposing
+current velocity when using this coarsest vertical resolution. Increasing the vertical resolution improved the
+results of the AC simulation, although H was over predicted at blocking compared to the measurements
+and the benchmark simulation. Results of the 4-layer benchmark Pump simulation were in good agreement
+with the measurements, apart from a slight spatial shift (approx. 0.15m) of the blocking location and nodal
+pattern.
+For test case R11, blocking was expected at x ≈ 0 m based on the energy balance with linear dispersion
+(Fig. 9b). In contrast, the energy balance with 2nd order dispersion predicted no blocking but transmission
+of energy for x > 0m. In the laboratory, partial reflections occurred at the blocking point with partial
+transmission of energy for x > 0. Both the AC and Pump simulations captured these patterns. Similar to
+case R1, simulations approximately converged when 4 layers were used. The 4-layer benchmark simulation
+was in best agreement with the observations, and captured both the spatial variability and magnitude of
+H. The 4-layer AC simulations overpredicted H near the linear blocking point for x > 0 m (similar to test
+case R1) and predicted weaker reflections resulting in a less pronounced nodal pattern for x < 0 m.
+These results show that the proposed extension of the model equations captured the overall patterns in
+the wave height that was observed in the laboratory and simulated by the benchmark model. Model results
+of both the AC and Pump simulations were found to be sensitive to the number of layers, indicating that
+the dispersive properties of the model affected the location of blocking and controlled the magnitude of wave
+reflections. Discrepancies in the blocking location at coarse vertical resolutions were larger for case R1 (with
+19
+
+case R1
+case R11
+0.10
+a)
+b)
+EBL
+(w)
+EBNL
+0.05
+2V AC
+4V AC
+4V Pump
+0.00
+2.0
+-1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+2.0
+-1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+x (m)
+x (m)a smaller wave period and thus larger kd compared to R11). This is consistent with the expected response
+based on the numerical dispersion relationship (Fig. 3): the relative absolute error in Ub compared to linear
+theory was 1.49% and 0.42% for R1 and R11 when using 2 layers, respectively, and < 0.25% when using 4
+layers.
+4.3.3. Results - wave breaking
+For larger incident wave heights (case B6 and B18), wave breaking on the opposing current was observed
+during the experiment in the narrow region of the flume (x = 0 − 5 m) and wave energy was transmitted
+beyond the blocking point from linear theory (Fig. 10). For case B6, the 2-layer AC simulation did not
+capture the transmission of wave energy beyond the blocking point, but showed signs of wave reflections
+near x = 0 m, resulting in an over prediction of the wave height for −2 < x < 0 m (Fig. 10a). Similar
+results were observed for the simulation with a Pump system (not shown). These results indicate that the
+2-layer simulations failed to capture the breaking of waves and transmission of energy beyond the linear
+blocking point for this particular test case. Increasing the number of vertical layers significantly improved
+the model results (as indicated by the 4 and 20-layer AC simulations, Fig. 10a). In particular, the 20V
+Pump benchmark simulation captured H throughout most of the domain, including the transmission of
+energy beyond the linear blocking point and the gradual decay of H for x > 0 m. The 20V AC simulation
+also captured part of this wave transmission but only up to x ≈ 1 m, and over predicted H near x = 0 m.
+For case B18, with a larger incident wave height and period, wave energy was transmitted beyond the
+linear blocking point for the 2-layer simulation with no sign of wave reflections (Fig. 10b). However, H was
+Figure 10: Comparisons between the measured and modelled (20-layer simulations) wave height for test cases B6 and B18 of the
+wave-current flume experiment of Chawla and Kirby (1999). The black circle markers indicate the experimental observations,
+and the coloured lines indicate the model predictions (light to dark blue, 2, 4 and 20-layer simulations with AC (Ambient
+Current); black, benchmark 20-layer simulations with pump system).
+The thin red lines show the results from an energy
+balance based on conservation of wave action using the linear dispersion relationship (dashed red line) and the 2nd order
+Stokes dispersion relationship (full red line).
+20
+
+case B6
+case B18
+0.50
+a
+b
+=
+EB,
+EBNL
+(w)
+0.25
+2V AC
+4V AC
+20V AC
+20V Pump
+0.00
+-4
+-2
+-4
+-2
+2
+4
+x (m)
+x (m)over predicted for x > −2m. Simulations with the Pump provided similar results (not shown). Increasing
+the number of vertical layers significantly improved the model results, and 20V AC simulations agreed well
+with 20V Pump simulations apart from a slight over prediction for x > −1 m. Both 20V models were also
+in satisfactory agreement with the observations, apart from an over prediction of H for x > 0 m.
+For the test cases with breaking waves, the model predictions were found to be sensitive to the vertical
+resolution. A relatively fine vertical resolution was found to be required to capture changes in the wave
+height. For case B6, a fine vertical resolution was required to prevent wave-reflections at the blocking point
+and to capture (part of) the transmission of energy for x > 0. In contrast, wave energy was transmitted
+beyond the linear blocking point at coarse resolutions for case B18.
+For this test case, higher vertical
+resolutions were required to better capture the shoaling of waves on the opposing current. The shoaling in
+2-layer simulations was similar to the linear energy balance, whereas shoaling in the case of more vertical
+layers was comparable to the nonlinear energy balance and the measurements. This suggests that for B18
+a higher vertical resolution is required to capture the effect of (nonlinear) amplitude dispersion.
+5. Discussion
+The results of this work demonstrated that the extended SWASH model was able to capture the dominant
+effects of currents on waves. Comparisons with linear theory and a spectral wave model showed that the
+model captured current-induced changes to the wave amplitude and length, and current-induced refraction.
+Comparisons with the laboratory experiment of Chawla and Kirby (1999, 2002) showed that the model
+reproduced the (partial) reflection of monochromatic waves on an opposing current near the blocking point
+in the case of small amplitude waves, and (partial) transmission and wave breaking in the case of larger
+amplitude waves. For these challenging test cases, model results were found to be sensitive to the number
+of vertical layers. In particular, a fine vertical resolution was required to capture the nonlinear shoaling,
+breaking and (partial) transmission of the large amplitude waves on the opposing current. Importantly, the
+results from the extended SWASH model were generally in good agreement with fully resolved benchmark
+simulations that intrinsically accounted for the wave-current interactions. This indicates that additional
+physics in the fully resolved SWASH model (e.g., vertical variations in the ambient flow, and the influence
+of waves on the ambient currents) did not significantly affect the wave dynamics in these test cases.
+Instead, this indicates that model-data discrepancies were largely inherited from the fully resolved model.
+For example, these could be related to the exclusion of span-wise flow effects, and shortcomings in the
+turbulence modelling (e.g., no wave breaking induced turbulence at the free-surface, incomplete description
+of turbulent boundary layers). To our knowledge, current state-of-the-art CFD models such as RANS and
+SPH-type models have not been widely used to simulate these nor similar laboratory experiments that
+consider such complex wave-current interactions. Only a few authors have used CFD for selected cases of
+21
+
+laboratory experiments (e.g., Olabarrieta et al., 2010; Teles et al., 2013; Chen and Zou, 2018; Yao et al.,
+2023) but not for a wide variety of conditions such as the reflective and breaking cases that were considered
+in this work. As such, we currently lack a clear benchmark that indicates how accurate fully resolved 3D
+models including more sophisticated turbulence models can capture wave-current interactions.
+6. Conclusions
+This study has demonstrated that the non-hydrostatic modelling approach can be extended to account
+for the effect of depth-uniform currents on the wave dynamics. By introducing a separation of scales and
+assuming vertically uniform mean currents, additional terms were derived that account for changes in the
+wave properties in the presence of spatially varying currents. These additional terms were included in the
+open-source SWASH model.
+A linear analysis of the model equations confirmed that the proposed model extension resolves the effect
+of currents on the linear wave properties (e.g., change in wavelength and group velocity). Comparisons of
+model predictions with linear wave theory further verified the numerical implementation. The extended
+SWASH model captured changes in the wavelength and amplitude in the presence of opposing and following
+currents for small amplitude waves. As a next step, we validated the model for more complex spatially
+varying flow fields: a vortex ring and a jet-like current.
+SWASH predictions were compared with the
+spectral wave model SWAN, including the Quasi-Coherent formulation to account for constructive and de-
+constructive wave interference effects. Comparisons of bulk wave parameters (significant wave height and
+mean wave direction) showed that the extended SWASH model was able to account for the current-induced
+refraction of both flow fields, and the resulting spatial variability in the wave height.
+Finally, we compared model predictions with a flume experiment that considered blocking and breaking
+of monochromatic waves on a strong opposing current. Although the model tended to overpredict the wave
+height, it was able to reproduce reflections of small amplitude waves, and breaking of larger amplitude
+waves.
+For breaking waves, model results were improved by increasing the vertical resolution (from 2
+to 20 layers). Results of the newly derived model were generally consistent with fully resolved SWASH
+simulations (in which a recirculating current was included through an inflow and outflow boundary at the
+bottom). This indicates that model-data discrepancies were largely inherited from the fully-resolved model
+and not introduced by missing physics in the extended model (e.g., no vertical variation of the ambient
+current, and no effect of waves on the ambient current).
+The findings of this work thereby demonstrated that phase-resolving models can be extended with
+additional terms to account for the major effect of ambient depth-uniform currents on the wave dynamics.
+This will allow models like SWASH to more accurately and efficiently simulate the wave dynamics in coastal
+environments where tidal and/or wind-driven currents are present.
+22
+
+Figure A.1: Changes to the height (panel a and c) and length (panel b and d) of a monochromatic wave (T = 10 s) on an
+opposing current (U = [−3, −1] m/s) as a function of the temporal resolution with a fixed grid resolution ∆x/L0 = 60 (panel
+a-b) and as a function of the horizontal grid resolution with a fixed temporal resolution ∆t = T/1000 (panel c-d). The full
+lines indicate the SWASH results and the dashed lines indicate the results according to linear wave theory. Results for U = −3
+m/s are printed in blue and results for U = −1 m/s in orange. For SWASH, the horizontal line with marker indicates the
+average change to the simulated wave height H in the current region, and the vertical lines with horizontal endings indicate
+the maximum and minimum H in the current region. The wave height and length are normalized by the incident wave height
+and length, respectively.
+Appendix A. Sensitivity study
+The behaviour of the SWASH model was found to be sensitive to the horizontal grid resolution ∆x and
+the time-step ∆t. To illustrate the sensitivity to the grid resolution, we consider a set of simulations of a
+T = 10 s monochromatic wave on a U = [−3, −1] m/s current for a range of horizontal grid and temporal
+resolutions. To study the influence of ∆t and ∆x separately, the first set considers simulations with fixed
+∆x = L0/60 for a range of ∆t, and the second set corresponds to several simulations with fixed ∆t = T/1000
+but for a range of ∆x.
+Changes to the wavelength L were not sensitive to either ∆x and ∆t. On the other hand, changes to
+the wave height H were sensitive to the model settings. The sensitivity was larger for the stronger current
+velocity.
+Modelled changes to H were less sensitive to the horizontal grid resolution, except for coarse
+resolutions (∆x/L0 < 40), with relatively weak improvement for ∆x/L0 ≤ 40 (Fig. A.1). Modelled changes
+to H were sensitive to the time-step, especially for U = −3 m/s. For this current velocity at larger time-
+steps, significant dissipation of wave energy occurred in the current region (as illustrated by the vertical
+lines in Fig. A.1 at smaller ∆t/T). For finer temporal resolutions, this non-physical dissipation reduced
+23
+
+3
+1.5
+a)
+b)
+(w) 
+2
+U=3 m/s
+(w)
+1.0
+“H/H
+U=1 m/s
+L/L o
+0.5
+0.0
+103
+104
+103
+104
+T/△t
+T/△t3
+1.5
+(
+(p
+(w)
+2
+1.0
+H/H
+107/7
+0.5
+0.0
+50
+100
+150
+50
+100
+150
+Lo/Ax
+Lo/Axand model results approximately converged to the solution of linear wave theory. This sensitivity to the
+horizontal grid and temporal resolution was primarily significant for strong opposing currents relative to
+the wave group velocity. For following currents and weak opposing currents the model results were not
+sensitive to ∆x and ∆t (as illustrated by the results for U = −1 m/s). Based on this sensitivity study, the
+optimal horizontal grid and temporal resolution for which model predictions were sufficiently converged was
+concluded to be ∆x = L0/100 and ∆t = T/1000.
+Appendix B. Re-circulating current
+To generate a re-circulating current in the model, we impose an inward and outward flux at the bottom
+at either side of the model domain. For this purpose, we have adopted the kinematic boundary condition
+as follows,
+wz=−d = −u∂d
+∂x − v ∂d
+∂y ± fs
+P
+W ,
+(B.1)
+where P is a discharge and W is the width of the region where the discharge is specified. By introducing an
+equal discharge of opposing sign in a region at either side of the numerical domain, a recirculating current
+is generated inside the domain. To reduce the spin-up time, we use a smoothing function fs to gradually
+ramp up the discharge from 0 to P. The smoothing function is defined as,
+fs = 0.5 (1 + tanh( t
+TS
+− 3)),
+(B.2)
+where TS is the smoothing period of the pump (taken as TS = 15 s in the simulations of this work).
+Appendix C. Linear semi-discrete analysis of the model equations
+The numerical dispersion relationship can be derived from the linearized and semi-discretized set of
+model equations (e.g., Cui et al., 2014; Bai and Cheung, 2013; Smit et al., 2014). Based on Smit et al.
+(2014), the linearized and semi-discretized SWASH equations extended with the additional terms for the
+wave-current interactions (on the right hand side) for N vertical layers reads,
+∂u′
+n− 1
+2
+∂t
++ g ∂ζ′
+∂x + 1
+2
+∂pnh,n
+∂x
++ 1
+2
+∂pnh,n−1
+∂x
+= −U
+∂u′
+n− 1
+2
+∂x
+,
+for n = 1...N,
+(C.1)
+∂wn + wn−1
+∂t
++ 2pnh,n − pnh,n−1
+∆z
+= −U ∂wn + wn−1
+∂x
+,
+for n = 1...N,
+(C.2)
+∂u′
+n− 1
+2
+∂x
++ wn − wn−1
+∆z
+= 0,
+for n = 1...N,
+(C.3)
+∂ζ′
+∂t + ∆z
+N
+�
+n=1
+∂u′
+n− 1
+2
+∂x
+= −U ζ′
+∂x.
+(C.4)
+The flow variables in the above set of equations are located on a staggered grid, with u′ located in a cell center
+(n − 1
+2) and w and pnh at a vertical cell face (n). Assuming a horizontal bottom (w0=0) and considering
+24
+
+the initial value problem in an infinite domain (with ∆z = d/N), we assume that the flow variables have
+a solution of the form y = ˆyexp(ikx − iωt) (where ˆy is the complex amplitude of a flow variable, k is the
+wavenumber and ω the absolute wave frequency). Substituting this into the above set of equations for each
+variable results in a system of equations of the form Aˆy = 0. The numerical dispersion relationship can
+subsequently found from Det(A) = 0 using symbolic algebra software. With the addition of an ambient
+current U, the numerical dispersion relationship provides a relationship between ω and k in the presence of
+a current with velocity U for N vertical layers. The relative group velocity can be found from the numerical
+dispersion relationship as cg,r = ∂σ
+∂k for an arbitrary current velocity U (with ω = σ + kU).
+25
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+30
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+DECOUPLING GENERALISED CONFIGURATION SPACES ON
+SURFACES
+LUCIANA BASUALDO BONATTO
+Abstract. The configuration space of k points on a manifold carries an action of
+its diffeomorphism group. The homotopy quotient of this action is equivalent to
+the classifying space of diffeomorphisms of a punctured manifold, and therefore ad-
+mits results about homological stability. Inspired by the works of Segal, McDuff,
+Bodigheimer, and Salvatore, we look at generalised configuration spaces where par-
+ticles have labels and even partially summable labels, in which points are allowed to
+collide whenever their labels are summable. These generalised configuration spaces
+also admit actions of the diffeomorphism group and we look at their homotopy quo-
+tients. Our main result is a decoupling theorem for these homotopy quotients on
+surfaces: in a range, their homology is completely described by the product of the
+moduli space of surfaces and a generalised configuration space of points in R∞.
+Using this result, we show these spaces admit homological stability with respect to
+increasing the genus, and we identify the stable homology. This can be interpreted
+as an Diff-equivariant homological stability for factorization homology. In addition,
+we use this result to study the group completion of the monoid of moduli spaces of
+configurations on surfaces.
+1. Introduction
+The ordered configuration space of k points on a smooth manifold M without bound-
+ary is defined as
+�Ck(M) := {(m1, . . . , mk) ∈ M k | mi ̸= mj if i ̸= j}.
+When M is a smooth manifold with boundary, we denote by �Ck(M) the space of ordered
+configurations of k points in its interior. The symmetric group Σk acts on this space
+by permuting the order of the k points. The configuration space of k points on M,
+denoted Ck(M), is the quotient �Ck(M)/Σk.
+We denote by Diff∂(M) the group of
+diffeomorphisms of a manifold M which fix a collar of its boundary.
+In this paper, we will focus on 2-dimensional manifolds and we denote by F k
+g,b an
+orientable surface of genus g, k punctures, and b ≥ 1 boundary components.
+The
+spaces Ck(Fg,b) admit an action of the group Diff∂(Fg,b), where a diffeomorphism φ
+acts by taking a collection of k points to its image via φ. Of interest here, is the Borel
+construction (homotopy quotient) of this action, denoted Ck(Fg,b)//Diff∂(Fg,b), to which
+we refer to as a moduli of configurations of k points in Fg,b. It is simple to show that
+Ck(Fg,b)//Diff∂(Fg,b) ≃ BDiff∂(F k
+g,b)
+(1.1)
+and in fact this relation is not only true for surfaces, but for any manifold with k
+punctures. In particular, this allows us to deduce homological stability results for these
+moduli of configurations of k points, directly from the known theorems for classifying
+spaces of punctured surfaces. For instance, when b ≥ 1 these spaces admit homological
+stability when increasing the genus and when increasing then number of points [Har85,
+Date: January 3, 2023.
+2020 Mathematics Subject Classification. 57R19, 55R80, 55R40, 55P47.
+This material is based upon work supported by CNPq (201780/2017-8) and by the NSF Grant No.
+DMS-1928930 while the author participated in a program hosted by the MSRI in 2022.
+1
+arXiv:2301.00093v1  [math.AT]  31 Dec 2022
+
+2
+LUCIANA BASUALDO BONATTO
+Har90, RW16, RW14]. Moreover, it was shown in [BT01] that the stable homology of this
+classifying space can be computed from the homology of BDiff∂(Fg,b)×B(Σk ≀GL+
+2 (R)),
+what is known as a decoupling theorem.
+In this paper, we study the analogue of this moduli space for generalised configuration
+spaces. As a first case, we look at labelled configurations: given a pointed space Z, the
+space of configurations in M with labels in Z is the quotient
+C(M; Z) :=
+�
+��
+k≥0
+�Ck(M) ×Σk Zk
+�
+�
+�
+∼
+under the relation (m1, . . . , mk; z1, . . . , zk) ∼ (m1, . . . , mk−1; z1, . . . , zk−1) if zk is the
+basepoint of Z. We can interpret this space geometrically by considering it as the space
+of particles in M where each particle is labelled by an element of Z, and a particle is
+allowed to disappear if labelled by the basepoint.
+This space has been of interest since the 70’s, appearing on the seminal works of
+May [May72] and Segal [Seg73]. It was noted that the space C(Rn; Z) can be given the
+structure of an (A∞-)monoid by taking multiplication to be roughly given by stacking
+configurations side by side [Seg73]. One of the main results about this space is what
+today is called a scanning map C(Rn; Z) → ΩnΣnZ which was shown by Segal to induce
+a weak-homotopy equivalence on group-completions. This idea has been generalised in
+many directions. For instance, B¨odigheimer [B¨od87] proved an analogous statement for
+configurations on general manifolds. In addition, similar results were proven for the
+case where the spaces of labels has extra structure, such as (partial) multiplications
+[McD75, Seg79, Gue95, Kal01, Sal01]. We discuss the later case in Section 1.1.
+More recently this labelled configuration space and scanning map argument have been
+expanded to sophisticated constructions in factorization homology [AFT17] on the one
+hand and, on the other, in the form of configuration spaces of manifolds, has been used
+to compute the stable homology of the moduli spaces of Riemann surfaces [MW07] and
+higher dimensional manifolds [GRW18, GRW17].
+Labelled configuration spaces also inherit an action of the diffeomorphism group.
+Even more, if Z is a pointed GL+
+2 (R)-space, we can define an action of Diff∂(Fg,b) on
+C(M; Z) where a diffeomorphism φ acts by taking a collection of k points to its image
+via φ, and the label z of a point w is taken to the label dwφ·z of the point φ(w). Unlike
+the case of configurations with a fixed number of points, C(Fg,b; Z)//Diff∂(Fg,b) is not
+in general equivalent to the classifying space of a diffeomorphism group. Hence we ask
+if it still has homological stability and if it admits an analogous decoupling theorem.
+For a surface Fg,b, with b ≥ 1, taking the boundary connected sum with the surface
+F1,1 induces a homomorphism Diff∂(Fg,b) → Diff∂(Fg+1,b), given by extending a map
+on Fg,b by the identity. We call this the stabilisation map and study what it induces on
+the moduli of configuration spaces:
+Theorem A. Let Z be a pointed GL+
+2 (R)-space and b ≥ 1. The stabilisation map on
+the Borel constructions
+s∗ : C(Fg,b; Z)//Diff∂(Fg,b) → C(Fg+1,b; Z)//Diff∂(Fg+1,b)
+induces a homology isomorphism in degrees ≤ 2
+3g.
+Moreover, we can determine precisely what the stable homology is:
+Theorem B. Let Z is a pointed connected GL+
+2 (R)-space. There is a map
+C(Fg,b; Z)//Diff∂(Fg,b) → Ω∞MTSO(2) × Ω∞Σ∞
+�
+EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z
+�
+which is compatible with the stabilisation maps and induces a homology isomorphism in
+degrees ≤ 2
+3g.
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+3
+In the above, EGL+
+2 (R) denotes the total space of a universal fibration for BGL+
+2 (R),
+we use (−)+ to denote adjoining a disjoint basepoint to a space, and −
+∧
+GL+
+2 (R)
+− denotes
+the quotient of the smash product of pointed topological GL+
+2 (R)-spaces by the diagonal
+action of GL+
+2 (R).
+Both of the results above are consequences of Theorem C below, which is an ana-
+logue of the decoupling theorem in [BT01]. It implies that the stable homology of this
+moduli of configuration spaces can be understood through a decoupling map τ × ε,
+which separates the points in the configurations from the underlying surfaces. The map
+τ : Ck(Fg,b)//Diff∂(Fg,b) → BDiff∂(Fg,b) forgets the data of the configurations, and ε
+forgets the underlying surface, but still remembers the points in the configuration and
+some local tangential data around them (for a detailed description of these maps see
+Section 3).
+Theorem C (Decoupling Theorem for Labelled Configurations). Let τ and ε be the
+maps described above. Then
+τ × ε : C(Fg,b; Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) × C(R∞; EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z)
+induces a homology isomorphism in degrees ≤ 2
+3g.
+This result may be interpreted in physical terms: in the high genus limit, the con-
+straints for the particles to stay on the underlying surface are lifted and the particles
+are now free.
+As a final application of Theorem C, we look at monoids of moduli of configurations
+on surfaces. Boundary connected sum induces a multiplication on the level of classifying
+spaces making
+�
+g BDiff∂(Fg,1) into a topological monoid. This important construction
+and its group completion have been central in the study of the stable homology of
+mapping class groups of surfaces [Mil86, Til00, MW07, GRW10, GRW18, GRW17].
+This gluing of surfaces induces also a multiplication on the Borel constructions
+C(Fg,1; Z)//Diff∂(Fg,1) × C(Fh,1; Z)//Diff∂(Fh,1) → C(Fg+h,1; Z)//Diff∂(Fg+h,1)
+making
+�
+g C(Fg,1; Z)//Diff∂(Fg,1) into a topological monoid. We study its group com-
+pletion.
+Corollary D. For any pointed GL+
+2 (R)-space Z, the decoupling map induces a weak
+equivalences on group completions
+ΩB
+��
+g C(Fg,1; Z)//Diff∂(Fg,1)
+�
+≃ ΩB
+��
+g BDiff∂(Fg,1)
+�
+×ΩBC(R∞; EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z).
+1.1. Configuration spaces with partially summable labels. The main result of
+this paper considers a more general type of configuration spaces with labels in a framed
+partial 2-monoid, where particles are allowed to collide if their labels are summable.
+This space has been explored in works such as [McD75, Seg79, Gue95, Kal01, Sal01]
+and, when the labels are E2-algebras (not partial), is equivalent to factorization ho-
+mology [AF15] and topological chiral homology [Lur09]. The first example of this con-
+struction can be seen in McDuff’s configuration spaces of positive and negative particles
+[McD75], where particles are labelled by “charges” and are allowed to collide whenever
+their charges are opposite. More generally, Salvatore [Sal01] defines partial 2-monoids,
+which are, in essence, spaces with a multiplication similar to an E2-algebra structure,
+but with the restriction that this multiplication does not need to be defined for every
+tuple of elements (see Definition 4.4). Whenever P is equipped with a compatible ac-
+tion of GL+
+2 (R), we call it a framed partial monoid, and we can define the space of
+configurations in Fg,b with partially summable labels in P, denoted CΣ(Fg,b; P). The
+definition CΣ(Fg,b; P) requires much more machinery then the case for non-summable
+
+4
+LUCIANA BASUALDO BONATTO
+labels, such as the Fulton-MacPherson operad, and yields more complicated models for
+configuration spaces. We discuss these constructions in Section 4.1. As before, these
+generalised configuration spaces admit an action of the diffeomorphism group and we
+study its Borel construction.
+Theorem E. Let P be a framed partial 2-monoid and b ≥ 1. The stabilisation map on
+the Borel constructions
+s∗ : CΣ(Fg,b; P)//Diff∂(Fg,b) → CΣ(Fg+1,b; P)//Diff∂(Fg+1,b)
+induces a homology isomorphism in degrees ≤ 2
+3g.
+While homological stability for configurations with summable labels with respect to
+increasing the number of points had been studied in [KM16], the above result is the first
+to look at stability with respect to increasing the genus. Theorem E can be interpreted
+as a Diff∂-equivariant homological stability result for factorisation homology.
+As in the case for labelled configurations, this result is a consequence of a decou-
+pling theorem for the space CΣ(Fg,b; P)//Diff∂(Fg,b), which is the main result of this
+paper. This is much more intricate than the decoupling for labelled configurations. The
+proof uses a semi-simplicial resolution of CΣ(Fg,b; P) developed in section 4.2, which
+we refer to as the disc model for configurations, denoted |DΣ(Fg,b; P)•| (Proposition
+4.13). This model makes explicit the connection between these spaces and factorization
+homology. In the decoupling context, we naturally encounter an analogue of this space
+with 2-dimensional discs with configurations embedded in R∞, we denote this space
+|D2
+Σ(R∞; P)•| (Definition 4.17). Using the Decoupling Theorem for Labelled Configu-
+rations (Theorem D) we then prove:
+Theorem F (Decoupling Theorem for Configurations with Summable Labels). For
+P a framed partial 2-monoid, there is a weak equivalence CΣ(Fg,b; P)//Diff∂(Fg,b) ≃
+|DΣ(Fg,b; P)•|//Diff∂(Fg,b) and the decoupling map
+|DΣ(Fg,b; P)•|//Diff∂(Fg,b) → BDiff∂(Fg,b) × |D2
+Σ(R∞; P)•|.
+induces a homology isomorphism in degrees ≤ 2
+3g.
+In future work we will discuss the homotopy type of the space |D2
+Σ(R∞; P)•| and
+its description as an infinite loop space. We conjecture that it is also equivalent to a
+configuration in R∞ with partially summable labels.
+Analogous to the case of labelled configurations, the spaces CΣ(Fg,1; P)//Diff∂(Fg,1)
+assemble into a topological monoid, and the decoupling theorem descends into its group
+completion.
+Corollary G. For any path-connected framed partial 2-monoid with unit P, the decou-
+pling map induce a homotopy equivalence
+ΩB(
+�
+g CΣ(Fg,1; P)//Diff∂(Fg,1)) ≃ ΩB(
+�
+g BDiff∂(Fg,1)) × ΩB(|D2
+Σ(R∞; P)•|).
+1.2. Outline of the paper. We start by recalling in Section 2 background results
+which will be used throughout the paper, especially on Section 4. This can be skipped
+and referred back to when necessary.
+In Section 3 we introduce labelled configuration spaces and prove Theorem C. Using
+this, we deduce Theorems A and B, and Corollary D.
+In Section 4 we discuss the case of configurations with summable labels, and prove
+the main results of the paper. We start by recalling in 4.1 the definitions of framed
+partial d-monoids and configuration spaces with partially summable labels. We then
+construct semi-simplicial resolutions for these spaces in Section 4.2. In Section 4.3, we
+use this disc model together with the Decoupling Theorem for Labelled Configurations
+(Theorem D) to prove Theorem F. Finally, we use this result to deduce Theorem E and
+Corollary G.
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+5
+Acknowledgements. I would like to thank Ulrike Tillmann for suggesting the prob-
+lem and for the many insightful conversations. In addition, I would like to thank David
+Ayala, Christopher Douglas, Jan Steinebrunner, and Nathalie Wahl for the helpful dis-
+cussions and comments.
+2. Preliminaries
+In this section we recall techniques and results on semi-simplicial spaces, which will
+be used in Section 4. The reader may skip this part and refer back to when necessary.
+For a detailed exposition of the concepts in this section see [ERW19].
+A semi-simplicial space is a functor ∆op
+inj → Top, where ∆inj is the category with ob-
+ject the linearly ordered sets [p] = {0 < · · · < p} and morphisms the injective monotone
+maps. We denote such a functor by X• and write Xp = X•({1, . . . , p}). The datum of
+a semi-simplicial space is equivalent to the collection of spaces Xp, p ≥ 0, together with
+face maps di : Xp → Xp−1 for i = 0, . . . , p, satisfying didj = dj−1di if i < j.
+Denote by ∆p the standard p-simplex
+∆p =
+�
+(t0, . . . , tp) ∈ Rp+1���
+p
+�
+i=1
+ti = 1 and ti ≥ 0 for all i
+�
+.
+To each morphism φ : [p] → [q] in ∆inj, there is a continuous map φ∗ : ∆p → ∆q such
+that φ∗(t0, . . . , tp) = (s0, . . . , sq) with sj = �
+i∈φ−1(j) ti. The geometric realisation of a
+semi-simplicial space X• is the quotient space
+|X•| :=
+��
+p
+Xp × ∆p
+� �
+∼
+where (x, φ∗t) ∼ (φ∗x, t), and φ is a morphism of ∆inj.
+2.1. Semi-simplicial nerve of a poset. Any topological poset (Q, <) defines a semi-
+simplicial space Q• by setting Qp to be the subspace of tuples (q0 < · · · < qp) ∈ Qp+1,
+and face maps
+di : Qp −→ Qp−1
+for 0 ≤ i ≤ p
+(q0 < · · · < qi < · · · < qp) �−→ (q0 < · · · < qi−1 < qi+1 < · · · < qp).
+We refer to Q• as the semi-simplicial nerve of the poset Q.
+Given a topological poset (Q, <), the space Q×Q can be equipped with a partial order
+where (q1, q2) < (q′
+1, q′
+2) if qi < q′
+i and qj ≤ q′
+j, for {i, j} = {1, 2}. We say that a pointed
+such Q is a partially ordered topological monoid if it is equipped with a multiplication
+− · − : Q × Q → Q
+which is strictly associative, unital and order preserving. In this case, the geometric
+realisation of the semi-simplicial nerve Q• is naturally endowed with a multiplication ·
+defined by
+�
+(q0 < · · · < qm; t0, . . . , tm) · (q0 < · · · < qk; t0, . . . , tk)
+�
+=
+= (q0 · q0 < · · · < q0 · qk < · · · < qm · q0 · · · < qm · qk; t0 · t, . . . , tm · t)
+where ti · t = tit0, . . . , titk, for all i = 0, . . . , m. It is straightforward to verify that this
+is well-defined.
+Lemma 2.1. For (Q, <, µ) a partially ordered topological monoid, (|Q•|, |µ|) is a topo-
+logical monoid. Moreover, any map of partially ordered topological monoids f : Q → Q′
+induces a map of topological monoids
+f∗ : |Q•| → |Q′
+•|.
+The proof is a straightforward computation and follows directly from the definitions.
+
+6
+LUCIANA BASUALDO BONATTO
+2.2. Spectral sequence. We quickly recall below the spectral sequence defined in
+[Seg68, Proposition 5.1] associated to a semi-simplicial space, which is the key for the
+homology argument used in the proof of Theorem 4.18 (see [ERW19, Section 1.4] for a
+detailed discussion).
+For any semi-simplicial space X•, the geometric realisation |X•| admits a filtration
+by its skeleta, with |X•|(0) = X0 and
+|X•|(q) = |X•|(q−1) ∪Xq×∂∆q Xq × ∆q.
+This filtration yields a spectral sequence
+E1
+p,q = Hp+q(|X•|(q), |X•|(q−1)) =⇒ Hp+q(|X•|)
+and by excision and the Kunneth isomorphism, the left-hand term can be re-written to
+give a spectral sequence with
+E1
+p,q ∼= Hp(Xq) =⇒ Hp+q(|X•|).
+Therefore a map of semi-simplicial spaces inducing a level-wise homology isomorphism
+gives an isomorphism of the first pages of the respective spectral sequences, and therefore
+a homology isomorphism between the geometric realisations.
+2.3. Semi-simplicial Resolutions. In the proof of the decoupling we will use a semi-
+simplicial resolution of the spaces of configurations with summable labels. Showing that
+we indeed have a resolution will be a direct application of [GRW14, Theorem 6.2]. For
+completeness, we quickly recall the statement of this result to clarify the conditions that
+will be checked in the proof of Proposition 4.13. We follow the notation and definitions
+of [GRW14].
+An augmented semi-simplicial space is a triple (X•, X−1, ε•), where X• is a semi-
+simplicial space, X−1 is a space and ε• is a collection of continuous maps εp : Xp →
+X−1 satisfying diεp = εp−1 for all p ≥ 0 and all face maps di.
+We also say that
+ε• : X• → X−1 is an augmentation for X•. It is simple to verify that an augmentation
+induces a continuous map |ε•| : |X•| → X−1.
+Definition 2.2. An augmented topological flag complex [GRW14, Definition 6.1] is an
+augmented semi-simplicial space ε : X• → X−1 such that
+(i) The map Xn → X0 ×X−1 · · ·×X−1 X0 taking an n-simplex to its (n+1) vertices
+is a homeomorphism onto its image, which is an open subset.
+(ii) A tuple (v0, . . . , vn) ∈ X0 ×X−1 · · ·×X−1 X0 lies in Xn if and only if (vi, vj) ∈ X1
+for all i < j.
+In other words, in an augmented topological flag complex, the space of n-simplices
+can be described as an open subspace of the (n + 1)-tuples of vertices with the same
+image under ε, and such a tuple forms an n-simplex if and only if the pairs of vertices
+are all 1-simplices. The result below is a criterion to determine when an augmented
+topological flag complex X• → X−1 induces a weak equivalence |X•| → X−1.
+Theorem 2.3 ([GRW14], Theorem 6.2). Let X• → X−1 be an augmented topological
+flag complex. Suppose that
+(i) The map ε : X0 → X−1 has local lifts of any map from a disc, i.e. given a map
+f : Dn → X−1, a point p ∈ ε−1(f(x)), there is an open neighbourhood U ⊂ Dn
+of x and a map F : U → X0 such that ε ◦ F = f|U and F(x) = p.
+(ii) ε : X0 → X−1 is surjective.
+(iii) For any p ∈ X−1 and any non-empty finite set {v1, . . . , vn} ∈ ε−1(p) there exists
+a v ∈ ε−1(p) with (v1, v) ∈ X1 for all i.
+Then |X•| → X−1 is a weak homotopy equivalence.
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+7
+3. Decoupling Labelled Configuration Spaces
+In this section, we recall the definition of configuration spaces with labels and in-
+troduce a model for the homotopy quotients C(Fg,b; Z)//Diff∂(Fg,b).
+With this, we
+construct the decoupling map of Theorem C. We then prove this result as Theorem
+3.2 and use it to prove Theorems A and B, and Corollary D. These are, respectively,
+Corollary 3.4, Corollary 3.5, and Corollary 3.7 below.
+Definition 3.1. Let M be a manifold and Z be a well-pointed space. The configuration
+space of M with labels in Z, denoted C(M; Z), is the quotient
+�
+k≥0 �Ck(M) ×
+Σk
+Zk/ ∼
+where �Ck(M) denotes the ordered configuration space of k points in M, and
+(m1, . . . , mk; z1, . . . , zk) ∼ (m1, . . . , mk−1; z1, . . . , zk−1)
+whenever zk is the basepoint of Z.
+If Z is a pointed GL+
+2 (R)-space, and M is a surface Fg,b, with b ≥ 1, the space
+C(Fg,b; Z) carries a natural action by the diffeomorphism group of Fg,b: for φ ∈ Diff∂(Fg,b)
+φ · (m1, . . . , mk; z1, . . . , zk) := (φ(m1), . . . , φ(mk); Dm1φ · z1, . . . , Dm1φ · zk).
+The basepoint relation is preserved by this action as Z is a pointed GL+
+2 (R)-space.
+The decoupling result is about the homotopy quotient of this action, that is, the Borel
+construction C(Fg,b; Z)//Diff∂(Fg,b).
+From now on, we denote by Fg,b a fixed orientable surface of genus g, b ≥ 1 bound-
+ary components, and pick once and for all a framing on Fg,b, that is, a section s of
+the frame bundle Fr(TFg,b) → Fg,b. We fix now a model for EDiff∂(Fg,b) that will
+be used to construct the decoupling map of Theorem 3.2. Let Emb(Fg,b, R∞) denote
+colimn→∞ Emb(Fg,b, Rn). This space has a free action of Diff∂(Fg,b) by precomposition
+and by [BF81] the quotient map Emb(Fg,b, R∞) → Emb(Fg,b, R∞)/Diff∂(Fg,b) has slices,
+hence it is a principal Diff∂(Fg,b)-bundle. Moreover, by Whitney’s embedding theorem,
+Emb(Fg,b, R∞) is weakly contractible and therefore it is a model for EDiff∂(Fg,b). We
+then let
+C(Fg,b; Z)//Diff∂(Fg,b) ≃ Emb(Fg,b, R∞)
+×
+Diff∂(Fg,b) C(Fg,b; Z).
+Analogously, we will take the model for BDiff∂(Fg,b) given by
+BDiff∂(Fg,b) ≃ Emb(Fg,b, R∞)/Diff∂(Fg,b).
+This can be interpreted as the space of abstract submanifolds of R∞ which are diffeo-
+morphic to Fg,b, but without a fixed diffeomorphism. Analogously, a point in C(Fg,b; Z)//
+Diff∂(Fg,b) consists of one such abstract manifold, together with a labelled configuration.
+The decoupling map will be the product of two maps
+τ : C(Fg,b; Z)//Diff∂(Fg,b) → BDiff∂(Fg,b)
+(3.1)
+ε : C(Fg,b; Z)//Diff∂(Fg,b) → C(R∞; EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z).
+(3.2)
+Recall that EGL+
+2 (R) is the total space of a universal fibration for BGL+
+2 (R), we use
+(−)+ to denote adjoining a disjoint basepoint to a space, and −
+∧
+GL+
+2 (R)
+− denotes the
+quotient of the smash product of pointed topological GL+
+2 (R)-spaces by the diagonal
+action of GL+
+2 (R).
+Intuitively, the map τ forgets the configuration, while ε forgets the underlying surface,
+but remembers the labelled configuration together with data on their tangent space on
+the submanifold.
+
+8
+LUCIANA BASUALDO BONATTO
+In details, τ is the classifying map for the homotopy quotient, and it is simply the
+one induced by the projection Emb(Fg,b, R∞)×Ck(Fg,b) → Emb(Fg,b, R∞).
+To define ε, we take as model for BGL+
+2 (R) the oriented Grassmanian manifold of
+2-dimensional oriented subsbaces of R∞, Gr+(2, ∞), and let EGL+
+2 (R) denote the total
+space of the universal GL+
+2 (R)-bundle over it. Then using the identification Fr(TR∞) ∼=
+R∞ ×EGL+
+2 (R), an embedding e : Fg,b �→ R∞ induces a map e∗ : Fr(TFg,b) → ESO(2)
+from the bundle of framings on TFg,b, taking a basis of TpFg,b to its image via Dpe. The
+map ε takes a point represented by a labelled configuration [m1, . . . , mk; z1, . . . , zk] and
+an embedding e : Fg,b �→ R∞ to the labelled configuration in R∞ given by
+[e(m1), . . . , e(mk); [e∗(s(m1)), z1], . . . , [e∗(s(mk)), zk]]
+where s(p) denotes the chosen oriented frame on p ∈ Fg,b. It is simple to verify that this
+indeed defines a continuous function to the configuration space C(R∞; (EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z).
+Theorem 3.2. Let τ and ε be the maps in (3.1). Then the decoupling map
+τ × ε : C(Fg,b; Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) × C(R∞; (EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z)
+induces a homology isomorphism in degrees ≤ 2
+3g.
+The proof of the result above will build upon a decoupling result for unlabelled
+configurations with a fixed number of points, which was first proved by [BT01] and
+generalised in [Han09, Bon22]. We show here a slight generalisation of the result which
+we will need in the proof. The space C(M; Z) is constructed as a quotient of the union
+of spaces �Ck(M) ×Σk Zk, and the Lemma below is about the decoupling map in each of
+these components.
+In fact, we will work on a slightly more general context which will be more convenient
+for the proof: let X be a well-pointed space with an action of the wreath product
+Σk ≀ GL+
+2 (R) (in the context above we were using X = Zk). The space �Ck(Fg,1) × X
+comes equipped with two actions: Σk acts diagonally by permuting the points in the
+configuration and by the action on X, and Diff∂(Fg,b) acts by
+φ · (m1, . . . , mk; x) = (φ(m1), . . . , φ(mk); (dm1φ, . . . , dmkφ)(x)).
+Note that the actions of Σk and Diff∂(Fg,1) on this space commute.
+As before, we have maps
+τk : ( �Ck(Fg,1) ×
+Σk
+X)//Diff∂(Fg,b) → BDiff∂(Fg,b)
+(3.3)
+εk : ( �Ck(Fg,1) ×
+Σk
+X)//Diff∂(Fg,b) → ( �Ck(R∞) × (EGL+
+2 (R))k)
+×
+Σk≀GL+
+2 (R)
+X.
+(3.4)
+Here τk is again simply the classifying map for the homotopy quotient, and εk is the
+map taking a point represented by [m1, . . . , mk; x] and an embedding e : Fg,b �→ R∞ to
+the class
+�
+[e(m1), . . . , e(mk); e∗(s(m1)), . . . , e∗(s(mk))], x
+�
+where s(p) denotes the chosen oriented frame on p ∈ Fg,b. We remark that replacing X
+by Zk one recovers precisely the definition of the maps τ and ε in (3.1).
+Lemma 3.3 ([BT01, Bon22]). For any Σk ≀GL+
+2 (R)-space X, let τk and εk be the maps
+defined in (3.3). Then
+τk×εk : ( �Ck(Fg,b)×
+Σk
+X)//Diff∂(Fg,b) → BDiff∂(Fg,b)×( �Ck(R∞)×(EGL+
+2 (R))k)
+×
+Σk≀GL+
+2 (R)
+X.
+induces a homology isomorphism in degrees ≤ 2
+3g.
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+9
+For completeness, we include a short proof of the above result. For details see [BT01,
+Bon22].
+Proof of Lemma 3.3. We start by reducing the proof to the case when X is a point. The
+projections
+( �Ck(Fg,b) ×
+Σk
+X)//Diff∂(Fg,b) → Ck(Fg,b)//Diff∂(Fg,b)
+BDiff∂(Fg,b) × ( �Ck(R∞) × (EGL+
+2 (R))k)
+×
+Σk≀GL+
+2 (R)
+X → BDiff∂(Fg,b) × Ck(R∞, BGL+
+2 (R))
+are both fibrations with fibre X, and the map τk × εk induces a map between the
+corresponding fibre sequences, which is the identity on the fibers. If the map between
+the base spaces induces a homology isomorphism in degrees ≤ 2
+3g, then by Zeeman’s
+Comparison Theorem [Zee57] applied to the Serre spectral sequences associated to these
+fibrations, so does τk × εk. Hence it is enough to show that the map between the base
+spaces
+τk × εk : Ck(Fg,b)//Diff∂(Fg,b) → BDiff∂(Fg,b) × Ck(R∞, BGL+
+2 (R))
+induces a homology isomorphism in degrees ≤ 2
+3g.
+By Palais’ Theorem [Pal60], the map εk is a fibration with fiber BDiff∂(Fg,b+n).
+Moreover, the map τk × εk induces a map of fibre sequences
+BDiff∂(Fg,b+n)
+Ck(Fg,b)//Diff∂(Fg,b)
+Ck(R∞, BGL+
+2 (R))
+BDiff∂(Fg,b)
+BDiff∂(Fg,b) × Ck(R∞, BGL+
+2 (R))
+Ck(R∞, BGL+
+2 (R))
+proj
+τk×εk
+εk
+where the leftmost map is the one induced by capping off the n extra boundary compo-
+nents by gluing discs. This map was shown to induce a homology isomorphism in the
+range ≤ 2
+3g [Har85, Iva87, Iva89, Iva93, Bol12, RW16]. Hence, by Zeeman’s Compar-
+ison Theorem [Zee57] applied to the Serre spectral sequences associated to these fibre
+sequences, the map between the total spaces also induces homology isomorphisms in the
+range ≤ 2
+3g.
+□
+Equipped with Lemma 3.3, we are now ready to prove Theorem 3.2.
+Proof of Theorem 3.2. The spaces C(Fg,b; Z) and C(R∞; (EGL+
+2 (R))+ ∧GL+
+2 (R) Z) con-
+sist of configuration with an arbitrary number of particles. However they have natural
+filtrations C≤k(−) by the subspaces of configurations with at most k points. These
+induce filtrations
+Xk := C≤k(Fg,b; Z)//Diff∂(Fg,b)
+Yk := BDiff∂(Fg,b) × C≤k(R∞; (EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z).
+that are preserved under the map τ × δ.
+We will inductively show the restrictions
+Xk → Yk are homology isomorphisms for all k.
+Since Z is a space with a good basepoint, the inclusions Xk−1 �→ Xk and Yk−1 �→ Yk
+are cofibrations. Their subquotients are
+Xk/Xk−1 = (EDiff∂(Fg,b))+
+∧
+Diff∂(Fg,b)
+�Ck(Fg,b)+ ∧
+Σk Z∧k
+and
+Yk/Yk−1 = (BDiff∂(Fg,b))+ ∧ �Ck(R∞)+ ∧
+Σk ((EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z)∧k.
+
+10
+LUCIANA BASUALDO BONATTO
+Comparing the spectral sequences associated to these filtrations, it is enough to show
+that the induced map on these subquotients is a homology isomorphism. Consider the
+map of cofibrations:
+EDiff∂(Fg,b)
+×
+Diff∂(Fg,b) Ck(Fg,b)
+EDiff∂(Fg,b)
+×
+Diff∂(Fg,b) ( �Ck(Fg,b) ×
+Σk
+Z∧k)
+Xk/Xk−1
+BDiff∂(Fg,b) × Ck(R∞, BGL+
+2 (R))
+BDiff∂(Fg,b) × ( �Ck(R∞) ×
+Σk
+((EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z)∧k)
+Yk/Yk−1
+τk×εk
+τk×εk
+By Lemma 3.3 with X = ∗ the left-hand map induces a homology isomorphism in
+degrees ≤ 2
+3g, and by the same result with X = Z∧n, so does the middle map. Then
+by the five lemma, the right-hand map also induces a homology isomorphism in degrees
+≤ 2
+3g, as required.
+□
+3.1. Homological Stability. Let Fg+1,b be a surface of genus g+1 and b ≥ 1 boundary
+components, which is obtained from Fg,b by a boundary connected sum with F1,1. Then
+extending diffeomorphisms by the identity on F1,1 gives a map of topological groups
+s : Diff∂(Fg,b) �→ Diff∂(Fg+1,b)
+which we refer to as the stabilisation map.
+Moreover, the inclusion Fg,1 �→ Fg+1,b
+induces a continuous map of labelled configuration spaces C(Fg,b; Z) → C(Fg+1,b; Z)
+which is s-equivariant. Together this implies we have an induced map on the Borel
+constructions:
+Corollary 3.4. For b ≥ 1, the stabilisation map on the Borel constructions
+s∗ : C(Fg,b; Z)//Diff∂(Fg,b) → C(Fg+1,b; Z)//Diff∂(Fg+1,b)
+induces a homology isomorphism in degrees ≤ 2
+3g.
+The above result is a corollary of Theorem 3.2, however care has to be taken with
+respect to the model we have used for the Borel constructions and classifying spaces.
+Namely, it is not clear how to define a stabilisation map on the level of embedding
+spaces Emb(Fg,b, R∞) → Emb(Fg+1,b, R∞) which induces the desired map on Borel
+constructions.
+This can be remedied by taking as model for EDiff∂(Fg,b) a weakly
+contractible subspace of Emb(Fg,b, R∞) that still has a free and has proper action of
+Diff∂(Fg,b) and in which the stabilisation is clear.
+Fix a boundary component of Fg,b and an embedding S1 �→ {0}×R∞. We denote by
+Emb∂(Fg,b, (−∞, 0]×R∞) the space of all extensions to an embedding of Fg,b which are
+standard on a collar neighbourhood of the marked boundary. By the same arguments
+as above, Emb∂(Fg,b, (−∞, 0] × R∞) is a model for EDiff∂(Fg,b), and the inclusion map
+Emb∂(Fg,b, (−∞, 0] × R∞) �→ Emb(Fg,b, R∞)
+is a Diff∂(Fg,b)-equivariant weak homotopy equivalence. Fixing an embedding e : F1,2 �→
+R∞ which restricts to the chosen embedding on a collar of the boundary, we get an
+inclusion
+Emb∂(Fg,b, (−∞, 0] × R∞) → Emb∂(Fg+1,b, (−∞, 0] × R∞)
+given by extending any embedding of Fg,b by e. This is clearly compatible with the
+stabilisation map.
+Proof of Corollary 3.4. Using as model for EDiff∂(Fg,b) the space Emb∂(Fg,b, (−∞, 0]×
+R∞) described above, we get a commutative diagram
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+11
+C(Fg,b; Z)//Diff∂(Fg,b)
+C(Fg+1,b; Z)//Diff∂(Fg+1,b)
+BDiff∂(Fg,b) × C(R∞; (EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z)
+BDiff∂(Fg+1,b) × C(R∞; (EGL+
+2 (R))+
+∧
+GL+
+2 (R)
+Z).
+s
+τ×ε
+τ×ε
+s×id
+By Theorem 3.2 the vertical maps induce homology isomorphisms in degrees ≤ 2
+3g,
+and by Harer’ Stability Theorem [Har85, Iva87, Iva89, Iva93, Bol12, RW16] and Kun-
+neth’s Theorem, so does the bottom map. Therefore the top map must also induce
+homology isomorphisms in degrees ≤ 2
+3g.
+□
+The Decoupling Theorem also allows us to identify what the homology stabilises to.
+Let C(F∞; Z)//Diff∂(F∞) and BDiff∂(F∞) be respectively
+C(F∞; Z)//Diff∂(F∞) := colim(C(F1,1; Z)//Diff∂(F1,1)
+s−→ C(F2,1; Z)//Diff∂(F2,1)
+s−→ . . . )
+BDiff∂(F∞) := colim(BDiff∂(F1,1
+s−→ BDiff∂(F2,1)
+s−→ . . . ).
+Corollary 3.5. For Z a pointed connected GL+
+2 (R)-space, the decoupling map induces
+τ × ε : C(F∞; Z)//Diff∂(F∞) → Ω∞MTSO(2) × Ω∞Σ∞
+�
+EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z
+�
+which is a homology isomorphism in all degrees.
+Proof. Theorem 3.2 and Corollary 3.4 imply that the decoupling map on the colimits
+τ × ε : C(F∞; Z)//Diff∂(F∞) → BDiff∂(F∞) × C(R∞, EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z)
+(3.5)
+induces a homology isomorphism. By [GTMW09, MW07], BDiff∂(F∞) admits a map to
+Ω∞MTSO(2) which is a homology isomorphism, and by [Seg73], the right-most config-
+uration space in (3.5) is homotopy equivalent to Ω∞Σ∞ �
+EGL+
+2 (R)+ ∧GL+
+2 (R) Z
+�
+.
+□
+3.2. Monoid of Moduli of Labelled Configuration Spaces. Gluing two surfaces
+Fg,1 and Fh,1 along part of their boundary defines a map of topological groups
+Diff∂(Fg,1) × Diff∂(Fh,1) → Diff∂(Fg+h,1).
+This can be made into an associative operation if we fix once and for all oriented surfaces
+Fg,b of genus g and one boundary component, and compatible with the stabilisation (see
+Section 3.1). Using these choices for our surfaces, we can see that the above map gives an
+associative operation in the collection of diffeomorphism groups of all Fg,1. An example
+of such surfaces and multiplication is depicted in Figure 1.
+µ
+,
+=
+Figure 1. Example of the map µ : MC(C)2 × MC(C)3 → MC(C)5
+where C is the space of colours, with white as the basepoint.
+Up to homotopy, the above gluing process is equivalent to gluing the boundary circles
+of surfaces Fg,1 and Fh,1, to two of the three boundary circles of F0,3, what is called the
+pair of pants multiplication. We choose to think of this multiplication in terms of the
+first description because in this way the product is strictly associative.
+This operation also induces a multiplication on the classifying spaces BDiff∂(Fg,1),
+which can be made associative by picking a convenient model. As in Section 3.1, we
+will use as EDiff∂(Fg,1) a certain subspace of Emb(Fg,1, R∞).
+Namely, we can fix
+
+12
+LUCIANA BASUALDO BONATTO
+embeddings δg : S1 �→ [0, g] × R∞ such that δg(S1) ∩ ({0} × R∞) = {0} × (−1, 1) × {0}
+and δg(S1) ∩ ({g} × R∞) = {g} × (−1, 1) × {0}. We denote by Embδg(Fg,1, [0, g] × R∞)
+the space of all extensions to an embedding of Fg,1 which are standard on a collar
+neighbourhood of the boundary. By the same arguments as above, Embδg(Fg,1, [0, g] ×
+R∞) is a model for EDiff∂(Fg,1), and the inclusion map
+Embδg(Fg,1, [0, g] × R∞) �→ Emb(Fg,1, R∞)
+is a Diff∂(Fg,1)-equivariant weak homotopy equivalence.
+By picking embeddings δg
+which are compatible with the stabilisation map, we can define a multiplication
+Embδg(Fg,1, [0, g] × R∞) × Embδh(Fh,1, [0, h] × R∞) → Embδg+h(Fg+h,1, [0, g + h] × R∞)
+by extending an embedding of Fg,1 by the translation of the embedding of Fh,1 in the
+first coordinate by g. This is clearly associative and it is compatible with the associative
+multiplication on the diffeomorphism groups.
+Taking as model for EDiff∂(Fg,1) the spaces Embδg(Fg,1, [0, g] × R∞) we obtain
+an associative multiplication on classifying spaces.
+This structure equips the space
+�
+g≥0BDiff∂(Fg,1) with the structure of a topological monoid, which we refer to as the
+surface monoid. This is equivalent to the one studied by Tillmann in [Til00], which was
+essential in the proof of the Madsen-Weiss Theorem [MW07].
+The operation on diffeomorphism groups and spaces EDiff∂(Fg,1) described above,
+together with the fixed identifications Fg+h,1 = Fg,1#∂Fh,1, induce an associative mul-
+tiplication also on the Borel constructions (see Figure 1)
+µ : C(Fg,1; Z)//Diff∂(Fg,1) × C(Fh,1; Z)//Diff∂(Fh,1) → C(Fg+h,1; Z)//Diff∂(Fg+h,1).
+Definition 3.6. We denote by MC(Z)g := C(Fg,1; Z)//Diff∂(Fg,1). The monoid of
+moduli of configurations labelled by Z is the topological monoid given by the disjoint
+union MC(Z) :=
+�
+g≥0MC(Z)g together with the operation µ.
+Theorem 3.7. For any pointed GL+
+2 (R)+-space Z, the decoupling map induces a weak
+equivalences on group completions
+ΩB
+��
+g C(Fg,1; Z)//Diff∂(Fg,1)
+�
+≃ ΩB
+��
+g BDiff∂(Fg,1)
+�
+×ΩB C(R∞; EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z).
+Segal showed in [Seg73] that the space Ω∞Σ∞(X) is the group completion of the con-
+figuration space C(R∞; X) seen as a topological monoid where the operation is roughly
+given by transposition. Using the inclusion Embδg(Fg,1, [0, g] × R∞) �→ Emb(Fg,b, R∞),
+we see that the decoupling map of Theorem 3.2 induces a map between the spaces using
+the current model. The proof of the above theorem will consist of showing that the
+decoupling induces a monoidal map between MC(Z) and the monoids Tillmann and
+Segal, and to show that this induces a homotopy equivalence on group completions.
+Lemma 3.8. The maps τ and ε defined in 3.1 are compatible with these monoidal
+structures.
+For the map τ, this result follows directly from the definition. For ε, some care has
+to be taken into making the configuration spaces into actual topological monoids (see
+[Seg73]). Namely, instead of C(R∞; X), we use the homotopy equivalent space
+C′(R∞, X) = {(c, t) ∈ C(R∞, X) × R : t ≥ 0, c ⊂ (0, t) × R∞}.
+The monoidal structure is given by juxtaposition, ie. (c, t), (c′, t′) �→ (c ∪ Tt(c′), t + t′)
+where Tt(−) is the map that translates a configuration by t on the first direction. The
+map τ of the decoupling theorem is then equivalent to the monoidal map MC(Z) →
+C′(R∞, X) taking an element of [e, c] ∈ MC(Z)g to (τ([e, c]), g).
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+13
+Proof of Theorem 3.7. It is enough to show that the map of monoids given by the decou-
+pling induces a homotopy equivalence on group completions. As the group completions
+are loop spaces, they are in particular simple and, by the Whitehead theorem for simple
+spaces, it suffices to show that it induces a homology equivalence on the group comple-
+tions. Both monoids are homotopy commutative, hence the group completion theorem
+[MS76] can be applied. Therefore it is enough to prove that the induced map on the
+limit spaces (defined in Section 3.1)
+τ × ε : C(F∞; Z)//Diff∂(F∞) → BDiff∂(F∞) × C(R∞; EGL+
+2 (R)+
+∧
+GL+
+2 (R)
+Z)
+(3.6)
+is a homology equivalence. This holds by Theorem 3.2 and Corollary 3.4.
+□
+4. Decoupling Configuration Spaces with Partially Summable Labels
+In this section we prove the main result of this paper, which is a decoupling result for
+configuration spaces with partially summable labels. In this case, the labelling space is
+equipped with a partial multiplication and the particles are allowed to collide whenever
+their labels can be multiplied. The space CΣ(M; P) of configurations in M with par-
+tially summable labels in P has been defined in [Sal01] and its definition requires more
+sophisticated tools such as the Fulton-MacPherson configuration spaces and operad. In
+section 4.1 we recall these definitions and the concept of a partial d-monoids.
+To prove the decoupling theorem for CΣ(M; P), we develop a semi-simplicial resolu-
+tion for this space in section 4.2, denoted |DΣ(M; P)•|. In Proposition 4.13 we show
+that indeed this space is weakly equivalent to CΣ(M; P). In the decoupling context, we
+naturally encounter another space of discs with configurations which is constructed in
+Definition 4.17.
+With these semi-simplicial spaces, we prove Theorem F, combining Corollary 4.14
+and Theorem 4.18. We then use this to deduce Theorem E and Corollary G, which are,
+respectively Corollaries 4.19 and 4.20.
+4.1. Fulton-MacPherson configuration space and partially summable labels.
+In this section we recall the concept of a configuration space with summable labels in a
+partial monoid as described in [Sal01].
+A partial d-monoid P is, in essence, a space with a continuous operation that is
+not defined on all collections of elements, but only on a subset of composable elements.
+The data defining such a partial monoid consists of the subset of composable elements,
+together with an operation on this subset which is associative. We are interested in
+partial monoids that moreover have the structure of an Ed-algebra. Essential to this
+construction is Fd, the Fulton-MacPherson operad in dimension d. This is a cofibrant
+replacement for the little d-discs operad and therefore will be used to make precise the
+notion of a Ed-partial monoid. Crucially for us, the Fulton-MacPherson operad is defined
+in terms of configuration spaces of points, in which the particles in the configuration are
+allowed to collide, but keeping track of the relative position of the particles before they
+collided and the order in which collided.
+We follow the definition of the Fulton-MacPherson configuration space Ck(M) of a
+manifold M as described in [Sin04]. To make it precise, we consider M as embedded in
+some Rm and we denote by k the set {1, . . . , k}. We record the directions of particles
+in a collision, by defining for (i, j) ∈ �C2(k), a map πi,j : �Ck(Rm) → Sm−1 sending
+a configuration (x1, . . . , xk) to the unit vector in the direction xi − xj. The order of
+collision is recorded by defining for (i, j, ℓ) ∈ �C3(k) the map si,j,ℓ : �Ck(Rm) → [0, ∞]
+sending (x1, . . . , xk) to |xi − xj|/|xi − xℓ|.
+
+14
+LUCIANA BASUALDO BONATTO
+Definition 4.1 ([Sin04], Definition 1.3). The Fulton-MacPherson configuration space
+Ck(M) of the manifold M is the closure of the image of the map
+i × (πi,j| �
+Ck(M)) × (si,j,k| �
+Ck(M)) : �Ck(M) −→ M k × (Sm−1)k(k−1) × [0, ∞]k(k−1)(k−2).
+The space Ck(M) is homotopy equivalent to the ordered configuration space �Ck(M)
+[Sin04, Corollary 4.5] and whenever M is compact, Ck(M) is a compactification of
+�Ck(M). Moreover, this construction is functorial with respect to embeddings [Sin04,
+Corollary 4.8], i.e. any embedding f : M �→ N induces an embedding f∗ : Ck(M) →
+Ck(N).
+The following result gives a convenient way to represent elements of Ck(M), which
+will be used throughout the chapter.
+Proposition 4.2 ([Sin04], Theorem 3.8). Each element in Ck(M) is uniquely deter-
+mined by:
+(1) A configuration of points P1, . . . , Pl in the interior of M, with 1 ≤ l ≤ n (we
+refer to these as the infinitesimal configuration),
+(2) For each 1 ≤ i ≤ l, a tree Ti with fi leaves (twigs), no bivalent vertices,
+so that �l
+i=1 fi = n, and for each vertex in Ti of valence m an element in
+Cm(TPiM)/G(d), where G(d) is the group of affine transformations of Rd gen-
+erated by translations and positive dilations.
+(3) A global ordering of the k leaves of the trees.
+This interpretation of the elements of Ck(M) also provides a way of describing the
+map f∗ : Ck(M) → Ck(N) induced by an embedding f : M �→ N into a d′-manifold N:
+it takes a point with infinitesimal configurations P1, . . . , Pl ∈ M to one with infinitesimal
+configurations f(P1), . . . , f(Pl) ∈ N, it preserves the trees and ordering of the leaves,
+but changes the labels of the vertices of the trees by taking a label ξ ∈ Cm(TPiM)/G(d)
+to the label DPif(ξ) ∈ Cm(Tf(Pi)N)/G(d′).
+The Fulton-MacPherson operad Fd is built out of subspaces of these configurations
+in Rd.
+Intuitively, for each k ≥ 0 the space Fd(k) is the subspace of the ordered
+configurations of k points in Rd, in which the points have collided at the origin.
+Definition 4.3. The Fulton-MacPherson operad in dimension d, denoted Fd, is defined
+by taking Fd(k) to be the pullback
+Fd(k)
+Ck(Rd)
+0
+(Rd)k.
+⌟
+In other words, it is the subspace of Ck(Rd) with infinitesimal configuration given by a
+single point at the origin.
+With the description of this space as in Proposition 4.2, the composition of this operad
+is given by grafting of trees. Pictorially, we will often represent elements of Fd(k) as
+trees of configurations such as in the rightmost picture of Figure 2.
+As shown in [Sal01], there exists a model structure on the category of topological op-
+erads in which Fd is a cofibrant replacement of the little d-discs operad. An algebra over
+Fd, called by Salvatore a d-monoid, consists of a space A together with Σk-equivariant
+maps
+Fd(k) × Ak → A
+which commute with the structure maps of Fd.
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+15
+10
+9
+8
+5
+1
+7
+6
+4
+3
+2
+11
+12
+10
+9
+8
+5
+1
+7
+6
+4
+3
+2
+11
+12
+8
+5
+1
+3
+9
+7
+2
+12
+10
+6
+4
+11
+Figure 2. Representation of an element in the space F2(12) in terms
+of a tree of infinitesimal configurations.
+Definition 4.4 ([Sal01], Definition 1.7 and 2.6). A partial d-monoid is a space P to-
+gether with monomorphisms of Σk-spaces
+i : Compk �→ Fd(k) ×Σk P k.
+and composition maps ρ : Compk → P such that
+(1) The unit η × P : P → Fd(1) × P factors uniquely through �η : P → Comp1 and
+the composition ρ ◦ �η is the identity idP ;
+(2) For f ∈ Fd(k) and [fj; pj] ⊂ Compmj, for j = 1, . . . , k, the element
+[f; ρ(f1; p1), . . . , ρ(fk; pk)] ∈ Fd(k) ×Σk P k
+belongs to Compk if and only if
+[µ(f; f1, . . . , fk); p1, . . . , pk]
+belongs to Compm1+···+mk. Moreover, if that is the case, then their image by ρ
+coincide.
+A pointed space (P, 0) is a partial d-monoid with unit if in addition it satisfies
+3. For every k there is an inclusion u : Fd(k) ×Σk
+�
+k P �→ Compk such that the
+composition with i◦u is the subspace inclusion, and ρ◦i : Fd(k)×Σk
+�
+k P → P
+is the projection onto the P coordinate.
+The definition above is better understood when in comparison to Ed-algebras, which
+Salvatore calls d-monoids. Given a d-monoid M and a infinitesimal configuration f ∈
+Fd(k), we can always compose any k elements of M via the composition rule described by
+f. In a partial d-monoid P, that is not the case. We instead are given a subset of the k-
+tuples of P which are composable via the operation described by a given f ∈ Fd(k). The
+space Compk should be then thought of as the pairs of possible composition rules in Fd
+together with the tuples of elements of P which can be composed with this composition
+rule. The map ρ is computes the results of compositions, whenever they are defined.
+Example 4.5.
+(a) Any space X admits the structure of a trivial partial d-monoid,
+by defining Comp1 = {[1, x] : x ∈ X} and Compk = ∅, for all k ̸= 1. In this
+case, the only composition rule that can be performed is the identity and no
+other collection of points in X is composable in any way.
+(b) When X is a space equipped with a basepoint ∗, we can define a unital partial
+d-monoid by setting Compk = Fd(k) × ∨kX and ρ(f, x) = xi, where xi is the
+unique non-basepoint coordinate, or ∗ otherwise. In this case, the basepoint
+acts as a unit and compositions are only defined when done with the unit.
+
+16
+LUCIANA BASUALDO BONATTO
+(c) Every d-monoid is trivially a partial d-monoid. In particular, every Ωd-space is
+a partial d-monoid.
+(d) The canonical inclusion id,n : Rd �→ Rd+n also allows us to construct a partial
+(d + n)-monoid P from a partial d-monoid by adding n trivial composition
+directions. We call this the naive upgrade of a partial d-monoid P and denote
+it Td,nP. The underlying space of Td,nP is P, and we define CompTd,nP
+k
+to be
+the image of
+CompP
+k �→ Fd(k) ×Σk P k
+(in,k)∗
+�−−−−→ Fd+n(k) ×Σk P k = Fd+n(k) ×Σk (Td,nP)k.
+In [Sal01], it was always assumed that the partial d-monoids were good, in the sense
+described below.
+Definition 4.6. A partial d-monoid P is good if for every k the inclusion CompP
+k �→
+Fd(k) ×Σk P k is a cofibration.
+From now on, we always assume partial d-monoids to be good and to have a unit.
+For future applications, we are further interested in partial d-monoids with compatible
+actions of GLd(R), so we introduce this concept here and perform our constructions in
+this more general setting.
+Recall from Proposition 4.2 that an element of Fd(k) is described by a tree with k
+ordered leaves and a decoration of the vertices by elements of �C|v|(Rd)/G(d). The action
+of GLd(R) on Rd induces an action on �C|v|(Rd)/G(d). This gives an action of this group
+on Fd(k) for every k, and it is simple to check that the operad maps µ are all GLd(R)
+equivariant.
+Definition 4.7 ([Sal01], Definition 4.3). The framed Fulton-MacPherson operad de-
+noted fFd, is the operad defined by fFd(k) = Fd(k) × GLd(R)k, with structure map
+�µ((x, g1, . . . , gk);(x1, g1
+1, . . . , gm1
+1 ), . . . , (xk, g1
+k, . . . , gmk
+k
+))
+= (µ(x; g1x1, . . . , gkxk), g1g1
+1, . . . , gkgmk
+k
+).
+The construction above is an instance of the construction A ⋊ G, the semi-direct
+product of an operad A and group G. This construction, and the proof that the above
+indeed defines an operad can be found in [SW03, Definition 2.1].
+Definition 4.8 ([Sal01], Definition 4.8). A framed partial d-monoid with unit is a
+pointed GLd(R)-space P together with monomorphisms of Σk ≀ GLd(R)-spaces
+i : fCompk �→ fFd(k) ×Σk P k.
+and GLd(R)-equivariant composition maps ρ : fCompk → P satisfying properties 1-3
+of Definition 4.4.
+The GLd(R)-bundle of frames on M induces a (GLd(R))k-bundle fCk(M) on Ck(M),
+acted on by Σk. This is called the framed configuration space of k points in M. Using
+the description of Proposition 4.2, an element of the space fCk(M) can be uniquely
+determined by an infinitesimal configuration in M with labelled trees, together with
+additional k frames of the tangent planes associated to the k leaves of the trees.
+Then the space of framed configurations fC(M) =
+�
+k fCk(M) is a right fFd-module,
+with GLd(R)-equivariant multiplication maps
+m : fCk(M) ×Σk (fFd(n1) × · · · × fFd(nk)) → fCn1+···+nk(M).
+defined by grafting the element of fFd(ni) on the i-th leaf of the element of fCk(M),
+for all i = 1, . . . , k and using the frame on the leaves to identify Cm(Rd)/G(d) with a
+configuration on the tangent space of M ([Sal01, Proposition 4.5]). It is simple to verify
+that any co-dimension zero embedding e : M �→ N induces a right fFd-homomorphism
+e∗ : fC(M) �→ fC(N).
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+17
+Definition 4.9 ([Sal01], Definition 4.14). Let P be a framed partial d-monoid, and let
+M be a manifold of dimension d. Then the space of configurations in M with partially
+summable labels in P, denoted CΣ(M; P) is defined as the co-equalizer of the following
+�
+k
+�
+fCk(M) ×Σk
+�
+�
+π∈Map(n,k)
+k�
+i=1
+fCompP
+π−1(i)
+��
+�
+k
+fCk(M) ×Σk P k
+(m×id)◦(id ×i)
+id ×ρk
+An element of CΣ(M; P) is then an equivalence class of elements in
+�
+k fCk(M) ×Σk
+P k. From the description of fCk(M) above, we can see that an element in CΣ(M; P)
+can be represented by an infinitesimal configuration w1, . . . , wl of ℓ < k points in M
+together with trees Ti, for i = 1, . . . , ℓ, where the vertices of Ti are labelled by elements
+in xi
+j ∈ fFd, and the leaves of the trees are labelled by elements of pi
+k ∈ P.
+The
+equivalence relation defining CΣ(M; P) implies that if some leaves labelled by p1, . . . , pk
+are departing from a vertex labelled by x ∈ fFd(k) and the composition ρ(x; p1, . . . , pk)
+is defined, then we identify this configuration with the one in which such leaves are
+removed and their vertex is replaces by a leaf labelled by ρ(x; p1, . . . , pk).
+For a framed partial d-monoid P and a co-dimension zero embedding f : M �→ N,
+we get an induced map
+f∗ :
+�
+k
+fCk(M) ×Σk P k →
+�
+k
+fCk(N) ×Σk P k.
+With the description above, a point in the domain with infinitesimal configurations
+w1, . . . , wℓ in M, with trees Ti, for i = 1, . . . , ℓ, where the vertices of Ti are labelled by
+elements xi
+j ∈ fFd, and the leaves of the trees are labelled by elements of pi
+k ∈ P, is
+taken to the point with infinitesimal configuration φ(x1), . . . , φ(xℓ), trees Ti, i = 1, . . . , ℓ,
+and corresponding labels
+dxiφ · xi
+j and dxiφ · pi
+k.
+Here we are using the standard actions of GLd(R) on fFd and P. By the equivariance
+condition in the definition of a framed partial monoid, the map preserves the equivalence
+classes described above. Therefore any such co-dimension zero embedding induces a map
+f∗ : CΣ(M; P) → CΣ(N; P).
+(4.1)
+Seeing the group Diff∂(M) as a subspace of Emb(M, M), the above construction
+shows that CΣ(M; P) admits an action of Diff∂(M). We will be interested in a decou-
+pling theorem for the space CΣ(M; P)//Diff∂(M).
+4.2. Disc models for configuration spaces with partially summable labels. Let
+M be a smooth compact d-manifold, possibly with boundary, and P a framed partial d-
+monoid. The group Σk ≀GLd(R) has a canonical inclusion into Diff∂(
+�
+k Rd). We consider
+Emb(
+�
+k Rd, M) to be a Σk ≀ GLd(R)-space with action given by pre-composition by the
+inverse, and CΣ(
+�
+k Rd; P) to be a Σk ≀ GLd(R)-space with action induced by the action
+of Diff∂(
+�
+k Rd), as described in the previous section.
+Definition 4.10 ([MT14]). Let Z be a pointed GLd(R)-space, The space of tubular
+configurations in M with labels in Z, denoted D(M; Z), is the quotient
+�
+��
+k≥0
+Emb(
+�
+k Rd, M)
+×
+Σk≀GLd(R) Zk
+�
+�
+�
+∼
+where (e1, . . . , ek; z1, . . . , zk) ∼ (e1, . . . , ek−1; z1, . . . , zk−1) whenever zk is the basepoint
+of Z, for ei : Rd �→ M and zi ∈ Z.
+
+18
+LUCIANA BASUALDO BONATTO
+The space D(M; Z) is equipped with an action of Diff∂(M): for ψ ∈ Diff∂(M), an
+embedding e :
+�
+k Rd �→ M, and z = (z1, . . . , zk) ∈ Zk, we define
+φ · [e, z] = [φ ◦ e; De(01)φ · z1, . . . , De(0k)φ · zk]
+where 0i denotes the origin of the i-th component of
+�
+k Rd.
+Lemma 4.11 ([MT14], Propositions 2.7, 2.8, and 3.6). The inclusion of the origin
+i : ∗ �→ Rd induces a Diff∂(M)-equivariant weak equivalence
+i∗ : D(M; Z)
+≃
+−→ C(M; Z).
+One can think of D(M; Z) as disc models for configuration spaces with labels. To
+construct a disc model for summable labels, we need much more structure:
+Definition 4.12. The space of surrounded configurations in M with summable labels
+in P, denoted DΣ(M; P), is the quotient
+�
+��
+k≥0
+Emb(
+�
+k Rd, M)
+×
+Σk≀GLd(R) CΣ(
+�
+k Rd; P)
+�
+�
+�
+∼
+where (e :
+�
+mRd → M, ξ) ∼ (e′ :
+�
+n Rd → M, ξ′) if there are injections k �→ m and k �→ n
+such that the induced inclusions i1 :
+�
+k Rd →
+�
+mRd and i2 :
+�
+k Rd →
+�
+n Rd satisfy
+ξ ⊂ Im i1, ξ′ ⊂ Im i2
+e ◦ i1 = e′ ◦ i2,
+and e∗(ξ) = e′
+∗(ξ′).
+Here e∗ denotes the map of configurations with summable labels induced by a codimen-
+sion zero embedding, as described in (4.1). Since the Fulton-MacPherson configuration
+spaces are functorial for co-dimension zero embeddings, we get a map
+p : DΣ(M; P) → CΣ(M; P)
+(4.2)
+taking a class (e, ξ) to the configuration e∗(ξ). By definition, this map does not depend
+on the choice of representative (e, ξ). The space DΣ(M; P) admits a partial ordering
+by declaring (e, ξ) < (e′, ξ′) if, for some representative of the classes, e(ξ) = e′(ξ′)
+and Im e ⊃ Im e′. We denote by DΣ(M; P)• the semi-simplicial nerve of the poset
+DΣ(M; P).
+By the definition of the partial order, the map p induces an augmentation DΣ(M; P)• →
+CΣ(M; P). In particular, DΣ(M; P)• is an augmented topological flag complex (Defini-
+tion 2.2), since the space of n-simplices is indeed an open subspace of the (n + 1)-tuples
+of vertices with the same image under the augmentation map, and the condition for
+a tuple to form an n-simplex is just given by checking the pairwise order relation. It
+might be helpful to keep in mind the following visualisation for elements in the space
+|DΣ(M; P)•|: any point can be expressed by a tuple
+((e0, ξ0) < · · · < (ek, ξk); t0, . . . , tk)
+t0 · · · + tk = 1.
+The poset construction implies that for such a tuple e0(ξ0) = · · · = ek(ξk) and Im e0 ⊃
+· · · ⊃ Im ek. We can visualise it as a collection of descending embedded discs around
+the configuration ei(ξi) in M. Each collection of embedded discs has weights adding
+up to 1 and when the weight associated to a collection of embeddings goes to zero,
+that collection disappears. The equivalence relation guarantees that we can always get
+a representative for the collection of discs such that each component has at least one
+particle of the configuration inside it. See Figure 3.
+Proposition 4.13. The map p : DΣ(M; P) → CΣ(M; P) induces a weak homotopy
+equivalence
+|DΣ(M; P)•| → CΣ(M; P).
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+19
+t0
+t2
+t1
+Figure 3. Element in a 2-simplex of |DΣ(F3)|.
+The proof of the above will be a direct application of Theorem 2.3 [GRW14, Theorem
+6.2].
+Proof. As discussed before, the augmented semi-simplicial space p : DΣ(M; P)• →
+CΣ(M; P) is an augmented topological flag complex. We need to verify that the hy-
+potheses of Theorem 2.3 are satisfied:
+(i) To see that p : DΣ(M; P) → CΣ(M; P) has local sections, take f : Dn →
+CΣ(M; P) and a point (e, ξ) ∈ p−1(f(x)). Then the image of e is an open subset
+of M containing f(x), and therefore the subspace V of configurations contained
+in Im e is an open subset of CΣ(M; P) containing f(x). Let U = f −1(V ), which
+is an open neighbourhood of x in Dn. Then the map F : U → DΣ(M; P) taking
+y to (e, e−1(f(y))) satisfies p ◦ F = f|U and F(x) = (e, ξ).
+(ii) p : DΣ(M; P) → CΣ(M; P) is surjective, since any configuration ξ admits a
+tubular neighbourhood e. Then (e, e−1(ξ)) is an element of DΣ(M; P) in the
+pre-image of ξ.
+(iii) For a configuration ξ ∈ CΣ(M; P) and a non-empty finite subset {(e1, ξ1), . . . , (ek, ξk)}
+in its pre-image, we can always find e an embedding of Rd’s containing ei(ξi)
+and contained in all ei. For instance by taking e to small open discs around the
+points in the configuration.
+Then by Theorem 2.3, the map |DΣ(M; P)•| → CΣ(M; P) is a weak homotopy equiva-
+lence.
+□
+The space DΣ(M; P) is equipped with an action of Diff∂(M): for ψ ∈ Diff∂(M), an
+embedding e :
+�
+k Rd �→ M, and ξ = (x1, . . . , xk; p1, . . . , pk) a configuration of points in
+�
+k Rd with labels in P, we define φ · [e, ξ] = [φ ◦ e, ξ]. It is simple to verify this action
+is well-de���ned and preserves the partial order in DΣ(M; P). It follows directly from
+the definition that the augmentation map p : DΣ(M; P) → CΣ(M; P) is Diff∂(M)-
+equivariant.
+Since the partial order in DΣ(M; P) is compatible with the Diff∂(M)-action, it in-
+duces a fibrewise partial order on DΣ(M; P) ×Diff∂(M) Emb(M, R∞) over BDiff∂(M).
+Then the semi-simplicial nerve of the poset DΣ(M; P) ×Diff∂(M) Emb(M, R∞) is simply
+is the fibrewise semi-simplicial nerve DΣ(M; P)• ×Diff∂(M) Emb(M, R∞).
+Corollary 4.14. The map DΣ(M; P) → CΣ(M; P) induces a weak homotopy equiva-
+lence
+|DΣ(M; P)•
+×
+Diff∂(M) Emb(M, R∞)| −→ CΣ(M; P)//Diff∂(M).
+The above follows directly from the definition of the definition of the partial order on
+DΣ(M; P) ×Diff∂(M) Emb(M, R∞) and Proposition 4.13.
+Corollary 4.14 allows us to use a disc model for the space of configurations with
+partially summable labels when proving the decoupling in this setting. We finish this
+section by introducing another space of discs and configurations which will be used in
+the decoupling.
+Definition 4.15. Let Z be a pointed GLd(R)-space and let M be a smooth compact
+manifold of dimension n > d. The space of d-tubular configurations in M with labels in
+
+20
+LUCIANA BASUALDO BONATTO
+Z, denoted Dd(M; Z), is the quotient
+�
+��
+k≥0
+Emb(
+�
+k Rd, M)
+×
+Σk≀GL+
+d (R)
+Zk
+�
+�
+�
+∼
+where (e1, . . . , ek; z1, . . . , zk) ∼ (e1, . . . , ek−1; z1, . . . , zk−1) whenever zk is the basepoint
+of Z, for ei : Rd �→ M and zi ∈ Z.
+The difference between the spaces Dd(M; Z) and D(M; Z) (Definition 4.10) is that on
+the former we look at embedded discs of a lower dimension than the ambient manifold
+M.
+Lemma 4.16. Let n > d, and denote by Ed,n denote the total space of the canonical
+GL+
+d (R)-bundle over the oriented Grassmanian Gr+(d, n). The inclusion of the origin
+i : ∗ �→ Rd induces a weak equivalence
+i∗ : Dd(Rn; Z)
+≃
+−→ C(M; (Ed,n)+
+∧
+GL+
+d (R)
+Z).
+This result should be seen as an analogue of Lemma 4.11 in the setting of d-tubular
+configurations in Rn.
+Proof. Recall that Emb(
+�
+k Rd, Rn) ≃ �Ck(Rn) × (Ed,n)k, where the map to �Ck(Rn) is
+induced by the inclusion of the origins. Then we get weak equivalences
+Emb(
+�
+k Rd, M)
+×
+(Σk≀GL+
+d (R))
+Zk
+≃
+−→ ( �Ck(Rn) × (Ed,n)k)
+×
+(Σk≀GL+
+d (R))
+Zk.
+These respect the equivalence relations and therefore induce a map
+i∗ : Dd
+Σ(Rn; P)
+≃
+−→ C(M; (Ed,n)+
+∧
+GL+
+d (R)
+Z).
+The proof then follows from the same arguments of [MT14, Propositions 2.7 and 2.8].
+□
+We also need an analogue of Definition 4.12 for the case of d-tubular configurations.
+Definition 4.17. Let M be a smooth compact manifold of dimension n > d. The space
+of d-surrounded configurations in M with summable labels in P, denoted Dd
+Σ(M; P), is
+the quotient
+�
+��
+k≥0
+Emb(
+�
+k Rd, M)
+×
+Σk≀GL+
+d (R)
+CΣ(
+�
+k Rd; P)
+�
+�
+�
+∼
+where (e :
+�
+mRd → M, ξ) ∼ (e′ :
+�
+n Rd → M, ξ′) if there are injections k �→ m and k �→ n
+such that the induced inclusions i1 :
+�
+k Rd →
+�
+mRd and i2 :
+�
+k Rd →
+�
+n Rd satisfy
+ξ ⊂ Im i1, ξ′ ⊂ Im i2 and e ◦ i1 = e′ ◦ i2.
+We equip this space with a partial ordering by declaring (e, ξ) < (e′, ξ′) if, for some
+representative of the classes, e(ξ) = e′(ξ′) and Im e ⊃ Im e′.
+We denote the semi-
+simplicial nerve of this poset by Dd
+Σ(M; P)•.
+4.3. The decoupling theorem. In this section we prove the main theorem of this
+paper, which is a decoupling result for CΣ(Fg,b; P)//Diff∂(Fg,b). As in Section 3, we take
+as a model for EDiff∂(Fg,b) the space Emb(Fg,b, R∞). For every g and b ≥ 0 we have
+τΣ × εΣ : DΣ(Fg,b; P) × Emb(Fg,b, R∞) −→ Emb(Fg,b, R∞) × D2
+Σ(R∞; P)
+((e :
+�
+k R2 �→ Fg,b, ξ), f : Fg,b �→ R∞) �−→ (f, (f ◦ e, ξ)).
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+21
+This map is Diff∂(Fg,b)-equivariant with respect to the diagonal action on the domain
+and the action on Emb(Fg,b, R∞) on the target, and it preserves the poset structures.
+Hence it induces a map τΣ × εΣ fitting into the following diagram
+|DΣ(Fg,b; P)•
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞)|
+|BDiff∂(Fg,b) × D2
+Σ(R∞; P)•|
+CΣ(Fg,b; P)
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞)
+BDiff∂(Fg,b) × |D2
+Σ(R∞; P)•|
+≃
+τΣ×εΣ
+≃
+Theorem 4.18. The map
+τΣ × εΣ : |DΣ(Fg,b; P)•
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞)| → |BDiff∂(Fg,b) × D2
+Σ(R∞; P)•|
+induces a homology isomorphism in degrees ≤ 2
+3g.
+Theorem 4.18 and Corollary 4.14 imply Theorem F.
+The proof of the above, will consist on showing that τΣ × εΣ is a level-wise homology
+isomorphism of semi-simplicial spaces in degrees ≤ 2
+3g and then show that this implies
+that the same holds on the geometric realisations, as done in [GRW17, Section 4]. To do
+this, we will use the spectral sequence recalled in Section 2.2, Lemma 4.11, and Theorem
+3.2, the decoupling result for the space of non-colliding configurations with labels.
+Throughout the proof, it will be helpful to keep in mind Figure 4.
+Figure 4. Correspondence of (4.3) between an element of DΣ(F3; P)2
+(top) and D(F3, Z2) (bottom). The dotted regions represent the em-
+bedded R2’s and the arrows indicate their labels.
+Proof of Theorem 4.18. We start by showing that
+DΣ(Fg,b; P)•
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞) → BDiff∂(Fg,b) × D2
+Σ(R∞; P)•
+induces level-wise homology isomorphisms in degrees ≤ 2
+3g.
+Let Zp be the subspace of DΣ(R2; P)p consisting of those tuples e = ((e0, ξ0), . . . , (ep, ξp))
+with e0 = id. This space is pointed by the unique class on the empty configuration. We
+
+%
+%22
+LUCIANA BASUALDO BONATTO
+show there is a Diff∂(Fg,b)-equivariant homeomorphism
+DΣ(Fg,b; P)p ∼= D(Fg,b; Zp)
+(4.3)
+where D(Fg,b; Zp) is the space of tubular configurations with labels in Zp (see Definition
+4.10). Let [(e0, ξ0) < · · · < (ep, ξp)] denote the an element in DΣ(Fg,b; P)p. Then, by
+definition e0(ξ0) = · · · = ep(ξp) and Im e0 ⊃ · · · ⊃ Im ep. Denote by ij : R2 �→
+�
+k R2 the
+map induced by the inclusion {j} �→ {1, . . . , k}, for 1 ≤ j ≤ k. The map DΣ(Fg,b; P)p →
+DΣ(Fg,b; Zp) takes a sequence ((e0, ξ0), . . . , (ep, ξp)), to the class represented by the
+embedding e0 and and the label associated to the jth component R2 ⊂
+�
+k R2 given by
+((id, ξ0), ((e0 ◦ ij)−1 ◦ e1, ξ1) . . . , ((e0 ◦ ij)−1 ◦ ep, ξp)) ∈ Zp.
+For intuition behind this homeomorphism, see Figure 4. It is simple to explicitly con-
+struct an inverse for this map and check this is a Diff∂(Fg,b)-equivariant homeomor-
+phism.
+By Lemma 4.11, the inclusion of the origins defines Diff∂(Fg,b)-equivariant map
+D(Fg,b; Zp) → C(Fg,b; Zp) which is a weak-equivalence of Diff∂(Fg,b)-spaces. Together
+with the homeomorphism (4.3) this implies that
+DΣ(Fg,b; P)p
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞)
+≃
+−→ C(Fg,b; Zp)
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞).
+(4.4)
+Similarly, we now show that
+D2
+Σ(R∞; P)p
+≃
+−→ C(R∞; (ESO(2))+
+∧
+SO(2) Zp).
+(4.5)
+By the same argument as above, we get a homeomorphism
+D2
+Σ(R∞; P)p ∼= D2(R∞; Zp)
+where D2(R∞; Zp) is the space of d-tubular configurations with labels in Zp as in Def-
+inition 4.15. By Lemma 4.16, the inclusion of the origins induces a weak homotopy
+equivalence D2(R∞; Zp) ≃ C(R∞; (ESO(2))+ ∧SO(2) Zp) which implies the homotopy
+equivalence (4.5).
+Then we have a commutative diagram
+DΣ(Fg,b; P)p
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞)
+C(Fg,b; Zp)
+×
+Diff∂(Fg,b) Emb(Fg,b, R∞)
+BDiff∂(Fg,b) × D2
+Σ(R∞; P)p
+BDiff∂(Fg,b) × C(R∞; ESO(2)+
+∧
+SO(2) Zp)
+≃
+(τΣ×εΣ)p
+τ×ε
+≃
+where the top and bottom maps are weak equivalences by (4.4) and (4.5), respectively.
+The right-hand map induces homology isomorphisms in degrees ≤ 2
+3g by Theorem 3.2
+with Z = Zp. Therefore so does the map (τΣ × εΣ)p.
+This implies that the map between the spectral sequences associated to the semi-
+simplicial spaces X• = DΣ(Fg,b; P)•×Diff∂(Fg,b)Emb(Fg,b, R∞), and Y• = BDiff∂(Fg,b)×
+D2
+Σ(R∞; P)•
+E1
+p,q = Hq(Xp)
+Hp+q(|X•|)
+E′1
+p,q = Hq(Yp)
+Hp+q(|Y•|).
+τΣ×εΣ
+induces an isomorphism on the E1-pages for all q ≤ 2
+3g, and therefore the right-hand
+map is also an isomorphism in such degrees.
+□
+
+DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES
+23
+As in the case for labelled configuration spaces, the Decoupling Theorem for Sum-
+mable Labels, allows us to deduce homological stability results.
+Corollary 4.19. For b ≥ 1, the map induced by gluing F1,1 along the boundary
+CΣ(Fg,b; Z)//Diff∂(Fg,b) → CΣ(Fg+1,b; Z)//Diff∂(Fg+1,b)
+induces a homology isomorphism in degrees ≤ 2
+3g.
+The proof follows from the same arguments as in the proof of Corollary 3.4.
+4.4. Monoids of configurations on surfaces with partially summable labels.
+In this section, we show that Theorem 4.18 implies a splitting for the group completion
+of the monoid of configuration on surfaces with partially summable labels, analogous to
+Corollary 3.7.
+As in Section 3.2, gluing two surfaces Fg,1 and Fh,1 along part of their boundary
+defines an associative multiplication
+BDiff∂(Fg,1) × BDiff∂(Fh,1) → BDiff∂(Fg+h,1).
+For (P, 0) a framed partial 2-monoid with unit, the operation on diffeomorphism
+groups and spaces EDiff∂(Fg,1) described above, together with the fixed identifications
+Fg+h,1 = Fg,1#∂Fh,1, induce an associative multiplication also on the Borel construc-
+tions
+µ : CΣ(Fg,1; P)//Diff∂(Fg,1)×CΣ(Fh,1; P)//Diff∂(Fh,1) → CΣ(Fg+h,1; P)//Diff∂(Fg+h,1).
+Analogous to the construction of Chapter 3, we denote the associated Borel construction
+by
+MCΣ(P)g = CΣ(Fg,1; P)//Diff∂(Fg,1).
+This multiplication makes MCΣ(P) =
+�
+g≥0MCΣ(P)g into a topological monoid, which
+we refer to as the monoid of configurations with summable labels in P.
+On the other hand, the poset D2
+Σ(R∞; P) can be made into a partially ordered topo-
+logical monoid, using the same strategy Segal used to define a topological monoid equiv-
+alent to C(R∞, X), as we recalled in Section 3.2.
+Corollary 4.20. For any path-connected framed partial 2-monoid with unit P, there is
+a homotopy equivalence
+ΩB(MCΣ(P)) ≃ ΩB
+��
+g BDiff∂(Fg,1)
+�
+× ΩB |D2
+Σ(R∞; P)•|.
+The proof of the above result consists of constructing a zig-zag of monoids
+(4.6)
+�
+g≥0
+|DΣ(Fg,1; P)•//Diff∂(Fg,1)|
+�
+g≥0
+BDiff∂(Fg,1) × |D2
+Σ(R∞; P)•|
+MCΣ(P)
+≃
+τΣ×εΣ
+The vertical arrow is induced by Corollary 4.14 while the horizontal arrow is induced
+by the decoupling map. We describe the monoidal structures on the top spaces using a
+general construction for posets recalled in Section 2.1.
+Proof of Corollary 4.20. The space �
+g≥0
+|DΣ(Fg,1; P)•//Diff∂(Fg,1)| is homotopy equiva-
+lent to the geometric realisation of the semi-simplicial nerve of the poset
+�
+g≥0
+DΣ(Fg,1; P)//Diff∂(Fg,1)
+
+24
+LUCIANA BASUALDO BONATTO
+with partial order defined component-wise. As before, gluing surfaces along part of their
+boundary makes
+�
+g≥0DΣ(Fg,1; P)//Diff∂(Fg,1) into a partially ordered topological monoid.
+Moreover, one can verify that the augmentations DΣ(Fg,1; P)• → CΣ(Fg,1; P) induce a
+map of monoids. Together with Lemma 2.1, this gives a map of monoids
+�
+g≥0
+|DΣ(Fg,1; P)•//Diff∂(Fg,1)| → MCΣ(P).
+(4.7)
+On the other hand, the decoupling map gives a map of topological monoids
+�
+g≥0
+|DΣ(Fg,1; P)•//Diff∂(Fg,1)| →
+�
+g≥0
+BDiff∂(Fg,1) × |D2
+Σ(R∞; P)•|
+(4.8)
+just as in Lemma 3.8.
+All that remains is to verify that the maps (4.7) and (4.8) induce homotopy equiva-
+lences on group completions. As the group completions are loop spaces, they are simple
+and, by Whitehead’s theorem, it suffices to show the maps induce homology equivalences
+on the group completions. All monoids are homotopy commutative, hence the group
+completion theorem [MS76] can be applied. It implies that it is enough to show that
+the induced maps on limit spaces
+|DΣ(F∞; P)•//Diff∂(F∞)| → CΣ(F∞; P)//Diff∂(F∞)
+(4.9)
+|DΣ(F∞; P)•//Diff∂(F∞)| → BDiff∂(F∞) × |D2
+Σ(R∞; P)•|
+(4.10)
+are homology equivalences. The first map is a weak equivalence by Corollary 4.14, while
+the second map is a homology equivalence by Theorem 4.18 and Corollary 4.19.
+□
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