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Condensed Matter Physics +2Department of Physics of Complex Systems +Weizmann Institute of Science, Rehovot 7610001, Israel +We propose a method to extract the mutual exchange statistics of the anyonic excitations of +a general Abelian fractional quantum Hall state, by comparing the tunneling characteristics of a +quantum point contact in two different experimental conditions. In the first, the tunneling current +between two edges at different chemical potentials is measured. In the second, one of these edges is +strongly diluted by an earlier point contact. We describe the case of the dilute beam in terms of a +time-domain interferometer between the anyons flowing along the edge and quasiparticle-quasihole +excitations created at the tunneling quantum point contact. In both cases, temperature is kept +large, such that the measured current is given to linear response. Remarkably, our proposal does +not require the measurement of current correlations, and allows us to carefully separate effects of +the fractional charge and statistics from effects of intra- and inter-edge interactions. +Introduction.— It has been almost four decades since +the initial proposal that the elementary quasiparticles +of fractional quantum Hall (FQH) systems obey anyonic +statistics [1]. Despite the apparent maturity of the field, +the pursuit to definitively observe the physical quanti- +ties and quantum numbers characterizing anyons [2, 3] is +constantly being reinvigorated [4–20]. In particular, early +2020 saw two major experimental steps forward: the ob- +servation of anyonic braiding in a Fabry-Perot interfer- +ometer [21], and demonstration of a so-called “anyon col- +lider” [22, 23] using cross-correlation measurements. +Here we show that anyonic statistics can be inferred di- +rectly from conductance measurements, without requir- +ing current correlation measurements or explicitly build- +ing an interferometer. The configuration we propose to +obtain this result consists of a quantum point contact +(QPC) between two edges of a general Abelian FQH state +which are driven out of equilibrium. The edges may be +driven off-equilibrium by one of three methods: inject- +ing a single quasiparticle into one of the edges; injecting +a Poissonian, dilute beam of quasiparticles into one of +the edges; and placing a finite bias voltage between the +edges. +Our proposed setup, shown in Fig. 1(a), allows a +smooth transition between the dilute Poissonian beam +and a full beam at finite bias voltage. +This is ob- +tained by tuning a second, injection QPC from fully open +(a differential conductance, Ginj ≡ dIinj/dV , satisfying +Ginj/σxy → 0) to fully closed (Ginj/σxy → 1). We hence- +forth refer to these as the dilute and full limits, respec- +tively. +We propose sweeping Ginj through this range, and +measuring the ratio I/Iinj, where I is the measured cur- +rent after the tunneling QPC, and Iinj is the injected inci- +dent current, as defined in Fig. 1(a). Comparing the val- +ues at the dilute and full limits cancels out non-universal +constants, yielding the relation, +� I(T) +Iinj(T) +� +dilute += +νe2 +2πe∗ +1e∗ +2 +sin 2θ12 +� I(T) +Iinj(T) +� +full ++ Gdirect +Ginj +. (1) +Here, e∗ +1 is the tunneling quasiparticle charge, e∗ +2 the +injected quasiparticle charge, δ1 is the tunneling quasi- +particle scaling dimension, θ12 is the mutual statistics +phase between the injected and tunneling quasiparticles, +T is temperature, and Gdirect is a residual conductance +corresponding to direct tunneling [24–26] through both +QPCs. +The full crossover between these two limits is +shown schematically in Fig. 1(b). +The mechanism leading to this result is a time-domain +interferometer at the tunneling QPC which is created by +the dilute incident beam. The interference is between two +processes, in which a quasiparticle-quasihole excitation +occurs at the tunneling QPC either before or after the +arrival of an injected quasiparticle (see Fig. 2). A similar +physical picture has been shown in Refs. [25, 27, 28]. We +further find that this interference is sensitive to the mu- +tual statistics phase between the injected and the tunnel- +ing quasiparticles, θ12. We emphasize that these quasi- +particles are not necessarily of the same type, although +they must be supported by the same FQH liquid. +Since our focus is the interference of two amplitudes +which differ from one another by the orderings of events, +the key point of our analysis is the identification of the +phase differences between the two orderings. +We find +phase differences that are determined by the quasiparti- +cle charge e∗, which is a fraction of the electron charge +for non-integer values of ν [4–6]; the scaling dimension δ, +which defines the zero-temperature time correlations of +the quasiparticle via the relation ⟨ψ†(τ)ψ(0)⟩ ∼ τ −2δ +[29–32]; and the exchange statistics phase θ, which for +anyons take special values beyond the fermionic π and +the bosonic 2π [1–3]. +We are interested here in isolating the effect of θ from +the other two effects. +In particular, we would like to +separate it from the effect of δ. +For non-interacting +edges, in which all the modes propagate in the same di- +rection, 2πδ = θ; however, in general δ is affected by +non-universal factors, such as intra-edge and inter-edge +interactions, 1/f noise or neutral modes [33–38]. This in +stark contrast to the charge, exchange statistics phase, +or filling factor, which are universal. +We separate the effect of θ from that of δ by tuning +arXiv:2301.00021v1 [cond-mat.mes-hall] 30 Dec 2022 + +2 +(a) +𝜈 +𝐼 +𝑉 +𝑒1 +∗, 𝛿1 +𝑒2 +∗, 𝛿2 +Injection +QPC +Tunneling +QPC +𝐼inj +𝑢 +𝑑 +𝑎 +𝐷𝑎 +𝑆𝑢 +𝐷𝑢 +𝑆𝑑 +(b) +200 +400 +600 +800 +1000 +0.30 +0.35 +0.40 +0.45 +400 +800 +0.1 +0.2 +FIG. 1. (a) Two counter-propagating edge modes (u/d) of +a fractional quantum Hall droplet at filling factor ν are con- +nected by a quantum point contact, through which quasipar- +ticles of charge e∗ +1 and scaling dimension δ1 can tunnel. Cur- +rent is measured at the lower edge’s drain, denoted by I. A +current of Iinj is injected into the upper edge via a second, in- +jection QPC, e.g. from a third auxiliary edge mode (a). The +injection QPC is placed at a bias voltage of V , and allows +tunneling of quasiparticles of charge e∗ +2 and scaling dimension +δ2. All other sources and drains are grounded. (b) The ratio +between I/Iinj in the dilute case and I/Iinj in the full case, +as a function of temperature, for ν = e∗ +1/e = e∗ +2/e = 1/3, +and for different scaling dimensions δ1. For the dilute case, +we Iinj = 10pA, and assume kBT ≪ eV for all relevant tem- +peratures, such that the contribution from Gdirect to Eq. (1) +is negligible. In the full case, we use V = 10µV . Both cases +use ξ = 72mK, τc = 10−13s. When the dilute case satisfies +ℏIinj/e ≪ kBT ≪ eV ≪ ℏ/τc, and the full case satisfies +ℏIinj/e = νeV/2π ≪ kBT ≪ ℏ/τc, the ratio approaches an +asymptote that does not depend on scaling dimension, allow- +ing extraction of the mutual statistics θ12. Inset: I/Iinj for the +dilute and full cases as a function of temperature for δ1 = 1/6, +the canonical value for a Laughlin 1/3 state. +the system to a regime where δ only affects observables +through a non-universal prefactor, which then cancels out +in the ratio of currents given in Eq. (1). We arrive at this +regime by employing a careful ordering of the various +energy scales in the system, such that ℏIinj/e ≪ kBT +throughout the entire crossover of Ginj. +This ensures +that the current I is given to linear response in Iinj. We +present an analytic expression generalizing Eq. (1) out- +side of this regime in App A, Eq. (A5). +While in the full limit the edge that enters the tunnel- +ing QPC is in equilibrium at chemical potential V , at the +dilute limit we need the injection QPC to reflect only a +small fraction of the impinging electrons, such that the +resulting injection current is Poissonian and rare. Said +differently, the injected current in this limit must satisfy +Iinj ≪ σxyV . Furthermore, the beam must still be dilute +when arriving at the tunneling QPC. As such, the dis- +tance between the two QPCs must be sufficiently small +that no equilibration or dephasing occurs along the way. +Finally, we assume that tuning the injection QPC does +not affect the transparency of the tunneling QPC, to en- +sure that all non-universal constants are cancelled when +examining the ratio of the two limits. [39] +Easy extraction of θ12 requires Gdirect to be sub- +dominant (see Eq. (1)). Quantitatively, this is the case +if both kBT ≪ eV and 4δ1 < 2 are satisfied. These con- +straints result from the direct tunneling process being +dominated by short time scales. Naive theories describ- +ing quasiparticles may satisfy this condition even if the +aforementioned non-universal effects change the scaling +dimension quite significantly. For example, theory gives +δ = 1/2m for Laughlin quasiparticles. +Edge theory.— We now define the system’s Hamilto- +nian and derive the current. As shown by Wen, the edge +theory of a general Abelian FQH state can be described +by n-boson fields, φ(x, t) ≡ (φ1, φ2, · · · φn)T [2]. These +define the theory in conjunction with a charge vector, q, +which determines the electric charge carried by each bo- +son field, and the so called K-matrix, which determines +the commutation relations between the boson fields, +[φi(x), ∂x′φj(x′)] = i2π(K−1)ijδ(x − x′). +(2) +The filling factor is then given by ν = qT K−1q, and the +charge density is given by ρ = − 1 +2πq · ∂xφ. In terms of +these fields, the Hamiltonian of a single FQH edge mode +is given by +Hedge = 1 +4π +n +� +i,j=1 +ˆ +dx∂xφiVij∂xφj, +(3) +where ˆV is a positive definite matrix describing the ve- +locities of the modes and intra-edge interactions. These +edges support quasiparticles of the form ψl ∼ eil·φ, where +l is a vector of integers. The charge of these quasiparti- +cles is then given by e∗ +l = qT K−1l. +The configuration of Fig. 1(a) involves two edges, u +and d, tunnel-coupled by a QPC. This is described by +two copies of the Hamiltonian Hedge, time reversed with +regard to one another, as well as a tunneling term, HT , +which we treat as a perturbation. +Assuming only one +type of quasiparticle, denoted by the vector l1 and car- +rying charge e∗ +1, tunnels between the edges, this is given + +3 +by +HT = ξ +� +ˆA + ˆA†� +; ˆA(t) ≡ ei(l1·φ(u)(0,t)−l1·φ(d)(0,t)). (4) +Here, ξ is a small tunneling amplitude, which we assume +to be real, and φ(u) (φ(d)) are the bosonic field operators +on the upper (lower) edge. We project the auxiliary edge +a out of the Hamiltonian, as it is only used to “initialize” +the state of the edge u. +The current that tunnels from the upper edge to +the lower edge is then given by the operator, ˆIT (t) = +iξe∗ +1 +� +ˆA†(t) − ˆA(t) +� +. +Since the lower edge is grounded, +we henceforth identify I = ⟨ˆIT ⟩. Expanding to leading +order in ξ, the current is given by +I(t) = e∗ +1ξ2 +ˆ t +−∞ +dt′ �� +ˆA†(t), ˆA(t′) +� ++ +� +ˆA†(t′), ˆA(t) +�� +. +(5) +Here, [·, ·] denotes commutation, and expectation values +are calculated with respect to the Hamiltonian in the +absence of tunneling. +Deviation from Equilibrium.— It is clear from Eq. (5) +that one needs to derive correlation functions such as +⟨ ˆA†(t) ˆA(t′)⟩. In equilibrium, at temperature T, the sys- +tem is particle-hole symmetric, and the correlation func- +tions are given by [2, 40] +⟨ ˆA†(t) ˆA(t′)⟩0 = ⟨ ˆA(t) ˆA†(t′)⟩0 +(6) += +� +πTτc +sinh (πT |t − t′|) +�4δ1 +e−i2πδ1sgn(t−t′), +where δ1 is the scaling dimension of the quasiparticle l1, +and τc > 0 is a short time cutoff. +Two main features are carried over from Eq. (6) to the +correlation functions out of equilibrium - the exponen- +tial decay at time difference larger than ℏ/T, and the +phase e2πiδ1 associated with an interchange of the time +arguments. +We now consider two non-equibrium cases. In the first +we introduce a constant bias voltage V ≡ Vu − Vd be- +tween the edges. In the setup of Fig. 1(a), this corre- +sponds to a fully closed injection QPC, i.e. Iinj = σxyV . +The introduction of the voltages can be formally ab- +sorbed into the boson fields by use of a simple gauge +transformation, which maps φ(u/d)(x, t) �→ φ(u/d)(x, t)+ +K−1qVu/d (t ∓ x/v) /ℏ. +This accordingly modifies the +correlation functions by a phase factor +⟨ ˆA†(t) ˆA(t′)⟩full = ⟨ ˆA†(t) ˆA(t′)⟩0ei +e∗ +1 V +ℏ +(t−t′), +⟨ ˆA(t) ˆA†(t′)⟩full = ⟨ ˆA(t) ˆA†(t′)⟩0e−i +e∗ +1 V +ℏ +(t−t′). +(7) +In the second non-equilibrium driving, we consider in- +jecting a single quasiparticle, denoted by the vector l2, +into the upper edge at the location xinj < 0 and at time +tinj. This is shown schematically in Fig. 2(a). In view +of the commutation relations (2), the application of the +quasiparticle creation operator e−il2·φ(u)(xinj,tinj) on the +edge creates a soliton in each of the boson fields, +φ(u)(x, tinj) �→ φ(u)(x, tinj) − 2πK−1l2Θ (x − xinj) . (8) +We assume here the injection happens instantaneously. +This assumption will be relaxed to find the subleading +term of Eq. (1). +The fields at general times can then be obtained using +the equations of motion dictated by the Hamiltonian in +Eq. (3). If all modes are chiral with the same velocity v, +this amounts to replacing x−xinj → x−xinj −v (t − tinj). +The soliton thus arrives at the QPC, x = 0, at time +t0 ≡ tinj − xinj/v. +The c-number shift in the bosonic field of Eq. (8) leads +to a phase shift in the correlator Eq. (6). We see directly +from the definition of the operator ˆA in Eq. (4) that +⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ ˆA†(t) ˆA(t′)⟩0e2πil1K−1l2[Θ(t−t0)−Θ(t′−t0)], +⟨ ˆA(t) ˆA†(t′)⟩qp = ⟨ ˆA(t) ˆA†(t′)⟩0e−2πil1K−1l2[Θ(t−t0)−Θ(t′−t0)]. +(9) +The phase we obtain is the standard definition of +mutual braiding statistics between two quasiparticles, +θ12 ≡ πl1K−1l2 [2]. The expression in Eq. (9) shows +that the product gains a phase of e2iθ12sgn(t−t′) if the +arrival time t0 is between the times t′ and t, and a triv- +ial phase of 1 otherwise. We emphasize how naturally +this result came from the underlying theory: the only as- +sumptions necessary to obtain this are the commutation +relations, (2), and the existence of quasiparticles in the +edge’s excitation spectrum. +This result holds for different boson modes with differ- +ent velocities if all solitons arrive at the tunneling QPC +more or less concurrently, avoiding dephasing. This is +the case if |xinj|/∆v ≪ ℏ/T, where ∆v is the velocity +difference between the fastest and the slowest modes. +Time-domain interferometry.— The appearance of the +phase, θ12, can be understood as time-domain interfer- +ometry of the two distinct ±e∗ +1 quasiparticle-quasihole +excitations, before and after the injected e∗ +2 quasiparticle +arrives at the QPC. A similar physical picture has been +shown in Ref. [25, 27, 28]. +To show this we consider the configuration of a single +injected particle, as described in Fig. 2(a). In this case +the non-equilibrium correlation function takes the form, +⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ψl2(t0) ˆA†(t) ˆA(t′)ψ† +l2(t0)⟩0, +(10) +i.e., the expectation value is calculated with respect to +the state resulting from exciting the ground state |0⟩ with +a single quasiparticle. Here we omit the position variable +from the quasiparticle injection operator ψ† +l2(t0), and as- +sume it arrives at the tunneling QPC x = 0 at time t0. +The current in Eq. (5) is then given by integration +over multiple terms of the form in Eq. (10). We define +|t, t0⟩− ≡ ˆA(t)ψ† +l2(t0) |0⟩ and |t, t0⟩+ ≡ ˆA†(t)ψ† +l2(t0) |0⟩. + +4 +𝜈 +𝐼 +𝑉 +−𝑒1 +∗ +𝑒1 +∗ +𝑒2 +∗ +(a) +(b) +I Injection +Time +I Injection +II Arrival +III Pair +Time +I Injection +III Pair +II Arrival +III Pair +II Arrival +FIG. 2. Time-domain interferometry. (a) I A quasiparticle +is injected from the sourced, left edge, through the injection +QPC, and into the upper edge. II The injected quasiparti- +cle, by virtue of its chiral motion along the edge, arrives at +the tunneling QPC. III A quasiparticle-quasihole pair is cre- +ated at the tunneling QPC. (b) The two processes by which +charge carriers may ultimately arrive at the drain. The in- +jected quasiparticle arrives at the tunneling QPC either before +(upper panel) or after (lower panel) the creation quasiparticle- +quasihole pair. These two processes interfere, with a relative +phase dictated by the mutual statistics phase, ei2θ12. +Eq. (5) can now be re-written as +I ∝ − +ˆ t +−∞ +dt′ � +b=± +b +�� |t, t0⟩b + |t′, t0⟩b +��2. +(11) +The expression above involves two interference terms. +The term with b = − is an interference between cre- +ation of −e∗ +1 quasiholes on the upper edge at the QPC at +times t and t′. The two interfering processes are shown +schematically in Fig. 2(b). +As shown in the first row +of Eq. (9), these two processes are distinguished by a +non-trivial phase of ei2θ12 if the arrival time t0 is in be- +tween the quasiholes’ creation times, t′ < t0 < t. Com- +bined with the equilibrium correlation function Eq. (6), +one finds that this interference gives a term proportional +to cos (2θ12 − 2πδ). Using similar arguments, the term +with b = + in Eq. (11), gives an interference term pro- +portional to cos (2θ12 + 2πδ). +The total contribution +from the two terms in Eq. (11) is thus proportional to +sin (2θ12) sin (2πδ) [41]. +This interference happens entirely in the time domain, +and along only one edge. It is however crucial that this +edge be part of a two-dimensional bulk. This is important +both because the second edge is required to absorb the +leftover quasiparticle or quasihole resulting from the pair +creation at the QPC, and because the injected quasiparti- +cle must be created within a bulk FQH droplet. Further- +more, the bulk is intimately related to the edge through +bulk-edge correspondence. This dictates that the statisti- +cal phase contributing to time-domain interference along +a single edge, which our setup measures, is the same as +the phase obtained from spatial exchange. +It is easy to generalize this to injection of multiple +quasiparticles: as long as all injected quasiparticles are +mutually independent, each injected quasiparticle con- +tributes a phase of e2iθ12 if and only if the arrival time +at the point contact was between t′ and t. If we assume +this is a Poissonian process, with a quasiparticle injection +rate of Iinj/e∗ +2, we obtain for t > 0 +⟨ ˆA†(t) ˆA(0)⟩dilute +⟨ ˆA†(t) ˆA(0)⟩0 += +∞ +� +n=0 +(tIinj/e∗ +2)ne−tIinj/e∗ +2 +n! +e2inθ12 += e−tIinj/e∗ +2(1−e2iθ12). +(12) +This is precisely the result given in Refs. [23, 25] for injec- +tion along a single edge. Adding injected quasiparticles +to the lower edge and generalizing for t < 0 are straight- +forward using the same arguments. +Currents.— The effect of driving the system out of +equilibrium is completely encapsulated in the correlation +functions obtained above. +These can then be used to +derive any observable of interest, such as charge or heat +currents in any of the system’s drains, or their respective +auto- and cross-correlations. +For concreteness, we present the explicit results of such +a calculation for the charge current at the lower drain, +denoted as I in Fig. 1. We show that a simple cohort +of current measurements is sufficient to obtain the mu- +tual statistics θ12, without requiring correlation measure- +ments. +We focus on the regime where the temperature is large +compared to the injected current ℏIinj/ekBT. +For the +full limit, this assumption guarantees linear response in +the voltage and in the injected current, which in this +limit is Iinj = σxyV . For the dilute limit, the exponen- +tial suppression of the equilibrium correlation function at +times larger than ℏ/T, guarantees that the exponent in +Eq. (12) may be expanded to first order in Iinj. Conse- +quently, +⟨ ˆA†(t) ˆA(t′)⟩full/dilute +⟨ ˆA†(t) ˆA(t′)⟩0 +≈ 1 + iωf/d (t − t′) , +(13) +where the frequencies ωf/d are given by +ωf = e∗ +1V +ℏ += e∗ +1 +ℏ +Iinj +σxy +; +ωd = iIinj +e∗ +2 +� +1 − e2iθ12� +. +(14) +The zeroth order term corresponds to the equilibrium +state and does not contribute to the current. The ratio +of the two first order contributions is Eq. (1). +Explicit calculation of the resulting current in Eq. (5), +given in App. A, finds that +Ifull/dilute = 2πe∗ +1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) Re +� +ωf/d +� +, +(15) +where B(x, y) is the Euler Beta function. It is thus imme- +diately apparent that by focusing on the ratio between +the full and dilute beams, all dependence on δ1, T and ξ +drops out. Examining the ratio I/Iinj, and noting that +σxyℏ/e∗ +1e∗ +2 = νe2/2πe∗ +1e∗ +2 we thus obtain Eq. (1). + +5 +For general temperatures, the current can no longer be +treated as a linear response to the drive of the full or di- +lute beams. We hence obtain the typical power laws char- +acterizing tunneling in Luttinger liquids [2, 34, 42, 43]. +Comparing measurements of the full and dilute limits at +the low temperature limit T ≪ e∗V, Iinj can still give a +quantity related to the mutual statistics θ12, but will ex- +plicitly depend on the value of δ1. We present general +expressions for the current in this case in App. A. +For a fermionic θ12 = π, Eq. (15) gives no current at all +for a dilute electron beam. However, Landauer-Buttiker- +Imry scattering theory [44] tells us the current is given +by the product of the transparencies of the two QPCs +along the electron’s path, regardless of whether they are +close to full transmission or full reflection. This requires +accounting for the direct tunneling term in Eq. (1), which +now becomes the leading contribution. +We do this by accounting for the finite width of the +soliton. This leads to the required, Landauer-Buttiker- +Imry consistent result of Idilute = 4π2τ 2 +c ξ2Iinj. The phys- +ical intuition behind the requirement of a finite soliton +width is that tunneling without time-domain interferom- +etry, dubbed the direct tunneling process in [24, 25], is +dominated by short times. Performing these calculations +explicitly in App. B, we show that the ratio between +the first term in Eq. (1) and Gdirect is ∝ (Tτs)4δ1−2, +where τs is the soliton width. It has been shown [24, 25] +that τ −1 +s +∝ max{eV, kBT}; as such, to ensure Gdirect is +sub-dominant, the dilute limit must be measured when +kBT ≪ eV and 4δ1 < 2. +Several contemporary experimental setups use the +equivalent of non-interacting fermionic formulae to rea- +sonable success [45], corresponding to the limiting value +of 2δ1 = 1. In this case, the second term of Eq. (1) is +a numerical coefficient of order one, which may depend +solely on e∗, δ1 and θ12. +For non-interacting fermions, +this coefficient is easily found by comparing to known +Landauer-Buttiker-Imry scattering theory [44], but it is +straightforward to generalize. We discuss this coefficient +further in App. B. +Discussion.— We propose a simple method to extract +anyonic exchange statistics. +Our system consists only +of a single quantum Hall droplet with two QPCs, which +effectively create a time-domain interferometer, as can +be identified from current measurements. We thus avoid +both current correlation (or noise) measurements, and +the need for a real space interferometer, making the iden- +tification of the exchange statistics much more accessible +than existing experiments. All time-domain interferom- +etry is between pairs of an injected quasiparticle and a +tunneling quasiparticle, and occurs at the same edge, as +previously proposed in Ref. [25]. +Both the exchange statistics θ11 of the tunneling quasi- +particle, and θ22 of the injection quasiparticle, do not +appear in our derivation. Rather, it is the two particles’ +mutual statistics, θ12 that affect the modified correlation +functions, and hence, the physical observables. Likewise, +the scaling dimension and electric charge which directly +effect observables are only those of the tunneling quasi- +particle, δ1 and e∗ +1 (properties of the injected quasiparti- +cles may implicitly enter through the injection rate). +Only in the case where the injected and tunneling +quasiparticles are identical, l1 = l2, do we obtain ex- +change statistics for a single quasiparticle type. We re- +mark that this is indeed the case in the experiment of +Ref. [22], where all quasiparticles are Laughlin e∗ = e/3 +anyons, and subsequent recreations for the ν = 1/3 and +ν = 2/5 cases [26, 46, 47]. +Interestingly, a recent ex- +periment employing a similar setup, where the injected +quasiparticle was a e/3 anyon and the tunneling quasi- +particle was an electron, observed Andreev-like reflection +[48]. This is consistent with a mutual statistics phase of +θ12 = π, for which Eq. (1) gives no time-domain interfer- +ometry signal. +Acknowledgements.— We thank Tomer Alkalay, Moty +Heiblum, Changki Hong, June-Young Lee and H.-S. +Sim for insightful discussions and comments on the +manuscript. This work was partially supported by grants +from the ERC under the European Union’s Horizon 2020 +research and innovation programme (grant agreements +LEGOTOP No. 788715 and HQMAT No. 817799), the +DFG (CRC/Transregio 183, EI 519/7-1), the BSF and +NSF (2018643), the ISF Quantum Science and Technol- +ogy (2074/19). N.S. was supported by the Clore Scholars +Programme. +[1] D. Arovas, J. R. Schrieffer, and F. Wilczek, Fractional +Statistics and the Quantum Hall Effect, Physical Review +Letters 53, 722 (1984), publisher: American Physical So- +ciety. +[2] X. 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Ca- +vanna, U. Gennser, Y. Jin, A. Anthore, and F. Pierre, +Cross-Correlation Investigation of Anyon Statistics in the +ν = 1/3 and 2/5 Fractional Quantum Hall States (2022), +arXiv:2210.01054 [cond-mat]. +[48] P. Glidic, O. Maillet, C. Piquard, A. Aassime, A. Ca- +vanna, Y. Jin, U. Gennser, A. Anthore, and F. Pierre, +Quasiparticle Andreev scattering in the ν = 1/3 frac- +tional quantum Hall regime (2022), arXiv:2206.08068 +[cond-mat]. +Appendix A: Finite temperature current from time-domain interferometry +Here derive explicit expressions for the tunneling current I at finite temperature T. +This section neglects the +contribution Gdirect (see Eq. (1), which is discussed in App. B. We begin with the expression for the current in +Eq. (5). Writing this explicitly, +I = e∗ +1ξ2 +ˆ t +−∞ +dt′ +� � +ˆA†(t) ˆA(t′) +� +− +� +ˆA(t′) ˆA†(t) +� ++ +� +ˆA†(t′) ˆA(t) +� +− +� +ˆA(t) ˆA†(t′) +� � +. +(A1) +In the case where the edges are not driven out of equilibrium, we plug the equilibrium correlation functions Eq. (6), +and obtain I = 0, as expected. A similar expression can be written for the symmetrized current fluctuations, +�� +δ ˆIT (t), δ ˆIT (t′) +�� += (e∗ +1)2ξ2 +� � +ˆA†(t) ˆA(t′) +� ++ +� +ˆA(t′) ˆA†(t) +� ++ +� +ˆA†(t′) ˆA(t) +� ++ +� +ˆA(t) ˆA†(t′) +� � +, +(A2) +where we define δ ˆIT ≡ δ ˆIT − ⟨δ ˆIT ⟩. We do not focus on current fluctuations in this work, but note that our methods +reproduce the known results of Refs. [23, 25]. +We now want to obtain the current for each of the three methods of driving the two edges out of equilibrium. Each +of these leads to a corresponding multiplicative factor to the correlation functions. A finite bias voltage V , used for the +“full” beam, gives the correlation functions of Eq. (7); injection of a single quasiparticle gives the correlation functions +of Eq. (9); and a dilute, Poissonian beam of quasiparticles gives the correlation functions of Eq. (12). Plugging in +these appropriate correlation functions gives after minor algebra and changes of variables +Ifull = 2ie∗ +1ξ2 +ˆ ∞ +0 +d˜t sin +�e∗ +1V +ℏ +˜t +�� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +, +(A3a) +Idilute = 2ie∗ +1ξ2 +ˆ ∞ +0 +d˜t +sin +� +Iinj +e∗ +2 ˜t sin 2θ12 +� +exp +� +Iinj +e∗ +2 ˜t (1 − cos 2θ12) +� +� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +, +(A3b) +Iqp = 2ie∗ +1ξ2 +ˆ t +−∞ +dt′ sin (2θ12 [Θ (t − t0) − Θ (t′ − t0)]) +� � +πTτc +i sinh (πT (t − t′ − iτc)) +�4δ1 +− +� +πTτc +i sinh (πT (t′ − t − iτc)) +�4δ1� +. +(A3c) +We proceed using the identity, correct at the limit τc → 0, +i +� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� += +� +� +� +� +� +−2πτ 2 +c ∂˜tδ(˜t) +2δ1 = 1 +� +πT τc +sinh(πT |˜t|) +�4δ1 +2 sin (2πδ1)sgn(˜t) +2δ1 ̸= 1 +, (A4) +where δ(t) is the Dirac delta function. This identity is necessary to treat the case of δ1 = 1, which otherwise may lead +to divergent integrals. + +2 +Standard manipulations of these integrals then give results in terms of the Euler Beta function and the incomplete +Beta function, B (x; a, b) ≡ +´ x +0 ta−1(1 − t)b−1dt, B (a, b) ≡ B (1; a, b). We thus obtain the general results +Ifull = 2e∗ +1ξ2(2πT)4δ1−1τ 4δ1 +c +sinh +�e∗ +1V +2T +� +B +� +2δ1 + i e∗ +1V +2πT , 2δ1 − i e∗ +1V +2πT +� +(A5a) +Idilute = − +2π +cos (2πδ1) Γ (4δ1)e∗ +1ξ2(2πT)4δ1−1τ 4δ1 +c +Im +� +� +Γ +� +Iinj +e∗ +2 +1−cos(2θ12)+i sin(2θ12) +2πT ++ 2δ1 +� +Γ +� +Iinj +e∗ +2 +1−cos(2θ12)+i sin(2θ12) +2πT ++ 1 − 2δ1 +� +� +� +(A5b) +Iqp = 4e∗ +1ξ2(2πT)4δ1−1τ 4δ1 +c +sin (2θ12) sin (2πδ) B +� +e−2πT (t−t0); 1 + 2δ1, 1 − 4δ1 +� +. +(A5c) +Here, Γ(a) is the Euler Gamma function, satisfying B(a, b) = Γ(a)Γ(b) +Γ(a+b) , and Im[. . . ] denotes the imaginary part. +The high temperature and zero temperature limits of the full beam and dilute beam are then immediately repro- +ducible. For e∗V, ℏIinj/e∗ +2 ≪ kBT, one expands to leading order in e∗V/T and Iinj/e∗ +2T, one obtains +Ifull ≈ 2πe∗ +1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) e∗ +1V +ℏ , +(A6a) +Idilute ≈ 2πe∗ +1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) Iinj +e∗ +2 +sin (2θ12) . +(A6b) +We thus see that the mutual statistics are immediately extractable from the dilute case. While this expression does +depend on the non-universal ξ and δ, as well as the temperature, these are all encoded in a prefactor which appears +in the full case as well. We can hence lose this unwanted prefactor by examining the ratio between the two cases. +For T ≪ e∗V, Iinj/e∗ +2, we use the identities +lim +x→∞ Γ (x + a) = Γ (x) xa, +lim +x→∞ sinh(πx)B (a + ix, a − ix) = +π +Γ(2a)x2a−1, +to obtain +Ifull ≈ 2πe∗ +1ξ2τ 4δ1 +c +Γ (4δ1) +�e∗ +1V +ℏ +�4δ1−1 +, +(A7a) +Idilute ≈ − +2πe∗ +1ξ2τ 4δ1 +c +cos (2πδ1)Γ (4δ1) +�Iinj +e∗ +2 +�4δ1−1 +Im +�� +1 − cos (2θ12) + i sin (2θ12) +�4δ1−1� +. +(A7b) +By tuning 2δ1 → 1, we again obtain an expression from which the mutual statistics are easily extractable, with an +identical non-universal prefactor appearing in both the full and dilute cases. However, once the scaling dimension is +tuned to this critical value, the contribution from time-domain interferometry no longer dominates the direct tunneling +process, as can be seen from the calculation of Gdirect in App. B. +We note that for temperatures larger than the source voltage, one has to account for injection of both quasiparticles +and quasiholes through the injection QPC. This can be done by modifying the Poissonian correlation function in +Eq. (12) according to +⟨ ˆA†(t) ˆA(0)⟩dilute +⟨ ˆA†(t) ˆA(0)⟩0 += e−tIinj/e∗ +2(1−e2iθ12) +→ e−tIqp +inj/e∗ +2(1−e2iθ12)e−tIqh +inj/e∗ +2(1−e−2iθ12), +(A8) +where Iqp +inj is the injection rate of quasiparticles, and Iqh +inj is the injection rate of quasiholes. This is a similar expression +to the three QPC setup considered in [23] and [25]. Performing the same algebra as in this section, and identifying +Iinj ≡ Iqp +inj − Iqh +inj, one then reproduces Eq. (A6) for the high temperature limit. +Finally, it is instructive to consider the current due to the injection of a single quasiparticle at time t0, which +was obtained in Eq. (A5c). In this case we must examine the explicit temperature dependence, as tunneling of a +single quasiparticle may be relevant, and we lack any other energy scale to serve as a cutoff for the RG flow of the +process. This current exhibits a power-law decay for t − t0 ≪ 1/πT, consistent with the orthogonality catastrophe +that characterizes injection into Luttinger liquid edges. For 2δ1 = 1, this results in ⟨ˆIT ⟩qp ∝ δ (t − t0). This gives +some intuition as to what makes the 2δ1 = 1 case so unique - the QPC just scatters the incident particle with some +probability, without inducing any long-time correlations, resulting in the direct tunneling process. + +3 +Appendix B: Finite soliton width: restoring Landauer-Buttiker-Imry for electrons and subleading corrections +The results of App. A are seemingly inconsistent with the known non-interacting electron limits. Indeed, inserting +e∗ +1 = e∗ +2 = e, 2δ1 = 2δ2 = 1 and θ12 = π into these results would indicate that the dilute electron beam gives no +current at all. This is in direct contrast with the intuition of Landauer-Buttiker-Imry scattering theory, which would +indicate that the current should be given by the product of the transparencies of the two QPCs along the electron’s +path, regardless of whether they are close to full transmission or full reflection. +The culprit of this result is a peculiarity of soliton physics. The boson field φ is compact under φ �→ φ + 2π. As +such, a soliton of height 2πK−1l2 would appear to leave the boson field completely unperturbed if K−1l2 is an integer. +This corresponds precisely to electron injection operators [2]. As such, our soliton description is ill-equipped to treat +electrons without modifications. +We solve this issue by introducing a finite width to the soliton, τs. To fully recreate the known non-interacting +result, it is crucial to maintain an order of limits such that the soliton width is larger than the short-time cutoff, τc. +We note that we still take care to ensure that τs < 1/T, (Iinj/e∗ +2)−1, i.e. the solitons are still narrow compared to +the larger time scales in the problem. Previous works [24, 25], performing a full Keldysh calculation, have shown the +soliton width (refered to in the cited papers as the temporal width) is given by the voltage, h/e∗V , if eV > kBT, +and by the inverse temperature ℏ/kBT if eV ≲ kBT; as such, the dilute limit must be measured in the regime +Iinj/e∗ +2 ≪ kBT ≪ eV . +Formally, this means that injecting a quasiparticle into the upper edge at the location x0 and time t0 transforms +the boson field according to +φ(u)(x, t0) �→ φ(u)(x, t0) − 2πK−1l2 +� 1 +π tan−1 +�x − x0 +τs +� +− 1 +2 +� +. +Accordingly, the correlation functions of Eq. (9) are now replaced with +⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ ˆA†(t) ˆA(t′)⟩0 exp +� +2iθ12 +π +� +tan−1 +�t − t0 +τs +� +− tan−1 +�t′ − t0 +τs +��� +, +⟨ ˆA(t) ˆA†(t′)⟩qp = ⟨ ˆA(t) ˆA†(t′)⟩0 exp +� +−2iθ12 +π +� +tan−1 +�t − t0 +τs +� +− tan−1 +�t′ − t0 +τs +��� +. +(B1) +One indeed sees that at the limit τc → 0, one reproduces the immediate soliton results from the main text. +To find the correlation function in the presence of a dilute, Poissonian beam of injected quasiparticles, we now +must sum over the number of injected quasiparticles, in a manner similar to Eq. (12). However, this is now trickier, +for two reasons. First, the accumulated phase explicitly depends on the time of the injected quasiparticle. Second, +injected quasiparticles outside of the window [0, t] can still affect the correlation function, due to the long tails of the +finite-width solitons. +So generalizing the methods that lead to Eq. (12), the correlation function now changes to define +⟨ ˆA†(t) ˆA(0)⟩fw +⟨ ˆA†(t) ˆA(0)⟩0 += +� +n +� +(t + 2cτc) Iinj +e∗ +2 +�n +e +−(t+2cτc) +Iinj +e∗ +2 +n! +�ˆ t+cτc +−cτc +dt0P (Particle injected at t0) e2i θ12 +π [tan−1( +t−t0 +τs )−tan−1( +0−t0 +τs )] +�n +. +(B2) +Here c is some unitless cutoff, chosen such that injected quasiparticles affect the correlation function only if they are +injected in the window [−cτc, t + cτc], which we will eventually take to be infinite. The probability of injection at a +particular time t0 is given by +P (Particle injected at t0) = +Iinj/e∗ +2e−Iinjt0/e∗ +2 +´ t+cτc +−cτc dt0Iinj/e∗ +2e−Iinjt0/e∗ +2 . +(B3) +Performing this sum, and re-defining this integration with unitless variables, we find that the new correlation function +is given in integral form by +⟨ ˆA†(t) ˆA(0)⟩fw +⟨ ˆA†(t) ˆA(0)⟩0 += exp +� +− (t + 2cτc) Iinj +e∗ +2 +� +1 − Iθ12 +�Iinj +e∗ +2 +τs, t +2τs +��� +, +Iθ12 (a, b) ≡ +a +2 sinh (a(b + c)) +ˆ b+c +−b−c +dxe−axe2i θ12 +π [tan−1(x+b)−tan−1(x−b)]. +(B4) + +4 +By plugging this new correlation function into the expression for the current in Eq. (5), one now finds +Idilute = 2ie∗ +1ξ2 +ˆ ∞ +0 +d˜t +sin +� +(˜t + 2cτc) Iinj +e∗ +2 Im +� +Iθ12 +� +Iinj +e∗ +2 τs, +t +2τs +��� +exp +� +(˜t + 2cτc) Iinj +e∗ +2 Re +� +1 − Iθ12 +� +Iinj +e∗ +2 τs, +t +2τs +��� +× +� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +. +(B5) +Careful re-application of the limit τc → 0 indeed replicates our previous result in Eq. (12). +For general θ12, the integral Iθ12 (a, b) is difficult to solve analytically. In the main text, this is circumvented by +taking the limit τc → 0, allowing use of Eq. (A4), in conjunction with replacing (˜t+2cτc) Iinj +e∗ +2 Iθ12 +� +Iinj +e∗ +2 τs, +t +2τs +� +→ −i˜tωd. +However, as noted previously, fermionic exchange statistics corresponding to values of θ12 that are integer multiples +of π lead to ωd = 0, and hence give a vanishing current. As such, Eq. (B5) must be calculated in full while retaining +a finite τc. +To simplify these expressions, we assume that +� +Iinj +e∗ +2 +� +is significantly larger than any other time scale in the system. +This makes sense from a physical perspective as well, as it corresponds to the assumption that injection is sufficiently +rare such that solitons do not overlap. In this case, one can assume the probability of injection which appears in +Eqs. (B2),(B3) is approximately uniform, i.e. P (Particle injected at t0) ≈ 1/(t + 2cτc). One can now safely take the +limit c → ∞ without artificial divergences, giving the simpler result, +⟨ ˆA†(t) ˆA(0)⟩fw +⟨ ˆA†(t) ˆA(0)⟩0 += exp +�Iinj +e∗ +2 +ˆ ∞ +−∞ +dt0 +� +e2i θ12 +π [tan−1( +t−t0 +τs )−tan−1( +0−t0 +τs )] − 1 +�� +. +(B6) +Since we undertook this endeavor with the explicit goal of finding the correct result for non-interacting electrons, +we wish to find this integral for 2δ1 = 1, θ12 = π, and e∗ +1 = e∗ +2 = e. This value of θ12 allows one to significantly +simplify Eq. (B6) using trignometric identities; plugging the resulting correlation function in Eq. (A1), we obtain +Iθ12=π +dilute = 2ie∗ +1ξ2 +ˆ ∞ +0 +d˜t +sin +� +Iinj +e∗ +2 +2π˜t(2τs)2 +˜t2+(2τs)2 +� +exp +� +Iinj +e∗ +2 +2π˜t2(2τs) +˜t2+(2τs)2 +� +� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +. +(B7) +As can be seen in Eq. (A4), the expression in the curled brackets is approximately zero for ˜t > τc. We can thus +approximate the total integral as the contribution from short times, ˜t ≤ τc ≪ 1/πT. To leading order, this will be +given by +Iθ12=π,2δ1=1 +dilute +≈ 2ie∗ +1ξ2τ 2 +c +ˆ ∞ +0 +d˜tIinj +e∗ +2 +2π˜t(2τs)2 +˜t2 + (2τs)2 +� � +1 +i˜t + τc +�2 +− +� +1 +−i˜t + τc +�2� += +(2τs)2 +(2τs + τc)2 4π2ξ2τ 2 +c Iinj. +(B8) +Now taking the limit τc ≪ τs, we compare to the electron case in, say, Eq. (15) or Eq. (A5). We find that the +result we expect for non-interacting electrons is indeed 4π2ξ2τ 2 +c Iinj. This is consistent with - the current is linear in +the injected current, and in the transparency of the tunneling QPC (which is given by ξ2τ 2 +c ). +For general values of θ12 and δ1 this integral is more difficult to solve analytically. However, it is possible to re-write +Eq. (B6) as +⟨ ˆA†(t) ˆA(0)⟩fw +⟨ ˆA†(t) ˆA(0)⟩0 += exp Iinj +e∗ +2 +� +sin (2θ12) t + fθ12 (t, τc)) +� +, +(B9) +fθ12 (t, τc)) ∝ +� +� +� +� +� +t +t ≲ τs +τs +t ≫ τs, θ12 ̸= π +(τs)2/t +t ≫ τs, θ12 = π. +(B10) +Plugging this into the general expression for the current, and expanding to linear response in Iinj +e∗ +2 we find + +5 +Idilute = 2ie∗ +1ξ2 Iinj +e∗ +2 +ˆ ∞ +0 +d˜t +� +sin (2θ12) t + fθ12 (t, τc)) +�� � +πThwτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +. +(B11) +The term proportional to sin (2θ12), as discussed at length above, is the main interest of this paper. This is calculated in +Eq. (A6). We see there that the time scales in the system contribute a leading term of the form ∝ (ξτc)2(Tτc)4δ1−2Iinj. +The term proportional to fθ12 (t, τc)) contains several contributions: at short times (˜t ∼ τc), we obtain a con- +tribution of order (ξτc)2; at long times (˜t ∼ 1/πT) we obtain a contribution of order (ξτc)2(τs/τc) (Tτc)4δ1−1 for +θ12 ̸= π and (ξτc)2(τs/τc) (Tτc)4δ1 for θ12 = π; and at intermediate times (˜t ∼ τs) we obtain contributions of order +(ξτc)2(τs/τc)1−4δ1 and (ξτc)2(τs/τc)2−4δ1. +We compare these contributions to the coefficients of Eq. (15) or Eq. (A6), which give the time-domain interferom- +etry process, which is of order (ξτc)2 (Tτc)4δ1−2. Utilizing τc ≪ τs ≪ 1/πT, we see that the long time contribution +is always subdominant, but the short time dominates for 2δ1 ≥ 1 - consistent with both Eq. (B8) and the known +electron result. This is consistent with our physical intuition: direct tunneling dominates short times, which give +the main contribution for 2δ1 ≥ 1, whereas time-domain interferometry dominates long times, which give the main +contribution for 2δ1 < 1. +Finally, if we indeed assume 2δ1 < 1, the intermediate time contribution dominates the entire direct process. In +this case, the ratio between the time-domain interferometry process and the direct process is given by ∝ (Tτs)4δ1−2. +This again confirms that we must have a soliton width smaller than the inverse temperature to ensure time-domain +interferometry +This method is also what we use to calculate the current for an almost full beam, i.e. σxy − Ginj ≪ 1. Since +in this case, the beam can be treated as a conjoined full beam of fractional quasiparticles with a dilute beam of +e∗ = e holes, we have 2θ12 = 2πn regardless of the tunneling quasiparticles. Defining the injection rate of holes as +Iholes +inj +≡ σxyV − Iinj, we combine the full beam correlation function of Eq. (7) and the regularized Poissonian hole +injection to obtain described in this section +I|Ginj−σxy|≪1 = 2ie∗ +1ξ2 +ˆ ∞ +0 +d˜t +sin +� +e∗ +1V +ℏ ˜t − +Iholes +inj +e +2π˜t(2τs)2 +˜t2+(2τs)2 +� +exp +� Iholes +inj +e +2π˜t2(2τs) +˜t2+(2τs)2 +� +× +� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +. +(B12) +In the relevant limits, the same methods as previously mention allow us to approximate the exponent in the +denominator as 1, and to expand the sine in the numerator. We thus have the sum of two linear responses, one in in +e∗ +1V +ℏ +and one in − +Iholes +inj +e +. Taking, as in the Landauer-Buttiker-Imry case, the limit τs ≫ τc, i.e. a soliton width that is +larger than the short time cutoff, this can be re-written as +I|Ginj−σxy|≪1 ≈ 2ie∗ +1ξ2 +ˆ ∞ +0 +d˜t +� +e∗ +1V +ℏ +− 2π Iholes +inj +e +� +˜t +� � +πTτc +i sinh +� +πT +�˜t − iτc +�� +�4δ1 +− +� +πTτc +i sinh +� +πT +� +−˜t − iτc +�� +�4δ1� +. +(B13) +Identifying +� +e∗ +1V +ℏ +− 2π +Iholes +inj +e +� += 2π +e +� +σxyV − Iholes +inj +� +≡ 2π +e Iinj, we see that this is precisely the same integral that we +had in Eq. (A3a) for the full beam case, with the replacement σxyV → Iinj. We note that we used here σxy = ee∗/h, +which is correct only for Laughlin edge states, ν = 1/m; this is valid as Laughlin edges are the outer level of heirarchal +FQH fluids, and thus are the states of interest for nearly full closed QPCs. + diff --git a/0NAyT4oBgHgl3EQfPPZ4/content/tmp_files/load_file.txt b/0NAyT4oBgHgl3EQfPPZ4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1428e090f79ae49b224c2de4d028062f55732162 --- /dev/null +++ b/0NAyT4oBgHgl3EQfPPZ4/content/tmp_files/load_file.txt @@ -0,0 +1,718 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf,len=717 +page_content='Anyon statistics through conductance measurements of time-domain interferometry Noam Schiller1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Yotam Shapira2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Ady Stern1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' and Yuval Oreg1 1Department of Condensed Matter Physics 2Department of Physics of Complex Systems Weizmann Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Rehovot 7610001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Israel We propose a method to extract the mutual exchange statistics of the anyonic excitations of a general Abelian fractional quantum Hall state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' by comparing the tunneling characteristics of a quantum point contact in two different experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In the first, the tunneling current between two edges at different chemical potentials is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In the second, one of these edges is strongly diluted by an earlier point contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We describe the case of the dilute beam in terms of a time-domain interferometer between the anyons flowing along the edge and quasiparticle-quasihole excitations created at the tunneling quantum point contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In both cases, temperature is kept large, such that the measured current is given to linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Remarkably, our proposal does not require the measurement of current correlations, and allows us to carefully separate effects of the fractional charge and statistics from effects of intra- and inter-edge interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— It has been almost four decades since the initial proposal that the elementary quasiparticles of fractional quantum Hall (FQH) systems obey anyonic statistics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Despite the apparent maturity of the field, the pursuit to definitively observe the physical quanti- ties and quantum numbers characterizing anyons [2, 3] is constantly being reinvigorated [4–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In particular, early 2020 saw two major experimental steps forward: the ob- servation of anyonic braiding in a Fabry-Perot interfer- ometer [21], and demonstration of a so-called “anyon col- lider” [22, 23] using cross-correlation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Here we show that anyonic statistics can be inferred di- rectly from conductance measurements, without requir- ing current correlation measurements or explicitly build- ing an interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The configuration we propose to obtain this result consists of a quantum point contact (QPC) between two edges of a general Abelian FQH state which are driven out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The edges may be driven off-equilibrium by one of three methods: inject- ing a single quasiparticle into one of the edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' injecting a Poissonian, dilute beam of quasiparticles into one of the edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' and placing a finite bias voltage between the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Our proposed setup, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1(a), allows a smooth transition between the dilute Poissonian beam and a full beam at finite bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is ob- tained by tuning a second, injection QPC from fully open (a differential conductance, Ginj ≡ dIinj/dV , satisfying Ginj/σxy → 0) to fully closed (Ginj/σxy → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We hence- forth refer to these as the dilute and full limits, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We propose sweeping Ginj through this range, and measuring the ratio I/Iinj, where I is the measured cur- rent after the tunneling QPC, and Iinj is the injected inci- dent current, as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Comparing the val- ues at the dilute and full limits cancels out non-universal constants, yielding the relation, � I(T) Iinj(T) � dilute = νe2 2πe∗ 1e∗ 2 sin 2θ12 � I(T) Iinj(T) � full + Gdirect Ginj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1) Here, e∗ 1 is the tunneling quasiparticle charge, e∗ 2 the injected quasiparticle charge, δ1 is the tunneling quasi- particle scaling dimension, θ12 is the mutual statistics phase between the injected and tunneling quasiparticles, T is temperature, and Gdirect is a residual conductance corresponding to direct tunneling [24–26] through both QPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The full crossover between these two limits is shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The mechanism leading to this result is a time-domain interferometer at the tunneling QPC which is created by the dilute incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The interference is between two processes, in which a quasiparticle-quasihole excitation occurs at the tunneling QPC either before or after the arrival of an injected quasiparticle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A similar physical picture has been shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [25, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We further find that this interference is sensitive to the mu- tual statistics phase between the injected and the tunnel- ing quasiparticles, θ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We emphasize that these quasi- particles are not necessarily of the same type, although they must be supported by the same FQH liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Since our focus is the interference of two amplitudes which differ from one another by the orderings of events, the key point of our analysis is the identification of the phase differences between the two orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We find phase differences that are determined by the quasiparti- cle charge e∗, which is a fraction of the electron charge for non-integer values of ν [4–6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' the scaling dimension δ, which defines the zero-temperature time correlations of the quasiparticle via the relation ⟨ψ†(τ)ψ(0)⟩ ∼ τ −2δ [29–32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' and the exchange statistics phase θ, which for anyons take special values beyond the fermionic π and the bosonic 2π [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We are interested here in isolating the effect of θ from the other two effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In particular, we would like to separate it from the effect of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For non-interacting edges, in which all the modes propagate in the same di- rection, 2πδ = θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' however, in general δ is affected by non-universal factors, such as intra-edge and inter-edge interactions, 1/f noise or neutral modes [33–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This in stark contrast to the charge, exchange statistics phase, or filling factor, which are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We separate the effect of θ from that of δ by tuning arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='00021v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='mes-hall] 30 Dec 2022 2 (a) 𝜈 𝐼 𝑉 𝑒1 ∗, 𝛿1 𝑒2 ∗, 𝛿2 Injection QPC Tunneling QPC 𝐼inj 𝑢 𝑑 𝑎 𝐷𝑎 𝑆𝑢 𝐷𝑢 𝑆𝑑 (b) 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='45 400 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (a) Two counter-propagating edge modes (u/d) of a fractional quantum Hall droplet at filling factor ν are con- nected by a quantum point contact, through which quasipar- ticles of charge e∗ 1 and scaling dimension δ1 can tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Cur- rent is measured at the lower edge’s drain, denoted by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A current of Iinj is injected into the upper edge via a second, in- jection QPC, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' from a third auxiliary edge mode (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The injection QPC is placed at a bias voltage of V , and allows tunneling of quasiparticles of charge e∗ 2 and scaling dimension δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' All other sources and drains are grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (b) The ratio between I/Iinj in the dilute case and I/Iinj in the full case, as a function of temperature, for ν = e∗ 1/e = e∗ 2/e = 1/3, and for different scaling dimensions δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For the dilute case, we Iinj = 10pA, and assume kBT ≪ eV for all relevant tem- peratures, such that the contribution from Gdirect to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In the full case, we use V = 10µV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Both cases use ξ = 72mK, τc = 10−13s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' When the dilute case satisfies ℏIinj/e ≪ kBT ≪ eV ≪ ℏ/τc, and the full case satisfies ℏIinj/e = νeV/2π ≪ kBT ≪ ℏ/τc, the ratio approaches an asymptote that does not depend on scaling dimension, allow- ing extraction of the mutual statistics θ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Inset: I/Iinj for the dilute and full cases as a function of temperature for δ1 = 1/6, the canonical value for a Laughlin 1/3 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' the system to a regime where δ only affects observables through a non-universal prefactor, which then cancels out in the ratio of currents given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We arrive at this regime by employing a careful ordering of the various energy scales in the system, such that ℏIinj/e ≪ kBT throughout the entire crossover of Ginj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This ensures that the current I is given to linear response in Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We present an analytic expression generalizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1) out- side of this regime in App A, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' While in the full limit the edge that enters the tunnel- ing QPC is in equilibrium at chemical potential V , at the dilute limit we need the injection QPC to reflect only a small fraction of the impinging electrons, such that the resulting injection current is Poissonian and rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Said differently, the injected current in this limit must satisfy Iinj ≪ σxyV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Furthermore, the beam must still be dilute when arriving at the tunneling QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' As such, the dis- tance between the two QPCs must be sufficiently small that no equilibration or dephasing occurs along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Finally, we assume that tuning the injection QPC does not affect the transparency of the tunneling QPC, to en- sure that all non-universal constants are cancelled when examining the ratio of the two limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [39] Easy extraction of θ12 requires Gdirect to be sub- dominant (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Quantitatively, this is the case if both kBT ≪ eV and 4δ1 < 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' These con- straints result from the direct tunneling process being dominated by short time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Naive theories describ- ing quasiparticles may satisfy this condition even if the aforementioned non-universal effects change the scaling dimension quite significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For example, theory gives δ = 1/2m for Laughlin quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Edge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— We now define the system’s Hamilto- nian and derive the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' As shown by Wen, the edge theory of a general Abelian FQH state can be described by n-boson fields, φ(x, t) ≡ (φ1, φ2, · · · φn)T [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' These define the theory in conjunction with a charge vector, q, which determines the electric charge carried by each bo- son field, and the so called K-matrix, which determines the commutation relations between the boson fields, [φi(x), ∂x′φj(x′)] = i2π(K−1)ijδ(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (2) The filling factor is then given by ν = qT K−1q, and the charge density is given by ρ = − 1 2πq · ∂xφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In terms of these fields, the Hamiltonian of a single FQH edge mode is given by Hedge = 1 4π n � i,j=1 ˆ dx∂xφiVij∂xφj, (3) where ˆV is a positive definite matrix describing the ve- locities of the modes and intra-edge interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' These edges support quasiparticles of the form ψl ∼ eil·φ, where l is a vector of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The charge of these quasiparti- cles is then given by e∗ l = qT K−1l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The configuration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1(a) involves two edges, u and d, tunnel-coupled by a QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is described by two copies of the Hamiltonian Hedge, time reversed with regard to one another, as well as a tunneling term, HT , which we treat as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Assuming only one type of quasiparticle, denoted by the vector l1 and car- rying charge e∗ 1, tunnels between the edges, this is given 3 by HT = ξ � ˆA + ˆA†� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' ˆA(t) ≡ ei(l1·φ(u)(0,t)−l1·φ(d)(0,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (4) Here, ξ is a small tunneling amplitude, which we assume to be real, and φ(u) (φ(d)) are the bosonic field operators on the upper (lower) edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We project the auxiliary edge a out of the Hamiltonian, as it is only used to “initialize” the state of the edge u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The current that tunnels from the upper edge to the lower edge is then given by the operator, ˆIT (t) = iξe∗ 1 � ˆA†(t) − ˆA(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Since the lower edge is grounded, we henceforth identify I = ⟨ˆIT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Expanding to leading order in ξ, the current is given by I(t) = e∗ 1ξ2 ˆ t −∞ dt′ �� ˆA†(t), ˆA(t′) � + � ˆA†(t′), ˆA(t) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5) Here, [·, ·] denotes commutation, and expectation values are calculated with respect to the Hamiltonian in the absence of tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Deviation from Equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— It is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5) that one needs to derive correlation functions such as ⟨ ˆA†(t) ˆA(t′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In equilibrium, at temperature T, the sys- tem is particle-hole symmetric, and the correlation func- tions are given by [2, 40] ⟨ ˆA†(t) ˆA(t′)⟩0 = ⟨ ˆA(t) ˆA†(t′)⟩0 (6) = � πTτc sinh (πT |t − t′|) �4δ1 e−i2πδ1sgn(t−t′), where δ1 is the scaling dimension of the quasiparticle l1, and τc > 0 is a short time cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Two main features are carried over from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (6) to the correlation functions out of equilibrium - the exponen- tial decay at time difference larger than ℏ/T, and the phase e2πiδ1 associated with an interchange of the time arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We now consider two non-equibrium cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In the first we introduce a constant bias voltage V ≡ Vu − Vd be- tween the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In the setup of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1(a), this corre- sponds to a fully closed injection QPC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Iinj = σxyV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The introduction of the voltages can be formally ab- sorbed into the boson fields by use of a simple gauge transformation, which maps φ(u/d)(x, t) �→ φ(u/d)(x, t)+ K−1qVu/d (t ∓ x/v) /ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This accordingly modifies the correlation functions by a phase factor ⟨ ˆA†(t) ˆA(t′)⟩full = ⟨ ˆA†(t) ˆA(t′)⟩0ei e∗ 1 V ℏ (t−t′), ⟨ ˆA(t) ˆA†(t′)⟩full = ⟨ ˆA(t) ˆA†(t′)⟩0e−i e∗ 1 V ℏ (t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (7) In the second non-equilibrium driving, we consider in- jecting a single quasiparticle, denoted by the vector l2, into the upper edge at the location xinj < 0 and at time tinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In view of the commutation relations (2), the application of the quasiparticle creation operator e−il2·φ(u)(xinj,tinj) on the edge creates a soliton in each of the boson fields, φ(u)(x, tinj) �→ φ(u)(x, tinj) − 2πK−1l2Θ (x − xinj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (8) We assume here the injection happens instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This assumption will be relaxed to find the subleading term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The fields at general times can then be obtained using the equations of motion dictated by the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' If all modes are chiral with the same velocity v, this amounts to replacing x−xinj → x−xinj −v (t − tinj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The soliton thus arrives at the QPC, x = 0, at time t0 ≡ tinj − xinj/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The c-number shift in the bosonic field of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (8) leads to a phase shift in the correlator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We see directly from the definition of the operator ˆA in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (4) that ⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ ˆA†(t) ˆA(t′)⟩0e2πil1K−1l2[Θ(t−t0)−Θ(t′−t0)], ⟨ ˆA(t) ˆA†(t′)⟩qp = ⟨ ˆA(t) ˆA†(t′)⟩0e−2πil1K−1l2[Θ(t−t0)−Θ(t′−t0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (9) The phase we obtain is the standard definition of mutual braiding statistics between two quasiparticles, θ12 ≡ πl1K−1l2 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (9) shows that the product gains a phase of e2iθ12sgn(t−t′) if the arrival time t0 is between the times t′ and t, and a triv- ial phase of 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We emphasize how naturally this result came from the underlying theory: the only as- sumptions necessary to obtain this are the commutation relations, (2), and the existence of quasiparticles in the edge’s excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This result holds for different boson modes with differ- ent velocities if all solitons arrive at the tunneling QPC more or less concurrently, avoiding dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is the case if |xinj|/∆v ≪ ℏ/T, where ∆v is the velocity difference between the fastest and the slowest modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Time-domain interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— The appearance of the phase, θ12, can be understood as time-domain interfer- ometry of the two distinct ±e∗ 1 quasiparticle-quasihole excitations, before and after the injected e∗ 2 quasiparticle arrives at the QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A similar physical picture has been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [25, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' To show this we consider the configuration of a single injected particle, as described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In this case the non-equilibrium correlation function takes the form, ⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ψl2(t0) ˆA†(t) ˆA(t′)ψ† l2(t0)⟩0, (10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=', the expectation value is calculated with respect to the state resulting from exciting the ground state |0⟩ with a single quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Here we omit the position variable from the quasiparticle injection operator ψ† l2(t0), and as- sume it arrives at the tunneling QPC x = 0 at time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5) is then given by integration over multiple terms of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We define |t, t0⟩− ≡ ˆA(t)ψ† l2(t0) |0⟩ and |t, t0⟩+ ≡ ˆA†(t)ψ† l2(t0) |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 4 𝜈 𝐼 𝑉 −𝑒1 ∗ 𝑒1 ∗ 𝑒2 ∗ (a) (b) I Injection Time I Injection II Arrival III Pair Time I Injection III Pair II Arrival III Pair II Arrival FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Time-domain interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (a) I A quasiparticle is injected from the sourced, left edge, through the injection QPC, and into the upper edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' II The injected quasiparti- cle, by virtue of its chiral motion along the edge, arrives at the tunneling QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' III A quasiparticle-quasihole pair is cre- ated at the tunneling QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (b) The two processes by which charge carriers may ultimately arrive at the drain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The in- jected quasiparticle arrives at the tunneling QPC either before (upper panel) or after (lower panel) the creation quasiparticle- quasihole pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' These two processes interfere, with a relative phase dictated by the mutual statistics phase, ei2θ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5) can now be re-written as I ∝ − ˆ t −∞ dt′ � b=± b �� |t, t0⟩b + |t′, t0⟩b ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (11) The expression above involves two interference terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The term with b = − is an interference between cre- ation of −e∗ 1 quasiholes on the upper edge at the QPC at times t and t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The two interfering processes are shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' As shown in the first row of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (9), these two processes are distinguished by a non-trivial phase of ei2θ12 if the arrival time t0 is in be- tween the quasiholes’ creation times, t′ < t0 < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Com- bined with the equilibrium correlation function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (6), one finds that this interference gives a term proportional to cos (2θ12 − 2πδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Using similar arguments, the term with b = + in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (11), gives an interference term pro- portional to cos (2θ12 + 2πδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The total contribution from the two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (11) is thus proportional to sin (2θ12) sin (2πδ) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This interference happens entirely in the time domain, and along only one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' It is however crucial that this edge be part of a two-dimensional bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is important both because the second edge is required to absorb the leftover quasiparticle or quasihole resulting from the pair creation at the QPC, and because the injected quasiparti- cle must be created within a bulk FQH droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Further- more, the bulk is intimately related to the edge through bulk-edge correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This dictates that the statisti- cal phase contributing to time-domain interference along a single edge, which our setup measures, is the same as the phase obtained from spatial exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' It is easy to generalize this to injection of multiple quasiparticles: as long as all injected quasiparticles are mutually independent, each injected quasiparticle con- tributes a phase of e2iθ12 if and only if the arrival time at the point contact was between t′ and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' If we assume this is a Poissonian process, with a quasiparticle injection rate of Iinj/e∗ 2, we obtain for t > 0 ⟨ ˆA†(t) ˆA(0)⟩dilute ⟨ ˆA†(t) ˆA(0)⟩0 = ∞ � n=0 (tIinj/e∗ 2)ne−tIinj/e∗ 2 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' e2inθ12 = e−tIinj/e∗ 2(1−e2iθ12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12) This is precisely the result given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [23, 25] for injec- tion along a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Adding injected quasiparticles to the lower edge and generalizing for t < 0 are straight- forward using the same arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— The effect of driving the system out of equilibrium is completely encapsulated in the correlation functions obtained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' These can then be used to derive any observable of interest, such as charge or heat currents in any of the system’s drains, or their respective auto- and cross-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For concreteness, we present the explicit results of such a calculation for the charge current at the lower drain, denoted as I in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We show that a simple cohort of current measurements is sufficient to obtain the mu- tual statistics θ12, without requiring correlation measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We focus on the regime where the temperature is large compared to the injected current ℏIinj/ekBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For the full limit, this assumption guarantees linear response in the voltage and in the injected current, which in this limit is Iinj = σxyV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For the dilute limit, the exponen- tial suppression of the equilibrium correlation function at times larger than ℏ/T, guarantees that the exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12) may be expanded to first order in Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Conse- quently, ⟨ ˆA†(t) ˆA(t′)⟩full/dilute ⟨ ˆA†(t) ˆA(t′)⟩0 ≈ 1 + iωf/d (t − t′) , (13) where the frequencies ωf/d are given by ωf = e∗ 1V ℏ = e∗ 1 ℏ Iinj σxy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' ωd = iIinj e∗ 2 � 1 − e2iθ12� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (14) The zeroth order term corresponds to the equilibrium state and does not contribute to the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The ratio of the two first order contributions is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Explicit calculation of the resulting current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5), given in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A, finds that Ifull/dilute = 2πe∗ 1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) Re � ωf/d � , (15) where B(x, y) is the Euler Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' It is thus imme- diately apparent that by focusing on the ratio between the full and dilute beams, all dependence on δ1, T and ξ drops out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Examining the ratio I/Iinj, and noting that σxyℏ/e∗ 1e∗ 2 = νe2/2πe∗ 1e∗ 2 we thus obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 5 For general temperatures, the current can no longer be treated as a linear response to the drive of the full or di- lute beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We hence obtain the typical power laws char- acterizing tunneling in Luttinger liquids [2, 34, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Comparing measurements of the full and dilute limits at the low temperature limit T ≪ e∗V, Iinj can still give a quantity related to the mutual statistics θ12, but will ex- plicitly depend on the value of δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We present general expressions for the current in this case in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For a fermionic θ12 = π, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (15) gives no current at all for a dilute electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' However, Landauer-Buttiker- Imry scattering theory [44] tells us the current is given by the product of the transparencies of the two QPCs along the electron’s path, regardless of whether they are close to full transmission or full reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This requires accounting for the direct tunneling term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1), which now becomes the leading contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We do this by accounting for the finite width of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This leads to the required, Landauer-Buttiker- Imry consistent result of Idilute = 4π2τ 2 c ξ2Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The phys- ical intuition behind the requirement of a finite soliton width is that tunneling without time-domain interferom- etry, dubbed the direct tunneling process in [24, 25], is dominated by short times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Performing these calculations explicitly in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' B, we show that the ratio between the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1) and Gdirect is ∝ (Tτs)4δ1−2, where τs is the soliton width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' It has been shown [24, 25] that τ −1 s ∝ max{eV, kBT};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' as such, to ensure Gdirect is sub-dominant, the dilute limit must be measured when kBT ≪ eV and 4δ1 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Several contemporary experimental setups use the equivalent of non-interacting fermionic formulae to rea- sonable success [45], corresponding to the limiting value of 2δ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In this case, the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1) is a numerical coefficient of order one, which may depend solely on e∗, δ1 and θ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For non-interacting fermions, this coefficient is easily found by comparing to known Landauer-Buttiker-Imry scattering theory [44], but it is straightforward to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We discuss this coefficient further in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— We propose a simple method to extract anyonic exchange statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Our system consists only of a single quantum Hall droplet with two QPCs, which effectively create a time-domain interferometer, as can be identified from current measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We thus avoid both current correlation (or noise) measurements, and the need for a real space interferometer, making the iden- tification of the exchange statistics much more accessible than existing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' All time-domain interferom- etry is between pairs of an injected quasiparticle and a tunneling quasiparticle, and occurs at the same edge, as previously proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Both the exchange statistics θ11 of the tunneling quasi- particle, and θ22 of the injection quasiparticle, do not appear in our derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Rather, it is the two particles’ mutual statistics, θ12 that affect the modified correlation functions, and hence, the physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Likewise, the scaling dimension and electric charge which directly effect observables are only those of the tunneling quasi- particle, δ1 and e∗ 1 (properties of the injected quasiparti- cles may implicitly enter through the injection rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Only in the case where the injected and tunneling quasiparticles are identical, l1 = l2, do we obtain ex- change statistics for a single quasiparticle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We re- mark that this is indeed the case in the experiment of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [22], where all quasiparticles are Laughlin e∗ = e/3 anyons, and subsequent recreations for the ν = 1/3 and ν = 2/5 cases [26, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Interestingly, a recent ex- periment employing a similar setup, where the injected quasiparticle was a e/3 anyon and the tunneling quasi- particle was an electron, observed Andreev-like reflection [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is consistent with a mutual statistics phase of θ12 = π, for which Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1) gives no time-domain interfer- ometry signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='— We thank Tomer Alkalay, Moty Heiblum, Changki Hong, June-Young Lee and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Sim for insightful discussions and comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This work was partially supported by grants from the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreements LEGOTOP No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 788715 and HQMAT No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 817799), the DFG (CRC/Transregio 183, EI 519/7-1), the BSF and NSF (2018643), the ISF Quantum Science and Technol- ogy (2074/19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' was supported by the Clore Scholars Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Arovas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Gennser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Anthore, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Pierre, Quasiparticle Andreev scattering in the ν = 1/3 frac- tional quantum Hall regime (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='08068 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Appendix A: Finite temperature current from time-domain interferometry Here derive explicit expressions for the tunneling current I at finite temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This section neglects the contribution Gdirect (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (1), which is discussed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We begin with the expression for the current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Writing this explicitly, I = e∗ 1ξ2 ˆ t −∞ dt′ � � ˆA†(t) ˆA(t′) � − � ˆA(t′) ˆA†(t) � + � ˆA†(t′) ˆA(t) � − � ˆA(t) ˆA†(t′) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A1) In the case where the edges are not driven out of equilibrium, we plug the equilibrium correlation functions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (6), and obtain I = 0, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A similar expression can be written for the symmetrized current fluctuations, �� δ ˆIT (t), δ ˆIT (t′) �� = (e∗ 1)2ξ2 � � ˆA†(t) ˆA(t′) � + � ˆA(t′) ˆA†(t) � + � ˆA†(t′) ˆA(t) � + � ˆA(t) ˆA†(t′) � � , (A2) where we define δ ˆIT ≡ δ ˆIT − ⟨δ ˆIT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We do not focus on current fluctuations in this work, but note that our methods reproduce the known results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' [23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We now want to obtain the current for each of the three methods of driving the two edges out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Each of these leads to a corresponding multiplicative factor to the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A finite bias voltage V , used for the “full” beam, gives the correlation functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' injection of a single quasiparticle gives the correlation functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' and a dilute, Poissonian beam of quasiparticles gives the correlation functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Plugging in these appropriate correlation functions gives after minor algebra and changes of variables Ifull = 2ie∗ 1ξ2 ˆ ∞ 0 d˜t sin �e∗ 1V ℏ ˜t �� � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A3a) Idilute = 2ie∗ 1ξ2 ˆ ∞ 0 d˜t sin � Iinj e∗ 2 ˜t sin 2θ12 � exp � Iinj e∗ 2 ˜t (1 − cos 2θ12) � � � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A3b) Iqp = 2ie∗ 1ξ2 ˆ t −∞ dt′ sin (2θ12 [Θ (t − t0) − Θ (t′ − t0)]) � � πTτc i sinh (πT (t − t′ − iτc)) �4δ1 − � πTτc i sinh (πT (t′ − t − iτc)) �4δ1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A3c) We proceed using the identity, correct at the limit τc → 0, i � � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� = � � � � � −2πτ 2 c ∂˜tδ(˜t) 2δ1 = 1 � πT τc sinh(πT |˜t|) �4δ1 2 sin (2πδ1)sgn(˜t) 2δ1 ̸= 1 , (A4) where δ(t) is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This identity is necessary to treat the case of δ1 = 1, which otherwise may lead to divergent integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 2 Standard manipulations of these integrals then give results in terms of the Euler Beta function and the incomplete Beta function, B (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' a, b) ≡ ´ x 0 ta−1(1 − t)b−1dt, B (a, b) ≡ B (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We thus obtain the general results Ifull = 2e∗ 1ξ2(2πT)4δ1−1τ 4δ1 c sinh �e∗ 1V 2T � B � 2δ1 + i e∗ 1V 2πT , 2δ1 − i e∗ 1V 2πT � (A5a) Idilute = − 2π cos (2πδ1) Γ (4δ1)e∗ 1ξ2(2πT)4δ1−1τ 4δ1 c Im � � Γ � Iinj e∗ 2 1−cos(2θ12)+i sin(2θ12) 2πT + 2δ1 � Γ � Iinj e∗ 2 1−cos(2θ12)+i sin(2θ12) 2πT + 1 − 2δ1 � � � (A5b) Iqp = 4e∗ 1ξ2(2πT)4δ1−1τ 4δ1 c sin (2θ12) sin (2πδ) B � e−2πT (t−t0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 1 + 2δ1, 1 − 4δ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A5c) Here, Γ(a) is the Euler Gamma function, satisfying B(a, b) = Γ(a)Γ(b) Γ(a+b) , and Im[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' ] denotes the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The high temperature and zero temperature limits of the full beam and dilute beam are then immediately repro- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For e∗V, ℏIinj/e∗ 2 ≪ kBT, one expands to leading order in e∗V/T and Iinj/e∗ 2T, one obtains Ifull ≈ 2πe∗ 1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) e∗ 1V ℏ , (A6a) Idilute ≈ 2πe∗ 1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) Iinj e∗ 2 sin (2θ12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A6b) We thus see that the mutual statistics are immediately extractable from the dilute case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' While this expression does depend on the non-universal ξ and δ, as well as the temperature, these are all encoded in a prefactor which appears in the full case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We can hence lose this unwanted prefactor by examining the ratio between the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For T ≪ e∗V, Iinj/e∗ 2, we use the identities lim x→∞ Γ (x + a) = Γ (x) xa, lim x→∞ sinh(πx)B (a + ix, a − ix) = π Γ(2a)x2a−1, to obtain Ifull ≈ 2πe∗ 1ξ2τ 4δ1 c Γ (4δ1) �e∗ 1V ℏ �4δ1−1 , (A7a) Idilute ≈ − 2πe∗ 1ξ2τ 4δ1 c cos (2πδ1)Γ (4δ1) �Iinj e∗ 2 �4δ1−1 Im �� 1 − cos (2θ12) + i sin (2θ12) �4δ1−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A7b) By tuning 2δ1 → 1, we again obtain an expression from which the mutual statistics are easily extractable, with an identical non-universal prefactor appearing in both the full and dilute cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' However, once the scaling dimension is tuned to this critical value, the contribution from time-domain interferometry no longer dominates the direct tunneling process, as can be seen from the calculation of Gdirect in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We note that for temperatures larger than the source voltage, one has to account for injection of both quasiparticles and quasiholes through the injection QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This can be done by modifying the Poissonian correlation function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12) according to ⟨ ˆA†(t) ˆA(0)⟩dilute ⟨ ˆA†(t) ˆA(0)⟩0 = e−tIinj/e∗ 2(1−e2iθ12) → e−tIqp inj/e∗ 2(1−e2iθ12)e−tIqh inj/e∗ 2(1−e−2iθ12), (A8) where Iqp inj is the injection rate of quasiparticles, and Iqh inj is the injection rate of quasiholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is a similar expression to the three QPC setup considered in [23] and [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Performing the same algebra as in this section, and identifying Iinj ≡ Iqp inj − Iqh inj, one then reproduces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A6) for the high temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Finally, it is instructive to consider the current due to the injection of a single quasiparticle at time t0, which was obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In this case we must examine the explicit temperature dependence, as tunneling of a single quasiparticle may be relevant, and we lack any other energy scale to serve as a cutoff for the RG flow of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This current exhibits a power-law decay for t − t0 ≪ 1/πT, consistent with the orthogonality catastrophe that characterizes injection into Luttinger liquid edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For 2δ1 = 1, this results in ⟨ˆIT ⟩qp ∝ δ (t − t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This gives some intuition as to what makes the 2δ1 = 1 case so unique - the QPC just scatters the incident particle with some probability, without inducing any long-time correlations, resulting in the direct tunneling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' 3 Appendix B: Finite soliton width: restoring Landauer-Buttiker-Imry for electrons and subleading corrections The results of App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' A are seemingly inconsistent with the known non-interacting electron limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Indeed, inserting e∗ 1 = e∗ 2 = e, 2δ1 = 2δ2 = 1 and θ12 = π into these results would indicate that the dilute electron beam gives no current at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is in direct contrast with the intuition of Landauer-Buttiker-Imry scattering theory, which would indicate that the current should be given by the product of the transparencies of the two QPCs along the electron’s path, regardless of whether they are close to full transmission or full reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The culprit of this result is a peculiarity of soliton physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The boson field φ is compact under φ �→ φ + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' As such, a soliton of height 2πK−1l2 would appear to leave the boson field completely unperturbed if K−1l2 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This corresponds precisely to electron injection operators [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' As such, our soliton description is ill-equipped to treat electrons without modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We solve this issue by introducing a finite width to the soliton, τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' To fully recreate the known non-interacting result, it is crucial to maintain an order of limits such that the soliton width is larger than the short-time cutoff, τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We note that we still take care to ensure that τs < 1/T, (Iinj/e∗ 2)−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' the solitons are still narrow compared to the larger time scales in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Previous works [24, 25], performing a full Keldysh calculation, have shown the soliton width (refered to in the cited papers as the temporal width) is given by the voltage, h/e∗V , if eV > kBT, and by the inverse temperature ℏ/kBT if eV ≲ kBT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' as such, the dilute limit must be measured in the regime Iinj/e∗ 2 ≪ kBT ≪ eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Formally, this means that injecting a quasiparticle into the upper edge at the location x0 and time t0 transforms the boson field according to φ(u)(x, t0) �→ φ(u)(x, t0) − 2πK−1l2 � 1 π tan−1 �x − x0 τs � − 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Accordingly, the correlation functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (9) are now replaced with ⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ ˆA†(t) ˆA(t′)⟩0 exp � 2iθ12 π � tan−1 �t − t0 τs � − tan−1 �t′ − t0 τs ��� , ⟨ ˆA(t) ˆA†(t′)⟩qp = ⟨ ˆA(t) ˆA†(t′)⟩0 exp � −2iθ12 π � tan−1 �t − t0 τs � − tan−1 �t′ − t0 τs ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B1) One indeed sees that at the limit τc → 0, one reproduces the immediate soliton results from the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' To find the correlation function in the presence of a dilute, Poissonian beam of injected quasiparticles, we now must sum over the number of injected quasiparticles, in a manner similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' However, this is now trickier, for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' First, the accumulated phase explicitly depends on the time of the injected quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Second, injected quasiparticles outside of the window [0, t] can still affect the correlation function, due to the long tails of the finite-width solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' So generalizing the methods that lead to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12), the correlation function now changes to define ⟨ ˆA†(t) ˆA(0)⟩fw ⟨ ˆA†(t) ˆA(0)⟩0 = � n � (t + 2cτc) Iinj e∗ 2 �n e −(t+2cτc) Iinj e∗ 2 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' �ˆ t+cτc −cτc dt0P (Particle injected at t0) e2i θ12 π [tan−1( t−t0 τs )−tan−1( 0−t0 τs )] �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B2) Here c is some unitless cutoff, chosen such that injected quasiparticles affect the correlation function only if they are injected in the window [−cτc, t + cτc], which we will eventually take to be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The probability of injection at a particular time t0 is given by P (Particle injected at t0) = Iinj/e∗ 2e−Iinjt0/e∗ 2 ´ t+cτc −cτc dt0Iinj/e∗ 2e−Iinjt0/e∗ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B3) Performing this sum, and re-defining this integration with unitless variables, we find that the new correlation function is given in integral form by ⟨ ˆA†(t) ˆA(0)⟩fw ⟨ ˆA†(t) ˆA(0)⟩0 = exp � − (t + 2cτc) Iinj e∗ 2 � 1 − Iθ12 �Iinj e∗ 2 τs, t 2τs ��� , Iθ12 (a, b) ≡ a 2 sinh (a(b + c)) ˆ b+c −b−c dxe−axe2i θ12 π [tan−1(x+b)−tan−1(x−b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B4) 4 By plugging this new correlation function into the expression for the current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (5), one now finds Idilute = 2ie∗ 1ξ2 ˆ ∞ 0 d˜t sin � (˜t + 2cτc) Iinj e∗ 2 Im � Iθ12 � Iinj e∗ 2 τs, t 2τs ��� exp � (˜t + 2cτc) Iinj e∗ 2 Re � 1 − Iθ12 � Iinj e∗ 2 τs, t 2τs ��� × � � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B5) Careful re-application of the limit τc → 0 indeed replicates our previous result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For general θ12, the integral Iθ12 (a, b) is difficult to solve analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In the main text, this is circumvented by taking the limit τc → 0, allowing use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A4), in conjunction with replacing (˜t+2cτc) Iinj e∗ 2 Iθ12 � Iinj e∗ 2 τs, t 2τs � → −i˜tωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' However, as noted previously, fermionic exchange statistics corresponding to values of θ12 that are integer multiples of π lead to ωd = 0, and hence give a vanishing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' As such, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B5) must be calculated in full while retaining a finite τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' To simplify these expressions, we assume that � Iinj e∗ 2 � is significantly larger than any other time scale in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This makes sense from a physical perspective as well, as it corresponds to the assumption that injection is sufficiently rare such that solitons do not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In this case, one can assume the probability of injection which appears in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B2),(B3) is approximately uniform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' P (Particle injected at t0) ≈ 1/(t + 2cτc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' One can now safely take the limit c → ∞ without artificial divergences, giving the simpler result, ⟨ ˆA†(t) ˆA(0)⟩fw ⟨ ˆA†(t) ˆA(0)⟩0 = exp �Iinj e∗ 2 ˆ ∞ −∞ dt0 � e2i θ12 π [tan−1( t−t0 τs )−tan−1( 0−t0 τs )] − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B6) Since we undertook this endeavor with the explicit goal of finding the correct result for non-interacting electrons, we wish to find this integral for 2δ1 = 1, θ12 = π, and e∗ 1 = e∗ 2 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This value of θ12 allows one to significantly simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B6) using trignometric identities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' plugging the resulting correlation function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A1), we obtain Iθ12=π dilute = 2ie∗ 1ξ2 ˆ ∞ 0 d˜t sin � Iinj e∗ 2 2π˜t(2τs)2 ˜t2+(2τs)2 � exp � Iinj e∗ 2 2π˜t2(2τs) ˜t2+(2τs)2 � � � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B7) As can be seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A4), the expression in the curled brackets is approximately zero for ˜t > τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We can thus approximate the total integral as the contribution from short times, ˜t ≤ τc ≪ 1/πT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' To leading order, this will be given by Iθ12=π,2δ1=1 dilute ≈ 2ie∗ 1ξ2τ 2 c ˆ ∞ 0 d˜tIinj e∗ 2 2π˜t(2τs)2 ˜t2 + (2τs)2 � � 1 i˜t + τc �2 − � 1 −i˜t + τc �2� = (2τs)2 (2τs + τc)2 4π2ξ2τ 2 c Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B8) Now taking the limit τc ≪ τs, we compare to the electron case in, say, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (15) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We find that the result we expect for non-interacting electrons is indeed 4π2ξ2τ 2 c Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is consistent with - the current is linear in the injected current, and in the transparency of the tunneling QPC (which is given by ξ2τ 2 c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' For general values of θ12 and δ1 this integral is more difficult to solve analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' However, it is possible to re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B6) as ⟨ ˆA†(t) ˆA(0)⟩fw ⟨ ˆA†(t) ˆA(0)⟩0 = exp Iinj e∗ 2 � sin (2θ12) t + fθ12 (t, τc)) � , (B9) fθ12 (t, τc)) ∝ � � � � � t t ≲ τs τs t ≫ τs, θ12 ̸= π (τs)2/t t ≫ τs, θ12 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B10) Plugging this into the general expression for the current, and expanding to linear response in Iinj e∗ 2 we find 5 Idilute = 2ie∗ 1ξ2 Iinj e∗ 2 ˆ ∞ 0 d˜t � sin (2θ12) t + fθ12 (t, τc)) �� � πThwτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B11) The term proportional to sin (2θ12), as discussed at length above, is the main interest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is calculated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We see there that the time scales in the system contribute a leading term of the form ∝ (ξτc)2(Tτc)4δ1−2Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' The term proportional to fθ12 (t, τc)) contains several contributions: at short times (˜t ∼ τc), we obtain a con- tribution of order (ξτc)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' at long times (˜t ∼ 1/πT) we obtain a contribution of order (ξτc)2(τs/τc) (Tτc)4δ1−1 for θ12 ̸= π and (ξτc)2(τs/τc) (Tτc)4δ1 for θ12 = π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' and at intermediate times (˜t ∼ τs) we obtain contributions of order (ξτc)2(τs/τc)1−4δ1 and (ξτc)2(τs/τc)2−4δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We compare these contributions to the coefficients of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (15) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A6), which give the time-domain interferom- etry process, which is of order (ξτc)2 (Tτc)4δ1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Utilizing τc ≪ τs ≪ 1/πT, we see that the long time contribution is always subdominant, but the short time dominates for 2δ1 ≥ 1 - consistent with both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B8) and the known electron result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This is consistent with our physical intuition: direct tunneling dominates short times, which give the main contribution for 2δ1 ≥ 1, whereas time-domain interferometry dominates long times, which give the main contribution for 2δ1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Finally, if we indeed assume 2δ1 < 1, the intermediate time contribution dominates the entire direct process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' In this case, the ratio between the time-domain interferometry process and the direct process is given by ∝ (Tτs)4δ1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' This again confirms that we must have a soliton width smaller than the inverse temperature to ensure time-domain interferometry This method is also what we use to calculate the current for an almost full beam, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' σxy − Ginj ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Since in this case, the beam can be treated as a conjoined full beam of fractional quasiparticles with a dilute beam of e∗ = e holes, we have 2θ12 = 2πn regardless of the tunneling quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Defining the injection rate of holes as Iholes inj ≡ σxyV − Iinj, we combine the full beam correlation function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (7) and the regularized Poissonian hole injection to obtain described in this section I|Ginj−σxy|≪1 = 2ie∗ 1ξ2 ˆ ∞ 0 d˜t sin � e∗ 1V ℏ ˜t − Iholes inj e 2π˜t(2τs)2 ˜t2+(2τs)2 � exp � Iholes inj e 2π˜t2(2τs) ˜t2+(2τs)2 � × � � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B12) In the relevant limits, the same methods as previously mention allow us to approximate the exponent in the denominator as 1, and to expand the sine in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We thus have the sum of two linear responses, one in in e∗ 1V ℏ and one in − Iholes inj e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' Taking, as in the Landauer-Buttiker-Imry case, the limit τs ≫ τc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' a soliton width that is larger than the short time cutoff, this can be re-written as I|Ginj−σxy|≪1 ≈ 2ie∗ 1ξ2 ˆ ∞ 0 d˜t � e∗ 1V ℏ − 2π Iholes inj e � ˜t � � πTτc i sinh � πT �˜t − iτc �� �4δ1 − � πTτc i sinh � πT � −˜t − iτc �� �4δ1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (B13) Identifying � e∗ 1V ℏ − 2π Iholes inj e � = 2π e � σxyV − Iholes inj � ≡ 2π e Iinj, we see that this is precisely the same integral that we had in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' (A3a) for the full beam case, with the replacement σxyV → Iinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' We note that we used here σxy = ee∗/h, which is correct only for Laughlin edge states, ν = 1/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} +page_content=' this is valid as Laughlin edges are the outer level of heirarchal FQH fluids, and thus are the states of interest for nearly full closed QPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAyT4oBgHgl3EQfPPZ4/content/2301.00021v1.pdf'} diff --git a/29AzT4oBgHgl3EQfR_sP/content/tmp_files/2301.01223v1.pdf.txt b/29AzT4oBgHgl3EQfR_sP/content/tmp_files/2301.01223v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1a0ec57c4c78100a9bd67817643d6f6ceb58e8f --- /dev/null +++ b/29AzT4oBgHgl3EQfR_sP/content/tmp_files/2301.01223v1.pdf.txt @@ -0,0 +1,2691 @@ +EXPLOREADV: TOWARDS EXPLORATORY ATTACK FOR +NEURAL NETWORKS +by +LUO TIANZUO +A THESIS SUBMITTED FOR THE DEGREE OF +MASTER OF COMPUTING +in +COMPUTER SCIENCE +in the +GRADUATE DIVISION +of the +NATIONAL UNIVERSITY OF SINGAPORE +2022 +Supervisor: +Associate Professor KHOO Siau Cheng +arXiv:2301.01223v1 [cs.CR] 1 Jan 2023 + +Contents +Abstract +iii +List of Figures +iv +List of Tables +v +1 +Introduction +1 +1.1 +Deep learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.2 +Adversarial Example . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +1.2.1 +Lp-norm distance . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.2.2 +Minimal Adversarial Perturbation . . . . . . . . . . . . . . . +4 +1.3 +Limitation of previous work +. . . . . . . . . . . . . . . . . . . . . . +5 +1.4 +Our work +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.5 +Thesis Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2 +Related Work +9 +2.1 +DeepFool attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.1.1 +Binary classifier . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.1.2 +Multi-class classifier +. . . . . . . . . . . . . . . . . . . . . . +10 +2.1.3 +Extend to Lp-norm . . . . . . . . . . . . . . . . . . . . . . . +11 +2.1.4 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +2.2 +Brendel & Bethge (BB) attack . . . . . . . . . . . . . . . . . . . . . +12 +2.3 +Imperceptible attack . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +2.4 +Localized (Patch) attack . . . . . . . . . . . . . . . . . . . . . . . . +14 +2.5 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3 +ExploreADV +16 +3.1 +The Basic Form: L∞ Attack . . . . . . . . . . . . . . . . . . . . . . +17 +3.2 +With Focus: Regional Attack +. . . . . . . . . . . . . . . . . . . . . +17 +3.3 +Imperceptible Attack: Variance Map +. . . . . . . . . . . . . . . . . +17 +3.3.1 +Non-adaptive Imperceptible Attack . . . . . . . . . . . . . . +19 +3.3.2 +Adaptive Imperceptible Attack +. . . . . . . . . . . . . . . . +19 +3.4 +Vulnerability Estimation: Importance Map . . . . . . . . . . . . . . +20 +3.4.1 +Integrated Gradients . . . . . . . . . . . . . . . . . . . . . . +21 +3.4.2 +SmoothGrad . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +i + +3.4.3 +Correction Coefficient . . . . . . . . . . . . . . . . . . . . . . +23 +3.4.4 +Efficiency of the method . . . . . . . . . . . . . . . . . . . . +23 +3.5 +The System as a Whole +. . . . . . . . . . . . . . . . . . . . . . . . +24 +3.6 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +4 +Experiments & Results +26 +4.1 +Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +4.2 +Evaluation of L∞ attack . . . . . . . . . . . . . . . . . . . . . . . . +28 +4.2.1 +Discussion on the result of BB and ExploreADV . . . . . . . +29 +4.3 +Evaluation of Variance Map based Imperceptible attack . . . . . . . +29 +4.3.1 +Discussion on imperceptibility . . . . . . . . . . . . . . . . . +30 +4.4 +Evaluation of Importance Map based Vulnerable Region Estimation +31 +4.4.1 +Discussion on the generality of regions +. . . . . . . . . . . . +33 +4.5 +Evaluation of System Usability +. . . . . . . . . . . . . . . . . . . . +33 +4.6 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +5 +Conclusion and Future Work +36 +5.1 +Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5.2 +Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . +36 +5.2.1 +Attack For Image Segmentation/Object Detection . . . . . . +36 +5.2.2 +Interpretable Attack +. . . . . . . . . . . . . . . . . . . . . . +37 +Bibliography +38 +A Distance Metric +43 +A.1 SSIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +A.2 CIEDE2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +B System test Instructions +44 +ii + +Abstract +ExploreADV: Towards exploratory attack for Neural Networks +by +LUO Tianzuo +Master of Computing in Computer Science +National University of Singapore +Although deep learning has made remarkable progress in processing various +types of data such as images, text and speech, they are known to be susceptible +to adversarial perturbations: perturbations specifically designed and added to the +input to make the target model produce erroneous output. Most of the existing +studies on generating adversarial perturbations attempt to perturb the entire input +indiscriminately. In this paper, we propose ExploreADV, a general and flexible +adversarial attack system that is capable of modeling regional and imperceptible +attacks, allowing users to explore various kinds of adversarial examples as needed. +We adapt and combine two existing boundary attack methods, DeepFool and +Brendel&Bethge Attack, and propose a mask-constrained adversarial attack system, +which generates minimal adversarial perturbations under the pixel-level constraints, +namely “mask-constraints”. +We study different ways of generating such mask- +constraints considering the variance and importance of the input features, and show +that our adversarial attack system offers users good flexibility to focus on sub-regions +of inputs, explore imperceptible perturbations and understand the vulnerability of +pixels/regions to adversarial attacks. We demonstrate our system to be effective +based on extensive experiments and user study. +Keywords— Neural network, Adversarial example, Regional attack, Imperceptible +attack, Mask constraint, Vulnerability estimation +iii + +List of Figures +1.1 +Artificial neural network architecture [2] . . . . . . . . . . . . . . . . . +2 +1.2 +Process of Adversarial Attack . . . . . . . . . . . . . . . . . . . . . . . +3 +1.3 +Adversarial Images on MNIST dataset . . . . . . . . . . . . . . . . . . +5 +1.4 +Regional and Imperceptible sticker on a truck. . . . . . . . . . . . . . . +7 +2.1 +Schematic of Brendel & Bethge (BB) attack [3] +. . . . . . . . . . . . . +12 +3.1 +Workflow of our proposed ExploreADV. +. . . . . . . . . . . . . . . . . +16 +3.2 +Rectangle Shaped Region. . . . . . . . . . . . . . . . . . . . . . . . . . +19 +3.3 +Arbitrary Shaped Region. +. . . . . . . . . . . . . . . . . . . . . . . . . +19 +3.4 +Using SmoothGrad to remove noise in importance maps generated by +Integrated Gradients [47]. +. . . . . . . . . . . . . . . . . . . . . . . . . +22 +4.1 +L∞ and Imperceptible attack on MNIST. . . . . . . . . . . . . . . . . . +30 +4.2 +L∞ and Imperceptible attack on CIFAR10. . . . . . . . . . . . . . . . . +31 +4.3 +Change of ratioIG+S(β) with respect to k. . . . . . . . . . . . . . . . . . +33 +4.4 +Change of time cost with respect to k. +. . . . . . . . . . . . . . . . . . +33 +4.5 +The Post-Study System Usability Questionnaire (Version 3) [27] . . . . +34 +B.1 The Post-Study System Usability Questionnaire (Version 3) +. . . . . . +46 +iv + +List of Tables +4.1 +Datasets and Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.2 +Average L∞ norm for different attacks +. . . . . . . . . . . . . . . . . . +28 +4.3 +Average execution time (seconds) for different attacks . . . . . . . . . . +28 +4.4 +Different Measures on the adversarial examples generated by L∞ attack +and Imperceptible attack. +. . . . . . . . . . . . . . . . . . . . . . . . . +31 +4.5 +Average rmin and average ratioh for different heuristics. . . . . . . . . . +32 +v + +CHAPTER 1. INTRODUCTION +Chapter 1 +Introduction +In recent years, deep learning has become a critical role in a variety of domains +such as computer vision [51], natural language processing [36], speech and audio +processing [40], etc. It has made significant breakthroughs, especially in the tasks +of image classification [24, 17, 19], segmentation [29, 16] and object detection +[13, 12, 41], where deep learning has achieved high accuracy and even exceeded +human performance. Szegedy et al. [50] found an intriguing property of deep +neural networks, that it is possible to arbitrarily change the network’s prediction by +applying an imperceptible and non-random perturbation to the test image. In this +work, we propose a novel system to study such examples, also known as “adversarial +examples”. +1.1 +Deep learning +Deep learning is part of a broader family of machine learning methods [25]. It +uses artificial neural network composed of a large number of neurons with activation +functions to perform representation learning of data. Deep neural network can +automatically learn the explicit and implicit features of the original data without +relying on expert knowledge. +A typical artificial neural network architecture is shown in Figure 1.1. Each +of the neurons receives input signal from previous layer and performs weighted +connection, then processes the output of the neuron through an activation function +and transmits the signal to next layer, thus constructing a deep neural network +structure. It can be formally expressed as shown in Equation 1.1. +y = hn(...h2(w2 · h1(w1 · x + b1) + b2)) +(1.1) +where x and y are the input and output of the network, wi, bi and hi are the +respective weights, biases and activation functions in the ith layer of the network, i += 1, 2, ..., n, where n is the number of hidden layers in the network. +Even though deep neural network has achieved remarkable results by simulating +the structure of human brain neural network, the way deep neural networks work +1 + +CHAPTER 1. INTRODUCTION +Figure 1.1: Artificial neural network architecture [2] +is still quite different from human cognition and lack of interpretability, making it +difficult to guarantee the credibility of its output. +The growing use of deep neural networks has raised concerns about their security +and reliability. Szegedy et al. [50] found that deep neural networks are highly +vulnerable to image samples with specific perturbations, and called such image +samples with adversarial perturbations as “adversarial examples”. +1.2 +Adversarial Example +An adversarial example refers to the input sample formed by adding specifically +designed perturbations to an original sample, which can make the well-trained deep +learning model give erroneous outputs. Specifically, In the field of computer vision, +an adversarial example is usually an image formed by adding slight perturbations +to the input image that are difficult to be perceived by human vision, resulting in +incorrect prediction from the model, e.g. identifying a panda as a gibbon. +Adversarial attack is the procedure of generating adversarial examples in order +to fool a deep learning model. Figure 1.2 demonstrates the process of adversarial +attacks. A variety of attack algorithms have been proposed to generate adversarial +examples [50, 14, 33, 35, 6]. Without loss of generality, we formally define an +adversarial example in the context of image classification problem. +Definition 1 (Adversarial Example). Given an input image x ∈ Rd, and a score- +based image classifier f : Rd �→ RK that maps x to a set of K labels S = {1, 2, ..., K} +according to: +ˆy(x) = arg max +k∈S +fk(x) +(1.2) +2 + +Input layer +Hidden layers +Output layer +2 +hi +h? +hn +0 +Input 1 +Output 1 +Input 2 +Output n +Input nCHAPTER 1. INTRODUCTION +Figure 1.2: Process of Adversarial Attack +where fk(x) is the score function for label k ∈ S, ˆy(x) ∈ S is the predicted label for +input x. +The collection of adversarial examples with respect to x and f is defined as: +{x′ | d(x, x′) < ϵ, ˆy(x′) ̸= ˆy(x)} +(1.3) +where d(x, x′) is the distance (a measure to be discussed in § 1.2.1) between the +adversarial example and the original input, and will be bounded by a small predefined +constant ϵ. Each adversarial example x′ can be considered as a combination of the +original image x and an adversarial perturbation δ, i.e., x′ = x + δ. When using +Lp-norms as distance metric, d(x, x′) = ∥x′ − x∥p = ∥δ∥p < ϵ. +In order to keep the adversarial image perceptually close to the original image, +good distance metrics to measure the perceptual similarity between two images +are important. Ideally, smaller distance represents closer similarity with respect to +human perception. As it is difficult to quantitatively measure human perception, in +many classical adversarial attack algorithms [50, 14, 37, 35, 6], Lp-norm distance is +applied. +1.2.1 +Lp-norm distance +To measure the distance between an image x and its adversarial image x′, Lp- +norm distance is defined by the Lp-norm of the pixel value difference ∥x′ − x∥p, i.e., +the Lp-norm of the adversarial perturbation: ∥δ∥p. The definition of Lp-norm is +shown in Definition 2. +3 + +DOrmc SSScatDE +awesanatt +orolnal Input a +real label +"panda" +Deep Learning +Classifier f +adversarial label +"gibbon" +acyarsarial +erturbarton d +meCHAPTER 1. INTRODUCTION +Definition 2 (Lp-norm). Given a vector δ = (δ1, δ2, ..., δn) in the n-dimensional +real vector space Rn, and a real number p ≥ 1, the Lp-norm of δ is defined by: +∥δ∥p = ( +n +� +i=1 +|δi|p) +1 +p +(1.4) +In practise, L0, L1, L2, and L∞-norm distances are commonly used: +• L1 (Manhattan distance): ∥δ∥1 = �n +i=1|δi| +• L2 (Euclidean distance): ∥δ∥2 = +��n +i=1|δi|2 +• L∞ (Chebyshev distance): ∥δ∥∞ = maxi|δi| +L0-norm is special, it’s defined as ∥δ∥0 = �n +i=1{1 | δi ̸= 0}, which counts the +number of non-zero pixel-value differences. It is actually not a norm because it does +not satisfy absolute homogeneity1. +From the definition, it can be noticed that Lp-norm distance is only related to +the pixel value differences δ, it is not affected by the actual pixel values in the clean +image x or its adversarial image x′. +1.2.2 +Minimal Adversarial Perturbation +More recent attention has focused on finding the minimal adversarial perturbation, +also known as the robustness of model at point x [35]. They typically use the Lp-norm +distances as the distance metric, and try to find the minimal perturbation necessary +to change the prediction of the model. The minimal adversarial perturbation with +respect to the Lp-norm is defined as: +arg min +δ +∥δ∥p, δ ∈ {δ | ˆy(x + δ) ̸= ˆy(x)} +(1.5) +The optimization problem in Equation 1.5 is NP-complete for non-linear and +non-convex classifiers [22]. In practice, it is often approximated by different attack +algorithms, either by using some heuristics [14, 35, 10] or by solving minimization +problems [50, 6]. +Specifically, when considering minimal adversarial perturbation with respect +to L∞-norm, we call the perturbation size the robust radius [54], as defined in +Definition 3. +Definition 3 (Robust Radius). Given an input image x ∈ Rd, and a score-based +image classifier with prediction function ˆy(x). The robust radius r∗ ∈ R of the +classifier on x is defined as: +r∗ = min(r) +s.th. ∃x′ ˆy(x′) ̸= ˆy(x) and |x′ +i − xi| ≤ r for i = 1, ..., d +(1.6) +1Given a vector space V , a function f : X �→ R satisfies absolute homogeneity if f(λv) = |λ|f(v) +for all v ∈ V and λ ∈ R, where |λ| denotes the absolute value of the scalar λ [39]. +4 + +CHAPTER 1. INTRODUCTION +1.3 +Limitation of previous work +Existing attacks are not perceptually constrained. While most adversarial +attack algorithms try to find adversarial perturbation with small Lp-norms, it is +argued that using Lp-norms to measure the perceptual similarity between two images +is neither necessary nor sufficient [44]. When using Lp-norms as the distance metric, it +implies the assumption that perturbations on different pixels in an image are equally +important for human eyes. However, as Liu et al. [28] suggests, perturbations become +less perceptible in the regions with high spatial variation, and more perceptible +in smooth regions. As shown in Figure 1.3, the adversarial images found by some +existing adversarial attack algorithms [35], while aiming to have small Lp-norm, +appear perceptually blurred and unrealistic with perceptible noise. +Figure 1.3: Adversarial Images on MNIST dataset. These adversarial images +are supposed to represent the digits 7, 2, 1, 0 and 4, and are predicted as 9, 0, 6, 3 +and 9, yet they look blurred and unrealistic. +Studies also showed that the adversarial examples found by some existing methods +neither faithfully simulate physical objects nor resemble natural images [30, 53]. +To develop methods to find adversarial examples perceptually closer to the original +image, better perceptual distance metrics are needed to evaluate the effective of +the methods. There are other image similarity distance metrics proposed, such as +CIEDE2000 [32] and SSIM [52], that can supplement the Lp-norms for measuring +perceptual similarity. Details of the metrics can be found in Appendix A. +Existing attacks are not suitable for modeling real-world threats. Re- +cent research also suggests that adversarial examples can be generalized to the real +world. Sharif et al. [45] showed that face recognition systems can be fooled by people +wearing adversarially constructed eyeglass frames. Brown et al. [5] create a method +to generate “adversarial patches” that can be printed and added to the scene to fool +a classifier. While such “physical-world” attacks may seem practical for real-world +ML systems, it is currently not suitable to be modeled by most of the existing +attacks which aim to modify the whole image indiscriminately — “physical-world” +attacks on an entire scene is usually not feasible. As a result, some attack methods +seek to perturb only few pixels [37] or a small region in the image [45]. Nevertheless, +these attacks often do not restrict themselves to imperceptible perturbations, the +resulting adversarial perturbations are often clearly visible. +To overcome these limitations and take advantage of the power of existing adver- +5 + +pred: 9 +pred: 0 +pred: 6 +pred: 3 +pred: 9 +2CHAPTER 1. INTRODUCTION +sarial attack methods, a natural approach is to properly constrain the adversarial +perturbations during the attack. For example, the perturbation can be constrained +to be: +• Regional. Keep pixels unperturbed outside the target region. +• Imperceptible. Reduce perturbations of pixels in regions where such pertur- +bations are more perceptible. +1.4 +Our work +In this paper, we propose ExploreADV, a general and flexible adversarial attack +system that is capable of modeling regional and imperceptible attacks. A novel type +of constraint, namely “mask-constraint” is proposed in our system. The system +allows users to explore different types of threat models based on their interest, e.g. +they can decide for an image the region they want to focus on, whether they want +the perturbation to be imperceptible, and how many pixels / how large a region +they want to perturb. For example, to test the reliability of the vision model on a +self-driving system, one may want to know if a maliciously designed sticker on a truck +would deceive the system, and whether such sticker can be designed imperceptible +so that people would not notice any anomaly before a potential accident happens. +Such exploration are possible in our system, as shown in Figure 1.4. +We propose an idea of mask-constraint in our system to enable such flexibility, +as defined in Definition 4. +Definition 4 (Mask-constraint). Given a clean image x = (x1, x2, ..., xd) with +d pixels, a mask-constraint contains a set of non-negative constant constraints +E = (ϵ1, ϵ2, ..., ϵd), where ϵi ∈ [0, 1] is the constraint on the ith pixel xi. Each ϵi +indicates the maximum allowable absolute perturbation on xi, where 0 means no +perturbation allowed. An adversarial image x′ = (x′ +1, x′ +2, ..., x′ +d) found under the +mask-constraint is limited in the closed interval xi − ϵi ≤ x′ +i ≤ xi + ϵi for each pixel +x′ +i. +With the mask-constraints in our system, we formulate the problem of finding +adversarial perturbation as: +Problem (Adversarial perturbation under mask-constraint). Given an im- +age classification neural network N with prediction function ˆy(x), an image x = +(x1, x2, ..., xd), a mask-constraint E = (ϵ1, ϵ2, ..., ϵd) on each pixel of x, and a real +number p ≥ 1. Find a perturbation δ = (δ1, δ2, ..., δd) satisfying {δi | − ϵi ≤ δi ≤ ϵi}, +such that when adding the perturbation to the image, the model’s prediction on the +resulting image x′ = x + δ is different from its prediction on the original image: +ˆy(x′) ̸= ˆy(x). Moreover, the Lp-norm of the perturbation ∥δ∥p is minimized. +In this thesis, we propose a novel approach to solve this problem. We adapt and +combine two existing adversarial attack methods, DeepFool [35] and Brendel&Bethge +6 + +CHAPTER 1. INTRODUCTION +Figure 1.4: Regional and Imperceptible sticker on a truck. We illustrate the +adversarial examples found by our system by adding perturbation to a small region +of a truck. From left to right. left column - original image, and the regional mask +indicating the region to apply attack (white area) where the black area remains +unperturbed. middle column - normal adversarial examples that is perturbed region- +ally, and the adversarial perturbations, right column - imperceptible adversarial +examples that is perturbed regionally, and the adversarial perturbations. +[4] Attack, which will be introduced in detail in Chapter 2, and adapt them to work +under our mask-constraint setting. DeepFool is first applied to yield a preliminary +adversarial example under the mask-constraint. If a preliminary adversarial example +is found, it is used as a starting point for Brendel&Bethge attack, which can +then minimize the Lp-norm of the perturbation under the same mask-constraint, if +DeepFool fails to find an adversarial example, the system terminates and returns +no result, an adversarial example might not exist or might be found with more +iterations of DeepFool. We later show that the integration of the mask-constraint +makes regional and imperceptible adversarial perturbations possible in our system. +To facilitate users to automatically generate imperceptible perturbations and +select pixels/regions to perturb, we study two types of maps reflecting the variance +and importance of pixels in this work: +Variance map The variance map measures the spatial variation of the image by +calculating the variance of pixel values in a small neighbourhood. It is helpful +when imperceptibility is desired, as perturbations in regions with high spatial +variation are less perceptible than those in smooth regions. Our system use a +variance map based method to generate constraints on pixels according to their +variance, allowing less perturbation for pixels in regions with low variance. +7 + +clean image +regional +regional &imperceptible +prediction:truck +prediction: automobile +prediction:automobile +regional mask +perturbation +perturbationCHAPTER 1. INTRODUCTION +Importance map The importance map measures the importance of each pixel to +changing the prediction of the classifier. It is helpful when there is a need +to perturb only a subset of all pixels. Our system use an importance map +based method to estimate the vulnerability of pixels/regions in the image to +adversarial attacks, and select pixels/regions with high vulnerability to apply +perturbations. +It is worth noticing that there are many ways to generate these maps, and our +methods are not restricted to any single implementation of them. There has been a +few attempts to make use of such variance map [9] and importance map [1], but we +adapt or improve their methods in this work. +In this work, we only consider L∞-norm as it is commonly used to access model +robustness [46], so our system can be used to estimate the robust radius under the +mask-constraint. Yet, our method is orthogonal to the selection of Lp-norms and +can be easily extended to other norms. +Our main contributions can be summarized as follows: +• We propose a novel adversarial attack system with mask-constraints, which +is more general than existing attacks because it can limit the adversarial +perturbation to any sub-region of the whole image, and limit the perturbation +magnitude on any pixel of the image independently. +• We adopt and combine two existing adversarial attack methods, DeepFool +and Brendel&Bethge Attack, and show that the resulting method generates +adversarial perturbations with small L∞ norm that are comparable to the +adversarial perturbations generated by the state of the art L∞ attack methods +[10, 38]. +• We study different ways to automatically generate mask-constraints in our +system by considering the variance and importance of the pixels in the image, +which provides the user with much flexibility to explore various kinds of +adversarial examples conveniently, to generate regional and imperceptible +adversarial perturbations as they need. +• We suggest ways to enhance variance map and importance map based method. +We propose to adaptively loosen the variance map based mask-constraint to +generate imperceptible perturbations for models with different robustness, and +to add a correction coefficient to the importance map to better estimate pixel +vulnerability. +1.5 +Thesis Synopsis +The rest of this thesis is organized as follows. In Chapter 2, we conduct a +literature review on related works. +Chapter 3 provides details of our method. +Extensive experimental results are depicted in Chapter 4. We conclude the entire +thesis as well as discuss further directions for future research in Chapter 5. +8 + +CHAPTER 2. RELATED WORK +Chapter 2 +Related Work +In this section, we review some prior work of adversarial attacks that are related +to ours. +2.1 +DeepFool attack +Considering that deep neural network is extremely vulnerable to adversarial ex- +amples, Moosavi-Dezfooli et al. [35] proposed a method called DeepFool, which aims +to calculate the minimal perturbation with respect to L2-norm (extendable to other +Lp-norms) necessary to change the classifier’s decision, as shown in Equation 2.1. +δ∗ = arg min +δ +∥δ∥2 subject to ˆy(x + δ) ̸= ˆy(x) +(2.1) +where x is an image and δ∗ is the minimum perturbation. ˆy is the prediction function +as in Equation 1.2. +Since a multi-class classifier can be viewed as an aggregation of binary classifiers, +they first introduced an efficient algorithm to find adversarial examples for binary +classifiers, then extended it to the multi-class case. +2.1.1 +Binary classifier +For the binary classifier, assume ˆy(x) = sign(f(x)), where f : Rd �→ R represents +an arbitrary scalar-valued image classifier, sign(·) is the sign function that extracts +the sign of a real number. +2.1.1.1 +Affine classifier +Given a binary affine classifier f(x) = ωTx+b, the corresponding affine hyperplane +F = {x|ωTx + b = 0}. Then the minimum perturbation δ∗ to change the classifier’s +prediction on the original sample x0 is equal to the orthogonal projection of x0 onto +9 + +CHAPTER 2. RELATED WORK +the affine hyperplane F, which is given by the closed-form formula: +δ∗(x0) = arg min +δ +∥δ∥2 +subject to sign(f(x0 + δ)) ̸= sign(f(x0)) +=f(x0) +∥ω∥2 +2 +ω +(2.2) +2.1.1.2 +General classifier +When f is a general binary differentiable classifier, DeepFool adopts an iterative +procedure to estimate the minimum perturbation δ. +At each iteration i, f is +linearized around the current point xi, and the minimal perturbation δi is computed +as: +arg min +δi +∥δi∥2 +such that f(xi) + ∇f(xi)Tδi = 0 +(2.3) +2.1.2 +Multi-class classifier +For the multi-class classifier, assume ˆy(x) = arg maxk∈S fk(x), where f : Rd �→ +RK represents an arbitrary score-based image classifier, fk(x) is the score function +of f(x) for corresponds to label k. +2.1.2.1 +Affine classifier +Given an affine classifier f(x) = ωTx+b. The minimum perturbation that spoofs +the classifier can be overridden in the following way: +δ∗(x0) = arg min +δ +∥δ∥2 +s.th. ∃k : fˆy(x0)(x0 + δ) ≤ fk(x0 + δ) += |fl(x0) − fˆy(x0)(x0)| +∥ωl − ωˆy(x0)∥2 +2 +(ωl − ωˆy(x0)) +(2.4) +where l is the class different from ˆy(x0) with the closest hyperplane of the boundary, +as shown in Equation 2.5. +l = arg min +k̸=ˆy(x0) +|fk(x0) − fˆy(x0)(x0)| +∥ωk − ωˆy(x0)∥2 +(2.5) +2.1.2.2 +General classifier +In general case of multi-class classifiers, DeepFool attack pushes the original +sample toward the decision boundary with each round of perturbation until it +crosses the decision boundary to form an adversarial example or reach the maximum +allowable iterations, as shown in Algorithm 1. +10 + +CHAPTER 2. RELATED WORK +Algorithm 1: DeepFool: multi-class case +Input: image x, classifier f, maximum iterations max_iter +Output: perturbation ˆδ +1: Initialize x0 ← x, i ← 0 +2: while ˆy(xi) = ˆy(x0) and i < max_iter do +3: +for k ̸= ˆy(x0) do +4: +f′ +k ← fk(xi) − fˆy(x0)(xi) +5: +ω′ +k ← ∇f′ +k +6: +ˆl ← arg mink̸=ˆy(x0) +|f′ +k| +∥ω′ +k∥2 +7: +δi ← +|f′ +ˆl| +∥ω′ +ˆl∥2 +2 ω′ +ˆl +8: +xi+1 ← xi + δi +9: +i ← i + 1 +10: return ˆδ = � +i δi +2.1.3 +Extend to Lp-norm +Although DeepFool was proposed based on L2-norm, it can simply be adapted +to find minimal adversarial perturbations for any Lp-norms (p ∈ {∞} ∪ [1, ∞)), by +substituting the update steps in line 6 and line 7 in Algorithm 1 with the following +steps respectively: +ˆl ← arg min +k̸=ˆy(x0) +|f ′ +k| +∥ω′ +k∥q +(2.6) +δi ← +|f ′ +ˆl| +∥ω′ +ˆl∥q +q |ω′ +ˆl|q−1 ⊙ sign(ω′ +ˆl) +(2.7) +where ⊙ is the element-wise multiplication, q = +p +p−1. +Note that for p = 2 as in Algorithm 1, the steps in line 6 and line 7 are also +equivalent to the steps in Equation 2.6 and Equation 2.7. When p = 2, q = +2 +2−1 = 2, +Equation 2.6 matches line 6 of Algorithm 1, and the right half of Equation 2.7 can +be written as |ω′ +ˆl| ⊙ sign(ω′ +ˆl), which equals to ω′ +ˆl in line 7 of Algorithm 1. +For L∞-norm, the steps become: +ˆl ← arg min +k̸=ˆy(x0) +|f ′ +k| +∥ω′ +k∥1 +(2.8) +δi ← +|f ′ +ˆl| +∥ω′ +ˆl∥1 +sign(ω′ +ˆl) +(2.9) +2.1.4 +Summary +DeepFool uses a local linear approximation of the classifier to estimate the +optimal step towards the decision boundary. Compared with FGSM [14] attack, +DeepFool attack generates adversarial examples with smaller perturbation, and close +to the decision boundary. +11 + +CHAPTER 2. RELATED WORK +Figure 2.1: Schematic of Brendel & Bethge (BB) attack [3] +2.2 +Brendel & Bethge (BB) attack +Brendel et al. [4] developed Brendel & Bethge (BB) attack, a new set of gradient- +based adversarial attacks to find local minimal adversarial examples. The scheme +of the attack is shown in Figure 2.1. It starts from a randomly-drawn adversarial +example that is far away from the clean image (left in Figure 2.1), and first performs +a 10-step binary search to reach the decision boundary (middle in Figure 2.1), then +walk along the decision boundary to minimize the Lp distance to the clean image +(right in Figure 2.1). At each step k, they solve a constrained optimization problem +to find the optimal descent step δk within a trust radius r, and add it to the current +image ˜xk−1. Finally, ˜xk is returned, with the Lp-norm of the adversarial perturbation +minimized. +The constrained optimization problem is defined as: +arg min +δ +∥x − ˜xk−1 − δk∥p +s.th. 0 ≤ ˜xk−1 + δk ≤ 1 ∧ bkTδk = ck ∧ ∥δk∥2 +2 ≤ r +(2.10) +where x is the clean image, ˜xk−1 is the perturbed image after step k-1, δk is +the perturbation for step k, bk is the direction of the local decision boundary, ck +is the distance to the decision boundary. The decision boundary is defined by a +differentiable equality constraint adv(˜x) = 0, where adv(·) is the adversarial criterion +function. +Let ft(˜x) be the log-probability for label t on the current input ˜x, y being the +true label for the clean image x, then for targeted attack: +adv(˜x) = fy(˜x) − ft(˜x) +(2.11) +When adv(˜x) = 0, the log-probability of y is equal to the log-probability t. For +untargeted attack: +adv(˜x) = min +t,t̸=y(fy(˜x) − ft(˜x)) +(2.12) +When adv(˜x) = 0, the log-probability of y is equal to the log-probability of the class +with the highest log-probability among the other classes. +12 + +clean image +clean image +frog +clean image +frogCHAPTER 2. RELATED WORK +2.3 +Imperceptible attack +Luo et al. +[31] discovered, by investigating the human visual system, that +human eyes are more sensitive to perturbation in flat areas than textured areas. +Therefore, the texture features around a pixel should be taken into consideration +when attempting to add adversarial perturbation to the pixel. Texture features of +an image x can be quantified by the notion of variance, as shown in Equation 2.13. +SD(xi) = +�� +xk∈Si(xk − µ)2 +n2 +(2.13) +where SD(xi) represents the standard deviation of the pixel values among an n × n +neighborhood Si of pixel xi. Considering the impact of perturbation intensity on +human vision, they introduced the notion of “perturbation sensitivity” to measure +how much “attention” will be drawn by adding per “unit” perturbation on a pixel. +Sen(xi) = 1/SD(xi) +(2.14) +They further defined a distance metric based on this notion of “perturbation +sensitivity”, as shown in Equation 2.15. +d(x, x′) = +m +� +i=1 +|δi| ∗ Sen(xi) +(2.15) +where d(x, x′) denotes the distance between the adversarial example x′ = (x′ +1, x′ +2, ..., x′ +m) +and the original one x = (x1, x2, ..., xm), m is the total number of pixels and |δi| is +the intensity of perturbation on pixel xi. +They then proposed a targeted attack, which attempts to mis-classify an image +into a specific target class. They used a greedy algorithm to select pixels with +lower sensitivity and higher impact (larger gradient) in order to maximize the gap +between the probability of the target class and the maximal probability of all other +classes, and perturb the pixels in an iterative manner under certain constraint +d(x, x′) ≤ dmax. +Croce et al. [9] proposed sparse and imperceivable adversarial attacks, with +locally adaptive component-wise constraints on the perturbation. They defined the +constraint on each pixel as: +σij = κ +� +min(σ(x) +ij , σ(y) +ij ) +(2.16) +where κ is a hyper-parameter set by users, σ(x) +ij , σ(y) +ij are the standard deviation of +each color channel in x- and y-axis directions with the two immediate neighboring +pixels and the original pixel, i ∈ [1, d] represents each of the d pixels, j ∈ [1, 3] +represents each color channel. Given the constraints on each pixel, the adversarial +example generated under the constraint can be expressed in the following form: +x′ +ij = (1 + λiσij)xij, +with +− κ ≤ λi ≤ κ +(2.17) +They took min of σ(x) +ij +and σ(y) +ij +so that pixels along coordinate-aligned edges are +not changed. +13 + +CHAPTER 2. RELATED WORK +2.4 +Localized (Patch) attack +Instead of perturbing the whole image, some attack methods try to perturb only +a localized region in the image, or sometimes referred to as “patch attack” because +the perturbed region resembles a patch attached to the image. +Karmon et al. [21] created localized and visible adversarial noise (LaVAN) that +cover only 2% of the pixels in the image without covering the main object. In +this method, visible noise is added to a local position of the image to produce an +adversarial example. To confine the noise δ to a small area over the image x, they +used a mask m ∈ {0, 1}n, and the noised image is defined as (1 − m) ⊙ x + m ⊙ δ, +where ⊙ is element-wise multiplication. It is worth noticing that the noise is used +to replace the area rather than be added to it. As the term “visible” suggests, the +adversarial examples generated by this method are aimed to be easily detected by +human. +Dia et al. [11] proposed localized uncertainty attacks, which is a novel class of +adversarial attacks that creates adversarial examples against deep learning models +through replacement of uncertain regions in the original inputs. Instead of per- +turbing inputs indiscriminately, they utilize the uncertainty associated with the +predictions of the classifier and focus only on carefully chosen regions, to yield +better imperceptibility. They proposed to use a binary-valued (with values in {0, 1}) +mask ω with the same size of the input x, and apply the perturbation to the region +indicated by ω ⊙ x to generate adversarial examples. They trained a feed-forward +network called mask model to learn a distribution pv(x) over all possible masks ω +that, when drawing a mask from ω ∼ pv(x) and applying the perturbation to the +region ω ⊙ x, the predictive uncertainty of the classifier is maximized. Ideally, masks +that lead to greater uncertainty should have a higher probability in the distribution +pv(x). +Bai et al. [1] proposed Inconspicuous Adversarial Patch Attack (IAPA), which +generates inconspicuous adversarial patches using GANs. Compared with other +adversarial patches attacks, IAPA causes less modifications, and has higher chance +to evade detection from human. In their approach, they first used a GradCAM +[43] based “vulnerability map” to decide perceptual sensitivity of the victim model. +Based on the vulnerability map M, they then choose the area with the highest +importance as the victim area for patch attack. However, since GradCAM can only +work for Convolutional Neural Network (CNN) based models, their approach can +not be applied to Non-CNN based models. +2.5 +Summary +We introduced DeepFool attack and Brendel&Bethge (BB) attack in this chapter, +providing methods of finding adversarial examples and minimizing the Lp-norm of +the adversarial perturbation, which are adopted in our system and integrated with +mask-constraint. +14 + +CHAPTER 2. RELATED WORK +Two works on imperceptible attacks are introduced, which showed that variance of +image provides useful guidance on generating imperceptible adversarial perturbations. +However, they both use heuristics to select pixels and apply perturbation, so they +have no control on which pixels are selected. Also, they didn’t give sufficient results +to quantitatively evaluate the imperceptibility of their attack. +Three Localized (Patch) attacks are introduced, but they have different problems +that make them not suitable for our need. LaVAN generate adversarial examples +easy to be detected by human, localized uncertainty attack neither have control on +the attack region nor consider imperceptibility, IAPA suggests a good way to find +vulnerable regions, but only applies to convolutional models. +15 + +CHAPTER 3. EXPLOREADV +Chapter 3 +ExploreADV +In this section, we introduce the framework and formulation of our proposed +adversarial perturbation generation system named ExploreADV. The complete +workflow of our system is shown in Figure 3.1. The system requires a clean image x +and a Deep Learning classifier f. A mask generator is first used to generate a mask- +constraint E. The generator takes the clean image x and a series of parameterized +user specification as inputs, and either generate the mask-constraint automatically +or prompt the user to specify it through a GUI. Details of the mask generator +interface can be found in § 3.2 and § 3.5. A preliminary adversarial example x′ +is then generated by using DeepFool attack under the mask-constraint E. Finally, +Brendel&Bethge (BB) attack is used to minimize the Lp-norm distance between x′ +and x and get the optimized adversarial example x′′ under the same constraint E. +Figure 3.1: Workflow of our proposed ExploreADV. +16 + +maske-oonstraine E +Mask +Generator +doan iage 2 +DeepFool +Brendel&Bethge +Attack +Attack +Deep Learning Classifier f +real label +adv label +adv label +"4" +"9" +"9"CHAPTER 3. EXPLOREADV +3.1 +The Basic Form: L∞ Attack +Our system performs L∞ attack based on the algorithms of DeepFool and Brendel +& Bethge (BB) attacks, as we believe these two algorithms are good complement +for each other based on our experiment results in § 4.3. DeepFool L∞ attack finds +adversarial example near the decision boundary but does not guarantee to have +minimal L∞ distance from the clean image. BB L∞ attack finds adversarial example +on the decision boundary with minimized L∞ distance from the clean image, but +relies heavily on the quality of the starting point. The attack method we use in our +system is a combination of these two brilliant methods, as shown in Algorithm 2. +In summary, our system finds an adversarial example on the decision boundary +with minimal L∞ distance from the clean image, by first searching for a small +adversarial example close to the decision boundary (line 2 to line 14 in Algorithm 2). +It then minimizes the L∞ distance from the clean image while the subject image +remains an adversarial example (line 21 to line 28 in Algorithm 2). Different from +the original methods where the constraint on each pixel is homogeneous i.e. each +adversarial example should stay within the L∞ ball {x + δ | ∥δ∥∞ ≤ ϵ}, we use a +general mask-constraint as defined in Definition 4 so that the adversarial examples +are constrained in {x + δ | ∀δi∈δ |δi| ≤ ϵi}. +3.2 +With Focus: Regional Attack +One major advantage of our system is the flexibility to add perturbation to +any sub-region of the clean image, making our system able to not only simulate +whole-image attacks, but also simulate regional attacks that are more practical in +the physical world. +In order to allow users to define their own mask-constraint in a convenient and +unified way, we implemented a Graphical User Interface to enable users to indicate +regions on which they want to focus. As shown in Figure 3.2 and Figure 3.3, a user +can specify the target region for an attack either by clicking and dragging on the +screen to select a rectangle shaped region, or by clicking and brushing to select an +arbitrary shaped region. +3.3 +Imperceptible Attack: Variance Map +Images with small L∞ distances do not necessarily resemble one another. Fol- +lowing the same idea of imperceptible attacks [31, 9] introduced in § 2.3, our system +is also capable of generating imperceptible adversarial perturbations. This can be +easily achieved with the mask-constraint of our system. For our imperceptible attack, +we adapted the variance map proposed by Croce et al. [9], as in Equation 2.16. +17 + +CHAPTER 3. EXPLOREADV +Algorithm 2: Attack procedure of ExploreADV. +Global: loosen rate of the constraint λ +Global: per # of iteration to loosen the constraint T +Input: clean image x, classifier f, mask constraint E, maximum iterations +max_iter +Output: adversarial image x′ +1: Procedure Attack(n): +2: +Initialize x0 ← x, i ← 0 +3: +while ˆy(xi) = ˆy(x0) and i < max_iter do +4: +for k ̸= ˆy(x0) do +5: +f′ +k ← fk(xi) − fˆy(x0)(xi) +6: +ω′ +k ← ∇f′ +k +7: +ˆl ← arg mink̸=ˆy(x0) +|f′ +k| +∥ω′ +k∥1 +8: +δi ← +|f′ +ˆl| +∥ω′ +ˆl∥1 sign(ω′ +ˆl) +9: +if i > 0 and i mod T = 0 then +10: +for ϵ in E do +11: +ϵ ← ϵ ∗ λ +12: +xi+1 ← clip(xi + δi, E) +13: +i ← i + 1 +14: +x′ ← xi +15: +if ˆy(x′) = ˆy(x0) then +16: +return None // Attack Failed +17: +else +18: +x′ ← Optimize(x, f, x’, C) +19: +return x′ +Input: clean image x, classifier f, starting point ˜x0, mask constraint E +Output: optimized adversarial image x′′ +20: Function Optimize(x, f, ˜x0, C): +21: +Initialize i ← 0, b0 ← 0 +22: +while i 1, it represents the number of +pixels allowed to be perturbed. +3.6 +Summary +In this chapter, we introduced our system in detail. We adopted and combined two +existing attack methods – DeepFool and Brendel&Bethge attack. We integrated them +24 + +CHAPTER 3. EXPLOREADV +with our mask-constraint. We proposed to use variance-based mask-constraint to +generate imperceptible adversarial perturbations and adaptively loosen the constraint +to handle more robust models. We formulated the problem of finding a vulnerable +region within an image and proposed an importance map method to approximately +but efficiently solve this problem. We proposed to use Integrated Gradients with +SmoothGrad to generate importance maps; the proposed method is applicable to +any deep learning models. Finally, we further proposed a correction coefficient to fix +a technical glitch in using importance maps for vulnerability estimation. +25 + +CHAPTER 4. EXPERIMENTS & RESULTS +Chapter 4 +Experiments & Results +To access the effectiveness of ExploreADV, we conducted a series of experiments +using several neural networks for image classification on different datasets. We also +conduct user study to evaluate users’ perceived satisfaction of our system. +4.1 +Experimental Setup +Datasets. We consider three commonly-used datasets for image classification +and adversarial robustness benchmarking, including MNIST [26], CIFAR10 [23] and +STL10 [8]. +Models. We select a variety of models to thoroughly evaluate attacks under +different conditions. Most of the models are pre-trained models obtained from +ERAN’s github page1. For MNIST, we consider the following two models: M1, a 9- +layer undefended fully-connected network with 1610 units and ReLU activation; M2, +a 3-layer undefended convolutional network with 3,604 units and ReLU activation. +For CIFAR10, we consider four models: C1, a 6-layer undefended fully-connected +network with 3000 units and ReLU activation; C2, a 3-layer undefended convolutional +network with 7,144 units and Sigmoid activation; C3, a 6-layer convolutional network +with 62,464 units and ReLU activation, adversarial trained using DiffAI [34]; and +C4, a 19-layer residual network with 558K units and ReLU activation, adversarial +trained using PGD [33]. For STL10, we used the pre-trained 6-layer convolutional +network S1 from this github repository2, which takes normalized inputs with values +ranging from -1 to 1. The models are listed in Table 4.1. +Attacks. For L∞ attack, we compare our method against different state-of- +the-art attacks for finding adversarial perturbations with minimum L∞ norm: the +DeepFool attack [35], the Brendel & Bethge (BB) attack [4], the Fast Adaptive +Boundary (FAB) attack [10], and the Fast Minimum-norm (FMN) attack [38]. We +used the open-source implementations of DeepFool and FAB from AdverTorch3, and +1https://github.com/eth-sri/eran +2https://github.com/aaron-xichen/pytorch-playground +3https://github.com/BorealisAI/advertorch +26 + +CHAPTER 4. EXPERIMENTS & RESULTS +dataset +image size +model +architecture +#layers +#units +activation +training defense +accuracy +MNIST +28*28 +M1 +fully connected +9 +1610 +ReLU +None +0.95 +M2 +convolutional +3 +3604 +ReLU +None +0.98 +CIFAR10 +32*32 +C1 +fully connected +6 +3000 +ReLU +None +0.56 +C2 +convolutional +3 +5704 +Sigmoid +None +0.55 +C3 +convolutional +6 +48064 +ReLU +DiffAI +0.51 +C4 +residual +19 +558K +ReLU +PGD +0.82 +STL10 +96*96 +S1 +convolutional +5 +652K +ReLU +None +0.77 +Table 4.1: Datasets and Models. +implementations of BB and FMN from Foolbox4. In the other experiments, we only +use the attack method of our system. +Hyper-parameters. To ensure a fair comparison, we used similar default set- +tings for each attack. We report here the hyper-parameters used in each experiments. +For L∞ attack, we configured each attack to have 100% Attack Success Rate for all +models and all datasets, we set the max allowable perturbation size ϵ = 1.0 for all +attacks, the hyper-parameter configurations for each attack are detailed below. +DeepFool. We set the max number of iterations to be 50, and overshoot be 0.02. +FAB. We set the max number of iterations to be 100, bound of step bias +αmax = 0.1, extrapolation step η = 1.05, backward step β = 0.9, with no random +restarts. +BB. We used the default initial attack of BB, called Random-Noise attack, which +randomly draws 1,000 directions to search for adversarial examples and takes 1,000 +binary search steps to blend adversarial example and original image in each direction. +The adversarial example with the smallest Lp distance to the original image is +selected as the starting point. We set the number of iterations to be 100, number +of binary search steps to be 10, the trust radius decays every 20 iterations with a +coefficient of 0.5. +FMN. We set the number of iterations to be 100, number of binary search steps +to be 10, the decaying step size γ starting from 0.05. +ExploreADV For DeepFool, we set the max number of iterations to be 50, and +overshoot be 0.02. For BB, we set the number of iterations to be 100, number of +binary search steps to be 10, the trust radius decays every 20 iterations with a +coefficient of 0.5. +For Imperceptible attack, we used the L∞ attack of ExploreADV with the same +configuration described above. For adaptive loosening of the constraint, we set the +loosen_rate to be 1.2, and loosen the constraint every 10 iterations. +For Vulnerable Region estimation, we also used the L∞ attack of ExploreADV +with the same configuration described above to estimate the robust radius. +Metrics. For L∞ attack, we report the average L∞ norm of the perturbations. +For Imperceptible attack, we report the Attack Success Rate (ASR) for each attack, +the L0, L2 and L∞ norm of the perturbations, the structural similarity (SSIM) +between original images and adversarial examples, and perceptual color distance +(CIEDE2000) between original images and adversarial examples for colored datasets +4https://github.com/bethgelab/foolbox +27 + +CHAPTER 4. EXPERIMENTS & RESULTS +(CIFAR10, STL10). For Vulnerable Region estimation, we report the minimal robust +radius among all regions for an image and average it over 100 images, we also report +the robust radius of the regions found by different importance map based method. +To measure users’ satisfaction of our system, we issued and collected some PSSUQs +(Post-Study System Usability Questionnaire) [27] to measure the System Usefulness +(SYSUSE), Information Quality (INFOQUAL) and Interface Quality (INTERQUAL) +of our system. +4.2 +Evaluation of L∞ attack +To evaluate the performance of ExploreADV in L∞ attack, we first compare it +to state-of-the-art L∞ adversarial attacks. We calculate over the first 100 correctly +classified images for each dataset , the average L∞ norm of the adversarial perturba- +tions generated by each attack on the whole image, where smaller average L∞ norm +indicates better performance, as shown in Table 4.2. FAB is not evaluated for S1 +because their official implementation does not support input values ranging from -1 +to 1. +DEEPFOOL +FAB +BB +FMN +ExploreADV +M1 +0.07423 +0.06379 +0.06358 +0.06153 +0.06182 +M2 +0.1837 +0.1571 +0.1558 +0.1692 +0.1533 +C1 +0.01328 +0.008917 +0.009386 +0.008186 +0.008179 +C2 +0.01135 +0.01082 +0.01236 +0.01169 +0.01074 +C3 +0.01731 +0.01433 +0.01424 +0.01397 +0.01370 +C4 +0.04297 +0.03434 +0.03629 +0.03477 +0.03368 +S1 +0.005805 +- +0.004537 +0.004575 +0.004366 +Table 4.2: Average L∞ norm for different attacks +The results show that by combining the strength of DeepFool and Brendel & +Bethge attack, ExploreADV achieves comparable results to the state of the art +methods on different datasets, model architectures and training methods. +We also compare the execution time taken by each attack. Table 4.3 shows +the average execution time on one image for different attacks, the experiment was +done on the ResNet18 model C4 on CIFAR10 dataset. ExploreADV took moderate +execution time among all the attacks, which is acceptable considering its specific +objective in generating minimal L∞ perturbations. +DEEPFOOL +FAB +BB +FMN +ExploreADV +Time +1.97 +29.43 +19.22 +2.96 +7.62 +Table 4.3: Average execution time (seconds) for different attacks +28 + +CHAPTER 4. EXPERIMENTS & RESULTS +4.2.1 +Discussion on the result of BB and ExploreADV +Although ExploreADV uses the same method for minimizing the L∞-norm of +the perturbation as that of BB, it outperforms BB by having shorter execution time, +which we attribute to the quality of the starting point and the method to find it. +As mentioned in § 4.1, by default, BB use a Random-Noise attack, BB will +randomly draw 1,000 directions to search for adversarial examples and take 1,000 +binary search steps to blend adversarial example and original image in each direction. +In each step, it needs to run a forward pass of neural network, resulting in 1,000,000 +forward passes in total, while DeepFool only runs at most 50 forward passes, which +makes it a lot faster. +The Random-Noise attack used by BB is also not efficient, as the number of all +possible directions is 2N, where N is the number of pixels in the image. The search +space is so large that it is infeasible to cover the optimal one in 1,000 randomly +drawn directions. On the contrary, DeepFool utilizes local gradient information to +search for only the most promising direction. +4.3 +Evaluation of Variance Map based Impercep- +tible attack +We evaluate here the effectiveness of our variance map based imperceptible +attack by comparing it with our L∞ attack which represents normal Lp-norm based +attacks. We assess the imperceptibility of an adversarial attack according to the +SSIM [52] and CIEDE2000 [32] measures between the adversarial image generated +by the attack and the original image, which are mentioned in § 1.3. We also provide +image examples for human assessor to assess the effectiveness of our imperceptible +attack. +We first illustrate the differences between adversarial examples found by our +L∞ and imperceptible attacks. This serves to dispel a common misconception +that adversarial attack images is naturally imperceptible, and vice versa. The +imperceptible attack we use here is the adaptive version introduced in § 3.3. Some +examples are shown in Figure 4.1 and Figure 4.2. The adversarial examples found +by L∞ attack, while have small L∞ distance (epsilon) to the original images, do not +resemble the original images because of the noise introduced by the perturbation at +low variance regions. The imperceptible attack that is constrained by the variance +map, generates adversarial examples that are more similar to the original images +according to human perception. +We further quantitatively evaluate the effective of our imperceptible attack, +over 100 correctly classified image samples on each dataset, as shown in Table 4.4. +We compare the unconstrained L∞ attack with the non-adaptive imperceptible +attack and adaptive imperceptible attack introduced in § 3.3. The adversarial +examples found by L∞ attack, though having small L∞ norm, have smaller structural +similarity (SSIM) and larger color difference (CIEDE2000) compared to imperceptible +attacks. Non-adaptive imperceptible attack result in smaller L0, L2, SSIM and +29 + +CHAPTER 4. EXPERIMENTS & RESULTS +Figure 4.1: L∞ and Imperceptible attack on MNIST. We illustrate the differ- +ences of the adversarial examples found by L∞ attack and Imperceptible attack. +From top to bottom in each group. top row - original image, middle row - ad- +versarial examples found by L∞ attack, and the adversarial perturbations, bottom +row - adversarial examples found by Imperceptible attack, and the adversarial +perturbations. +CIEDE2000 measures, indicating smaller perceptual distance between original and +adversarial images, however, it often significantly reduces the attack success rate. +The adaptive imperceptible attack finds a balance between the attack success rate and +the imperceptibility of the perturbation, which generates adversarial examples with +slightly larger perceptual distance to the original image comparing with non-adaptive +imperceptible attack, but greatly improves the attack success rate. +4.3.1 +Discussion on imperceptibility +The difficulty in attaining imperceptible attack depends on the robustness of the +model and the input region allowed for perturbation. When the model has weak +robustness and the region allowed for perturbation is large, it can be easy to derive +an imperceptible attack, even without considering variance map. So, in Table 4.4, +we can find that in some cases, e.g. C3 and S1, the SSIM measures on the L∞ attack +is also high, which means the adversarial perturbations generated by L∞ attack can +also be quite imperceptible. The advantage of our variance map based imperceptible +attack is not well demonstrated in such cases. Our method is more powerful when +the model has relatively strong robustness, e.g., M1 and M2 in Table 4.4, or when +the perturbation is limited to a small region where normal Lp-norm based attacks +can not achieve imperceptibility. +30 + +clean +clean +clean +pred: 1 +pred: 4 +pred: 4 +minimaladv +Difference +minimal adv +Difference +minimaladv +Difference +pred: 6 +epsilon:0.16 +pred: 9 +epsilon: 0.14 +pred: 8 +epsilon:0.085 +imperceptible adv +Difference +imperceptible adv +Difference +imperceptibleadv +Difference +pred: 8 +epsilon: 0.53 +pred: 9 +epsilon:0.51 +pred: 8 +epsilon:0.15 +ICHAPTER 4. EXPERIMENTS & RESULTS +Figure 4.2: L∞ and Imperceptible attack on CIFAR10. We illustrate the +differences of the adversarial examples found by L∞ attack and Imperceptible +attack. From top to bottom in each group. top row - original image, middle row - +adversarial examples found by L∞ attack, and the adversarial perturbations, bottom +row - adversarial examples found by Imperceptible attack, and the adversarial +perturbations. +model +Attack +Attack Success Rate +L0 +L2 +L∞ +SSIM +CIEDE2000 +M1 +L∞ +100% +592.04 +2.21 +0.06 +0.82 +- +Imperc +24% +140.71 +1.44 +0.13 +0.98 +- +Imperc-Adap +60% +133.63 +4.36 +0.34 +0.93 +- +M2 +L∞ +100% +517.82 +9.47 +0.15 +0.71 +- +Imperc +14% +145.79 +1.73 +0.13 +0.97 +- +Imperc-Adap +52% +136.56 +5.56 +0.41 +0.91 +- +C3 +L∞ +100% +2815.62 +0.81 +0.01 +0.98 +70.50 +Imperc +89% +2759.44 +0.62 +0.02 +0.99 +58.60 +Imperc-Adap +99% +2731.90 +0.94 +0.03 +0.99 +68.71 +C4 +L∞ +100% +3051.14 +4.66 +0.03 +0.94 +152.87 +Imperc +58% +2937.72 +1.54 +0.04 +0.98 +92.23 +Imperc-Adap +93% +2900.11 +4.02 +0.11 +0.96 +137.07 +S1 +L∞ +100% +27236.80 +0.74 +0.01 +0.99 +151.30 +Imperc +100% +24803.66 +0.72 +0.01 +1.00 +97.43 +Imperc-Adap +100% +24803.66 +0.72 +0.01 +1.00 +97.50 +Table 4.4: Different Measures on the adversarial examples generated by L∞ attack +and Imperceptible attack. +4.4 +Evaluation of Importance Map based Vulner- +able Region Estimation +We evaluate the effectiveness of our proposed importance map based method for +finding the vulnerable region to adversarial attack. We estimate for an input image +31 + +clean +clean +clean +pred: ship +pred: ship +pred:airplane +minimal ady +Difference +minimal adv +Difference +minimal adv +Difference +pred: airplane +epsilon: 0.05 +pred:automobile +epsilon:0.013 +pred: bird +epsilon:0.029 +imperceptible adv +Difference +imperceptible adv +Difference +imperceptibleadv +Difference +pred: airplane +epsilon: 0.3 +pred:automobile +epsilon:0.015 +pred: bird +epsilon:0.06CHAPTER 4. EXPERIMENTS & RESULTS +rmin +ratioGradCAM +ratioGradCAM++ +ratioIG+S (ours) +ratioIG+S(β) (ours) +M1 +0.2060 +- +- +1.28 +1.30 +M2 +0.2610 +2.16 +1.97 +1.21 +1.19 +C1 +0.05169 +- +- +1.24 +1.21 +C2 +0.05595 +2.95 +3.76 +1.80 +1.65 +C3 +0.05304 +2.70 +2.42 +2.38 +2.31 +C4 +0.1046 +3.16 +4.61 +1.52 +1.46 +S1 +0.1105 +11.96 +12.80 +2.40 +3.04 +Table 4.5: Average rmin and average ratioh for different heuristics. GradCAM +and GradCAM++ only works for Convolutional Networks, so not applicable to M1 +and C1. +x ∈ Rd the vulnerability of a region specified by a binary mask ω ∈ {0, 1}d using +the minimal L∞ norm of the adversarial perturbation on this region, i.e., the robust +radius of the region. The robust radius is estimated using our L∞ attack. +We conducted experiment by focusing on imposing rectangular regions of fixed +10 × 10 size on different datasets and models. To determine the robust radius within +such a rectangular region for an image, We employed sliding window technique, +sliding the rectangular region across the entire image and applying our L∞ attack +on the region, and estimating the minimum robust radius among all the adversarial +perturbations found. The final minimum robust radius is denoted as rmin. +Next we computed the robust radius rh from the regions selected by the importance +map generated using different heuristics h, and computed the ratio ratioh = +rh +rmin +for each heuristic; this ratio illustrates the effectiveness of a heuristic in identifying +regions with high vulnerability. We considered four different heuristics with respect +to four importance maps, the GradCAM map, the GradCAM++ map, the Intergrat- +edGradients+SmoothGrad map (IG+S) and the IntergratedGradients+SmoothGrad +map with correction coefficient (IG+S(β)). +GradCAM [43] is a method that uses gradient coming back to last Convolutional +layer of CNN to assign importance weights to input pixels, with the same objective +of Integrated Gradients, it also can be used to generate the importance map. +GradCAM++ [7] is an improved version of GradCAM with similar usage. We +compare the average ratioh over 100 image samples on each dataset and model (10 +image samples for STL10 due to time efficiency) for the different heuristics, as shown +in Table 4.5. When no adversarial example is found in a region, the robust radius is +recorded as 1.0, which is the length of the valid range of pixel value. +We find that our method often select relatively good regions in affordable time. +For C4 (image size 32 * 32), exhaustive applying L∞ attack on a sliding window +on one image costs more than 5,000 seconds, while our importance map method +costs 34 seconds. For S1 (image size 96 * 96), exhaustive search on one image costs +more than 50,000 seconds, while our importance map method costs 28 seconds. +The result also shows that importance map generated using Integrated gradient +with SmoothGrad is good in estimating vulnerability, which is demonstrated by the +comparison with GradCAM and GradCAM++. +32 + +CHAPTER 4. EXPERIMENTS & RESULTS +Figure 4.3: +Change of ratioIG+S(β) +with respect to number of candidates +k. +Figure 4.4: Change of time cost with +respect to number of candidates k. +We also experimented on improving our method by calculating on more regions, +instead of selecting only the top 1 region with respect to importance score, we try to +find the top k candidates and then selects the one actually returns smaller distance. +We experiment on the IG+S(β) map we proposed, and show how the ratioIG+S(β) +and time cost change with the number of candidates k, as shown in Figure 4.3 and +Figure 4.4. +The ratioIG+S(β) can be largely reduced as the growing of k, while the time cost +grows linearly, which suggests that we can select a suitable k (e.g. k=20) to balance +the computing efficiency and the estimation accuracy of vulnerability. +4.4.1 +Discussion on the generality of regions +Although we used a set of rectangular regions in our experiments to show the +effective of our method, we can actually use regions of arbitrary shape: it can be any +subset of all pixels in the image, and the set of pixels may or may not be clustered +together. For example, it is possible to consider the set of regions each containing +exactly M pixels, where M = 1, 2, ..., N, where N is the total number of pixels in the +image. Such a set has cardinality +�N +M +� +, which can be quite large. Nevertheless, using +our importance map based method, the most vulnerable region can be determined +by simply selecting the M pixels with the highest importance score. +4.5 +Evaluation of System Usability +To evaluate the usability of our system. We created some instructions for testing +our system, detailed in Appendix B, and asked 4 volunteers to use our system +and fill out the Post-Study System Usability Questionnaire (PSSUQ) [27], which is +widely used to measure users’ perceived satisfaction of a website, software, system or +product at the end of a study. We used the third version of PSSUQ, which consists +33 + +M1 +2.2 +M2 +C1 +C2 +2.0 +C3 +C4 +1.8 +1.6 +1.4 +1.2 +1.0 +1357 +10 +15 +20 +30 +40 +50 +kM1 +700 +M2 +C1 +C2 +600 +C3 +C4 +ime(seconds) +500 +400 +300 +200 +100 +0 +1357 +10 +15 +20 +30 +40 +50 +kCHAPTER 4. EXPERIMENTS & RESULTS +of 16 questions and follows a 7-point Likert Scale (+ NA option) between Strongly +Agree to Strongly Disagree, as shown in Figure 4.5. +Figure 4.5: The Post-Study System Usability Questionnaire (Version 3) [27] +The overall result is calculated by averaging the scores from the 7 points of the +scale. PSSUQ also has 3 sub-scales, namely system usefulness, information quality, +and interface quality. +• Overall: the average scores of questions 1 to 16 +• System Usefulness (SYSUSE): the average scores of questions 1 to 6 +• Information Quality (INFOQUAL): the average scores of questions 7 to 12 +• Interface Quality (INTERQUAL): the average scores of questions 13 to 15 +PSSUQ score starts with 1 (strongly agree) and ends with 7 (strongly disagree), +with 4 being neutral. The lower the score, the better the performance and satisfaction. +34 + +ThePost-StudySystemUsabilityQuestionnaire +Strongly +Strongly +Version 3 +agree +disagree +1 +2 +3 +4 +5 +6 +7 +NA +1 +Overall, I am satisfied with how easy it is to use this +0 +0 +system. +2 +It was simple to use this system. +O +0 +3 +I was able to complete thetasks and scenarios quickly +using this system. +4 +I felt comfortable using this system. +0 +5 +It was easy to learn to use this system. +0 +0 +0 +6 +I believeI could become productive quickly using this +0 +system. +7 +The system gave error messages that clearly told me +0 +howtofixproblems. +8 +Whenever I made a mistake using the system, I could +O +recover easily and quickly. +The information (such as online help, on-screen +6 +messages, and other documentation)provided with +0 +this system was clear. +10 +It was easy to find the information I needed +0 +11 +The information was effective in helping me complete +C +the tasks and scenarios. +12 +The organization of information on the system +0 +screens was clear. +13 +The interface* of this system was pleasant. +0 +0 +14 +I liked using the interface of this system. +0 +15 +Thissystemhas all thefunctionsandcapabilitiesI +1 +expectittohave +16 +Overall, I am satisfied with this system. +0 +*The"interface"includesthose itemsthatyouuseto interactwith the system.Forexample,some componentsof theCHAPTER 4. EXPERIMENTS & RESULTS +However, a score below 4 does not indicate that the system have performed above +average. To help interpreting the PSSUQ scores, Sauro and Lewis [42] analyzed the +means of PSSUQ scores with data collected from 21 studies and 210 participants, as +shown below: +• SYSUSE: 2.80 +• INFOQUAL: 3.02 +• INTERQUAL: 2.49 +• Overall: 2.82 +The PSSUQ scores of our system calculated base on the questionnaire results we +collected are: +• SYSUSE: 1.71 +• INFOQUAL: 1.92 +• INTERQUAL: 1.92 +• Overall: 1.81 +which shows that the usability of our system is relatively good. +4.6 +Summary +In this chapter, we evaluated the effectiveness of our system in three tasks: L∞ +attack, imperceptible attack, and vulnerable region estimation. The experiment +results showed that our method finds close or smaller L∞-norm adversarial examples +compared with some state of the art L∞ attacks. We showed our system is able to +generate more imperceptible adversarial perturbations compared with normal L∞ +attack, and the adaptive loosening of the constraint can successfully increase the +Attack Success Rate. We also showed that our proposed importance map method +based on IntergratedGradients and SmoothGrad is able to identified vulnerable +regions in the image, and our proposed correction coefficient can be used to improve +the accuracy of vulnerability estimation. Finally, we evaluated the usability of our +system based on users’ feedback in the Post-Study System Usability Questionnaire, +and found that our system performs above the average. +35 + +CHAPTER 5. CONCLUSION AND FUTURE WORK +Chapter 5 +Conclusion and Future Work +5.1 +Conclusion +We propose ExploreADV, a flexible adversarial perturbation generation system +for deep learning models. We utilize various techniques in our system to generate +adversarial perturbations with minimal L∞ norm, and provide flexibility with +respect to regional and imperceptible adversarial perturbations. Different from many +previously proposed L∞ attacks which perturb the whole inputs indiscriminately, +we propose to use mask-constraints to generalize our attack to more scenarios, e.g. +“physical-world”. We show that out system is suitable for modeling various kinds of +attacks, like imperceptible attack and regional attack. We also proposed variance +map and importance map based methods to automatically generate imperceptible +perturbation and approximately estimate the vulnerability of pixels/regions in an +image. Extensive experiments show that our system is comparable to state of the +art methods in terms of L∞ attack, is effective on generating imperceptible and +regional adversarial perturbations, and can identify regions with high vulnerability. +User studies of our system showed that our system also has good usability. +5.2 +Future Research Directions +Research on understanding robustness and adversarial examples of neural net- +works is still in its infancy. In the following, we discuss a few future research +directions. +5.2.1 +Attack For Image Segmentation/Object Detection +Besides image classification, image segmentation and object detection are among +the mainstream problems in modern deep learning applications [29, 16, 13, 12, 41]. +While many efforts on adversarial attack and defence have been spent on image +classification networks, there are not enough attention drawn to image segmenta- +tion and object detection networks. Nevertheless, image segmentation and object +detection are closely related to some safe-critical systems, e.g. face detection [18], +36 + +CHAPTER 5. CONCLUSION AND FUTURE WORK +medical imaging [49] and autonomous driving [15]. Research on attack for image +segmentation/object detection are certainly needed to identify potential threats. It +would be interesting to see if regional and imperceptible attack can be achieved in +these scenarios. +5.2.2 +Interpretable Attack +While our method of identifying vulnerable region can provide information +about the weakness of a model, specifically, which pixels are unstable when facing +adversarial attack, the perturbations generated by the attack methods remain +uninterpretable to human, which means the cause and effect can not be determined. +Why such adversarial perturbations that are incomprehensible to humans can +cause the model to produce erroneous output? 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Fergus, “Intriguing properties of neural networks”, arXiv preprint +arXiv:1312.6199, 2013. +[51] +A. Voulodimos, N. Doulamis, A. Doulamis, and E. Protopapadakis, “Deep +learning for computer vision: A brief review”, Computational intelligence and +neuroscience, vol. 2018, 2018. +[52] +Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality +assessment: From error visibility to structural similarity”, IEEE transactions +on image processing, vol. 13, no. 4, pp. 600–612, 2004. +41 + +BIBLIOGRAPHY +[53] +J. Yosinski, J. Clune, A. Nguyen, T. Fuchs, and H. Lipson, “Understanding +neural networks through deep visualization”, arXiv preprint arXiv:1506.06579, +2015. +[54] +R. Zhai, C. Dan, D. He, H. Zhang, B. Gong, P. Ravikumar, C.-J. Hsieh, and +L. Wang, “Macer: Attack-free and scalable robust training via maximizing +certified radius”, arXiv preprint arXiv:2001.02378, 2020. +42 + +APPENDIX A. DISTANCE METRIC +Appendix A +Distance Metric +A.1 +SSIM +The structural similarity index measure (SSIM) is a perception-based metric +to measure the similarity between two images. Given two images x and y, the +structural similarity of the two images can be calculated as follows: +SSIM(x, y) = (2µxµy + C1) + (2σxy + C2) +(µ2 +x + µ2 +y + C1)(σ2 +x + σ2 +y + C2) +(A.1) +where µx is the mean of x, µy is the mean of y, σ2 +x is the variance of x, σ2 +y is +the variance of y, σxy is the covariance of x and y, C1 = (k1L)2, C2 = (k2L)2 are +constants used to maintain stability, where L is the dynamic range of the pixel-values +(typically 2#bits per pixel−1), k1 = 0.01 and k2 = 0.03 by default. Structural similarity +ranges from -1 to 1. When two images are identical, the value of SSIM is equal to 1. +A.2 +CIEDE2000 +CIEDE2000 is the latest ∆E∗ color difference formula developed by the Interna- +tional Commission on Illumination (CIE). The pixel-wise perceptual color distance +is calculated as: +∆E∗ +00 = +� +( ∆L′ +kLSL +)2 + ( ∆C′ +kCSC +)2 + ( ∆H′ +kHSH +)2 + RT +∆C′ +kCSC +∆H′ +kHSH +(A.2) +where ∆L′, ∆C′, ∆H′ denotes the distance between pixel values in the three channels, +L (lightness), C (chroma) and H (hue), SL, SC, SH and RT are weighting functions +used to compensate color space uniformity, kL, kC, kH are weighting factors (usually +unity) relative to experimental conditions. +43 + +APPENDIX B. SYSTEM TEST INSTRUCTIONS +Appendix B +System test Instructions +Hi, thank you for willing to test our adversarial attack system! +Given an image classifier and an input image, an adversarial attack generates +small adversarial perturbation on the input image to make the classifier produce +wrong output. +The whole process should take you about 15 minutes. After you finish it, we +would like you to provide feedback according to you experience during the testing. +To begin testing follow these steps: +1. Setup the environment on your local machine. +a) Ensure that you have a python>=3.7 installed as the default python +version on your local machine. +b) Go to a local directory where you want to install our system. +c) Clone the github repository by running +git clone https://github.com/SUSTC11612405/ExploreADV.git +d) cd into the source root +cd ExploreADV +e) Setup a virtual environment (Optional). +pip3 install virtualenv +python3 -m virtualenv venv +source venv/bin/activate +f) Install the dependencies. +pip3 install -r requirements.txt +2. Start running the system, you may need to download the datasets when you +run the system for the first time. +a) Take a look a the usage +python run_attack.py -h +b) Try to run the normal L∞ attack +python run_attack.py +44 + +APPENDIX B. SYSTEM TEST INSTRUCTIONS +c) Try to run the attack on 30% of the pixels +python run_attack.py –ratio 0.3 +d) Try to run the imperceptible attack +python run_attack.py –imperceptible +e) Try to run attack on specified region using region selector. A GUI should +show up for you to specify the region for attack, please feel free to click +any button and play with the GUI. +python run_attack.py –region select +f) Try to switch dataset and models, and explore any adversarial examples +as you like. For example: +python run_attack.py –dataset cifar10 –path_model ./model- +s/cifar10_convBigRELU_DiffAI.onnx –region select –imperceptible +python run_attack.py –region select –ratio 0.1 +3. Congratulations on completing the tasks! Hope you had fun. Now, we would +like to ask you to fill out the following PSSUQ questionnaire to evaluate our +system. Thank you again for your effort! +45 + +APPENDIX B. SYSTEM TEST INSTRUCTIONS +Figure B.1: The Post-Study System Usability Questionnaire (Version 3) +46 + +ThePost-StudySystemUsabilityQuestionnaire +Strongly +Strongly +Version 3 +agree +disagree +1 +2 +3 +4 +5 +6 +7 +NA +1 +Overall, I am satisfied with how easy it is to use this +0 +0 +system. +2 +It was simple to use this system. +O +0 +3 +I was able to complete thetasks and scenarios quickly +using this system. +4 +I felt comfortable using this system. +0 +5 +It was easy to learn to use this system. +0 +0 +0 +6 +I believeI could become productive quickly using this +0 +system. +7 +The system gave error messages that clearly told me +0 +howtofixproblems. +8 +Whenever I made a mistake using the system, I could +O +recover easily and quickly. +The information (such as online help, on-screen +6 +messages, and other documentation)provided with +0 +this system was clear. +10 +It was easy to find the information I needed +0 +11 +The information was effective in helping me complete +C +the tasks and scenarios. +12 +The organization of information on the system +0 +screens was clear. +13 +The interface* of this system was pleasant. +0 +0 +14 +I liked using the interface of this system. +0 +15 +Thissystemhas all thefunctionsandcapabilitiesI +1 +expectittohave +16 +Overall, I am satisfied with this system. +0 +*The"interface"includesthose itemsthatyouuseto interactwith the system.Forexample,some componentsof the \ No newline at end of file diff --git a/29AzT4oBgHgl3EQfR_sP/content/tmp_files/load_file.txt b/29AzT4oBgHgl3EQfR_sP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23c7f3a4515ee448c342815b33a78594104870ef --- /dev/null +++ b/29AzT4oBgHgl3EQfR_sP/content/tmp_files/load_file.txt @@ -0,0 +1,2778 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf,len=2777 +page_content='EXPLOREADV: TOWARDS EXPLORATORY ATTACK FOR NEURAL NETWORKS by LUO TIANZUO A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF COMPUTING in COMPUTER SCIENCE in the GRADUATE DIVISION of the NATIONAL UNIVERSITY OF SINGAPORE 2022 Supervisor: Associate Professor KHOO Siau Cheng arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='01223v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='CR] 1 Jan 2023 Contents Abstract iii List of Figures iv List of Tables v 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Deep learning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Minimal Adversarial Perturbation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Our work .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 6 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 8 2 Related Work 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 DeepFool attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Brendel & Bethge (BB) attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Imperceptible attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Localized (Patch) attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 14 3 ExploreADV 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 The Basic Form: L∞ Attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 With Focus: Regional Attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Imperceptible Attack: Variance Map .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Non-adaptive Imperceptible Attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Adaptive Imperceptible Attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Vulnerability Estimation: Importance Map .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Integrated Gradients .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 SmoothGrad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 22 i 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Correction Coefficient .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Efficiency of the method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 The System as a Whole .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='6 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Discussion on imperceptibility .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 Evaluation of System Usability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='6 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 35 5 Conclusion and Future Work 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Future Research Directions .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Attack For Image Segmentation/Object Detection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Interpretable Attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 37 Bibliography 38 A Distance Metric 43 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 SSIM .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 43 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 CIEDE2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 43 B System test Instructions 44 ii Abstract ExploreADV: Towards exploratory attack for Neural Networks by LUO Tianzuo Master of Computing in Computer Science National University of Singapore Although deep learning has made remarkable progress in processing various types of data such as images, text and speech, they are known to be susceptible to adversarial perturbations: perturbations specifically designed and added to the input to make the target model produce erroneous output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Most of the existing studies on generating adversarial perturbations attempt to perturb the entire input indiscriminately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In this paper, we propose ExploreADV, a general and flexible adversarial attack system that is capable of modeling regional and imperceptible attacks, allowing users to explore various kinds of adversarial examples as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We adapt and combine two existing boundary attack methods, DeepFool and Brendel&Bethge Attack, and propose a mask-constrained adversarial attack system, which generates minimal adversarial perturbations under the pixel-level constraints, namely “mask-constraints”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We study different ways of generating such mask- constraints considering the variance and importance of the input features, and show that our adversarial attack system offers users good flexibility to focus on sub-regions of inputs, explore imperceptible perturbations and understand the vulnerability of pixels/regions to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We demonstrate our system to be effective based on extensive experiments and user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Keywords— Neural network, Adversarial example, Regional attack, Imperceptible attack, Mask constraint, Vulnerability estimation iii List of Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Artificial neural network architecture [2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Process of Adversarial Attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 3 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Change of time cost with respect to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 The Post-Study System Usability Questionnaire (Version 3) [27] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 34 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 The Post-Study System Usability Questionnaire (Version 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 46 iv List of Tables 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Datasets and Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Average L∞ norm for different attacks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Average execution time (seconds) for different attacks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Different Measures on the adversarial examples generated by L∞ attack and Imperceptible attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 Average rmin and average ratioh for different heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 32 v CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION Chapter 1 Introduction In recent years, deep learning has become a critical role in a variety of domains such as computer vision [51], natural language processing [36], speech and audio processing [40], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It has made significant breakthroughs, especially in the tasks of image classification [24, 17, 19], segmentation [29, 16] and object detection [13, 12, 41], where deep learning has achieved high accuracy and even exceeded human performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [50] found an intriguing property of deep neural networks, that it is possible to arbitrarily change the network’s prediction by applying an imperceptible and non-random perturbation to the test image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In this work, we propose a novel system to study such examples, also known as “adversarial examples”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Deep learning Deep learning is part of a broader family of machine learning methods [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It uses artificial neural network composed of a large number of neurons with activation functions to perform representation learning of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Deep neural network can automatically learn the explicit and implicit features of the original data without relying on expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' A typical artificial neural network architecture is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Each of the neurons receives input signal from previous layer and performs weighted connection, then processes the output of the neuron through an activation function and transmits the signal to next layer, thus constructing a deep neural network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It can be formally expressed as shown in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' y = hn(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='h2(w2 · h1(w1 · x + b1) + b2)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1) where x and y are the input and output of the network, wi, bi and hi are the respective weights, biases and activation functions in the ith layer of the network, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', n, where n is the number of hidden layers in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Even though deep neural network has achieved remarkable results by simulating the structure of human brain neural network, the way deep neural networks work 1 CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1: Artificial neural network architecture [2] is still quite different from human cognition and lack of interpretability, making it difficult to guarantee the credibility of its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The growing use of deep neural networks has raised concerns about their security and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [50] found that deep neural networks are highly vulnerable to image samples with specific perturbations, and called such image samples with adversarial perturbations as “adversarial examples”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Adversarial Example An adversarial example refers to the input sample formed by adding specifically designed perturbations to an original sample, which can make the well-trained deep learning model give erroneous outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Specifically, In the field of computer vision, an adversarial example is usually an image formed by adding slight perturbations to the input image that are difficult to be perceived by human vision, resulting in incorrect prediction from the model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' identifying a panda as a gibbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Adversarial attack is the procedure of generating adversarial examples in order to fool a deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 demonstrates the process of adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' A variety of attack algorithms have been proposed to generate adversarial examples [50, 14, 33, 35, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Without loss of generality, we formally define an adversarial example in the context of image classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Definition 1 (Adversarial Example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Given an input image x ∈ Rd, and a score- based image classifier f : Rd �→ RK that maps x to a set of K labels S = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', K} according to: ˆy(x) = arg max k∈S fk(x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2) 2 Input layer Hidden layers Output layer 2 hi h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' hn 0 Input 1 Output 1 Input 2 Output n Input nCHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2: Process of Adversarial Attack where fk(x) is the score function for label k ∈ S, ˆy(x) ∈ S is the predicted label for input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The collection of adversarial examples with respect to x and f is defined as: {x′ | d(x, x′) < ϵ, ˆy(x′) ̸= ˆy(x)} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3) where d(x, x′) is the distance (a measure to be discussed in § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1) between the adversarial example and the original input, and will be bounded by a small predefined constant ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Each adversarial example x′ can be considered as a combination of the original image x and an adversarial perturbation δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', x′ = x + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' When using Lp-norms as distance metric, d(x, x′) = ∥x′ − x∥p = ∥δ∥p < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In order to keep the adversarial image perceptually close to the original image, good distance metrics to measure the perceptual similarity between two images are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Ideally, smaller distance represents closer similarity with respect to human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' As it is difficult to quantitatively measure human perception, in many classical adversarial attack algorithms [50, 14, 37, 35, 6], Lp-norm distance is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Lp-norm distance To measure the distance between an image x and its adversarial image x′, Lp- norm distance is defined by the Lp-norm of the pixel value difference ∥x′ − x∥p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', the Lp-norm of the adversarial perturbation: ∥δ∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The definition of Lp-norm is shown in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 3 DOrmc SSScatDE awesanatt orolnal Input a real label "panda" Deep Learning Classifier f adversarial label "gibbon" acyarsarial erturbarton d meCHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION Definition 2 (Lp-norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Given a vector δ = (δ1, δ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', δn) in the n-dimensional real vector space Rn, and a real number p ≥ 1, the Lp-norm of δ is defined by: ∥δ∥p = ( n � i=1 |δi|p) 1 p (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4) In practise, L0, L1, L2, and L∞-norm distances are commonly used: L1 (Manhattan distance): ∥δ∥1 = �n i=1|δi| L2 (Euclidean distance): ∥δ∥2 = ��n i=1|δi|2 L∞ (Chebyshev distance): ∥δ∥∞ = maxi|δi| L0-norm is special, it’s defined as ∥δ∥0 = �n i=1{1 | δi ̸= 0}, which counts the number of non-zero pixel-value differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It is actually not a norm because it does not satisfy absolute homogeneity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' From the definition, it can be noticed that Lp-norm distance is only related to the pixel value differences δ, it is not affected by the actual pixel values in the clean image x or its adversarial image x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Minimal Adversarial Perturbation More recent attention has focused on finding the minimal adversarial perturbation, also known as the robustness of model at point x [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' They typically use the Lp-norm distances as the distance metric, and try to find the minimal perturbation necessary to change the prediction of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The minimal adversarial perturbation with respect to the Lp-norm is defined as: arg min δ ∥δ∥p, δ ∈ {δ | ˆy(x + δ) ̸= ˆy(x)} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5) The optimization problem in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 is NP-complete for non-linear and non-convex classifiers [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In practice, it is often approximated by different attack algorithms, either by using some heuristics [14, 35, 10] or by solving minimization problems [50, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Specifically, when considering minimal adversarial perturbation with respect to L∞-norm, we call the perturbation size the robust radius [54], as defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Definition 3 (Robust Radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Given an input image x ∈ Rd, and a score-based image classifier with prediction function ˆy(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The robust radius r∗ ∈ R of the classifier on x is defined as: r∗ = min(r) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' ∃x′ ˆy(x′) ̸= ˆy(x) and |x′ i − xi| ≤ r for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', d (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='6) 1Given a vector space V , a function f : X �→ R satisfies absolute homogeneity if f(λv) = |λ|f(v) for all v ∈ V and λ ∈ R, where |λ| denotes the absolute value of the scalar λ [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 4 CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Limitation of previous work Existing attacks are not perceptually constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' While most adversarial attack algorithms try to find adversarial perturbation with small Lp-norms, it is argued that using Lp-norms to measure the perceptual similarity between two images is neither necessary nor sufficient [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' When using Lp-norms as the distance metric, it implies the assumption that perturbations on different pixels in an image are equally important for human eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' However, as Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [28] suggests, perturbations become less perceptible in the regions with high spatial variation, and more perceptible in smooth regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' As shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3, the adversarial images found by some existing adversarial attack algorithms [35], while aiming to have small Lp-norm, appear perceptually blurred and unrealistic with perceptible noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3: Adversarial Images on MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' These adversarial images are supposed to represent the digits 7, 2, 1, 0 and 4, and are predicted as 9, 0, 6, 3 and 9, yet they look blurred and unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Studies also showed that the adversarial examples found by some existing methods neither faithfully simulate physical objects nor resemble natural images [30, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' To develop methods to find adversarial examples perceptually closer to the original image, better perceptual distance metrics are needed to evaluate the effective of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' There are other image similarity distance metrics proposed, such as CIEDE2000 [32] and SSIM [52], that can supplement the Lp-norms for measuring perceptual similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Details of the metrics can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Existing attacks are not suitable for modeling real-world threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Re- cent research also suggests that adversarial examples can be generalized to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Sharif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [45] showed that face recognition systems can be fooled by people wearing adversarially constructed eyeglass frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [5] create a method to generate “adversarial patches” that can be printed and added to the scene to fool a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' While such “physical-world” attacks may seem practical for real-world ML systems, it is currently not suitable to be modeled by most of the existing attacks which aim to modify the whole image indiscriminately — “physical-world” attacks on an entire scene is usually not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' As a result, some attack methods seek to perturb only few pixels [37] or a small region in the image [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Nevertheless, these attacks often do not restrict themselves to imperceptible perturbations, the resulting adversarial perturbations are often clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' To overcome these limitations and take advantage of the power of existing adver- 5 pred: 9 pred: 0 pred: 6 pred: 3 pred: 9 2CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION sarial attack methods, a natural approach is to properly constrain the adversarial perturbations during the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' For example, the perturbation can be constrained to be: Regional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Keep pixels unperturbed outside the target region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Imperceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Reduce perturbations of pixels in regions where such pertur- bations are more perceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Our work In this paper, we propose ExploreADV, a general and flexible adversarial attack system that is capable of modeling regional and imperceptible attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' A novel type of constraint, namely “mask-constraint” is proposed in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The system allows users to explore different types of threat models based on their interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' they can decide for an image the region they want to focus on, whether they want the perturbation to be imperceptible, and how many pixels / how large a region they want to perturb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' For example, to test the reliability of the vision model on a self-driving system, one may want to know if a maliciously designed sticker on a truck would deceive the system, and whether such sticker can be designed imperceptible so that people would not notice any anomaly before a potential accident happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Such exploration are possible in our system, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We propose an idea of mask-constraint in our system to enable such flexibility, as defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Definition 4 (Mask-constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Given a clean image x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', xd) with d pixels, a mask-constraint contains a set of non-negative constant constraints E = (ϵ1, ϵ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', ϵd), where ϵi ∈ [0, 1] is the constraint on the ith pixel xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Each ϵi indicates the maximum allowable absolute perturbation on xi, where 0 means no perturbation allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' An adversarial image x′ = (x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', x′ d) found under the mask-constraint is limited in the closed interval xi − ϵi ≤ x′ i ≤ xi + ϵi for each pixel x′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' With the mask-constraints in our system, we formulate the problem of finding adversarial perturbation as: Problem (Adversarial perturbation under mask-constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Given an im- age classification neural network N with prediction function ˆy(x), an image x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', xd), a mask-constraint E = (ϵ1, ϵ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', ϵd) on each pixel of x, and a real number p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Find a perturbation δ = (δ1, δ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', δd) satisfying {δi | − ϵi ≤ δi ≤ ϵi}, such that when adding the perturbation to the image, the model’s prediction on the resulting image x′ = x + δ is different from its prediction on the original image: ˆy(x′) ̸= ˆy(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Moreover, the Lp-norm of the perturbation ∥δ∥p is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In this thesis, we propose a novel approach to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We adapt and combine two existing adversarial attack methods, DeepFool [35] and Brendel&Bethge 6 CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4: Regional and Imperceptible sticker on a truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We illustrate the adversarial examples found by our system by adding perturbation to a small region of a truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' From left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' left column - original image, and the regional mask indicating the region to apply attack (white area) where the black area remains unperturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' middle column - normal adversarial examples that is perturbed region- ally, and the adversarial perturbations, right column - imperceptible adversarial examples that is perturbed regionally, and the adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [4] Attack, which will be introduced in detail in Chapter 2, and adapt them to work under our mask-constraint setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' DeepFool is first applied to yield a preliminary adversarial example under the mask-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' If a preliminary adversarial example is found, it is used as a starting point for Brendel&Bethge attack, which can then minimize the Lp-norm of the perturbation under the same mask-constraint, if DeepFool fails to find an adversarial example, the system terminates and returns no result, an adversarial example might not exist or might be found with more iterations of DeepFool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We later show that the integration of the mask-constraint makes regional and imperceptible adversarial perturbations possible in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' To facilitate users to automatically generate imperceptible perturbations and select pixels/regions to perturb, we study two types of maps reflecting the variance and importance of pixels in this work: Variance map The variance map measures the spatial variation of the image by calculating the variance of pixel values in a small neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It is helpful when imperceptibility is desired, as perturbations in regions with high spatial variation are less perceptible than those in smooth regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Our system use a variance map based method to generate constraints on pixels according to their variance, allowing less perturbation for pixels in regions with low variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 7 clean image regional regional &imperceptible prediction:truck prediction: automobile prediction:automobile regional mask perturbation perturbationCHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' INTRODUCTION Importance map The importance map measures the importance of each pixel to changing the prediction of the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It is helpful when there is a need to perturb only a subset of all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Our system use an importance map based method to estimate the vulnerability of pixels/regions in the image to adversarial attacks, and select pixels/regions with high vulnerability to apply perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It is worth noticing that there are many ways to generate these maps, and our methods are not restricted to any single implementation of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' There has been a few attempts to make use of such variance map [9] and importance map [1], but we adapt or improve their methods in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In this work, we only consider L∞-norm as it is commonly used to access model robustness [46], so our system can be used to estimate the robust radius under the mask-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Yet, our method is orthogonal to the selection of Lp-norms and can be easily extended to other norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Our main contributions can be summarized as follows: We propose a novel adversarial attack system with mask-constraints, which is more general than existing attacks because it can limit the adversarial perturbation to any sub-region of the whole image, and limit the perturbation magnitude on any pixel of the image independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We adopt and combine two existing adversarial attack methods, DeepFool and Brendel&Bethge Attack, and show that the resulting method generates adversarial perturbations with small L∞ norm that are comparable to the adversarial perturbations generated by the state of the art L∞ attack methods [10, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We study different ways to automatically generate mask-constraints in our system by considering the variance and importance of the pixels in the image, which provides the user with much flexibility to explore various kinds of adversarial examples conveniently, to generate regional and imperceptible adversarial perturbations as they need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We suggest ways to enhance variance map and importance map based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We propose to adaptively loosen the variance map based mask-constraint to generate imperceptible perturbations for models with different robustness, and to add a correction coefficient to the importance map to better estimate pixel vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 Thesis Synopsis The rest of this thesis is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In Chapter 2, we conduct a literature review on related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Chapter 3 provides details of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Extensive experimental results are depicted in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' We conclude the entire thesis as well as discuss further directions for future research in Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 8 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK Chapter 2 Related Work In this section, we review some prior work of adversarial attacks that are related to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 DeepFool attack Considering that deep neural network is extremely vulnerable to adversarial ex- amples, Moosavi-Dezfooli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [35] proposed a method called DeepFool, which aims to calculate the minimal perturbation with respect to L2-norm (extendable to other Lp-norms) necessary to change the classifier’s decision, as shown in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' δ∗ = arg min δ ∥δ∥2 subject to ˆy(x + δ) ̸= ˆy(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1) where x is an image and δ∗ is the minimum perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' ˆy is the prediction function as in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Since a multi-class classifier can be viewed as an aggregation of binary classifiers, they first introduced an efficient algorithm to find adversarial examples for binary classifiers, then extended it to the multi-class case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Binary classifier For the binary classifier, assume ˆy(x) = sign(f(x)), where f : Rd �→ R represents an arbitrary scalar-valued image classifier, sign(·) is the sign function that extracts the sign of a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Affine classifier Given a binary affine classifier f(x) = ωTx+b, the corresponding affine hyperplane F = {x|ωTx + b = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Then the minimum perturbation δ∗ to change the classifier’s prediction on the original sample x0 is equal to the orthogonal projection of x0 onto 9 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK the affine hyperplane F, which is given by the closed-form formula: δ∗(x0) = arg min δ ∥δ∥2 subject to sign(f(x0 + δ)) ̸= sign(f(x0)) =f(x0) ∥ω∥2 2 ω (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 General classifier When f is a general binary differentiable classifier, DeepFool adopts an iterative procedure to estimate the minimum perturbation δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' At each iteration i, f is linearized around the current point xi, and the minimal perturbation δi is computed as: arg min δi ∥δi∥2 such that f(xi) + ∇f(xi)Tδi = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Multi-class classifier For the multi-class classifier, assume ˆy(x) = arg maxk∈S fk(x), where f : Rd �→ RK represents an arbitrary score-based image classifier, fk(x) is the score function of f(x) for corresponds to label k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 Affine classifier Given an affine classifier f(x) = ωTx+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The minimum perturbation that spoofs the classifier can be overridden in the following way: δ∗(x0) = arg min δ ∥δ∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' ∃k : fˆy(x0)(x0 + δ) ≤ fk(x0 + δ) = |fl(x0) − fˆy(x0)(x0)| ∥ωl − ωˆy(x0)∥2 2 (ωl − ωˆy(x0)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4) where l is the class different from ˆy(x0) with the closest hyperplane of the boundary, as shown in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' l = arg min k̸=ˆy(x0) |fk(x0) − fˆy(x0)(x0)| ∥ωk − ωˆy(x0)∥2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 General classifier In general case of multi-class classifiers, DeepFool attack pushes the original sample toward the decision boundary with each round of perturbation until it crosses the decision boundary to form an adversarial example or reach the maximum allowable iterations, as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 10 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK Algorithm 1: DeepFool: multi-class case Input: image x, classifier f, maximum iterations max_iter Output: perturbation ˆδ 1: Initialize x0 ← x, i ← 0 2: while ˆy(xi) = ˆy(x0) and i < max_iter do 3: for k ̸= ˆy(x0) do 4: f′ k ← fk(xi) − fˆy(x0)(xi) 5: ω′ k ← ∇f′ k 6: ˆl ← arg mink̸=ˆy(x0) |f′ k| ∥ω′ k∥2 7: δi ← |f′ ˆl| ∥ω′ ˆl∥2 2 ω′ ˆl 8: xi+1 ← xi + δi 9: i ← i + 1 10: return ˆδ = � i δi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Extend to Lp-norm Although DeepFool was proposed based on L2-norm, it can simply be adapted to find minimal adversarial perturbations for any Lp-norms (p ∈ {∞} ∪ [1, ∞)), by substituting the update steps in line 6 and line 7 in Algorithm 1 with the following steps respectively: ˆl ← arg min k̸=ˆy(x0) |f ′ k| ∥ω′ k∥q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='6) δi ← |f ′ ˆl| ∥ω′ ˆl∥q q |ω′ ˆl|q−1 ⊙ sign(ω′ ˆl) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='7) where ⊙ is the element-wise multiplication, q = p p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Note that for p = 2 as in Algorithm 1, the steps in line 6 and line 7 are also equivalent to the steps in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='6 and Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' When p = 2, q = 2 2−1 = 2, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='6 matches line 6 of Algorithm 1, and the right half of Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='7 can be written as |ω′ ˆl| ⊙ sign(ω′ ˆl), which equals to ω′ ˆl in line 7 of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' For L∞-norm, the steps become: ˆl ← arg min k̸=ˆy(x0) |f ′ k| ∥ω′ k∥1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='8) δi ← |f ′ ˆl| ∥ω′ ˆl∥1 sign(ω′ ˆl) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Summary DeepFool uses a local linear approximation of the classifier to estimate the optimal step towards the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Compared with FGSM [14] attack, DeepFool attack generates adversarial examples with smaller perturbation, and close to the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 11 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1: Schematic of Brendel & Bethge (BB) attack [3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 Brendel & Bethge (BB) attack Brendel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [4] developed Brendel & Bethge (BB) attack, a new set of gradient- based adversarial attacks to find local minimal adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The scheme of the attack is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It starts from a randomly-drawn adversarial example that is far away from the clean image (left in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1), and first performs a 10-step binary search to reach the decision boundary (middle in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1), then walk along the decision boundary to minimize the Lp distance to the clean image (right in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' At each step k, they solve a constrained optimization problem to find the optimal descent step δk within a trust radius r, and add it to the current image ˜xk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Finally, ˜xk is returned, with the Lp-norm of the adversarial perturbation minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The constrained optimization problem is defined as: arg min δ ∥x − ˜xk−1 − δk∥p s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 0 ≤ ˜xk−1 + δk ≤ 1 ∧ bkTδk = ck ∧ ∥δk∥2 2 ≤ r (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='10) where x is the clean image, ˜xk−1 is the perturbed image after step k-1, δk is the perturbation for step k, bk is the direction of the local decision boundary, ck is the distance to the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The decision boundary is defined by a differentiable equality constraint adv(˜x) = 0, where adv(·) is the adversarial criterion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Let ft(˜x) be the log-probability for label t on the current input ˜x, y being the true label for the clean image x, then for targeted attack: adv(˜x) = fy(˜x) − ft(˜x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='11) When adv(˜x) = 0, the log-probability of y is equal to the log-probability t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' For untargeted attack: adv(˜x) = min t,t̸=y(fy(˜x) − ft(˜x)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='12) When adv(˜x) = 0, the log-probability of y is equal to the log-probability of the class with the highest log-probability among the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 12 clean image clean image frog clean image frogCHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Imperceptible attack Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [31] discovered, by investigating the human visual system, that human eyes are more sensitive to perturbation in flat areas than textured areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Therefore, the texture features around a pixel should be taken into consideration when attempting to add adversarial perturbation to the pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Texture features of an image x can be quantified by the notion of variance, as shown in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' SD(xi) = �� xk∈Si(xk − µ)2 n2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='13) where SD(xi) represents the standard deviation of the pixel values among an n × n neighborhood Si of pixel xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Considering the impact of perturbation intensity on human vision, they introduced the notion of “perturbation sensitivity” to measure how much “attention” will be drawn by adding per “unit” perturbation on a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Sen(xi) = 1/SD(xi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='14) They further defined a distance metric based on this notion of “perturbation sensitivity”, as shown in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' d(x, x′) = m � i=1 |δi| ∗ Sen(xi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='15) where d(x, x′) denotes the distance between the adversarial example x′ = (x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', x′ m) and the original one x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=', xm), m is the total number of pixels and |δi| is the intensity of perturbation on pixel xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' They then proposed a targeted attack, which attempts to mis-classify an image into a specific target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' They used a greedy algorithm to select pixels with lower sensitivity and higher impact (larger gradient) in order to maximize the gap between the probability of the target class and the maximal probability of all other classes, and perturb the pixels in an iterative manner under certain constraint d(x, x′) ≤ dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Croce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [9] proposed sparse and imperceivable adversarial attacks, with locally adaptive component-wise constraints on the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' They defined the constraint on each pixel as: σij = κ � min(σ(x) ij , σ(y) ij ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='16) where κ is a hyper-parameter set by users, σ(x) ij , σ(y) ij are the standard deviation of each color channel in x- and y-axis directions with the two immediate neighboring pixels and the original pixel, i ∈ [1, d] represents each of the d pixels, j ∈ [1, 3] represents each color channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Given the constraints on each pixel, the adversarial example generated under the constraint can be expressed in the following form: x′ ij = (1 + λiσij)xij, with − κ ≤ λi ≤ κ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='17) They took min of σ(x) ij and σ(y) ij so that pixels along coordinate-aligned edges are not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 13 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='4 Localized (Patch) attack Instead of perturbing the whole image, some attack methods try to perturb only a localized region in the image, or sometimes referred to as “patch attack” because the perturbed region resembles a patch attached to the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Karmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [21] created localized and visible adversarial noise (LaVAN) that cover only 2% of the pixels in the image without covering the main object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In this method, visible noise is added to a local position of the image to produce an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' To confine the noise δ to a small area over the image x, they used a mask m ∈ {0, 1}n, and the noised image is defined as (1 − m) ⊙ x + m ⊙ δ, where ⊙ is element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It is worth noticing that the noise is used to replace the area rather than be added to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' As the term “visible” suggests, the adversarial examples generated by this method are aimed to be easily detected by human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Dia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [11] proposed localized uncertainty attacks, which is a novel class of adversarial attacks that creates adversarial examples against deep learning models through replacement of uncertain regions in the original inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Instead of per- turbing inputs indiscriminately, they utilize the uncertainty associated with the predictions of the classifier and focus only on carefully chosen regions, to yield better imperceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' They proposed to use a binary-valued (with values in {0, 1}) mask ω with the same size of the input x, and apply the perturbation to the region indicated by ω ⊙ x to generate adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' They trained a feed-forward network called mask model to learn a distribution pv(x) over all possible masks ω that, when drawing a mask from ω ∼ pv(x) and applying the perturbation to the region ω ⊙ x, the predictive uncertainty of the classifier is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Ideally, masks that lead to greater uncertainty should have a higher probability in the distribution pv(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [1] proposed Inconspicuous Adversarial Patch Attack (IAPA), which generates inconspicuous adversarial patches using GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Compared with other adversarial patches attacks, IAPA causes less modifications, and has higher chance to evade detection from human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In their approach, they first used a GradCAM [43] based “vulnerability map” to decide perceptual sensitivity of the victim model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Based on the vulnerability map M, they then choose the area with the highest importance as the victim area for patch attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' However, since GradCAM can only work for Convolutional Neural Network (CNN) based models, their approach can not be applied to Non-CNN based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5 Summary We introduced DeepFool attack and Brendel&Bethge (BB) attack in this chapter, providing methods of finding adversarial examples and minimizing the Lp-norm of the adversarial perturbation, which are adopted in our system and integrated with mask-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 14 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' RELATED WORK Two works on imperceptible attacks are introduced, which showed that variance of image provides useful guidance on generating imperceptible adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' However, they both use heuristics to select pixels and apply perturbation, so they have no control on which pixels are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Also, they didn’t give sufficient results to quantitatively evaluate the imperceptibility of their attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Three Localized (Patch) attacks are introduced, but they have different problems that make them not suitable for our need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' LaVAN generate adversarial examples easy to be detected by human, localized uncertainty attack neither have control on the attack region nor consider imperceptibility, IAPA suggests a good way to find vulnerable regions, but only applies to convolutional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 15 CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' EXPLOREADV Chapter 3 ExploreADV In this section, we introduce the framework and formulation of our proposed adversarial perturbation generation system named ExploreADV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The complete workflow of our system is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The system requires a clean image x and a Deep Learning classifier f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' A mask generator is first used to generate a mask- constraint E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The generator takes the clean image x and a series of parameterized user specification as inputs, and either generate the mask-constraint automatically or prompt the user to specify it through a GUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Details of the mask generator interface can be found in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 and § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' A preliminary adversarial example x′ is then generated by using DeepFool attack under the mask-constraint E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Finally, Brendel&Bethge (BB) attack is used to minimize the Lp-norm distance between x′ and x and get the optimized adversarial example x′′ under the same constraint E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1: Workflow of our proposed ExploreADV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 16 maske-oonstraine E Mask Generator doan iage 2 DeepFool Brendel&Bethge Attack Attack Deep Learning Classifier f real label adv label adv label "4" "9" "9"CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' EXPLOREADV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='1 The Basic Form: L∞ Attack Our system performs L∞ attack based on the algorithms of DeepFool and Brendel & Bethge (BB) attacks, as we believe these two algorithms are good complement for each other based on our experiment results in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' DeepFool L∞ attack finds adversarial example near the decision boundary but does not guarantee to have minimal L∞ distance from the clean image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' BB L∞ attack finds adversarial example on the decision boundary with minimized L∞ distance from the clean image, but relies heavily on the quality of the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' The attack method we use in our system is a combination of these two brilliant methods, as shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In summary, our system finds an adversarial example on the decision boundary with minimal L∞ distance from the clean image, by first searching for a small adversarial example close to the decision boundary (line 2 to line 14 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' It then minimizes the L∞ distance from the clean image while the subject image remains an adversarial example (line 21 to line 28 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Different from the original methods where the constraint on each pixel is homogeneous i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' each adversarial example should stay within the L∞ ball {x + δ | ∥δ∥∞ ≤ ϵ}, we use a general mask-constraint as defined in Definition 4 so that the adversarial examples are constrained in {x + δ | ∀δi∈δ |δi| ≤ ϵi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 With Focus: Regional Attack One major advantage of our system is the flexibility to add perturbation to any sub-region of the clean image, making our system able to not only simulate whole-image attacks, but also simulate regional attacks that are more practical in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' In order to allow users to define their own mask-constraint in a convenient and unified way, we implemented a Graphical User Interface to enable users to indicate regions on which they want to focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' As shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='2 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3, a user can specify the target region for an attack either by clicking and dragging on the screen to select a rectangle shaped region, or by clicking and brushing to select an arbitrary shaped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3 Imperceptible Attack: Variance Map Images with small L∞ distances do not necessarily resemble one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Fol- lowing the same idea of imperceptible attacks [31, 9] introduced in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='3, our system is also capable of generating imperceptible adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' This can be easily achieved with the mask-constraint of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' For our imperceptible attack, we adapted the variance map proposed by Croce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' [9], as in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' 17 CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' EXPLOREADV Algorithm 2: Attack procedure of ExploreADV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' Global: loosen rate of the constraint λ Global: per # of iteration to loosen the constraint T Input: clean image x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' classifier f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' mask constraint E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' maximum iterations max_iter Output: adversarial image x′ 1: Procedure Attack(n): 2: Initialize x0 ← x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' i ← 0 3: while ˆy(xi) = ˆy(x0) and i < max_iter do 4: for k ̸= ˆy(x0) do 5: f′ k ← fk(xi) − fˆy(x0)(xi) 6: ω′ k ← ∇f′ k 7: ˆl ← arg mink̸=ˆy(x0) |f′ k| ∥ω′ k∥1 8: δi ← |f′ ˆl| ∥ω′ ˆl∥1 sign(ω′ ˆl) 9: if i > 0 and i mod T = 0 then 10: for ϵ in E do 11: ϵ ← ϵ ∗ λ 12: xi+1 ← clip(xi + δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' E) 13: i ← i + 1 14: x′ ← xi 15: if ˆy(x′) = ˆy(x0) then 16: return None // Attack Failed 17: else 18: x′ ← Optimize(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' x’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' C) 19: return x′ Input: clean image x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' classifier f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' starting point ˜x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' mask constraint E Output: optimized adversarial image x′′ 20: Function Optimize(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' ˜x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' C): 21: Initialize i ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQfR_sP/content/2301.01223v1.pdf'} +page_content=' b0 ← 0 22: while i 3.5 V in the FeFET, together with noisy +signals before breakdown, are recovered during the Vg backward scan, resulting in repeatable Ih-Vg +and Ie-Vg characteristics. These results imply that, although rapidly increasing Ih is an indication that +breakdown is going to be triggered, the permanent degradation still does not occur yet in this +condition and occurs when Ih increases in a step-wise manner, which can be observed in Figure 4D at +Vg = 4.1 V. +The analysis above suggests that Ih is a convenient indicator for determining appropriate operating +range of Vg. Figure 6 shows the I-Vg characteristics of the FeFET when Vg was kept below 3.5 V. In +this Vg range, the ferroelectric hysteresis can still be achieved with a satisfactory memory window of +1.7 V while Ih is suppressed to under the detection limit. Note that Isub at negative Vg is due to gate- +induced drain leakage (GIDL), which is unrelated to gate leakage currents. Although Ih does not +necessarily imply to device degradation as discussed in Figure 5, hole tunneling back is flowing and +leads to a higher probability that breakdown is triggered; therefore, the operating condition with high +Ih should be avoided. The reliability of FeFETs operating in this way is notably improved and we +cannot observe breakdown under electrical stress for a practically long time (> 105 s). + + + Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + + +5 +3.4 +Time-dependent dielectric breakdown: Constant voltage stress and interrupted test +TDDB tests with a carrier separation setup were carried out to gain more insights into the breakdown +behavior of FeFETs. Ie(t) and Ih(t) under constant voltage stress (CVS) as a function of stress time t +are shown in Figures 7A and 7B for nonferro-FETs and FeFETs, respectively. Both Ie(t) and Ih(t) of +the FeFET increase with time, which is in the opposite direction of Ie(t) of nonferro-FETs in the early +stage. Note that Ih of nonferro-FETs is so low that cannot be measured until breakdown, indicating +that there is less hole tunneling back in nonferro-FETs. We call the behavior of FeFETs having Ie(t) +increasing with time as a SILC-like behavior, as stress-induced leakage current (SILC) refers to a +phenomenon that a leakage current increases with electrical stress. This SILC-like behavior of Ie(t) of +FeFETs can be fitted with a power-law function to be +e ∝ +I +t , independent of Vg stress, as displayed +in Figures 7C. Increasing gate current over time becomes positive feedback to the damage in the gate +insulator, leading to breakdown when Ie is raised to the order of A/cm2. The Ie and Ih levels that +trigger breakdown are almost independent of the stress voltage Vg. +Time-to-breakdown tBD under CVS are summarized in Figures 7D and 7E for nonferro-FETs and +FeFETs, respectively. Not only the breakdown at lower Vg than nonferro-FETs but also tBD more +sensitive to Vg can be observed for FeFETs, with tBD of approximately 103 s at Vg = 3.75 V reduced to +approximately 10-1 s at Vg = 4.2 V. The results of charge-to-breakdown QBD for electrons Qe = +e( ) +∫ I t dt and holes Qe = +h( ) +∫ I +t dt are summarized in Figures 7F and 7G for non-ferro FETs and +FeFETs, respectively. An obvious difference in the QBD-Vg properties in FeFETs and nonferro-FETs +can be observed. While the total electron fluence Qe of nonferro-FETs at which the breakdown of +HfO2 gate insulators occurs has only a weak dependence on stress voltage (note that Qh could not be +extracted as Ih was too low), the total electron Qe and hole fluences Qh at which FeFETs reach +breakdown vary in a wide range, implying that the total fluence is not a factor that is responsible for +the trigger of breakdown of HZO insulators in FeFETs. Figure 7H shows the ratio of Qe/Qh at +different stress voltages. It is interesting that the electron-to-hole ratio of QBD of FeFETs is almost +constant independent of stress voltage. This behavior is remarkably different from conventional +SiO2-gate MOSFETs, where the hole fluence Qh triggers gate dielectric breakdown and the Qe/Qh +ratio is not a constant (Chen et al., 1986; Schuegraf et al., 1994a). This finding indicates that the gate +dielectric breakdown mechanism in FeFETs should be different from SiO2-gate MOSFETs. We +could not compare with nonferro-FETs as Qh was below the detection limit, so further investigation +of the Qe/Qh ratio in nonferro-FETs is needed to specify whether or not the constant Qe/Qh ratio is a +unique feature of FeFETs. Further studies of what physical parameters trigger the breakdown of HZO +insulators in FeFETs would provide a clearer understanding of the interaction between the leakage +current and gate dielectric breakdown event in FeFETs. +We have observed from Figure 7B that gate leakage increases with stress time, as similar to a SILC- +like behavior. Here, we investigate the device behavior during the increase of gate leakage current. +Figures 8A and 8C show the I-Vg characteristics before and after a CVS at 4 V for 10 s shown in +Figure 8B. Although Ie(t) and Ih(t) increase by approximately 100 times during the 10-s CVS test, it +is found that an only small change of the I-Vg characteristics can be observed after stress. This +implies that increases of Ie(t) and Ih(t) in FeFETs are not similar to typical SILC, where increasing +current cannot be easily recovered: increasing currents in FeFETs can be recovered after releasing the +stress. + +Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + +6 +This peculiar behavior of the gate leakage current is further investigated by applying interrupt pulses +during TDDB tests. Figure 9A displays a voltage waveform when TDDB tests stressed at Vg = 4 V +were interrupted by Vg = 0 V for 1 s every stress time of ts. Figures 9B,C show Ie(t) and Ih(t) for each +stress cycle when ts = 10 s (cycles of 4 V for 10 s and 0 V for 1 s). Ie(t) and Ih(t) increase cycle by +cycle regardless of interrupts by 0 V, implying that electrical stress keeps accumulated. Figure 9D +summarizes the time-to-breakdown tBD (excluding interrupt time at 0 V). tBD independent of interrupt +frequency indicates that the interrupts at 0 V have no significant effect on tBD. On the other hand, +interrupting with negative voltage of Vg = -4 V is different. Figure 9E displays a voltage waveform +when interrupted by Vg = -4 V for 1 s every stress time ts. Figures 9F,G illustrate that the SILC-like +gate leakage current is recovered after interrupted with Vg = -4 V for 1 s: increasing Ie(t) and Ih(t) are +recovered back almost to Ie(t = 0) and Ih(t = 0), respectively, in every cycle. Note that only the +current at the first cycle was slightly different because the polarization state of pristine devices is +different. This is in agreement with the repeatable Ig-Vg and Isub-Vg in Figure 5. Due to the recovery +of SILC-like behavior, applying negative voltage interruption in this way helps extend the time-to- +breakdown tBD by more than an order of magnitude, as summarized in Figure 9H. +3.5 +Mechanism under voltage stress +The behavior of stress recovery by negative interrupt pulses can be found as well in HfO2 nonferro- +FETs, as shown in Figures 10A,B. These facts suggest that although the leakage current and +breakdown voltage of HfO2 nonferro-FETs and HZO FeFETs are different in detail due to +differences in crystallinity or defect density, the fundamental mechanisms of the breakdown and +recovery behavior should be generally similar in HfO2-based materials, for instance, same type of +defect generation. +Considering the above findings, we propose the mechanism under high Vg stress, shown in Figure 11. +Typically, SILC as well as noisy gate leakage current (PBD; progressive breakdown) under electrical +stress before hard breakdown are attributed to the generation of defects such as oxygen vacancies +(Olivo et al., 1988; Rofan et al., 1991; DiMaria et al., 1995; Degraeve et al, 1995). On the other hand, +the recovery and repeatable behavior of apparently degraded gate leakage currents observed in +FeFETs suggests that the defect redistribution should be the main contribution of apparently +degraded characteristics rather than the generation of new defects. These defects are redistributed +again after applying an opposite voltage pulse, recovered to the condition close to the initial one +before stress. This model is supported by the fact that oxygen vacancies can move during the voltage +cycling (Pešić et al., 2016; Florent et al., 2018b). However, if the stress is large enough for defects to +move to the condition that triggers hard breakdown, suddenly increasing current generates a huge +density of defects, which forms a permanent conduction path and results in the failure of the device. +Then, the recovery is no longer available for devices that reach the breakdown condition. +Such a memory operation that the polarization states are frequently switched in a bipolar manner can +help extend the device lifetime in terms of breakdown failure. In other words, not only the +improvement in the material aspect but also choosing an appropriate memory operation is important +for the reliability of FeFETs. Whereas bipolar operation is favorable to improving the breakdown- +limited endurance, the memory-window-limited endurance has been reported to have the opposite +behavior: memory window narrowing is degraded in a bipolar operation faster than in a unipolar +operation (Yurchuk et al, 2014). These findings address that the ideal writing operation on the +aspects of breakdown and MW narrowing are different. Thus, the endurance tests for evaluating the +real lifetime should be carefully designed. Conventional endurance tests of FeFETs using bipolar + + + Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + + +7 +stress evaluates only one aspect of device endurance, resulting in underestimation of gate dielectric +breakdown and overestimation of MW narrowing. +4 +Conclusion +We investigated the behavior of stress-induced degradation and gate dielectric breakdown in FeFETs +with ferroelectric HZO as gate dielectrics on Si substrates. It was observed that gate dielectric +breakdown in FeFETs is dominated by the breakdown in the HZO layer, not in the IL. Increasing +gate and substrate hole currents under stress, due to the defect movement in HZO, were observed +before gate dielectric breakdown occurs. These increasing currents are not a permanent phenomenon: +temporary degradation is recovered by applying opposite voltage because of defect redistribution. We +found that continuous electrical stress with the same polarity leads to easier hard breakdown, whereas +bipolar stress frequently recovers the device distribution and help extend the time-to-breakdown. +Because bipolar stress suppresses the breakdown-limited endurance while accelerates the memory +window-limited endurance, accurate endurance tests should be carried out to correctly evaluate the +endurance characteristics of FeFETs in practical memory operations. +5 +Conflict of Interest +The authors declare that the research was conducted in the absence of any commercial or financial +relationships that could be construed as a potential conflict of interest. +6 +Author Contributions +K.T. and S.T. conceived and proposed the main concepts. K.T. fabricated devices and characterized +the electrical properties. K.T., M.T. and S.T analyzed the data and contributed to the in-depth +discussion. K.T. and S.T. wrote the manuscript. All authors contributed to the discussions regarding +the manuscript. +7 +Funding +This paper is based on results obtained from a project, JPNP16007, commissioned by New Energy +and Industrial Technology Development Organization (NEDO) as well as JST CREST Grant Number +JPMJCR20C3 by the Japan Science and Technology Agency (JST). +8 +Data Availability Statement +The raw data supporting the conclusion of this article will be made available by the authors, without +undue reservation. +9 +References +Böscke, T. S., Müller, J., Bräuhaus, D., Schröder, U., Böttger, U. (2011a). Ferroelectricity in hafnium +oxide thin films. Appl. Phys. Lett. 99, 102903. doi: 10.1063/1.3634052 +Böscke, T. S., Müller, J. Bräuhaus, D., Schröder, U., Böttger, U. 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(2014). +“Origin of the endurance degradation in the novel HfO2-based 1T ferroelectric nonvolatile +memories,” in Proc. 2014 IEEE International Reliability Physics Symposium (IRPS), 2E.5.1-2E.5.5. +doi: 10.1109/IRPS.2014.6860603 +Yurchuk, E., Müller, J., Muller, S., Paul, J., Pesic, M., Bentum, R.v., et al. (2016). Charge-trapping +phenomena in HfO2-based FeFET-type nonvolatile memories. IEEE Trans. Electron Devices 63, +3501-3507. doi: 10.1109/TED.2016.2588439 + + + +Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + +12 + +FIGURE 1 | (A) Fabrication process flow. TEM images of (B) HfO2 nonferro-FET and (C) HZO +FeFET. HZO was crystallized whereas HfO2 remained amorphous. + +FIGURE 2 | Schematic band diagram of HZO/IL/Si gate stack (A) when there is no interface charge +trapping and (B) when there is a large amount of interface charge trapping, where 90% of induced +electrons are trapped. Here, the Si band was scaled in the depth direction by 1:100 ratio to make the +band bending clear. + +FIGURE 3 | Characteristics of Ig, Id, Is, and Isub for (A) HfO2 nonferro-FET and (B) HZO FeFET +with L/W = 10/100 µm. Ig of a HZO FeFET is around 103 times higher than that of a HfO2 nonferro- +FET. Steeply increasing substrate current Isub can be found in FeFETs. (C) Leakage current path in +ferroelectric HZO gate insulator. + +A +nonferro-FET +FeFET +TiN +TiN +HfO2 +710 nm +HZO +B +c +D +nonferro-FET +FeFET +S +Si +Si +D +TiN +TiN +Preparationof gate insulator +O Grow IL by HPM +OALD300℃ +10 nm +HfO2 +HZO +(TEMAHf + H,O) x 135 cycles +for 10-nm HfO2 +(TEMAHf + H,O + +0.7 nm +IL +IL +TEMAZr + H,O) x 71 cycles +Si +Si +for 10-nm Hfo.5Zro.5O2 (HZO) +O TiN by sputtering +5 nm +5 nm +O Anneal at 400℃, 30 s (N2)A +IDEAL FeFET +B +ACTUAL FeFET +without interface trap +with large interface trap +Si +Si +TiN +TiN +Free +O +electrons +Trapped +HZO +electrons +HZO +IL +IL +ILBreakdown +>FerroelectricBreakdownA +nonferro-FET +B +FeFET +c +10~3 +N!I +103 +Current (A) +10 +5 +E +10~5 +. +HZO +10' +Is +107 +-Id +10-9 +Si +Isub +o Defects +Tunneling +10-13 +.2 +2 +0 +2 +3 +4 +-1 +0 +2 +3 +4 +- Trap-assisted tunneling +-1 +Vg (M) +Vg (v) + Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + + +13 + +FIGURE 4 | (A) Schematic of carrier separation measurement for analyzing gate current. (B) Current +components in gate current. Inversion electron tunneling flows through S/D, while tunneling of +valence-band electrons and generated holes appears as substrate current. Electron-component (blue +lines), hole-component (red lines), and total (circle symbols) gate currents of (C) nonferro-FET and +(D) FeFET when Vg was scanned from 0 V until the breakdown point. The electron component +dominates the gate current while the hole component rapidly increases near the breakdown voltage. +Gate currents after breakdown for (E) nonferro-FET and (F) FeFET, showing ohmic characteristics. +Band diagrams and expected gate current components at (G) low Vg, (H) Vg < VBD, and (I) Vg > VBD. + + +FIGURE 5 | Repeatedly measured electron and hole components of Ig in the FeFET. Repeatable +current implies that it is not a behavior of permanent trap generation. + +FIGURE 6 | Characteristics of Ig, Id, Is, and Isub for HZO FeFET with L/W = 10/100 µm when the Vg +ranged is limited below 3.5 V. No substrate current Isub is observed at positive Vg. + +c +nonferro-FET +D +(i) +FeFET +Carrierseparation +A +B +0 +102 +102 +Symbols: +(A/cm²) +100 +100 +Total I。 +(A/cm² +G +Ie +10-2 +Symbols: Total I. +10-2 +Ve +① (ii) + 10-4 + 10-4 +n +10-6 +10-6 +(i) Inversion electron tunneling +(ii)Valence-bandelectron tunneling +10~8 +10-8 +I. +0 +1 +2 +3 +4 +6 +0 +2 +3 +4 +5 +6 +(ili)Tunnelingbackofgeneratedholes +Vg (V) +Vg (M) +E +F +120 +120 +100 +100 +G +H +(A/cm²) +21 +80 +/cm +80 +60 +Si +60 +A +Si +40 +F +10 +40 +Si +TiN +TiN +Ih +TiN +20 +20 +00 +1 +2 +3 +4 +1 +2 +4 +Vg (V) +Vg(V) +V.<3V +3V4V100 +(A/cm²) +lh +1st V. +scan +-Ih 2nd V. + scan +107 +-lh. + scan +106 + scan +Ih +108 +0 +1 +2 +3 +4 +Vg (V)10 +Current (A) +10 +10 +10 +.9 +Isub +10° +10 +13 +2 +-1 +0 +2 +3 +4 +Vg (V)Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + +14 + + +FIGURE 7 | TDDB results with carrier separation for (A) HfO2 nonferro-FET at Vg = 4.7 V and (B) +HZO FeFET at Vg = 3.9 V with L/W = 100/100 µm. A SILC-like behavior, with current increasing +with stress time, can be observed in FeFETs. (C) TDDB of FeFET at different stress voltage Vg. +Electron current increases with time by approximately +e ∝ +I +t . The electron and hole current levels +at breakdown have weak dependency on Vg. Time-to-breakdown of (D) nonferro-FET and (E) FeFET +under constant voltage stress. The FeFET has a stronger dependence on Vg. Charge-to-breakdown +QBD for (F) nonferro-FET and (G) FeFET under constant voltage stress. QBD in the FeFET strongly +depends on stressing voltage, whereas QBD in the nonferro-FET is almost constant. (H) Qe/Qh ratio at +breakdown condition for FeFET. + + +FIGURE 8 | (A) I-Vg characteristics of FeFET before CVS. (B) Electron and hole components of +gate leakage current under CVS at Vg = 4 V for 10 s. (C) I-Vg characteristics of FeFET after CVS. +Although gate current increases during CVS, it has a negligible effect on I-Vg characteristics. + + +A +B +c +102 +I.@BD = 2~5 A/cm2 +Current (A/cm²) +10° +10° +Ie +Ih +3.8 V +—Ih 3.9 V +Current ( +10-2 +T。 +Ih 4.0 V +10-4 +In +-Ih 4.1 v +104 += 4.7 V +I。 +Ih 4.15 V +10-6FV +10° +100 +101 +102 +10° +10° +101 +102 +101 +100 +101 +10 +103 +Time (s) +Time (s) +D +F +H +Time-to-breakdown (s) +104 +104 +103 +103 +Qe +103 +(C/cm²) +10 +102 +10 +Qe +Qh +10° +101 +10% +.. +0 +? +101 +10° +00 +10 +107 +.01 +10° +4.5 +4.75 +5 +5.253.5 +3.75 +4 +4.25 +4.5 +4.5 +4.75 +5.25 +3.5 +3.75 +4 +4.25 +4.5 +3.5 +4 +4.5 +Vg(v) +Vg(v) +V(v) +Vg(V) +Vg (V)A +B +c +10° +10-3 +Id +E +Id +Isub +Isub +Current (A) +10 +10° +Current (A) +10 +10 +10 +10 +10° +T +10 +10-11 +10-11 +10 +CVS at 4 V, 10 s +10-13 +E +10 +13 +-2 +-1 +0 +2 +3 +4 +10-1 +100 +101 +2 +-1 +0 +2 +3 +4 +Vg (v) +Time (s) +Vg (v) + Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + + +15 + +FIGURE 9 | (A) Applied voltage scheme with repeating stress of 4 V for time ts and 0 V for 1 s. +(B,C) Electron and hole components of gate leakage current at each 4-V stress cycle for ts = 10 s +when current is plotted in (B) log scale and (C) linear scale. Between each stress cycle, tests were +interrupted by 0 V for 1 s. (D) Total stress time (excluding 0 V interruption duration) before +breakdown for different time ts of 4-V stress. (E) Applied voltage scheme when the interrupted +voltage is -4 V for 1 s. (F,G) Electron and hole components of gate leakage current at each 4-V stress +cycle for ts = 10 s, which were interrupted at -4 V for 1 s between cycles, when current is plotted in +(F) log scale and (G) linear scale. (H) Total stress time (excluding -4 V interruption duration) before +breakdown for different time ts of 4-V stress. + + + + +FIGURE 10 | (A) Applied voltage scheme with repeating stress of 4.7 V for time ts and -4.7 V for 1 +s. (B) Total stress time before breakdown of HfO2 nonferro-FETs. CVS indicates experiments +without recovery pulses. + +le +I +1st stress cycle +2nd stress cycle +Te +Ih +3rd stress cycle +Te +Ih +4th stress cycle +B +104 +c +A + (A/cm²) +Time-to-breakdown +103 +V +100 +4 V +3 +10 +oV +Current ( +1 s +1 s +1 s +10-4 +00 +106 +101 +0 +10-1 +10 +101 +10-1 +100 +10 +0 +10 +20 +30 +CVS +Time (s) +Time (s) +Stress time per cycle ts (s) +F +102 +G +H +E +Time-to-breakdown +103 +4 +上 +A +100 +g. +4 V +3 +[Φ] +102 +2 +Q0 +-4 V +1 s +1s1s +106 +101 +10-1 +100 +101 +10- +100 +101 +0 +1020 +30 +CVS +Stress time per cycle ts (s) +Time (s) +Time (s)B +55 +TiN +S +10 +HfO2 +A +S +Si +D +10 +4.71 +OXO +-4.7 V +1 s +1 s +1S +10 +0 +20 +40 +60 +CVS +Stress time per cycle ts (s)Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress + +16 + +FIGURE 11 | Mechanism under electrical stress. The SILC-like behavior is attributed to the +redistribution of defects rather than permanent defect generation as recovery is observed. Too much +stress will trigger breakdown. + +Voltage stress +More voltage stress +TiN +TiN +TiN +81 +HZO +OZH +OZH +Si +Si +Si +o Defects +Breakdown +Opposite voltage pulse \ No newline at end of file diff --git a/3dE2T4oBgHgl3EQfOAYN/content/tmp_files/load_file.txt b/3dE2T4oBgHgl3EQfOAYN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b4987ca7f432a960ebd67d3b6ad6bee938de8c5 --- /dev/null +++ b/3dE2T4oBgHgl3EQfOAYN/content/tmp_files/load_file.txt @@ -0,0 +1,1037 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf,len=1036 +page_content='Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress Kasidit Toprasertpong1*, Mitsuru Takenaka1, Shinichi Takagi1 1 Department of Electrical Engineering and Information Systems, the University of Tokyo, Tokyo, Japan Correspondence: Kasidit Toprasertpong toprasertpong@mosfet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='jp Keywords: Ferroelectrics, MOSFET, reliability, oxide breakdown, substrate hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Abstract Breakdown is one of main failure mechanisms that limit write endurance of ferroelectric devices using hafnium oxide-based ferroelectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In this study, we investigate the gate current and breakdown characteristics of Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5O2/Si ferroelectric field-effect transistors (FeFETs) by using carrier separation measurements to analyze electron and hole leakage currents during time-dependent dielectric breakdown (TDDB) tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Rapidly increasing substrate hole currents and stress-induced leakage current (SILC)-like electron currents can be observed before the breakdown of the ferroelectric gate insulator of FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This apparent degradation under voltage stress is recovered and the time-to-breakdown is significantly improved by interrupting the TDDB test with gate voltage pulses with the opposite polarity, suggesting that defect redistribution, rather than defect generation, is responsible for the trigger of hard breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 1 Introduction HfO2-based ferroelectric thin films have been actively employed in recent electron device research thanks to their CMOS compatibility, established know-how on the fabrication process, and high scalability of thickness to 10 nm or lower (Böscke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2011a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Migita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Schroeder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Ferroelectric field-effect transistors (FeFETs) with HfO2-based ferroelectric thin films as gate insulators have received considerable attention, not only because of the maturity of the HfO2 deposition technology in the advanced transistor process, but also because of their low energy consumption, high speed, and satisfactory retention during their operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' HfO2-based FeFETs have been investigated as promising devices for low-power nonvolatile memory (Böscke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Trentzsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Dünkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Florent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021) and non-von Neumann computing applications (Jerry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Dutta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Matsui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Despite their excellent properties, one of the most crucial issues to be dealt with towards the practical use of HfO2-based FeFETs is the write endurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' There are two major mechanisms that have been reported to determine the write endurance of FeFETs: the memory window narrowing and gate dielectric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The memory window narrowing refers to a phenomenon where a separation of Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 2 the threshold voltages of the two states (high and low threshold voltage states) becomes gradually smaller and eventually becomes zero after certain operating cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The polarization states are no longer able to be read out through threshold voltages and FeFETs lose a capability as memory devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' On the other hand, gate dielectric breakdown refers to a situation where the gate insulator experiences hard breakdown under a certain amount of electrical stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Hard breakdown makes gate insulators conductive, electrically connects the gate and channel, and causes FeFETs to lose their function as field-effect transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Memory window narrowing and gate dielectric breakdown originate from different physics and occur almost independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' therefore, the write endurance of FeFETs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' a number of write operations before failure, is determined by the mechanism that leads to earlier failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The dominant mechanism depends on the device property and the operation scheme of each specific device and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Write endurance of state-of-the-art FeFETs is typically dominated by the memory window narrowing (Böscke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Yurchuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Trentzsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Dünkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Florent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018) because of the presence of large density of trapped charges in the vicinity of the interfacial layer (IL) between HfO2 and Si (Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2020a), while there are only a few reports showing that endurance of FeFETs is limited by gate dielectric breakdown (Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' That is, the FeFET operation so far usually reaches failure because of memory window narrowing before gate dielectric breakdown occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' thus, there is still a poor understanding of the gate dielectric breakdown mechanism in HfO2-based FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' On the other hand, a lot of effort has been put on the material and device-structure engineering such that there have already been some reports in recent years demonstrating FeFET memory devices with remarkably suppressed memory window narrowing (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In such devices with suppressed memory window narrowing, gate dielectric breakdown may become a dominant mechanism that limits write endurance and play a crucial role in device reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Furthermore, there are some applications of FeFETs using new-concept computing that are insensitive to memory window narrowing, such as reservoir computing (Nako et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In such applications, gate dielectric breakdown will be a dominant endurance-limiting mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Therefore, gaining an understanding of the mechanism of gate dielectric breakdown is important to improve the overall write endurance characteristics of HfO2- based FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In this study, we investigate the breakdown characteristics and the stress-induced degradation behavior as well as the underlying physical mechanism in Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5O2 (HZO)/IL/Si FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The carrier separation measurement and interrupted stress for time-dependent dielectric breakdown (TDDB) evaluation are employed to analyze the physical mechanism underlying gate dielectric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 2 Sample Preparation The process flow is shown in Figure 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' We fabricated n-channel non-ferroelectric FETs (called here as nonferro-FET) with a paraelectric HfO2 gate insulator and FeFETs with a ferroelectric HZO gate insulator on p-type Si substrates with a moderate doping concentration of 4×1015 cm-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' After the source and drain (S/D) regions were doped by phosphorus ion implantation and annealed to activate dopants, the Si substrates were cleaned by hydrochloric-peroxide mixture (HPM)-last cleaning process to grow a high-quality SiO2 IL (Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' For FeFETs, 10-nm-thick ferroelectric HZO was deposited by atomic layer deposition (ALD) using at using tetrakis(ethylmethylamino)hafnium (TEMAH), tetrakis(ethylmethylamino)zirconium (TEMAZ), and H2O at 300°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' For nonferro-FETs, 10-nm-thick HfO2 was deposited in a similar way but without Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 3 TEMAZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' TiN was deposited as gate metal by sputtering and Al:Si was deposited as S/D contacts by thermal evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Samples were annealed at 400°C for 30 s in a N2 atmosphere to crystalized the ferroelectric phase in FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The nonferro-FETs were also annealed at the same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Except the ALD step, both samples were processed simultaneously in the same chamber to ensure the same device condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 1B and 1C show transmission electron microscopic (TEM) images of the gate stacks of a nonferro-FET and a FeFET, respectively, indicating that HZO was crystallized whereas HfO2 remained amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The IL thickness was similar in the both samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 3 Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='1 Band diagram and breakdown position Before we discuss the experimental results of the leakage and breakdown behaviors, we examine the band diagram of the HZO (10 nm)/IL (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='7 nm)/Si gate stack and the possible gate leakage path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 2A depicts an example of an ideal band diagram of the HZO/IL/Si gate stack at 3 V when HZO has ferroelectric polarization of 10 µC/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Due to high ferroelectric polarization, most literature considers a band diagram with a strong electric field across the IL, which significantly pulls down the band position HZO, as shown in Figure 2A (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Yurchuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Mulaosmanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In such a case, the breakdown of the IL is supposed to determine the gate dielectric breakdown of FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' However, it has been reported that a large density of trapped charges near the HZO/IL interface electrically screens the polarization and suppresses the electric field across the IL (Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Toprasertpong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2022c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 2B depicts the band diagram with ferroelectric polarization of 10 µC/cm2 and 90% (Ichihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2020) of induced electrons are trapped at the HZO/IL interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' It can be seen that the band of HZO is not at such a low energy position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This fact indicates that electrons have to tunnel through a thick HZO layer and thus the breakdown of HZO is necessary to describe the gate breakdown failure of FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='2 Device characteristics The I-Vg characteristics of the nonferro-FET and FeFET are shown in Figures 3A and 3B, respectively, for gate current Ig, drain current Id, source current Is, and substrate current Isub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' A gate length L is 10 µm and a gate width W is 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' As expected, the nonferro-FET exhibits the Id-Vg characteristics with clockwise hysteresis, which is a feature of electron trapping during Vg scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' On the other hand, the FeFET exhibits counterclockwise hysteresis, which is a feature of ferroelectricity, with a memory window of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Comparison of the I-Vg characteristics of the nonferro-FET and FeFET indicates interesting features on Ig and Isub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Gate current Ig in the HZO FeFET is much larger by several orders of magnitude than in nonferro-FETs having HfO2 with a similar physical thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This can be understood from the fact that the poly-crystallinity and a lot of defects such as oxygen vacancies in HZO can promote the gate leakage current, as shown in Figure 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' It is also found that the substrate current Isub in the FeFET rapidly increases by four orders of magnitude in a narrow range of Vg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='6 V to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='0 V during the forward Vg scan, which is in the same range that Ig also increases rapidly by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This finding suggests that a study of the behavior of Isub would be helpful in understanding the behavior of the gate leakage and gate dielectric degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The nonferro-FET in Figure 3A does not exhibit this Isub behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='3 Carrier Separation Measurements Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 4 Carrier separation measurements (Eitan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1985) were carried out to analyze the behavior of gate leakage and gate dielectric degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The electrical measurement tool (Keysight B1500A with high-resolution source/monitor unit modules) was connected with FETs in a way shown in Figure 4A, where Vd = Vs = Vsub = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The current detected at the S/D terminal corresponds to the electron component of gate current, denoted by Ie, while the current detected at the substrate corresponds to the hole component, denoted by Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' When Vg is larger than the threshold voltage, Ie corresponds to the tunneling current of inversion electrons from the Si substrate to the gate, whereas Ih corresponds to the sum of the tunneling current of valance-band electrons in the Si substrate to the gate (Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Schuegraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1994b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Shanware et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1999) and the tunneling back current of holes from the gate to the Si substrate (Schuegraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1994a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Schuegraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1994b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1995), as illustrated in Figure 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The results of the carrier separation measurements are shown in Figures 4C and 4D for the HfO2 nonferro-FET and HZO FeFET, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In these measurements, Vg of pristine samples was scanned from 0 V to the positive voltage where breakdown occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' It can be seen that tunneling of inversion electrons is the main contribution of Ig for both the nonferro-FETs and FeFET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Ih is found to be under detection limit in a low Vg regime, but it rapidly increases at Vg close to the breakdown voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The breakdown voltage VBD of the nonferro-FET is approximately 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='2 V, whereas the FeFET reaches hard breakdown much earlier at approximately VBD = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Earlier breakdown is contributed to more defects in HZO than those in HfO2, in agreement with larger gate current shown in Figures 3A and 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Hard breakdown of the nonferro-FET occurs at comparatively low Ih, whereas Ih of HZO FeFET keeps noisy until very high level of Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' After breakdown, the electrical properties of the gate insulators of both the devices become ohmic and dominated by electron current, as shown in Figures 4E and 4F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The band alignments are shown in Figures 4G-4I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' At small Vg, it is clear from the band alignment that electrons in the conduction band of Si can easily tunnel to the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' At Vg in the mid-range, both electrons in the valence band and holes generated at the gate can tunnel more easily, resulting in increasing Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' At large Vg, an electric field across HZO is so large that hole tunneling back can reach the valence band of HZO, resulting in large Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Increasing hole tunneling back consequently causes breakdown in the gate insulator, as the hole tunneling back is known to be the main cause of damage in the gate insulator (Schuegraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1994a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Schuegraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1994b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Takayanagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Results of repeated measurements of Ie and Ih in a Vg scan range of -2 V to 4 V are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' It is interesting that rapidly increasing Ih and Ie at Vg > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5 V in the FeFET, together with noisy signals before breakdown, are recovered during the Vg backward scan, resulting in repeatable Ih-Vg and Ie-Vg characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' These results imply that, although rapidly increasing Ih is an indication that breakdown is going to be triggered, the permanent degradation still does not occur yet in this condition and occurs when Ih increases in a step-wise manner, which can be observed in Figure 4D at Vg = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The analysis above suggests that Ih is a convenient indicator for determining appropriate operating range of Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 6 shows the I-Vg characteristics of the FeFET when Vg was kept below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In this Vg range, the ferroelectric hysteresis can still be achieved with a satisfactory memory window of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='7 V while Ih is suppressed to under the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Note that Isub at negative Vg is due to gate- induced drain leakage (GIDL), which is unrelated to gate leakage currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Although Ih does not necessarily imply to device degradation as discussed in Figure 5, hole tunneling back is flowing and leads to a higher probability that breakdown is triggered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' therefore, the operating condition with high Ih should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The reliability of FeFETs operating in this way is notably improved and we cannot observe breakdown under electrical stress for a practically long time (> 105 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='4 Time-dependent dielectric breakdown: Constant voltage stress and interrupted test TDDB tests with a carrier separation setup were carried out to gain more insights into the breakdown behavior of FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Ie(t) and Ih(t) under constant voltage stress (CVS) as a function of stress time t are shown in Figures 7A and 7B for nonferro-FETs and FeFETs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Both Ie(t) and Ih(t) of the FeFET increase with time, which is in the opposite direction of Ie(t) of nonferro-FETs in the early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Note that Ih of nonferro-FETs is so low that cannot be measured until breakdown, indicating that there is less hole tunneling back in nonferro-FETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' We call the behavior of FeFETs having Ie(t) increasing with time as a SILC-like behavior, as stress-induced leakage current (SILC) refers to a phenomenon that a leakage current increases with electrical stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This SILC-like behavior of Ie(t) of FeFETs can be fitted with a power-law function to be e ∝ I t , independent of Vg stress, as displayed in Figures 7C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Increasing gate current over time becomes positive feedback to the damage in the gate insulator, leading to breakdown when Ie is raised to the order of A/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The Ie and Ih levels that trigger breakdown are almost independent of the stress voltage Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Time-to-breakdown tBD under CVS are summarized in Figures 7D and 7E for nonferro-FETs and FeFETs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Not only the breakdown at lower Vg than nonferro-FETs but also tBD more sensitive to Vg can be observed for FeFETs, with tBD of approximately 103 s at Vg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='75 V reduced to approximately 10-1 s at Vg = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The results of charge-to-breakdown QBD for electrons Qe = e( ) ∫ I t dt and holes Qe = h( ) ∫ I t dt are summarized in Figures 7F and 7G for non-ferro FETs and FeFETs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' An obvious difference in the QBD-Vg properties in FeFETs and nonferro-FETs can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' While the total electron fluence Qe of nonferro-FETs at which the breakdown of HfO2 gate insulators occurs has only a weak dependence on stress voltage (note that Qh could not be extracted as Ih was too low), the total electron Qe and hole fluences Qh at which FeFETs reach breakdown vary in a wide range, implying that the total fluence is not a factor that is responsible for the trigger of breakdown of HZO insulators in FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 7H shows the ratio of Qe/Qh at different stress voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' It is interesting that the electron-to-hole ratio of QBD of FeFETs is almost constant independent of stress voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This behavior is remarkably different from conventional SiO2-gate MOSFETs, where the hole fluence Qh triggers gate dielectric breakdown and the Qe/Qh ratio is not a constant (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Schuegraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1994a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This finding indicates that the gate dielectric breakdown mechanism in FeFETs should be different from SiO2-gate MOSFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' We could not compare with nonferro-FETs as Qh was below the detection limit, so further investigation of the Qe/Qh ratio in nonferro-FETs is needed to specify whether or not the constant Qe/Qh ratio is a unique feature of FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Further studies of what physical parameters trigger the breakdown of HZO insulators in FeFETs would provide a clearer understanding of the interaction between the leakage current and gate dielectric breakdown event in FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' We have observed from Figure 7B that gate leakage increases with stress time, as similar to a SILC- like behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Here, we investigate the device behavior during the increase of gate leakage current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figures 8A and 8C show the I-Vg characteristics before and after a CVS at 4 V for 10 s shown in Figure 8B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Although Ie(t) and Ih(t) increase by approximately 100 times during the 10-s CVS test, it is found that an only small change of the I-Vg characteristics can be observed after stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This implies that increases of Ie(t) and Ih(t) in FeFETs are not similar to typical SILC, where increasing current cannot be easily recovered: increasing currents in FeFETs can be recovered after releasing the stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 6 This peculiar behavior of the gate leakage current is further investigated by applying interrupt pulses during TDDB tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 9A displays a voltage waveform when TDDB tests stressed at Vg = 4 V were interrupted by Vg = 0 V for 1 s every stress time of ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figures 9B,C show Ie(t) and Ih(t) for each stress cycle when ts = 10 s (cycles of 4 V for 10 s and 0 V for 1 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Ie(t) and Ih(t) increase cycle by cycle regardless of interrupts by 0 V, implying that electrical stress keeps accumulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 9D summarizes the time-to-breakdown tBD (excluding interrupt time at 0 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' tBD independent of interrupt frequency indicates that the interrupts at 0 V have no significant effect on tBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' On the other hand, interrupting with negative voltage of Vg = -4 V is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figure 9E displays a voltage waveform when interrupted by Vg = -4 V for 1 s every stress time ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Figures 9F,G illustrate that the SILC-like gate leakage current is recovered after interrupted with Vg = -4 V for 1 s: increasing Ie(t) and Ih(t) are recovered back almost to Ie(t = 0) and Ih(t = 0), respectively, in every cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Note that only the current at the first cycle was slightly different because the polarization state of pristine devices is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This is in agreement with the repeatable Ig-Vg and Isub-Vg in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Due to the recovery of SILC-like behavior, applying negative voltage interruption in this way helps extend the time-to- breakdown tBD by more than an order of magnitude, as summarized in Figure 9H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5 Mechanism under voltage stress The behavior of stress recovery by negative interrupt pulses can be found as well in HfO2 nonferro- FETs, as shown in Figures 10A,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' These facts suggest that although the leakage current and breakdown voltage of HfO2 nonferro-FETs and HZO FeFETs are different in detail due to differences in crystallinity or defect density, the fundamental mechanisms of the breakdown and recovery behavior should be generally similar in HfO2-based materials, for instance, same type of defect generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Considering the above findings, we propose the mechanism under high Vg stress, shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Typically, SILC as well as noisy gate leakage current (PBD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' progressive breakdown) under electrical stress before hard breakdown are attributed to the generation of defects such as oxygen vacancies (Olivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Rofan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' DiMaria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Degraeve et al, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' On the other hand, the recovery and repeatable behavior of apparently degraded gate leakage currents observed in FeFETs suggests that the defect redistribution should be the main contribution of apparently degraded characteristics rather than the generation of new defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' These defects are redistributed again after applying an opposite voltage pulse, recovered to the condition close to the initial one before stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' This model is supported by the fact that oxygen vacancies can move during the voltage cycling (Pešić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Florent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' However, if the stress is large enough for defects to move to the condition that triggers hard breakdown, suddenly increasing current generates a huge density of defects, which forms a permanent conduction path and results in the failure of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Then, the recovery is no longer available for devices that reach the breakdown condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Such a memory operation that the polarization states are frequently switched in a bipolar manner can help extend the device lifetime in terms of breakdown failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' In other words, not only the improvement in the material aspect but also choosing an appropriate memory operation is important for the reliability of FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Whereas bipolar operation is favorable to improving the breakdown- limited endurance, the memory-window-limited endurance has been reported to have the opposite behavior: memory window narrowing is degraded in a bipolar operation faster than in a unipolar operation (Yurchuk et al, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' These findings address that the ideal writing operation on the aspects of breakdown and MW narrowing are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Thus, the endurance tests for evaluating the real lifetime should be carefully designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Conventional endurance tests of FeFETs using bipolar Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 7 stress evaluates only one aspect of device endurance, resulting in underestimation of gate dielectric breakdown and overestimation of MW narrowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 4 Conclusion We investigated the behavior of stress-induced degradation and gate dielectric breakdown in FeFETs with ferroelectric HZO as gate dielectrics on Si substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' It was observed that gate dielectric breakdown in FeFETs is dominated by the breakdown in the HZO layer, not in the IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Increasing gate and substrate hole currents under stress, due to the defect movement in HZO, were observed before gate dielectric breakdown occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' These increasing currents are not a permanent phenomenon: temporary degradation is recovered by applying opposite voltage because of defect redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' We found that continuous electrical stress with the same polarity leads to easier hard breakdown, whereas bipolar stress frequently recovers the device distribution and help extend the time-to-breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Because bipolar stress suppresses the breakdown-limited endurance while accelerates the memory window-limited endurance, accurate endurance tests should be carried out to correctly evaluate the endurance characteristics of FeFETs in practical memory operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 5 Conflict of Interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 6 Author Contributions K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' conceived and proposed the main concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' fabricated devices and characterized the electrical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T analyzed the data and contributed to the in-depth discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' All authors contributed to the discussions regarding the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 7 Funding This paper is based on results obtained from a project, JPNP16007, commissioned by New Energy and Industrial Technology Development Organization (NEDO) as well as JST CREST Grant Number JPMJCR20C3 by the Japan Science and Technology Agency (JST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 8 Data Availability Statement The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 9 References Böscke, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', Müller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', Bräuhaus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', Schröder, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', Bentum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Charge-trapping phenomena in HfO2-based FeFET-type nonvolatile memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Electron Devices 63, 3501-3507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='1109/TED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='2588439 Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 12 FIGURE 1 | (A) Fabrication process flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' TEM images of (B) HfO2 nonferro-FET and (C) HZO FeFET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' HZO was crystallized whereas HfO2 remained amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' FIGURE 2 | Schematic band diagram of HZO/IL/Si gate stack (A) when there is no interface charge trapping and (B) when there is a large amount of interface charge trapping, where 90% of induced electrons are trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Here, the Si band was scaled in the depth direction by 1:100 ratio to make the band bending clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' FIGURE 3 | Characteristics of Ig, Id, Is, and Isub for (A) HfO2 nonferro-FET and (B) HZO FeFET with L/W = 10/100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Ig of a HZO FeFET is around 103 times higher than that of a HfO2 nonferro- FET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Steeply increasing substrate current Isub can be found in FeFETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' (C) Leakage current path in ferroelectric HZO gate insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' A nonferro-FET FeFET TiN TiN HfO2 710 nm HZO B c D nonferro-FET FeFET S Si Si D TiN TiN Preparationof gate insulator O Grow IL by HPM OALD300℃ 10 nm HfO2 HZO (TEMAHf + H,O) x 135 cycles for 10-nm HfO2 (TEMAHf + H,O + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='7 nm IL IL TEMAZr + H,O) x 71 cycles Si Si for 10-nm Hfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5O2 (HZO) O TiN by sputtering 5 nm 5 nm O Anneal at 400℃, 30 s (N2)A IDEAL FeFET B ACTUAL FeFET without interface trap with large interface trap Si Si TiN TiN Free O electrons Trapped HZO electrons HZO IL IL ILBreakdown >FerroelectricBreakdownA nonferro-FET B FeFET c 10~3 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='I 103 Current (A) 10 5 E 10~5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=" HZO 10' Is 107 Id 10-9 Si Isub o Defects Tunneling 10-13 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='2 2 0 2 3 4 1 0 2 3 4 Trap-assisted tunneling 1 Vg (M) Vg (v) Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 13 FIGURE 4 | (A) Schematic of carrier separation measurement for analyzing gate current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' (B) Current components in gate current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Inversion electron tunneling flows through S/D, while tunneling of valence-band electrons and generated holes appears as substrate current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Electron-component (blue lines), hole-component (red lines), and total (circle symbols) gate currents of (C) nonferro-FET and (D) FeFET when Vg was scanned from 0 V until the breakdown point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' The electron component dominates the gate current while the hole component rapidly increases near the breakdown voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Gate currents after breakdown for (E) nonferro-FET and (F) FeFET, showing ohmic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Band diagrams and expected gate current components at (G) low Vg, (H) Vg < VBD, and (I) Vg > VBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' FIGURE 5 | Repeatedly measured electron and hole components of Ig in the FeFET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' Repeatable current implies that it is not a behavior of permanent trap generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' FIGURE 6 | Characteristics of Ig, Id, Is, and Isub for HZO FeFET with L/W = 10/100 µm when the Vg ranged is limited below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' No substrate current Isub is observed at positive Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' c nonferro-FET D (i) FeFET Carrierseparation A B 0 102 102 Symbols: (A/cm²) 100 100 Total I。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' (A/cm² G Ie 10-2 Symbols: Total I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 10-2 Ve ① (ii) 10-4 10-4 n 10-6 10-6 (i) Inversion electron tunneling (ii)Valence-bandelectron tunneling 10~8 10-8 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content=' 0 1 2 3 4 6 0 2 3 4 5 6 (ili)Tunnelingbackofgeneratedholes Vg (V) Vg (M) E F 120 120 100 100 G H (A/cm²) 21 80 /cm 80 60 Si 60 A Si 40 F 10 40 Si TiN TiN Ih TiN 20 20 00 1 2 3 4 1 2 4 Vg (V) Vg(V) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfOAYN/content/2301.03742v1.pdf'} +page_content='<3V 3V 0 is defined as [150, 44] +퐽 훼 +푎+ 푓 (푥) = +1 +Γ(훼) +푥 +∫ +푎 +(푥 − 푡) 훼−1 푓 (푡)푑푡, +(1) +If 훼 = 0, 퐽0 +푎+ = 퐼, 퐼 is the identity operator. Here +Γ(훼) = +∞ +∫ +0 +exp(−훼)푢훼−1푑푢 +is the Euler Gamma function. This integral exists if 푓 (푡) is the locally integrable +function and for 푡 → 0 behaves like 푂(푡−휈) with 휈 < 훼. To get the strict mathematical +rigor it is possible to use the framework of the Lebesgue spaces of the summable +functions or the Sobolev spaces of the generalized functions [75]. +The RL integral is a generalization of Cauchy’s formula for an n-fold integral +푥 +∫ +푎 +푑푥1 +푥1 +∫ +푎 +푑푥2· · · +푥푛−1 +∫ +푎 +푑푥푛 = +1 +(푛 − 1)! +푥 +∫ +푎 +(푥 − 푡)푛−1푑푡 +using the relation +(푛 − 1)! = +푛−1 +� +푘=1 +푘 = Γ(푛). +The equation (1) is left-sided RL integral. The right-sided RL integral is written as +퐽 훼 +푏− 푓 (푥) = +1 +Γ(훼) +푏 +∫ +푥 +(푡 − 푥) 훼−1 푓 (푡)푑푡. +(2) +2 + +The RL integral is a case of the convolution integral of the Volterra type [149] +퐾 ∗ 푓 (푥) = +푏 +∫ +푎 +푘(푥 − 푡) 푓 (푡)푑푡. +The RL integral has the semi-group property (also called additivity law [99]): +퐽 훼 +푎+퐽훽 +푎+ 푓 (푥) = 퐽 훼+훽 +푎+ +푓 (푥), +훼 > 0, +훽 > 0 +which implies the commutative property [149]: 퐽훽 +푎+퐽 훼 +푎+ = 퐽 훼 +푎+퐽훽 +푎+. +The RL fractional integral coincides with the classical definition in the case 훼 ∈ N. +The fractional integration improves the smoothness of functions [50]. +Sometimes the RL integral could be expressed via the elementary functions, e.g., +퐽 훼 +푎+(푥 − 푎)휇 = +Γ(휇 + 1) +Γ(훼 + 휇 + 1) (푥 − 푎) 훼+휇. +A particular case of the RL fractional integrals is the Liouville fractional integrals +[82] that is obtained by transitions 푎 → −∞ and 푏 → ∞ in equations (1) and (2) as +퐽 훼 ++ 푓 (푥) = +1 +Γ(훼) +푥 +∫ +−∞ +(푥 − 푡) 훼−1 푓 (푡)푑푡, +퐽 훼 +− 푓 (푥) = +1 +Γ(훼) +∞ +∫ +푥 +(푡 − 푥) 훼−1 푓 (푡)푑푡. +1.2 +Riemann-Liouville Fractional Derivative +The left and the right Riemann-Liouville fractional derivatives are defined as [181] +퐷 훼 +푎+[ 푓 (푥)] = + + +1 +Γ(1 − 훼) +푑 +푑푥 +푥 +∫ +푎 +(푥 − 푡)−훼 푓 (푡)푑푡, +훼 ∈ (0, 1) +푑푓 (푥) +푑푡 +, +훼 = 1 +(3) +and +퐷 훼 +푏− [ 푓 (푥)] = + + +1 +Γ(1 − 훼) +푑 +푑푥 +푏 +∫ +푥 +(푡 − 푥)−훼 푓 (푡)푑푡, +훼 ∈ (0, 1) +푑푓 (푥) +푑푡 +, +훼 = 1 +(4) +Operator 퐷 훼 +푎+ is left-inverse meaning that 퐷 훼 +푎+퐽 훼 +푎+ = 퐼, 퐼 is the identity operator. Thus +퐷 훼 +푎+퐽 훼 +푎+ 푓 = 푓 but the unconditional semigroup property of fractional differentiation +in the RL sense does not hold: Diethelm [50] gives examples where 퐷 훼1 +푎+퐷 훼2 +푎+ 푓 = +퐷 훼2 +푎+퐷 훼1 +푎+ 푓 ≠ 퐷 훼1+훼2 +푎+ +푓 and 퐷 훼1 +푎+퐷 훼2 +푎+ 푓 ≠ 퐷 훼2 +푎+퐷 훼1 +푎+ 푓 = 퐷 훼1+훼2 +푎+ +푓 . +Atangana & Secer [13] presented tables of the RL derivatives of the trigonometric +and some special functions. The fractional RL derivative of the power function is +퐷 훼 +푎+푡휈 = +Γ(1 + 휈) +Γ(1 + 휈 − 훼) 푡휈−훼 +3 + +and, particular, the derivative of a constant 퐷 훼 +푎+1 = 푡−훼/Γ(1 − 훼). +Since the fractional RL derivative of a constant is not zero, thus the magnitude of +the fractional derivative changes with adding of the constant. +Jumarie [110] suggested a modification to remove this drawback. He started with a +fractional derivative (F-derivative) +푓 훼(푥) = lim +ℎ→0 +Δ훼 푓 (푥) +ℎ훼 +based on the fractional difference Δ훼 푓 (푥) of order 훼, +훼 ∈ ℜ, +0 < 훼 ≤ 1. +Jumarie proposed the modification of the fractional RL derivative +1 +Γ(1 − 훼) +푑 +푑푥 +푥 +∫ +푎 +(푥 − 푡)−훼( 푓 (푡) − 푓 (0))푑푡. +1.2.1 +Leibniz’ formula +The classical Leibnitz’ formula for the first-order derivative (i.e. when 푛 ∈ N) is +퐷푛[ 푓 (푥)푔(푥)] = +푛 +� +푘=0 +�푛 +푘 +� +[퐷푘푔(푥)퐷푛−푘 푓 (푥)] +where 푓 (푥) and 푔(푥) are the 푛-time differentiable functions. +The fractional derivatives violate the classical Leibnitz’ rule [215, 217]. General- +ization of the Lebnitz’ formula was developed by Osler [176, 177]. +The Leibniz’ formula for the differentiation of the product of the functions for the +RL operators for the functions that are analytic on (푎 − ℎ, 푎 + ℎ) is written as [50] +퐷푛 +푎+ [ 푓 푔](푥) = +⌊푛⌋ +� +푘=0 +�푛 +푘 +� +(퐷푘 +푎+ 푓 )(푥)(퐷푛−푘 +푎+ 푔)(푥) + +∞ +� +푘=[푛]+1 +�푛 +푘 +� +(퐷푘 +푎+ 푓 )(푥)(퐽푛−푘 +푎+ 푔)(푥). +where ⌊ +⌋ denotes the floor function. Jumarie studied the Leibniz’ formula for the +differentiation of the product of the non-differentiable functions [113]. +1.2.2 +Faá di Bruno formula (the chain rule) +For the functions 푓 and 푔 with a sufficient number of the derivatives and 푛 ∈ N +[50, 44, 218] +퐷푛[푔( 푓 (·))](푥) = +� +(퐷푘푔) +푛 +� +푚=1 +(퐷푚 푓 (푥)푏푚 +where the sum is over all partitions of {1, 2, . . . , 푛} and for each partition 푘 is its number +of the blocks and 푏 푗 is the number of the blocks with exactly 푗 elements. +Tarasov [216] analysed the simplified chain rules suggested by Jumarie [112, 111, +114] and found that these simplifications are not universally valid. +4 + +1.2.3 +Fractional Taylor expansion +The fractional Taylor expansion is written as [50, 188, 173] +푓 (푥) = +(푥 − 푎)푛−푚 +Γ(푛 − 푚 + 1) lim +푧→푎+ 퐽푚−푛 +푎 +푓 (푧)+ +푚−1 +� +푘=1 +(푥 − 푎)푘+푛−푚 +Γ(푘 + 푛 − 푚 + 1) lim +푧→푎+ 퐷푘퐽푚−푛 +푎 +푓 (푧) + 퐽푛 +푎퐷푛 +푎 푓 (푥). +1.2.4 +Symmetrised space derivative +Vermeersch & Shakouri [227] formulated the symmetrised space derivatives of the +fractional order between 1 and 2 and between 0 and 1: +• 1 < 훼 < 2. +The symmetrised space derivative of the function 푔(푥) that is +integrable over the entire real axis is +휕 훼푔 +휕|푥|훼 = 휕 +휕푥 +� +푤훼 ★ 휕푔 +휕푥 +� +where ★ denotes the convolution and 푤훼 is an unknown kernel function with the +Fourier image found to be 푊훼 = 1/|휉|2−훼. The Fourier inversion yields +푤훼 = 1 +2휋 +∞ +∫ +−∞ +푒푥푝( 푗휉푥)푑휉 +|휉|2−훼 += +|푥|−(훼−1) +2Γ(2 − 훼) cos[(2 − 훼) 휋 +2 ] . +Thus +휕 훼푔 +휕|푥|훼 = +1 +2Γ(2 − 훼) cos[(2 − 훼) 휋 +2 ] +휕 +휕푥 +∞ +∫ +−∞ +휕푔 +휕푥 (푥′)푑푥′ +|푥 − 푥′|훼−1 . +• 0 < 훼 < 1. The symmetrised space derivative of the function 푔(푥) is +휕 훼푔 +휕|푥|훼 = 푤훼 ★ 휕푔 +휕푥 . +The Fourier image of the kernel function 푤훼 is 푊훼 = 푗 · 푠푔푛(휉)/|휉|1−훼. Per- +forming the Fourier inversion, the authors get finally +휕 훼푔 +휕|푥|훼 = +−1 +2Γ(1 − 훼) cos[(1 − 훼) 휋 +2 ] +휕 +휕푥 +∞ +∫ +−∞ +−푠푔푛(푥) · 휕푔 +휕푥 (푥′)푑푥′ +|푥 − 푥′|훼 +. +In the case 훼 = 1/2 the fractional integrals and derivatives are also called semi- +integrals and semi-derivatives [207]. +5 + +1.3 +Caputo Fractional Derivative +The fractional derivatives in the Caputo sense on the left (퐶퐷 훼 +푎+) and on the right +(퐶퐷 훼 +푏−) are defined via the RL fractional integral [82] (퐶퐷 훼 +푎+ 푓 ) = (퐽푛−훼 +푎+ +푓 (푛))(푥) and +(−1)푛(퐶퐷 훼 +푏− 푓 ) = (퐽푛−훼 +푏− +푓 (푛))(푥). It was introduced independently in 1948 by M. +Caputo and by A.N. Gerasimov [65]; later by Dzherbashyan & Nersesian [56]. +The major difference of the Caputo fractional derivative is that the derivative act +first on the function and after the integral is evaluated while in the RL approach the +derivative act on the integral. +The Caputo fractional derivative is defined as [181] +퐷 훼 +★ 푓 (푡) = + + +1 +Γ(1 − 훼) +푡 +∫ +0 +(푡 − 푥)−훼 푑푓 (푥) +푑푡 +푑푡, +훼 ∈ (0, 1) +푑푓 (푥) +푑푡 +, +훼 = 1 +(5) +The definition of the Caputo derivative (5) is more restrictive than of the RL one (3, +4) since it requires the absolute integrability of the derivative 푑푓 (푥)/푑푡 [75]. +The Caputo fractional derivative can be considered as the regularization in the time +origin for the RL derivative [76] +퐷 훼 +★ 푓 (푡) = 퐷 훼 푓 (푡) − 푓 (0+) +푡−훼 +Γ(1 − 훼) +and satisfies the property of being zero when applied to a constant. +Yuan & Agrawal and Singh & Chatterjee suggested the alternative definitions of +the Caputo fractional derivative [50]. The first approach is based on the introduction of +the auxiliary bivariate function 휙 : (0, ∞) × [푎, 푏] → R as +휙(푤, 푥) = (−1) ⌊푛⌋ 2 sin 휋푛 +휋 +푤2푛−2⌈푛⌉+1 +∫ +푥 +푎 +푓 ( ⌈푛⌉) (휏)푒−(푥−휏)푤2푑휏 +where ⌈ +⌉ denote the ceiling function, and, finally +퐷푛 +★푎 푓 (푥) = +∞ +∫ +0 +휙(푤, 푥)푑푤. +The second approach is based on expressing the fractional derivative of the the +given function in the form of the integral over (0, ∞) with the integrand that can be +obtained as the solution of the first-order initial value problem +휕휙★(푤, 푥) +휕푥 += −푤 +1 +푛−⌈푛⌉−1 휙★(푤, 푥) + (−1) ⌊푛⌋ sin 휋푛 +휋(푛 − ⌈푛⌉ − 1) 푓 ( ⌈푛⌉) (푥) +with the initial condition 휙★(푤, 푎) = 0. Thus +휙★(푤, 푥) = (−1) ⌊푛⌋ sin 휋푛 +휋(푛 − ⌈푛⌉ − 1) +푥 +∫ +0 +푓 ( ⌈푛⌉) (휏) exp(−(푥 − 휏)푤 +1 +푛−⌈푛⌉−1)푑휏 +6 + +퐷푛 +★푎 푓 (푥) = +∞ +∫ +0 +휙★(푤, 푥)푑푤. +1.4 +Matrix Approach +Operations of the fractional calculus can be expressed by matrix [182, 184]. E.g., the +left-sided RL or Caputo derivative can be approximated in the nodes in the equidistant +discretization net with the help of the upper triangular strip matrix 퐵(훼) +푛 +as [182] +� +푣(훼) +푛 +푣(훼) +푛−1 . . . 푣(훼) +1 +푣(훼) +0 +�푇 += 퐵(훼) +푛 +[푣푛 푣푛−1 . . . 푣1 푣0]푇 +where +퐵(훼) +푛 += 1 +휏훼 + +휔(훼) +0 +휔(훼) +0 +. . . +. . . +휔(훼) +푛−1 +휔(훼) +푛 +0 +휔(훼) +0 +휔(훼) +0 +. . . +. . . +휔(훼) +푛−1 +. . . +. . . +0 +0 +0 +. . . +. . . +휔(훼) +0 + +Similarily, the right-hand RL or Caputo fractional derivative can be approximated with +the help of the corresponding lower triangular strip matrix. +1.5 +Caputo & Fabrizio Fractional Derivatives +Caputo & Fabrizio [26, 27] proposed the fractional derivatives without the singular +kernel [142] by replacing the kernel (푡 − 휏)−훼 with the function exp(−훼/(1 − 훼)) +that does not have singularity for 푡 = 휏 in the definition of the Caputo derivative and +replacing the factor 1/Γ(1 − 훼) with 푀(훼)/(1 − 훼). +E.g., the fractional time derivative for 훼 ∈ [0, 1] and function 푓 ∈ 퐿1(−∞, 푏) is +D 훼 +푡 푓 (푡) = 훼푀(훼) +1 − 훼 +푡 +∫ +−∞ +( 푓 (푡) − 푓 (휏)) exp +� +−훼(푡 − 휏) +1 − 훼 +� +푑휏 +where 푀(훼) is a normalization function such as 푀(0) = 푀(1) = 1. +1.6 +GC & GRL derivatives +Zhao & Luo [264] suggested to divide the fractional derivativewith different — singular +and non-singular — kernels (e.g., RL, Caputo, Caputo-Fabrizio, Atangana-Baleanu +[11]1 with the kernel +푘(푥, 훼) = 퐸훼 +� +− +훼 +1 − 훼푥 +� +, +Atangana-Gomez [12] with the kernel +푘(푥, 훼) = exp +� +− +훼 +1 − 훼푥2� +1The equation with the Atangana-Baleanu operator is related to the derivatives of distributed order [219]. +7 + +derivative with the stretched exponential kernel [210] (that is useful in the study of the +water diffusion in the human brain using the magnetic resonance imaging [19]) +푘(푥, 훼) = exp +� +− +훼 +1 − 훼푥훽� +, +훽 > 0, +훽 ≠ 1) +into two classes — GC (general, Caputo sense) and GRL (general, RL) derivatives +that obeys the the principles formulated by V. Volterra in his "general laws of heredity" +[228]: the linearity principle, the invariance principle, the fading memory principle, the +compatibility principle. The compatibility principle requires the validity of two limits: +퐷 훼 푓 (푥) → 푓 (푥) when 훼 → 0 and 퐷 훼 푓 (푥) → 푓 ′(푥) when 훼 → 1. +The principle of nonlocality was suggested by Tarasov [213]. +1.6.1 +GC derivatives +Zhao & Luo [264] defined the GC derivative by +퐷퐺퐶 +푎,훼 푓 (푥) = 푁(훼) +푥 +∫ +푎 +푘(푥 − 푡, 훼) 푑푓 (푡) +푑푡 +푑푡 +The fading memory principle requires that the remote time and position has less effect: +푘(푥 − 푡, 훼) decreases when 푥 increases and 푘(푥 − 푡, 훼) → 0 when 푥 → ∞. +The compatibility principle requires that 푁(훼)/푘(푥, 훼) → 1 when 훼 → 0 and +푁(훼)/푘(푥, 훼) → 훿(푥) when 훼 → 1. +1.6.2 +GRL derivatives +퐷푅퐿 +푎,훼 푓 (푥) = 푑 +푑푥 푁(훼) +푥 +∫ +푎 +푘(푥 − 푡, 훼) 푓 (푡)푑푡. +The restrictions on 푘(푥 − 푡, 훼) and 푁(훼) are the same. +1.7 +Marchaud-Hadamard Fractional Derivatives +Marchaud’s approach is based on the analytic coninuation of the fractional integrals +to the negative orders using the Hadamard’s finite parts of the divergent integrals +(Hadamard’s idea is to ignore the unbounded contribution to the integral and to assign +the value of the remaining — finite — expression [50]). +The Marchaud fractional derivative with the lower limit 푎 is +(푀 훼 +푎+ 푓 )(푥) = +푓 (푥) +Γ(1 − 훼)(푥 − 푎) 훼 + +훼 +Γ(1 − 훼) +푥 +∫ +푎 +푓 (푥) − 푓 (푦) +(푥 − 푦) 훼+1 푑푦 +and with the upper limit 푏 is +(푀 훼 +푏− 푓 )(푥) = +푓 (푥) +Γ(1 − 훼)(푏 − 푥) 훼 + +훼 +Γ(1 − 훼) +푏 +∫ +푥 +푓 (푥) − 푓 (푦) +(푥 − 푦) 훼+1 푑푦. +8 + +Marchard’s method is to extend the RL integral to 훼 < 0 +(퐽−훼 ++ +푓 )(푥) = +1 +Γ(−훼) +∞ +∫ +0 +푦−훼−1 푓 (푥 − 푦)푑푦 +(6) +and to substract the divergent part of the integral in (6) +∞ +∫ +휖 +푦−훼−1 푓 (푥 − 푦)푑푦 = 푓 (푥) +훼휖 휖 +to get finally +(푀 훼 ++ 푓 )(푥) = lim +휖 →0+ +1) +Γ(−훼) +∞ +∫ +휖 +푓 (푥) − 푓 (푦) +푦훼+1 +푑푦. +(7) +There are two approaches to extend the definition (7) to the case 훼 > 1 [99]: +1. To apply (7) to the 푛th derivative 푑푛 푓 /푑푥푛 for 푛 < 훼 < 푛 + 1. +2. To consider 푓 (푥 − 푦) − 푓 (푥) as the first-order difference and to generalize to 푛th +order difference (difference quotient) +(Δ푛 +ℎ 푓 )(푥) = (I − 푇ℎ)푛 푓 (푥) = +푛 +� +푘=0 +(−1)푘 +�푛 +푘 +� +푓 (푥 − 푘ℎ) +(8) +where I is the identity operator and 푇ℎ = 푓 (푥 − ℎ) is the translation operator. +Thus Marchard fractional derivative for 0 < 훼 < 푛 is written as +(푀 훼 ++ 푓 )(푥) = lim +휖 →0+ +1 +퐶훼,푛 +∞ +∫ +휖 +Δ푛 +푦 푓 (푥) +푦훼+1 푑푦 +where +퐶훼,푛 = +∞ +∫ +0 +(1 − 푒−푦)푛 +푦훼+1 +. +1.8 +Grünwald - Letnikov Derivative +The approach suggested independently by Grünwald in 1867 and Letnikov [132] in +1868 is based on the use the limits of the difference quotients (8) similar to the classical +definition of the derivatives for 푛 ∈ N, +푓 ∈ 퐶푛[푎, 푏], +푎 < 푥 ≤ 푏 +˜퐷푛 푓 (푥) = lim +ℎ→0 +Δ푛 +ℎ 푓 )(푥) +ℎ푛 +and extension to the case of the arbitrary 푛. +9 + +Since �푛 +푘 +� = 0 if 푛 ∈ N and 푛 < 푘 the expression (8) is equivalent to +(Δ푛 +ℎ 푓 )(푥) = +∞ +� +푘=0 +(−1)푘 +�푛 +푘 +� +푓 (푥 − 푘ℎ). +(9) +The series (9) is uniformly convergent for any bounded function if 푛 > 0 [69]. +The use of (9) introduce two problems [50]: the function 푓 needs to be defined on +(∞, 푏]; the function 푓 should be such that the series converges. +These problems are solved by the introduction a new function 푓 ★ +푓 ★ = +� +푓 (푥) +푥 ∈ [푎, 푏] +0 +푥 ∈ (−∞, 푎) +that is used instead of the original 푓 . +It is also assumed that in the tending to zero ℎ takes only the Grünwald-Letnikov +fractional derivative of order 푛 defined as +˜퐷푛 +푎 = lim +푁→∞ +Δ푛 +ℎ푁 푓 (푥) +ℎ푛 +푁 += lim +푁→∞ +푁 +� +푘=0 +(−1)푘 +�푛 +푘 +� +푓 (푥 − 푘ℎ푁 ). +(10) +The Grünwald-Letnikov derivative is called pointwise or strong depending on +whether the limit is taken pointwise or in the norm of a suitable Banach space [99]. +The binomial coefficient can be generalized to the non-integer arguments +(−1) 푗 +�푞 +푗 +� += (−1) 푗 +Γ(푞 + 1) +Γ( 푗 + 1)Γ(푞 − 푗 + 1) = +Γ( 푗 − 푞) +Γ(−푞)Γ( 푗 + 1) . +The (left-sided) Grünwald-Letnikov derivative could be written as (푛ℎ = 푥 − 푎) +˜퐷 훼 +푎 푓 (푥) = lim +ℎ→0 +1 +ℎ훼 +⌊푛⌋ +� +푘=0 +(−1)푘 Γ(훼 + 1) 푓 (푥 − 푘ℎ) +Γ(푘 + 1)Γ(훼 − 푘 + 1) +and right-sided (푛ℎ = 푏 − 푥) as +˜퐷 훼 +푏 푓 (푥) = lim +ℎ→0 +1 +ℎ훼 +⌊푛⌋ +� +푘=0 +(−1)푘 Γ(훼 + 1) 푓 (푥 + 푘ℎ) +Γ(푘 + 1)Γ(훼 − 푘 + 1) . +The Grünwald-Letnikov integral of the order 푛 of the function 푓 is written as +˜퐽푛푎 푓 (푥) = +1 +Γ(푛) lim +푁→∞ ℎ푛 +푁 +푁 +� +푘=0 +Γ(푛 + 푘) +Γ(푘 + 1) 푓 (푥 − 푘ℎ푁 ). +1.9 +Riesz Fractional Operators +The fractional integral of the order 훼 in the Riesz sense (also known as the Riesz +potential) is defined by the Fourier convolution product +(I 훼 푓 )(풙) = +∫ +R푛 +푲 훼(풙 − 흃) 푓 (흃)푑흃, +10 + +where 푅푒(훼) > 0. The Riesz kernel +푲 훼 = +1 +훾푛(훼) +� +∥풙∥훼−푛, +훼 − 푛 ≠ 0, 2, . . . , +∥풙∥훼−푛 ln +� +1 +∥풙 ∥ +� +, +훼 − 푛 = 0, 2, . . . +where 훾푛(훼) is defined by +훾푛(훼) +2훼휋 +푛 +2 Γ(훼/2) += +��Γ � 푛−훼 +2 +��−1 , +훼 − 푛 ≠ 0, 2, . . . , +(−1) +푛−훼 +2 2−1Γ � 훼−푛 +2 +� , +훼 − 푛 = 0, 2, . . . . +The Riesz fractional integral is [82] +(I훼 푓 )(풙) = +Γ +� +1−훼 +2 +� +2훼휋 +푛 +2 Γ(훼/2) +∞ +∫ +−∞ +푓 (휉)|푥 − 휉|훼−1푑휉. +The Riesz fractional derivative is [43] +퐷 훼[ 푓 (푥)] = − +1 +2 cos(훼휋/2) +1 +Γ(훼) +푑푛 +푑푥푛 + +푥 +∫ +−∞ +(푥 − 휉)푛−훼푛−1 푓 (휉)푑휉 + +∞ +∫ +푥 +(푥 − 휉)푛−훼푛−1 푓 (휉)푑휉 + +. +The Riesz derivative is the generalization of the Laplace operator [254] +(−Δ) +훼 +2 = − +1 +2 cos(훼휋/2) +� 푑 훼 +푑푥 훼 + +푑 훼 +푑(−푥) 훼 +� +, +훼 ≠ 1. +The Riesz derivative could be expressed in terms of the Marchaud derivative +퐷 훼[ 푓 (푥)] = − +1 +2 cos(훼휋/2) [(푀 훼 ++ 푓 )(푥) + (푀 훼 +− 푓 )]. +The related Riesz-Feller derivative [78] has an additional parameter - "skewness" 휃 +퐷 훼 +휃 푓 (푥) = Γ(1 + 훼) +휋 +× + +sin +� +(훼 + 휃) 휋 +2 +� +∞ +∫ +0 +푓 (푥 + 휉) 푓 (푥) +휉1+훼 +푑휉 + sin +� +(훼 − 휃) 휋 +2 +� +∞ +∫ +0 +푓 (푥 − 휉) 푓 (푥) +휉1+훼 +푑휉 + +. +The allowed region of the parameters 훼 and 휃 turn out to be a diamond in the plane +{훼, 휃} with the vertices in the points (0,0), (1,1), (2,0), (1, -1) called the "Feller-Takayasu +diamond" [76, 159]. +11 + +1.10 +Weyl Fractional Derivative +The Weyl derivative is based on the generalization of the differentiation in the Fourier +space [69] — the integer derivative of the 푛th order (푖푘)푛 of the absolutely integrable +function on [−휋, 휋] presented as the Fourier series is extended to the noninteger 푛. +The Weyl fractional derivative is defined as [148] +퐷 훼 +± = + + +± 푑 +푑푥 [퐼1−훼 +± +푓 (푥)] +0 < 훼 < 1, +푑2 +푑푥2 [퐼2−훼 +± +푓 (푥)] +1 < 훼 < 2, +where the Weyl fractional integrals are (휇 > 0) +퐼 휇 ++ = +1 +Γ(휇) +푥 +∫ +−∞ +(푥 − 휒)휇−1 푓 (휒)푑휒. +1.11 +Erdélye-Kober Fractional Operators +The Erdélye-Kober integral for a well-behaved function 휙(푡) is defined as [143, 178] +퐼훾,휇 +휂 +휙(푡) = +휂 +Γ(휇) 푡−휂(휇+훾) +푡 +∫ +0 +휏휂(훾+1)−1(푡휂 − 휏휂)휇−1휙(휏)푑휏, +where 휇 > 0, +휂 > 0, +훾 ∈ R. +In the special case 훾 = 0, 휂 = 1 the Erdélye-Kober fractional integral is related to +the RL fractional integral of the order 휇 as +퐼0,휇 +1 +휙(푡) = 푡−휇 +Γ(휇) +푡 +∫ +0 +(푡 − 휏)휇−1휙(휏)푑휏 = 푡−휇퐽 휇휙(푡). +The Erdélye-Kober fractional derivative for 푛 − 1 < 휇 < 푛, +푛 ∈ N is defined as +퐷훾,휇 +휂 휙(푡) = +푛 +� +푗=1 +� +훾 + 푗 + 1 +휂 푡 푑 +푑푡 +� +(퐼훾+휇,푛−휇 +휂 +휙(푡)). +The Erdélye-Koberfractional derivative reduces to the identity operator when 휇 = 0 +퐷훾,0 +휂 휙(푡) = 휙(푡) +and for 휂 = 1 and 훾 = −휇 is related to the RL fractional derivative as +퐷훾,휇 +휂 휙(푡) = 푡휇퐷휇 +푅퐿휙(푡). +12 + +1.12 +Interpretation of Fractional Integral and Derivatives +The integer-orderand integrals have a clear physiscal and geometricalinterpretation that +simplify their use in practice. The numerous different interpretations of the fractional +derivatives and integrals have been proposed [100]: the probabilistic [208, 222, 221], +geometric [18, 17, 162, 40], physical interpretations [162, 40, 169, 194, 97]. +However, as noted Podlubny [183], "since the appearance of the idea of differ- +entiation and of arbitrary (not necessary integer) order there was not any acceptable +geometric and physical interpretation of these operations for more than 300 years". +Teneiro Machado [221] wrote the Günwald-Letnikov derivative of 푥(푡) as +퐷 훼[푥(푡)] = lim +ℎ→0 +� +1 +ℎ훼 +∞ +� +푘=0 +훾(훼, 푘)푥(푡 − 푘ℎ) +� +, +훾(훼, 푘) = (−1)푘 +Γ(훼 + 1) +푘!Γ(훼 − 푘 + 1) +where ℎ is the time increment. The author noted that +훾(훼, 0) = 1, +− +∞ +� +푘=0 +훾(훼, 푘) = 1 +thus the "present" (P) is constituted by 푥(푡) the probability 1 while the totality of the +"past/future"(PF) is constituted by the samples 푥(푡), 푥(푡−ℎ), 푥(푡−2ℎ), . . . ; each sample +is weighted with a probability −훾(훼, 푘). +Nigmatullin [169, 170] interpreted the fractional integral in terms of the fractal +Cantor set. The author considered the evolution of the state of the physical system +퐽(푡) = +푡 +∫ +0 +퐾(푡, 휏) 푓 (푡)푑휏 +where the memoryfunction 퐾(푡, 휏) 푓 (푡) accountsfor the loss of some states of th system; +the fractional index of integration equals the fractal dimension of the Cantor set. +Podlubly [183] and Podlubny et al. [185] suggested the geometrical interpreation of +the left-sided (equation (1)) and right-sided (equation (2)) RL integrals and of the RL +(equations (3) - (4)) and the Caputo (equation (5)) derivatives, as well as of the Riesz +potential that is the sum of the left-sided and right-sided RL fractional integrals +푅훼 +푏 푓 (푥) = +1 +Γ(훼) + +푥 +∫ +푎 +(푥 − 푡) 훼−1 푓 (푡)푑푡 + +푏 +∫ +푥 +(푡 − 푥) 훼−1 푓 (푡)푑푡 + +and of the Feller potential +Φ훼 푓 (푥) = 푐퐽 훼 +푎+ 푓 (푥) + 푑퐽 훼 +푏− 푓 (푥). +The geometric interpretation by Podlubny is based on additing the third dimension +푔푥(푡) = +1 +Γ(훼 + 1) [푥 훼 + (푥 − 푡) 훼] +13 + +to the pair (t, f (x)) and considering the three-dimensional line (푡, 푔푥(푡), 푓 (푡)) as the +top edge of the "fence" that gives shadow on the wall in the (g,f) plane. +Tarasov [57] proposed the ”informatic” (”computer science”) interpretation of the +RL and the Caputo derivatives of the non-integer orders using the reconstructions from +the infinite sequence of the derivatives of the integer orders. Such reconstructions atre +based on the Kotel’nikov theorem (also known as the sampling theorem) proved by +Vladimir Kotel’nikov in 1933 and also by Claude Shannon 1949: under the certain +restrictive conditions, function 푓 (푡) can be restored from its samples 푓 [푛] = 푓 (푛푇) +according to the Whittaker-Shannon interpolation formula. The author stressed that +infinity of the sequences of the integer derivatives plays a fundamental role in represen- +tation of the fractional derivatives that describe nonlocality and memory. +Gómez-Aguilar et al. [70] analysed the Caputo differentiation using the RC circuit +for which the fractional version of the Ohm’s law and Kirchhoff’s law are written as +푣(푡) = +1 +휎1−훾 +푑훾푞 +푑푡훾 , +푅 푑푞 +푑푡 + 1 +퐶 푞(푡) = 푣(푡) +where 푞 is the electric charge, 푣 is the voltage, 푅 is the resistance of the conductor, 퐶 +is the capacitance. The parameter 휎 is introduced in order to be consistent with the +dimensionality; it characterizes the fractional structures (the components that show the +intermediate behaviour between conservative (capacitor) and dissipative (resistor) [70]. +The authors derived the fractional differential equation for the RC circuit +푑훾 +푑푡훾 + 1 +휏훾 +푞(푡) = 퐶 +휏훾 +푣(푡), +휏훾 = 푅퐶 +휎1−훾 +where 휏훾 is the time constant. Gómez-Aguilar et al. claimed that the differentiation is +related to the memory effects that reflect the intrinsic dissipation in the system. +Sierociuk et al. [204] used the RC network to model the fractional order diffusion +based on the analogy between the heat and electrical conduction. The authors showed +that the equations for the capacitor and for the resistor in the transmission line could be +used to get the diffusion equation; the loosing of heat was represented by the additional +resistors connected parallel to capacitors. +Carpinteri et al. [29] considered the mechanical interpretation of the Marchaud +fractional derivative using the body springs connecting the non-adjacent points of the +body with the stiffness decaying with the distance between the material points. +1.13 +Local Fractional Derivatives +The fractional derivatives are nonlocal. Several researches introduced the local variants +[259] that are useful for study of the pointwise behaviour of the fractal and multifractal +functions that describe, e.g., the stress and deformation patterns in materials exhibiting +the fractal-like microstructure [29] or the velocity field of turbulent fluid [128]. +Kolwankar & Gangal [127, 128, 129, 126] defined the derivative via the RL deriva- +tive as +픇푞 푓 (푦) = lim +푥→푦 +퐷푞( 푓 (푥) − 푓 (푦)) +(푥 − 푦)푞 +if the limit exists and finite. +14 + +The local fractional Taylor formula is written as [242] +푓 (푥) = +푛 +� +푖=0 +푓 (푛) (푦) +Γ(1 + 푛) (푥 − 푦)푛 +픇훼 +Γ(푛! + 훼) (푥 − 푦) 훼 + 푅훼(푥, 푦). +Yang et al. [249, 266] used similar definition +픇(푘) 푓 (휏) = lim +휏→휏0 +푓 (휏) − 푓 (휏0) +휏푘 − 휏푘 +0 +. +Chen et al. +[39] proposed the local derivatives based on the integrals of the +difference-quotient (IDQ) or the singular of difference-quotient (SIDQ). For example, +the right SIDQ local derivative is +픇훼 푓 (푦) = +1 +Γ(1 − 훼) lim +ℎ→0+ +1 +∫ +0 +(1 − 푡)−훼 푓 (푡ℎ + 푦) − 푓 (푦) +ℎ훼 +푑푡. +The local fractional derivative is essentially the fractal derivative [36, 90]. +In +contrast to the purely analytical approach of the fractional calculus, the fractal calculus +follows the physical-geometric approach; to avoid confusion it is suggested to call the +latter the scaled calculus [175]. +The fractal ("Hausdorff") derivative on the time fractal is defined as [92] +휕 푓 +휕푡휎 = lim +푡퐵→푡퐴 +푓 (푡퐵) − 푓 (푡퐴) +(푡퐵) 휎 − (푡퐴) 휎 +where 휎 is the fractal dimension of time. +A more general definition is formulated as [37, 10] +휕 휏 푓 +휕푡휎 = lim +푡퐵→푡퐴 +푓 휏(푡퐵) − 푓 휏(푡퐴) +(푡퐵) 휎 − (푡퐴) 휎 +Since the fractal derivative is the local operator, the numerical solution of the +fractal derivative equations can be performed by the standard numerical techniques for +the integer-order differential equations [38]. The similar properties have the so called +"conformable" fractional derivatives. +1.13.1 +"Conformable" Fractional Derivative +Most fractional derivatives do not have the desirable properties [2, 118, 117]: the +derivative of a constant is not zero; they do not obey the product rule 퐷 훼( 푓 푔) = +푓 퐷 훼(푔) + 푔퐷 훼 푓 ; +they do not obey the quotient rule 퐷 훼( 푓 /푔) = (푔퐷 훼( 푓 ) − +푓 퐷 훼(푔))/푔2; they do not obey the chain rule 퐷 훼( 푓 푔) = 푓 훼(푔(푡)푔훼(푡); they do +not obey in general 퐷 훼퐷훽 푓 = 퐷 훼+훽 푓 . +Khalil et al.[118] and Katugampola [117, 116] suggested the so called "conformable" +limit based [7] derivatives +퐷 훼 푓 (푡) = lim +휖 →0 +푓 (푡 + 휖푡1−훼) − 푓 (푡) +휖 +, +0 < 훼 < 1, +15 + +and +퐷 훼 푓 (푡) = lim +휖 →0 +푓 (푡푒휖 푡−훼) − 푓 (푡) +휖 +, +0 < 훼 < 1. +Since the conformable derivative is the extension of the classical derivative defi- +nition, this derivative obeys the product rule, the quotient rule, the linearity property, +the zero derivative for the constant and are valid for some extensions of the classical +calculus such as the Rolle’s Theorem or Mean Value Theorem [105]. +2 +Tempered Fractional Calculus +Sabzikar et al. [195] suggested a variant of the fractional calculus where power laws are +tempered by the exponential factor. The random walks model with the exponentially +tempered power law jumps converges to a tempered stable motion [31]. This tempered +fractional diffusion is useful in the geophysical [157, 263] and financial [30] problems. +The authors considered two intervals for the parameter 훼: +• 0 < 훼 < 1. The tempered fractional derivative 휕 훼,휆 +푥 +is defined as the function +with the Fourier transform [(휆 + 푖푘) 훼 − 휆훼] ˆ푓 (푘) that in real space is written as +휕 훼,휆 +푥 +푓 (푥) = +훼 +Γ(1 − 훼) +∞ +∫ +0 +( 푓 (푥) − 푓 (푥 − 푦))푒−휆푦푦−훼−1푑푦. +The negative tempered fractional derivative 휕 훼,휆 +−푥 is defined as the function with +the Fourier transform [(휆 − 푖푘) 훼 − 휆훼] ˆ푓 (푘) that in real space is written as +휕 훼,휆 +−푥 푓 (푥) = +훼 +Γ(1 − 훼) +∞ +∫ +0 +( 푓 (푥) − 푓 (푥 + 푦))푒−휆푦푦−훼−1푑푦. +• 1 < 훼 < 2. The tempered fractional derivative 휕 훼,휆 +푥 +is defined as the function +with the Fourier transform [(휆 + 푖푘) 훼 − 휆훼 − 푖푘훼휆훼−1] ˆ푓 (푘) that in real space is +휕 훼,휆 +푥 +푓 (푥) = 훼(1 − 훼) +Γ(2 − 훼) +∞ +∫ +0 +( 푓 (푥 − 푦) − 푓 (푥) + 푦 푓 ′(푥))푒−휆푦푦−훼−1푑푦. +The negative tempered fractional derivative 휕 훼,휆 +푥 +is defined as the function with +the Fourier transform [(휆 − 푖푘) 훼 − 휆훼 + 푖푘훼휆훼−1] ˆ푓 (푘) that in real space is +휕 훼,휆 +−푥 푓 (푥) = 훼(1 − 훼) +Γ(2 − 훼) +∞ +∫ +0 +( 푓 (푥 + 푦) − 푓 (푥) − 푦 푓 ′(푥))푒−휆푦푦−훼−1푑푦. +16 + +Sabzikar et al. introduced the positive tempered integral as +ℑ훼,휆 ++ +푓 (푥) = +1 +Γ(훼) +푥 +∫ +−∞ +푓 (푢)(푥 − 푢) 훼−1푒−휆(푥−푢)푑푢 +and the negative tempered integral as +ℑ훼,휆 +− +푓 (푥) = +1 +Γ(훼) +푥 +∫ +−∞ +푓 (푢)(푢 − 푥) 훼−1푒−휆(푢−푥)푑푢 +called the RL tempered integrals since for 휆 = 0 they reduce to the usual RL integrals. +The authors defined the RL tempered fractional derivatives D 훼,휆 +± +as functions with +the Fourier transform (휆 ± 푖푘) 훼 ˆ푓 (푘) that can be expressed +D 훼,휆 +± +푓 (푥) = +� +휕 훼,휆 +±푥 푓 (푥) + 휆훼 푓 (푥) +0 < 훼 < 1 +휕 훼,휆 +±푥 푓 (푥) + 휆훼 푓 (푥) ± 훼휆훼−1 푓 ′(푥) +1 < 훼 < 2. +Evidently, integration and differentiation are the inverse operators: +D 훼,휆 +± +ℑ훼,휆 +± +푓 (푥) = 푓 (푥), +ℑ훼,휆 +± +D 훼,휆 +± +푓 (푥) = 푓 (푥). +The integration and differentiation operators have the semigroup property +ℑ훼,휆 +± +ℑ훽,휆 +± +푓 = ℑ훼+훽,휆 +± +푓 , +D 훼,휆 +± +D훽,휆 +± +푓 = D 훼+훽,휆 +± +푓 . +3 +Fractional Differential Equations +Generally, the fractal media could not be considered as continuous media. The use of +the non-integer dimensional spaces [174] is necessary to describe a fractal medium by +the continuous models [216]. The fractional differential equations [53, 167, 121, 50] are +non-local (i.e. could incorporate the effects of the memory and the spatial correlations) +and could be formulated in the distinct but mathematically equivalent forms. Mainardi +et al. [155] compared the fractional extensions of the standard Cauchy problem +휕푢(푥, 푡) +휕푡 += 휕2푢(푥, 푡) +휕푥2 +, +푥 ∈ R, +t ∈ R+ +0, +u(x, 0+) = u0(x). +(11) +The fundamental solution (or Green function) of (11), i.e. the solution subjected to +the initial condition 푢0(푥) = 훿(푥), is the Gaussian probability density function +푢(푥, 푡) = +1 +2√휋 푡−1/2푒−푥2/(4푡). +The Green function has the scaling property 푢(푥, 푡) = 푡1/2푈(푥/푡1/2), 푈(푥) is the +reduced Green function. +17 + +The Cauchy problem (11) is equivalent to the integro-differential equation +푢(푥, 푡) = 푢0(푥) + +푡 +∫ +0 +� 휕2푢(푥, 휏) +휕푥2 +� +푑휏 +where the initial condition is incorporated. +The fractional diffusion equation could be written with the use of the RL derivative +퐷1−훽 (훽 is the real number 0 < 훽 < 1) +휕푢(푥,푡) +휕푡 += 퐷1−훽 휕2푢(푥, 푡) +휕푥2 +(12) +or the Caputo derivative 퐷훽 +★ +퐷훽 +★푢(푥, 푡) = 휕2푢(푥, 푡) +휕푥2 +. +(13) +The equations (12) and (13) are equivalent to the equation based on the use of the +RL fractional integral of the order 훽 +푢(푥, 푡) = 푢0(푥) + 퐽훽 +� 휕2푢(푥, 휏) +휕푥2 +� +. +(14) +Note that the equation (12) could be obtained by differentiating (14), the equation +(14) can be derived by the fractional integration of (13). +The equation (12) was studied by Metzler et al. [158] and by Saichev & Zaslavsky +[196], the equation (14) by Gorenflo et al. [80, 79] and by Mainardi [146, 147], the +integrodifferentialequation (14) by Schneider & Wyss [200] using the Mellin transform. +Mainardi et al. [155] search for the fundamental solution of the equation (13) by +applying the sequence of the Fourier +F {푣(푥); 푘} = ˆ푣(푘) = +∞ +∫ +−∞ +푒푖푘푥푣(푥)푑푥, +푘 ∈ R +and the Laplace +L{푤(푡); 푠} = ˜푤(푠) = +∞ +∫ +0 +푒−푠푡푤(푡)푑푡, +푠 ∈ C +transforms. Thus the Green function in the Fourier-Laplace domain is determined by +ˆ˜푢(푘, 푠) = +푠훽−1 +푠훽 + 푘2 , +0 < 훽 ≤ 1, +R(푠) > 0, +푘 ∈ R. +(15) +There are two strategies to determine the Green function in the space-time domain +푢(푥, 푡) related to the order in performing inversions in the expression (15) [155]: +18 + +1) Invert the Fourier transform to get ˜푢(푥, 푠) and then invert the Laplace transform +[146, 147] or 2) invert the Laplace transform to get ˆ푢(푘, 푡) and then invert the Fourier +transform [81, 151]. +Nieto [168] studied the linear fractional differential equation with the spatial RL +derivative for initial or periodic boundary conditions and derived the maximumprinciple +using the properties of the Mittag-Leffler functions. Compte [42] and West et al. [235] +studied the equation for the hyperdiffusion (Lévy-flight diffusion) +휕푃 +휕푡 = 퐷(−Δ) +훾 +2 +where the fractional 푛-dimensional Laplace operator (−Δ) +훾 +2 is defined by its Fourier +transform with respect to the spatial variable [53] +F [(−Δ) +훾 +2 푔(푥)] = |휔|훾F [푔(푥)]. +Luchko [144] derived the maximum principle fortheinitial-boundary-valueproblem +for the time-fractional diffusion equation with Caputo derivative over the open bounded +domain 퐺 × (0.푇), 퐺 ⊂ 푅푛. +The equation could subjected to the complex transformation [137, 94, 138] 푠 = +푥푆/Γ(1 + 훼) to convert to a partial differential equation2 For example, the heat conduc- +tion equation (훼 is the fractal dimension of the fractal medium) +휕푇 +휕푡 = 휕 훼 +휕푥 훼 +� +휆 휕 훼푇 +휕푥 훼 +� +is converted into the equation +휕푇 +휕푡 = 휕 +휕푠 +� +휆 휕푇 +휕푠 +� +. +That could be further transformed by introduction of the Boltzmann variable [140] +휒 = 푠/√푡 = 푥 훼/√푡Γ(1 + 훼) into the ordinary differential equation +푑 +푑휒 +� +휆 푑푇 +푑휒 +� ++ 휒 +2 +푑푇 +푑휒 . +For the general fractional differential equation in the Jumarie’s modification of the +RL derivatives +푓 (푢, 푢훼 +푡 , 푢훽 +푥, 푢훾 +푦, 푢휆 +푧, 푢2훼 +푡 , 푢2훽 +푥 , 푢2훾 +푦 , 푢2휆 +푧 , . . . ) = 0 +He & Li [95] suggested the following transforms +푠 = +푞푡훼 +Γ(1 + 훼) , +푋 = +푝푥훽 +Γ(1 + 훽) , +푌 = +푘푦훾 +Γ(1 + 훾) , +푍 = +푙푧휆 +Γ(1 + 휆) +2Such transformation is possible in the multidimensional case if the variables 푡, 푥, 푦, 푧 obey the following +constraint [94] (푞, 푝, 푘, 푙 are constants) +휉 = +푞푡 훼 +Γ(1 + 훼) + +푝푥훽 +Γ(1 + 훽) + +푘푦훾 +Γ(1 + 훾) + +푙푧휆 +Γ(1 + 휆) . +19 + +thus converting the fractional derivatives into classical derivatives +휕 훼푢 +휕푡훼 = 푞 휕푢 +휕푠 , +휕훽푢 +휕푥훽 = 푝 휕푢 +휕푋 , +휕훾푢 +휕푦훾 = 푘 휕푢 +휕푌 , +휕휆푢 +휕푧휆 = 푙 휕푢 +휕푍 . +Babusci et al. [14] discussed relations between the differential equations and the +theories of the pseudo-operators [220, 202] and the generalized integral transforms. +3.1 +Distributed order differential equations +The distributed order differential equations (DODE) are a special class of the fractional +differential equations [165, 253, 33, 34, 35, 125, 24, 77, 153]. Chechkin et al. [32] +discussed the natural and the modifies forms of DODEs and noted that the latter in com- +bination with the continuity equation and the retarded linear response equation for the +flux exhibiting memory of the processes at the previous times admits a thermodynamic +interpretation. DODEs are used to describe the accelerating subdiffusion, decelerating +superdiffusion or transformation of the anomalous behaviour at the short times into the +normal behaviour at the long times. For example, Metzler & Klafter [159] consid- +ered the DODE for the description of the ultraslow diffusion with the logarithmic time +dependence �푥2(푡)� ∝ log푘 푡 including the so called Sinai diffusion (푘 = 4). +The concept of the distributed order differentiation is close to the variable order +fractional operators that are useful for the study of the viscoelasticity, the reaction +kinetics of proteins, the electrorheological fluids, the damage modelling [71, 41, 197]. +There are two approaches to the formulation of the distributed order differential +equations: 1) direct — a new variable does not assigned; 2) Independent variable +approach — the order is considered as a function of some independent variable. +Mainardi et al. [155] studied the fractional diffusion equation of distributed order +1 +∫ +0 +푏(훽)[퐷훽푢(푥, 푡)]푑훽 = 휕2푢(푥, 푡) +휕푥2 +, +푏(훽) ≥ 0, +1 +∫ +0 +푏(훽)푑훽 = 1 +(16) +with 푥 ∈ R, 푡 ≥ 0 and the initial condition 푢(푥, 0+) = 훿(푥). The weight function 푏(훽) +is called the order-density. The authors used the Fourier and Laplace transforms to get +the fundamental solution similar to a single-order case (15) + +1 +∫ +0 +푏(훽)푠훽푑훽 + +ˆ˜푢(푘, 푠) − +1 +∫ +0 +푏(훽)푠훽−1푑훽 = −푘2 ˆ˜푢(푘, 푠) +and +ˆ˜푢(푘, 푠) = +퐵(푠)/푠 +퐵(푠) + 푘2 , +푘 ∈ R, +B(s) = +1 +∫ +0 +b(훽)s훽d훽. +(17) +In the case of small 푘 the equation (17) can be approximated as +ˆ˜푢(푘, 푠) = 1 +푠 +� +1 − +푘2 +퐵(푠) + . . . +� +20 + +and the second moment is written as +˜휇2(푠) = − 휕2 +휕푘2 ˆ˜푢(푘 = 0, 푠) = +2 +퐵(푠) . +(18) +The special case of DODEs are the double-order fractional equations [32] +푏(훽) = 푏1훿(훽−훽1)+푏2훿(훽−훽2), +0 < 훽1 < 훽2 ≤ 1, +훽1 > 0, 훽2 > 0, +훽1+훽2 = 1. +Asymptotic behaviour of 휇2(푡) follows from (18) for cases of the slow diffu- +sion (the power-law growth, 푏(훽) = 푏1훿(훽 − 훽1) + 푏2훿(훽 − 훽2)) where ˜휇2(푠) = +2/(푏1푠훽1+1 + 푏2푠훽2+1) and the ultra-slow diffusion (the logarithmic growth, 푏(훽) = 1) +with ˜휇2(푠) = 2ln 푠/푠(푠 − 1). +The distributed order equations allow to describe the more complex media. The +time-fractional diffusion equation of the distributed order (16) is potentially more +flexible to represent the local phenomena while the space-fractional diffusion equation +of the distributive order is more suited to represent the variations in space [25]. +3.2 +Special Functions +There are special functions related to the differential equation similar to the classical +case (such as e.g., the Bessel and the cylindrical functions, the classical orthogonal +polynomials, Airy functions etc.) [130]. The most important functions in the fractional +calculus are the Mittag-Leffler function [86], the H-functions [98, 156, 124], the Wright +functions [154, 152], the generalized Lommel-Write functions [187]. +The Mittag- +Leffler function is even called the "Queen"-function of the fractional calculus [124]. +3.2.1 +Mittag-Leffler Functions +The eigenfunction of the RL derivatives are the solutions of the equation [99] +퐷 훼 +0+[ 푓 (푥)] = 휆 푓 (푥) +where 휆 is the eigenvalue. The eigenfunctions are 푓 (푥) = 푥1−훼퐸훼,훼(휆푥 훼) where +퐸훼,훽 = +∞ +� +푘=0 +푥퐾 +Γ(훼푘 + 훽) +(19) +is the generalized Mittag-Leffler function (also called the Wiman’s function [203]). +The more general eigenvalue equation for derivatives of the orders 훼 and 훽 is +퐷 훼,훽 +0+ [ 푓 (푥)] = 휆 푓 (푥) +The solution is [99] 푓 (푥) = 푥 (1−훽) (1−훼)퐸훼,훼+훽(휆푥 훼). +The special case is the equation 퐷 훼,1 +0+ [ 푓 (푥)] = 휆 푓 (푥) with eigenfunction 푓 (푥) = +퐸훼(휆푥 훼). +The one-parameter Mittag-Lefler function is the particular case of (19) for 훽 = 훼. +21 + +Evidently [50], +퐸0,1(푥) = +∞ +� +푘=0 +1 +Γ(1) = +∞ +� +푘=0 +푥푘 = +1 +1 − 푥 , +퐸1(푥) = +∞ +� +푘=0 +푥푘 +푘! = exp(푥). +There are other special cases such as [86] 퐸2(−푥2) = 퐸2,1(−푥2) = cos(푥); 퐸2(푥2) = +퐸2,1(푥2) = cosh(푥); for 푥 > 0 +퐸1/2(푥1/2) = 퐸1/2(푥1/2) = (1 + 푒푟 푓 (푥)) exp(푥2); for +푥 ∈ C and 푟 ∈ N +퐸1,푟 = +1 +푥푟−1 +� +exp(푥) − +푟−2 +� +푘=0 +푥푘 +푘! +� +; +퐸3(푥) = 1/2[푒푥1/3 + 2푒−1/2푥1/3 cos( +√ +3/2푥1/3)]; 퐸4(푥) = 1/2[cos(푥1/4) + cosh(푥1/4)] +where 푒푟 푓 (푥) = 2/√휋 +∫ 푥 +0 exp(−푡2)푑푡. +The Mittag-Leffler function 퐸1 satisfies the functional relation [50, 86] 퐸1(푥 − +푦) = 퐸1(푥)/퐸1(푦) and the relation between two Mittag-Leffler functions with different +parameters 퐸푛1,푛2 (푥) = 푥퐸푛1,푛1+푛2 (푥) + 1/Γ(푛2). Note that the frequently used relation +퐸훼(푎(푡 + 푠) 훼) = 퐸훼(푎푡훼)퐸훼(푎푠훼), +푡, 푠 ≥ 1 is valid only if 훼 = 0 or 훼 = 1 [179]. +Asymptotic expansions and integral representations of the Mittag-Leffler functions +could be found in the papers [74, 71, 86]. +Prabhakar [186] suggested the extension +퐸 훾 +훼,훽(푥) = +∞ +� +푛=0 +(훾)푛 +Γ(훼푛 + 훽) +푥푛 +푛! , +푅푒(훼) > 0, 푅푒(훽) > 0 +(훾)푛 is the Pochhammer symbol [203] (훾)0 = 1, (훾)푛 = 훾(훾 + 1)(훾 + 2) . . . (훾 + 푛 − 1). +The extension to the multi-index Mittag-Leffler functions [123, 124] +퐸( 1 +휌푖 ),(휇푖) (푥) = +∞ +� +푘=0 +푥푘 +Γ(휇1 + 푘/휌1) . . . Γ(휇푚 + 푘/휌푚) +is performed by replacing of the indices 훼 = 1/휌 and 훽 = 휇 by two sets of multi-indices +훼 → (1/휌1, 1/휌2, . . . , 1/휌푚) and 훽 → (휇1, 휇2, . . . , 휇푚). +There are a couple of related functions [166] +• Barret’s function +푈(푥, 휆) = +∞ +� +푘=1 +휆푘−1푥푘훼푖 +Γ(푘훼 − 푖 + 1); +• Rabotnov’s (fractional exponential) function [189, 21] +E훼(훽, 푥) = 푥 훼 +∞ +� +푛=0 +훽푛푥푛(훼+1) +Γ((푛 + 1)(1 + 훼)) . +22 + +3.2.2 +H Functions +The H-function of order (푚, 푛, 푝, 푞) ∈ N4 is defined via the Mellin-Barnes type contour +integral [53, 156] +퐻푚,푛 +푝,푞 (푧) = +1 +2휋푖 +∫ +L +H 푚,푛 +푝,푞 푧푠푑푠 +where 푧푠 = 푒푥푝[푠(푙푛|푧| + 푖푎푟푔푧)], +H 푚,푛 +푝,푞 = 퐴(푠)퐵(푠) +퐶(푠)퐷(푠) , +퐴(푠) = +푚 +� +푗=1 +Γ(푏 푗 − 훽 푗푠), +퐵(푠) = +푛 +� +푗=1 +Γ(1 − 푎 푗 + 훼푗푠), +퐶(푠) = +푞 +� +푗=푚+1 +Γ(1 − 푏 푗 + 훽 푗푠), +퐷(푠) = +푝 +� +푗=푛+1 +Γ(푎 푗 − 훼푗푠). +Here 푚, 푛, 푝, 푞 are integers satisfying 0 ≤ 푛 ≤ 푝, +1 ≤ 푚 ≤ 푞, 푚2 + 푛2 ≠ 0, +푎 푗( 푗 = 1, . . . , 푝), 푏 푗( 푗 = 1, . . . , 푞) are complex numbers. +The integration contour L could be chosen in different ways: +• L = L−푖∞,푖∞ chosen to go from −푖∞ +to +푖∞ leaving to the right all poles of +P(퐴) of the functions Γ in 퐴(푠) and to the left all poles of P(퐵) of the functions +Γ in 퐵(푠); +• L = L푖∞ is a loop beginning and ending at +∞ and encircling in the negative +direction all the poles of P(퐴); +• L = L−푖∞ is a loop beginning and ending at −∞ and encircling in the negative +direction all the poles of P퐵). +3.2.3 +Wright Functions +The Write function is defined by the series representation that is convergentin the whole +푧-complex plane [152, 72, 73, 120] +푊휆,휇(푧) = +∞ +� +푛=0 +푧푛 +푛!Γ(휆푛 + 휇) , +휆 > −1, 휇 ∈ C. +The integral representation of the Write function is written as +푊휆,휇(푧) = +1 +2휋푖 +∫ +퐻 푎 +푒휎+푧휎−휆 푑휎 +휎휇 +where 퐻푎 is the Hankel path (a loop that starts from −∞ along the lower side of the +negative real axis, encircles the circular area the origin with radius 휖 → 0 in the positive +sense, and ends at −∞ along the upper side of the negative real axis). +There are Write-type auxiliary functions 퐹휈(푧) = 푊−휈,0(푧), 푀휈(푧) = 푊−휈,1−휈(푧), +where 0 < 휈 < 1; these functions are related 퐹휈(푧) = 휈푧푀휈(푧). +23 + +The series representations of the auxiliary functions are +퐹휈(푧) = +∞ +� +푛=1 +(−푧)푛 +푛!Γ(−휈푛) = 1 +휋 +∞ +� +푛=1 +(−푧)푛−1 +푛! +Γ(휈푛 + 1) sin(휋휈푛) +and +푀휈(푧) = 퐹휈(푧) = +∞ +� +푛=0 +(−푧)푛 +푛!Γ[−휈푛 + (1 − 휈))] = 1 +휋 +∞ +� +푛=1 +(−푧)푛−1 +(푛 − 1)!Γ(휈푛) sin(휋휈푛). +4 +Solution of Fractional Differential Equations +4.1 +Analytical Methods +Numerous approximate analytical methods are known: +• the Adomian decomposition method (ADM) [131]; +• the combined Laplace-Adomian method (CLAM) [232]; +• the variational iteration method (VIM) [87, 237, 59, 239, 244]3 and its local +(LVIM) [248, 244, 236, 246] and fractional (using the fractional order Lagrange +multipliers) [119, 262] variants; +• the homotopy perturbation method (HPM)[1, 241, 251, 205, 190, 233] and its +modification [238] and local fractional variant (LFHPM) [247]; +• the differential transformation method [109, 6, 66]; +• the heat-balance integral method (HBIM)[101, 103, 102]; +• the fractional complex transform method (FCTM) [245, 137, 136, 93, 209]; +• the local fractional Fourier series method (FSM) [250, 261]; +• the modified simple equation method [106, 252]; +• the method of images (limited to special spatial symmetries); +• the Mellin integral transform method [145]; +• the local fractional decomposition method (LFDM) [5]; +• the fractional sub-equation method [260, 83]; +3VIM includes three steps to determine the variational iteration formula: +1. establishing the correction functional; +2. identifying the Lagrange multipliers; +3. determining the initial iteration. +The second step is the crucial one [236]. +24 + +• the Sumudu transform methods [231, 45] and its variant — the local fractional +homotopy perturbation Sumudu transform mehod [265]; +• the theta-method [8]; +• the Picard succesive approximation method (PSAM) [243, 240]; +• the local Laplace transforms. +Frequently analytical methods are variants of perturbation methods [226]). For +example, He [88, 89, 91] based his method to solve the general equation 퐴(푢)− 푓 (푟) = 0 +with the general differential operator 퐴 divided into linear 퐿 and nonlinear 푁 parts +퐿(푢) + 푁(푢) − 푓 (푟) = 0 on the approach of Liao [139] (who used the two-parameter +family of equations) by considering the one-parameter family (1− 푝)퐿(푢) + 푝푁(푢) = 0. +He constructed the homotopy 푣(푟, 푝) : Ω × [0, 1] → 푅 that satisfies +퐻(푣, 푝) = (1 − 푝)[퐿(푣) − 퐿(푣0)] + 푝[퐴(푣) − 푓 (푟)] = 0 +where the homotopy parameter 푝 ∈ [0, 1], 푣0 is the initial approximation. +Evidently, 퐻(푣, 0) = 퐿(푣) − 퐿(푣0) = 0 and 퐻(푣, 1) = [퐴(푣) − 푓 (푟)] = 0. +In topology, 퐿(푣) − 퐿(푣0) is called deformation. The homotopy parameter 푝 is +considered as a small parameter and the solution is written as a series +푣 = 푣0 + 푝푣1 + 푝2푣2 + 푝3푣3 + . . . +and when 푝 → 1 +푢 = lim +푝→1 푣 = 푣0 + 푣1 + 푣2 + 푣3 + . . . . +The Adomian decomposition method (ADM) [3, 54] does not use linearization, +perturbation or the Green’s functions. +The accuracy of the approximate analytical +solutions can be verified by the direct substitution. +The initial value is written as 퐿푢 + 푅푢 + 푁푢 = 푔 where 퐿 is the linear operator to +be inverted, 푅 is the linear remainder operator and 푁 is the nonlinear operator. Thus +퐿−1퐿푢 = 푢 − Φ, Φ incorporates the initial values. +The solution and nonlinear term are decomposed into series +푢 = +∞ +� +푛=0 +푢푛, +푁푢 = +∞ +� +푛=0 +퐴푛 +where 퐴푛 are the Adomian polynomials for 푁푢 = 푓 (푢) are [54] +퐴푛 = 1 +푛! +휕푛 +휕휆푛 푓 +� ∞ +� +푘=0 +푢푘휆푘 +� +, +푛 = 0, 1, 2, . . . . +Finally +∞ +� +푛=0 +푢푛 = Φ + 퐿−1푔 − 퐿−1 +� +푅 +∞ +� +푛=0 +푢푛 + +∞ +� +푛=0 +퐴푛 +� +. +25 + +The nonlinear term 푁푢(푥, 푡) can be also decomposed [238] as +푁푢 = +∞ +� +푛=0 +푝푛퐻푛 +where He’s polynomials are [171, 68] +퐻푛(푢0, 푢1, . . . , 푢푛) = 1 +푛! +휕푛 +휕푝푛 +� +푁 +� 푛 +� +푖=0 +푝푖푢푖 +�� +. +The fractional sub-equation method includes several steps [83]: +• Transformation of the nonlinear fractional equation in two variables 푥 and 푡 +퐷 훼 +푡 푢, 퐷 훼 +푥 푢, . . . ) = 0, +0 < 훼 ≤ 1, 퐷 훼 +푡 푢 and 퐷 훼 +푥 푢 are Jumarie modification of +the RL derivtives, using the travelling wave transformation 푢(푥, 푡) = 푢(휉), 휉 = +푥 + 푐푡, where 푐 is a constant to be determined, to the equation +푃(푢, 푐푢′, 푢′, 푐퐷 훼 +휉푢, 퐷 훼 +휉푢, . . . ) = 0. +(20) +• The solution of the equation (20) is assumed to have the form +푢(휉) = +−1 +� +푖=−푛 +푎푖휙푖 + 푎0 + +푛 +� +푖=1 +푎푖휙푖, +where 푎푖(푖 = −푛, −푛 + 1, . . . , 푛 − 1, 푛) are constants to be determined, 휙 = 휙(휉) +are functions that satisfy the following Riccati equation 퐷 훼 +휉 휙(휉) = 휎휙2(휉), 휎 is +a constant. +• Formulation of a set of overdetermined nonlinear algebraic equations for 푐 and +푎푖(푖 = −푛, −푛 + 1, . . . , 푛 − 1, 푛) [83]. +4.2 +Numerical Methods +Diethelm et al. [115] listed the requirement to the numerical methods that should be +convergent, consistent of some reasonable order ℎ푝, stable, reasonably inexpensive to +run, reasonably easy to program. +Numerous methods are used in practice: finite difference, finite elements, radial +basis functions, spectral methods, meshfree methods. The numerical methods for the +fractional differential equations usually are constructed by the modification of the meth- +ods for the ordinary differential equations but require significantly more computation +time and storage. The approximation of the fractional derivative needs the computation +of the convolution integral that requires to sample and multiply the behaviour of two +functions over the whole of the interval of integration leading to the operation count of +푂(푛2) where 푛 is number of sampling points [63]. +The reduction of the computational efforts is related to the fading memory property +of the fractional derivatives that allows to restrict the integration interval — using the +26 + +short memory principle [46, 49] (also fixed memory principle [63] and logarithmic +memory principle [85]), and using adaptive time stepping and basis selection [22]. +Numerous methods are used to solve the fractional differential equations in practice: +the finite difference [135, 199] (both the explicit, e.g. Euler [172] and the implicit +[163, 67], e.g., the Crank-Nicolson [211, 212] or the alternating direction implicit +[257, 258] schemes, compact schemes [229, 230, 64, 52]), the finite elements [104, +107, 256, 267, 192] (including least squres FEM [61], Galerkin FEM [206, 108], +discontinuous Galerkin FEM [164]), the spectral methods [23, 58, 51, 255], the meshfree +methods [55, 201] (including the radial basis functions methods that exploit cubic 휙 = +푟3, Gaussian 휙 = 푒푥푝(−푟2/푐2), multiquadrics 휙 = +√ +푐2 + 푟2 or inverse multiquadrics +휙 = 1/ +√ +푐2 + 푟2 functions [234, 180, 9]), Legendre wavelet collocation method [96]. +Bahuguna et al. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 Faá di Bruno formula (the chain rule) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 "Conformable" Fractional Derivative .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 15 2 Tempered Fractional Calculus 16 3 Fractional Differential Equations 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Distributed order differential equations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 20 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Mittag-Leffler Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 H Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='3 Wright Functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 23 ∗Ioffe Physical Technical Institute & SoftImpact, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', e-mail: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='zhmakin0@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='com 1 4 Solution of Fractional Differential Equations 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Analytical Methods .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 Numerical Methods .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 26 1 Fractional Derivatives Fractional calculus (FC) is now an efficient tool for problems in science and engineering [174, 160, 122, 198, 28, 181, 166, 214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The term "fractional" is kept for the historical reasons — it is a misnomer since the order can be real [76, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Historical survey of the development of FC starting from the letter by Gottfried Leibniz to Guillaume l’Hôpital (1695) including contributions by Joseph Liouville, BernhardRiemann, Niels Abel, Grünwald,Aleksey Letnikov,Gerasimov, Marcel Riesz, Magnus Mittag-Leffler, Paul Lévy, Raoul Nigmatullin, Yuri Rabotnov, Arthur Erdélyi and others during the XIX and XX centuries could be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [193, 99, 223, 225, 224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Lorenzo & Hartley [141] analysed the minimal set of criteria for the generalized calculus formulated by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Ross: analyticity: if 푓 (푧) is an analytic function of the complex variable 푧, the derivative 퐷휈 푧 푓 (푧) is an analytic function of 푧 and 휈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' backward compatibility: the operation 퐷휈 푧 푓 (푧) must produce the same result as ordinary differ- entiation when 휈 = 푛 is a positive integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the operation 퐷휈 푧 푓 (푧) must produce the same result as ordinary 푛-fold integration when 푛 is a negative integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 퐷휈 푧 푓 (푧) must vanish along with its 푛 − 1 derivatives at 푥 = 푐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' zero property: the operation of order zero leaves the function unchanged 퐷0 푧 푓 (푧) = 푓 (푥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' linearity: the fractional operators must be linear 퐷휈 푧 [푎 푓 (푥) + 푏푔(푥)] = 푎퐷휈 푧 푓 (푥) + 퐷휈 푧푔(푥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' composition (index law): the law of exponents for integration of arbitrary order 퐷휈 푧 퐷휇 푧 푓 (푥) = 퐷휈+휇 푧 푓 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional derivatives are based on the extension of the repeated integration and are defined either by the continuation of the fractional integral to the negative order or by the integer order derivatives of the fractional integrals [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' There is no unique definition [198, 181, 214] (and notation is not standardized [99, 43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' There are several kinds of definitions of the fractional derivatives (Riemann- Liouville, Caputo, Grünfeld-Letnikov, Riesz, Weyl, Marchaud, Caputo-Fabrizio, Yang, Chen, He and others) that are not equivalent [133, 134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The initial conditions for the Caputo derivative are expressed in terms of the initial values of the integer order deriva- tives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' for the zero initial conditions Riemann-Liouville, Caputo and Grünwald-Letnikov derivatives coincide [191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Most of the fractional derivatives are defined through the fractional integral thus derivatives inherent some non-local behaviour [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fol- lowing relation is valid fo all types of the fractional derivatives [254] 푑 훼+훽 푑푡훼+훽 = 푑 훼 푑푡훼 푑훽 푑푡훽 = 푑훽 푑푡훽 푑 훼 푑푡훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The most frequently used are the Riemann-Liouville (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', in the calculus), the Caputo (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', in the physics, the numerical computations) and Grünwald-Letnikov (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', in the signal processing, the engineering) fractional derivatives [4, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Grigoletto & de Oliveira [82] considered the generalization of the fundamental theorem of calculus — Fundamental Theorem of Fractional Calculus (FTFC) for the 1 cases of the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Baleanu & Fernandez [16] considered the possible classification of the fractional operators into broad classes under some restrictions and criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' In particularity, many operators could be considered as the special cases of [60] 퐴 푐 퐼 훼,훽 푥 푥 ∫ 푐 (푥 − 푡) 훼−1퐴 � (푥 − 푡)훽� 푓 (푡)푑푡, 퐴(푧) = ∞ � 푘=0 푎푘푧푘 where 푐 is a constant often taken as zero or ∞, 훼 and 훽 are complex parameters with positive real parts, and 퐴(푧) is a general analytic function whose coefficients 푎푘 ∈ C are permitted to depend on 훼 and 훽, 푥 as a real variable larger than 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Riemann-Liouville Fractional Integral Both the Riemann-Liouville (RL) and the Caputo fractional derivatives are based on the RL fractional integral that for any 훼 > 0 is defined as [150, 44] 퐽 훼 푎+ 푓 (푥) = 1 Γ(훼) 푥 ∫ 푎 (푥 − 푡) 훼−1 푓 (푡)푑푡, (1) If 훼 = 0, 퐽0 푎+ = 퐼, 퐼 is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Here Γ(훼) = ∞ ∫ 0 exp(−훼)푢훼−1푑푢 is the Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' This integral exists if 푓 (푡) is the locally integrable function and for 푡 → 0 behaves like 푂(푡−휈) with 휈 < 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' To get the strict mathematical rigor it is possible to use the framework of the Lebesgue spaces of the summable functions or the Sobolev spaces of the generalized functions [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The RL integral is a generalization of Cauchy’s formula for an n-fold integral 푥 ∫ 푎 푑푥1 푥1 ∫ 푎 푑푥2· · · 푥푛−1 ∫ 푎 푑푥푛 = 1 (푛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 푥 ∫ 푎 (푥 − 푡)푛−1푑푡 using the relation (푛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' = 푛−1 � 푘=1 푘 = Γ(푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The equation (1) is left-sided RL integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The right-sided RL integral is written as 퐽 훼 푏− 푓 (푥) = 1 Γ(훼) 푏 ∫ 푥 (푡 − 푥) 훼−1 푓 (푡)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (2) 2 The RL integral is a case of the convolution integral of the Volterra type [149] 퐾 ∗ 푓 (푥) = 푏 ∫ 푎 푘(푥 − 푡) 푓 (푡)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The RL integral has the semi-group property (also called additivity law [99]): 퐽 훼 푎+퐽훽 푎+ 푓 (푥) = 퐽 훼+훽 푎+ 푓 (푥), 훼 > 0, 훽 > 0 which implies the commutative property [149]: 퐽훽 푎+퐽 훼 푎+ = 퐽 훼 푎+퐽훽 푎+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The RL fractional integral coincides with the classical definition in the case 훼 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional integration improves the smoothness of functions [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Sometimes the RL integral could be expressed via the elementary functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', 퐽 훼 푎+(푥 − 푎)휇 = Γ(휇 + 1) Γ(훼 + 휇 + 1) (푥 − 푎) 훼+휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' A particular case of the RL fractional integrals is the Liouville fractional integrals [82] that is obtained by transitions 푎 → −∞ and 푏 → ∞ in equations (1) and (2) as 퐽 훼 + 푓 (푥) = 1 Γ(훼) 푥 ∫ −∞ (푥 − 푡) 훼−1 푓 (푡)푑푡, 퐽 훼 − 푓 (푥) = 1 Γ(훼) ∞ ∫ 푥 (푡 − 푥) 훼−1 푓 (푡)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 Riemann-Liouville Fractional Derivative The left and the right Riemann-Liouville fractional derivatives are defined as [181] 퐷 훼 푎+[ 푓 (푥)] = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1 Γ(1 − 훼) 푑 푑푥 푥 ∫ 푎 (푥 − 푡)−훼 푓 (푡)푑푡, 훼 ∈ (0, 1) 푑푓 (푥) 푑푡 , 훼 = 1 (3) and 퐷 훼 푏− [ 푓 (푥)] = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1 Γ(1 − 훼) 푑 푑푥 푏 ∫ 푥 (푡 − 푥)−훼 푓 (푡)푑푡, 훼 ∈ (0, 1) 푑푓 (푥) 푑푡 , 훼 = 1 (4) Operator 퐷 훼 푎+ is left-inverse meaning that 퐷 훼 푎+퐽 훼 푎+ = 퐼, 퐼 is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Thus 퐷 훼 푎+퐽 훼 푎+ 푓 = 푓 but the unconditional semigroup property of fractional differentiation in the RL sense does not hold: Diethelm [50] gives examples where 퐷 훼1 푎+퐷 훼2 푎+ 푓 = 퐷 훼2 푎+퐷 훼1 푎+ 푓 ≠ 퐷 훼1+훼2 푎+ 푓 and 퐷 훼1 푎+퐷 훼2 푎+ 푓 ≠ 퐷 훼2 푎+퐷 훼1 푎+ 푓 = 퐷 훼1+훼2 푎+ 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Atangana & Secer [13] presented tables of the RL derivatives of the trigonometric and some special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional RL derivative of the power function is 퐷 훼 푎+푡휈 = Γ(1 + 휈) Γ(1 + 휈 − 훼) 푡휈−훼 3 and, particular, the derivative of a constant 퐷 훼 푎+1 = 푡−훼/Γ(1 − 훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Since the fractional RL derivative of a constant is not zero, thus the magnitude of the fractional derivative changes with adding of the constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Jumarie [110] suggested a modification to remove this drawback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' He started with a fractional derivative (F-derivative) 푓 훼(푥) = lim ℎ→0 Δ훼 푓 (푥) ℎ훼 based on the fractional difference Δ훼 푓 (푥) of order 훼, 훼 ∈ ℜ, 0 < 훼 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Jumarie proposed the modification of the fractional RL derivative 1 Γ(1 − 훼) 푑 푑푥 푥 ∫ 푎 (푥 − 푡)−훼( 푓 (푡) − 푓 (0))푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Leibniz’ formula The classical Leibnitz’ formula for the first-order derivative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' when 푛 ∈ N) is 퐷푛[ 푓 (푥)푔(푥)] = 푛 � 푘=0 �푛 푘 � [퐷푘푔(푥)퐷푛−푘 푓 (푥)] where 푓 (푥) and 푔(푥) are the 푛-time differentiable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional derivatives violate the classical Leibnitz’ rule [215, 217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' General- ization of the Lebnitz’ formula was developed by Osler [176, 177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Leibniz’ formula for the differentiation of the product of the functions for the RL operators for the functions that are analytic on (푎 − ℎ, 푎 + ℎ) is written as [50] 퐷푛 푎+ [ 푓 푔](푥) = ⌊푛⌋ � 푘=0 �푛 푘 � (퐷푘 푎+ 푓 )(푥)(퐷푛−푘 푎+ 푔)(푥) + ∞ � 푘=[푛]+1 �푛 푘 � (퐷푘 푎+ 푓 )(푥)(퐽푛−푘 푎+ 푔)(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' where ⌊ ⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Jumarie studied the Leibniz’ formula for the differentiation of the product of the non-differentiable functions [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 Faá di Bruno formula (the chain rule) For the functions 푓 and 푔 with a sufficient number of the derivatives and 푛 ∈ N [50, 44, 218] 퐷푛[푔( 푓 (·))](푥) = � (퐷푘푔) 푛 � 푚=1 (퐷푚 푓 (푥)푏푚 where the sum is over all partitions of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푛} and for each partition 푘 is its number of the blocks and 푏 푗 is the number of the blocks with exactly 푗 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Tarasov [216] analysed the simplified chain rules suggested by Jumarie [112, 111, 114] and found that these simplifications are not universally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='3 Fractional Taylor expansion The fractional Taylor expansion is written as [50, 188, 173] 푓 (푥) = (푥 − 푎)푛−푚 Γ(푛 − 푚 + 1) lim 푧→푎+ 퐽푚−푛 푎 푓 (푧)+ 푚−1 � 푘=1 (푥 − 푎)푘+푛−푚 Γ(푘 + 푛 − 푚 + 1) lim 푧→푎+ 퐷푘퐽푚−푛 푎 푓 (푧) + 퐽푛 푎퐷푛 푎 푓 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='4 Symmetrised space derivative Vermeersch & Shakouri [227] formulated the symmetrised space derivatives of the fractional order between 1 and 2 and between 0 and 1: 1 < 훼 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The symmetrised space derivative of the function 푔(푥) that is integrable over the entire real axis is 휕 훼푔 휕|푥|훼 = 휕 휕푥 � 푤훼 ★ 휕푔 휕푥 � where ★ denotes the convolution and 푤훼 is an unknown kernel function with the Fourier image found to be 푊훼 = 1/|휉|2−훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Fourier inversion yields 푤훼 = 1 2휋 ∞ ∫ −∞ 푒푥푝( 푗휉푥)푑휉 |휉|2−훼 = |푥|−(훼−1) 2Γ(2 − 훼) cos[(2 − 훼) 휋 2 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Thus 휕 훼푔 휕|푥|훼 = 1 2Γ(2 − 훼) cos[(2 − 훼) 휋 2 ] 휕 휕푥 ∞ ∫ −∞ 휕푔 휕푥 (푥′)푑푥′ |푥 − 푥′|훼−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 0 < 훼 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The symmetrised space derivative of the function 푔(푥) is 휕 훼푔 휕|푥|훼 = 푤훼 ★ 휕푔 휕푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Fourier image of the kernel function 푤훼 is 푊훼 = 푗 · 푠푔푛(휉)/|휉|1−훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Per- forming the Fourier inversion, the authors get finally 휕 훼푔 휕|푥|훼 = −1 2Γ(1 − 훼) cos[(1 − 훼) 휋 2 ] 휕 휕푥 ∞ ∫ −∞ −푠푔푛(푥) · 휕푔 휕푥 (푥′)푑푥′ |푥 − 푥′|훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' In the case 훼 = 1/2 the fractional integrals and derivatives are also called semi- integrals and semi-derivatives [207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='3 Caputo Fractional Derivative The fractional derivatives in the Caputo sense on the left (퐶퐷 훼 푎+) and on the right (퐶퐷 훼 푏−) are defined via the RL fractional integral [82] (퐶퐷 훼 푎+ 푓 ) = (퐽푛−훼 푎+ 푓 (푛))(푥) and (−1)푛(퐶퐷 훼 푏− 푓 ) = (퐽푛−훼 푏− 푓 (푛))(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' It was introduced independently in 1948 by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Caputo and by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Gerasimov [65];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' later by Dzherbashyan & Nersesian [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The major difference of the Caputo fractional derivative is that the derivative act first on the function and after the integral is evaluated while in the RL approach the derivative act on the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Caputo fractional derivative is defined as [181] 퐷 훼 ★ 푓 (푡) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1 Γ(1 − 훼) 푡 ∫ 0 (푡 − 푥)−훼 푑푓 (푥) 푑푡 푑푡, 훼 ∈ (0, 1) 푑푓 (푥) 푑푡 , 훼 = 1 (5) The definition of the Caputo derivative (5) is more restrictive than of the RL one (3, 4) since it requires the absolute integrability of the derivative 푑푓 (푥)/푑푡 [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Caputo fractional derivative can be considered as the regularization in the time origin for the RL derivative [76] 퐷 훼 ★ 푓 (푡) = 퐷 훼 푓 (푡) − 푓 (0+) 푡−훼 Γ(1 − 훼) and satisfies the property of being zero when applied to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Yuan & Agrawal and Singh & Chatterjee suggested the alternative definitions of the Caputo fractional derivative [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The first approach is based on the introduction of the auxiliary bivariate function 휙 : (0, ∞) × [푎, 푏] → R as 휙(푤, 푥) = (−1) ⌊푛⌋ 2 sin 휋푛 휋 푤2푛−2⌈푛⌉+1 ∫ 푥 푎 푓 ( ⌈푛⌉) (휏)푒−(푥−휏)푤2푑휏 where ⌈ ⌉ denote the ceiling function, and, finally 퐷푛 ★푎 푓 (푥) = ∞ ∫ 0 휙(푤, 푥)푑푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The second approach is based on expressing the fractional derivative of the the given function in the form of the integral over (0, ∞) with the integrand that can be obtained as the solution of the first-order initial value problem 휕휙★(푤, 푥) 휕푥 = −푤 1 푛−⌈푛⌉−1 휙★(푤, 푥) + (−1) ⌊푛⌋ sin 휋푛 휋(푛 − ⌈푛⌉ − 1) 푓 ( ⌈푛⌉) (푥) with the initial condition 휙★(푤, 푎) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Thus 휙★(푤, 푥) = (−1) ⌊푛⌋ sin 휋푛 휋(푛 − ⌈푛⌉ − 1) 푥 ∫ 0 푓 ( ⌈푛⌉) (휏) exp(−(푥 − 휏)푤 1 푛−⌈푛⌉−1)푑휏 6 퐷푛 ★푎 푓 (푥) = ∞ ∫ 0 휙★(푤, 푥)푑푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='4 Matrix Approach Operations of the fractional calculus can be expressed by matrix [182, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', the left-sided RL or Caputo derivative can be approximated in the nodes in the equidistant discretization net with the help of the upper triangular strip matrix 퐵(훼) 푛 as [182] � 푣(훼) 푛 푣(훼) 푛−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 푣(훼) 1 푣(훼) 0 �푇 = 퐵(훼) 푛 [푣푛 푣푛−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 푣1 푣0]푇 where 퐵(훼) 푛 = 1 휏훼 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 휔(훼) 0 휔(훼) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 휔(훼) 푛−1 휔(훼) 푛 0 휔(훼) 0 휔(훼) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 휔(훼) 푛−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 휔(훼) 0 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb Similarily, the right-hand RL or Caputo fractional derivative can be approximated with the help of the corresponding lower triangular strip matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='5 Caputo & Fabrizio Fractional Derivatives Caputo & Fabrizio [26, 27] proposed the fractional derivatives without the singular kernel [142] by replacing the kernel (푡 − 휏)−훼 with the function exp(−훼/(1 − 훼)) that does not have singularity for 푡 = 휏 in the definition of the Caputo derivative and replacing the factor 1/Γ(1 − 훼) with 푀(훼)/(1 − 훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', the fractional time derivative for 훼 ∈ [0, 1] and function 푓 ∈ 퐿1(−∞, 푏) is D 훼 푡 푓 (푡) = 훼푀(훼) 1 − 훼 푡 ∫ −∞ ( 푓 (푡) − 푓 (휏)) exp � −훼(푡 − 휏) 1 − 훼 � 푑휏 where 푀(훼) is a normalization function such as 푀(0) = 푀(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='6 GC & GRL derivatives Zhao & Luo [264] suggested to divide the fractional derivativewith different — singular and non-singular — kernels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', RL, Caputo, Caputo-Fabrizio, Atangana-Baleanu [11]1 with the kernel 푘(푥, 훼) = 퐸훼 � − 훼 1 − 훼푥 � , Atangana-Gomez [12] with the kernel 푘(푥, 훼) = exp � − 훼 1 − 훼푥2� 1The equation with the Atangana-Baleanu operator is related to the derivatives of distributed order [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 7 derivative with the stretched exponential kernel [210] (that is useful in the study of the water diffusion in the human brain using the magnetic resonance imaging [19]) 푘(푥, 훼) = exp � − 훼 1 − 훼푥훽� , 훽 > 0, 훽 ≠ 1) into two classes — GC (general, Caputo sense) and GRL (general, RL) derivatives that obeys the the principles formulated by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Volterra in his "general laws of heredity" [228]: the linearity principle, the invariance principle, the fading memory principle, the compatibility principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The compatibility principle requires the validity of two limits: 퐷 훼 푓 (푥) → 푓 (푥) when 훼 → 0 and 퐷 훼 푓 (푥) → 푓 ′(푥) when 훼 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The principle of nonlocality was suggested by Tarasov [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 GC derivatives Zhao & Luo [264] defined the GC derivative by 퐷퐺퐶 푎,훼 푓 (푥) = 푁(훼) 푥 ∫ 푎 푘(푥 − 푡, 훼) 푑푓 (푡) 푑푡 푑푡 The fading memory principle requires that the remote time and position has less effect: 푘(푥 − 푡, 훼) decreases when 푥 increases and 푘(푥 − 푡, 훼) → 0 when 푥 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The compatibility principle requires that 푁(훼)/푘(푥, 훼) → 1 when 훼 → 0 and 푁(훼)/푘(푥, 훼) → 훿(푥) when 훼 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 GRL derivatives 퐷푅퐿 푎,훼 푓 (푥) = 푑 푑푥 푁(훼) 푥 ∫ 푎 푘(푥 − 푡, 훼) 푓 (푡)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The restrictions on 푘(푥 − 푡, 훼) and 푁(훼) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='7 Marchaud-Hadamard Fractional Derivatives Marchaud’s approach is based on the analytic coninuation of the fractional integrals to the negative orders using the Hadamard’s finite parts of the divergent integrals (Hadamard’s idea is to ignore the unbounded contribution to the integral and to assign the value of the remaining — finite — expression [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Marchaud fractional derivative with the lower limit 푎 is (푀 훼 푎+ 푓 )(푥) = 푓 (푥) Γ(1 − 훼)(푥 − 푎) 훼 + 훼 Γ(1 − 훼) 푥 ∫ 푎 푓 (푥) − 푓 (푦) (푥 − 푦) 훼+1 푑푦 and with the upper limit 푏 is (푀 훼 푏− 푓 )(푥) = 푓 (푥) Γ(1 − 훼)(푏 − 푥) 훼 + 훼 Γ(1 − 훼) 푏 ∫ 푥 푓 (푥) − 푓 (푦) (푥 − 푦) 훼+1 푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 8 Marchard’s method is to extend the RL integral to 훼 < 0 (퐽−훼 + 푓 )(푥) = 1 Γ(−훼) ∞ ∫ 0 푦−훼−1 푓 (푥 − 푦)푑푦 (6) and to substract the divergent part of the integral in (6) ∞ ∫ 휖 푦−훼−1 푓 (푥 − 푦)푑푦 = 푓 (푥) 훼휖 휖 to get finally (푀 훼 + 푓 )(푥) = lim 휖 →0+ 1) Γ(−훼) ∞ ∫ 휖 푓 (푥) − 푓 (푦) 푦훼+1 푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (7) There are two approaches to extend the definition (7) to the case 훼 > 1 [99]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' To apply (7) to the 푛th derivative 푑푛 푓 /푑푥푛 for 푛 < 훼 < 푛 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' To consider 푓 (푥 − 푦) − 푓 (푥) as the first-order difference and to generalize to 푛th order difference (difference quotient) (Δ푛 ℎ 푓 )(푥) = (I − 푇ℎ)푛 푓 (푥) = 푛 � 푘=0 (−1)푘 �푛 푘 � 푓 (푥 − 푘ℎ) (8) where I is the identity operator and 푇ℎ = 푓 (푥 − ℎ) is the translation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Thus Marchard fractional derivative for 0 < 훼 < 푛 is written as (푀 훼 + 푓 )(푥) = lim 휖 →0+ 1 퐶훼,푛 ∞ ∫ 휖 Δ푛 푦 푓 (푥) 푦훼+1 푑푦 where 퐶훼,푛 = ∞ ∫ 0 (1 − 푒−푦)푛 푦훼+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='8 Grünwald - Letnikov Derivative The approach suggested independently by Grünwald in 1867 and Letnikov [132] in 1868 is based on the use the limits of the difference quotients (8) similar to the classical definition of the derivatives for 푛 ∈ N, 푓 ∈ 퐶푛[푎, 푏], 푎 < 푥 ≤ 푏 ˜퐷푛 푓 (푥) = lim ℎ→0 Δ푛 ℎ 푓 )(푥) ℎ푛 and extension to the case of the arbitrary 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 9 Since �푛 푘 � = 0 if 푛 ∈ N and 푛 < 푘 the expression (8) is equivalent to (Δ푛 ℎ 푓 )(푥) = ∞ � 푘=0 (−1)푘 �푛 푘 � 푓 (푥 − 푘ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (9) The series (9) is uniformly convergent for any bounded function if 푛 > 0 [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The use of (9) introduce two problems [50]: the function 푓 needs to be defined on (∞, 푏];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the function 푓 should be such that the series converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' These problems are solved by the introduction a new function 푓 ★ 푓 ★ = � 푓 (푥) 푥 ∈ [푎, 푏] 0 푥 ∈ (−∞, 푎) that is used instead of the original 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' It is also assumed that in the tending to zero ℎ takes only the Grünwald-Letnikov fractional derivative of order 푛 defined as ˜퐷푛 푎 = lim 푁→∞ Δ푛 ℎ푁 푓 (푥) ℎ푛 푁 = lim 푁→∞ 푁 � 푘=0 (−1)푘 �푛 푘 � 푓 (푥 − 푘ℎ푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (10) The Grünwald-Letnikov derivative is called pointwise or strong depending on whether the limit is taken pointwise or in the norm of a suitable Banach space [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The binomial coefficient can be generalized to the non-integer arguments (−1) 푗 �푞 푗 � = (−1) 푗 Γ(푞 + 1) Γ( 푗 + 1)Γ(푞 − 푗 + 1) = Γ( 푗 − 푞) Γ(−푞)Γ( 푗 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The (left-sided) Grünwald-Letnikov derivative could be written as (푛ℎ = 푥 − 푎) ˜퐷 훼 푎 푓 (푥) = lim ℎ→0 1 ℎ훼 ⌊푛⌋ � 푘=0 (−1)푘 Γ(훼 + 1) 푓 (푥 − 푘ℎ) Γ(푘 + 1)Γ(훼 − 푘 + 1) and right-sided (푛ℎ = 푏 − 푥) as ˜퐷 훼 푏 푓 (푥) = lim ℎ→0 1 ℎ훼 ⌊푛⌋ � 푘=0 (−1)푘 Γ(훼 + 1) 푓 (푥 + 푘ℎ) Γ(푘 + 1)Γ(훼 − 푘 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Grünwald-Letnikov integral of the order 푛 of the function 푓 is written as ˜퐽푛푎 푓 (푥) = 1 Γ(푛) lim 푁→∞ ℎ푛 푁 푁 � 푘=0 Γ(푛 + 푘) Γ(푘 + 1) 푓 (푥 − 푘ℎ푁 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='9 Riesz Fractional Operators The fractional integral of the order 훼 in the Riesz sense (also known as the Riesz potential) is defined by the Fourier convolution product (I 훼 푓 )(풙) = ∫ R푛 푲 훼(풙 − 흃) 푓 (흃)푑흃, 10 where 푅푒(훼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Riesz kernel 푲 훼 = 1 훾푛(훼) � ∥풙∥훼−푛, 훼 − 푛 ≠ 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , ∥풙∥훼−푛 ln � 1 ∥풙 ∥ � , 훼 − 푛 = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' where 훾푛(훼) is defined by 훾푛(훼) 2훼휋 푛 2 Γ(훼/2) = ��Γ � 푛−훼 2 ��−1 , 훼 − 푛 ≠ 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , (−1) 푛−훼 2 2−1Γ � 훼−푛 2 � , 훼 − 푛 = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Riesz fractional integral is [82] (I훼 푓 )(풙) = Γ � 1−훼 2 � 2훼휋 푛 2 Γ(훼/2) ∞ ∫ −∞ 푓 (휉)|푥 − 휉|훼−1푑휉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Riesz fractional derivative is [43] 퐷 훼[ 푓 (푥)] = − 1 2 cos(훼휋/2) 1 Γ(훼) 푑푛 푑푥푛 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 푥 ∫ −∞ (푥 − 휉)푛−훼푛−1 푓 (휉)푑휉 + ∞ ∫ 푥 (푥 − 휉)푛−훼푛−1 푓 (휉)푑휉 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Riesz derivative is the generalization of the Laplace operator [254] (−Δ) 훼 2 = − 1 2 cos(훼휋/2) � 푑 훼 푑푥 훼 + 푑 훼 푑(−푥) 훼 � , 훼 ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Riesz derivative could be expressed in terms of the Marchaud derivative 퐷 훼[ 푓 (푥)] = − 1 2 cos(훼휋/2) [(푀 훼 + 푓 )(푥) + (푀 훼 − 푓 )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The related Riesz-Feller derivative [78] has an additional parameter - "skewness" 휃 퐷 훼 휃 푓 (푥) = Γ(1 + 훼) 휋 × \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 sin � (훼 + 휃) 휋 2 � ∞ ∫ 0 푓 (푥 + 휉) 푓 (푥) 휉1+훼 푑휉 + sin � (훼 − 휃) 휋 2 � ∞ ∫ 0 푓 (푥 − 휉) 푓 (푥) 휉1+훼 푑휉 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The allowed region of the parameters 훼 and 휃 turn out to be a diamond in the plane {훼, 휃} with the vertices in the points (0,0), (1,1), (2,0), (1, -1) called the "Feller-Takayasu diamond" [76, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='10 Weyl Fractional Derivative The Weyl derivative is based on the generalization of the differentiation in the Fourier space [69] — the integer derivative of the 푛th order (푖푘)푛 of the absolutely integrable function on [−휋, 휋] presented as the Fourier series is extended to the noninteger 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Weyl fractional derivative is defined as [148] 퐷 훼 ± = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 ± 푑 푑푥 [퐼1−훼 ± 푓 (푥)] 0 < 훼 < 1, 푑2 푑푥2 [퐼2−훼 ± 푓 (푥)] 1 < 훼 < 2, where the Weyl fractional integrals are (휇 > 0) 퐼 휇 + = 1 Γ(휇) 푥 ∫ −∞ (푥 − 휒)휇−1 푓 (휒)푑휒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='11 Erdélye-Kober Fractional Operators The Erdélye-Kober integral for a well-behaved function 휙(푡) is defined as [143, 178] 퐼훾,휇 휂 휙(푡) = 휂 Γ(휇) 푡−휂(휇+훾) 푡 ∫ 0 휏휂(훾+1)−1(푡휂 − 휏휂)휇−1휙(휏)푑휏, where 휇 > 0, 휂 > 0, 훾 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' In the special case 훾 = 0, 휂 = 1 the Erdélye-Kober fractional integral is related to the RL fractional integral of the order 휇 as 퐼0,휇 1 휙(푡) = 푡−휇 Γ(휇) 푡 ∫ 0 (푡 − 휏)휇−1휙(휏)푑휏 = 푡−휇퐽 휇휙(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Erdélye-Kober fractional derivative for 푛 − 1 < 휇 < 푛, 푛 ∈ N is defined as 퐷훾,휇 휂 휙(푡) = 푛 � 푗=1 � 훾 + 푗 + 1 휂 푡 푑 푑푡 � (퐼훾+휇,푛−휇 휂 휙(푡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Erdélye-Koberfractional derivative reduces to the identity operator when 휇 = 0 퐷훾,0 휂 휙(푡) = 휙(푡) and for 휂 = 1 and 훾 = −휇 is related to the RL fractional derivative as 퐷훾,휇 휂 휙(푡) = 푡휇퐷휇 푅퐿휙(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='12 Interpretation of Fractional Integral and Derivatives The integer-orderand integrals have a clear physiscal and geometricalinterpretation that simplify their use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The numerous different interpretations of the fractional derivatives and integrals have been proposed [100]: the probabilistic [208, 222, 221], geometric [18, 17, 162, 40], physical interpretations [162, 40, 169, 194, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' However, as noted Podlubny [183], "since the appearance of the idea of differ- entiation and of arbitrary (not necessary integer) order there was not any acceptable geometric and physical interpretation of these operations for more than 300 years".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Teneiro Machado [221] wrote the Günwald-Letnikov derivative of 푥(푡) as 퐷 훼[푥(푡)] = lim ℎ→0 � 1 ℎ훼 ∞ � 푘=0 훾(훼, 푘)푥(푡 − 푘ℎ) � , 훾(훼, 푘) = (−1)푘 Γ(훼 + 1) 푘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='Γ(훼 − 푘 + 1) where ℎ is the time increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The author noted that 훾(훼, 0) = 1, − ∞ � 푘=0 훾(훼, 푘) = 1 thus the "present" (P) is constituted by 푥(푡) the probability 1 while the totality of the "past/future"(PF) is constituted by the samples 푥(푡), 푥(푡−ℎ), 푥(푡−2ℎ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' each sample is weighted with a probability −훾(훼, 푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Nigmatullin [169, 170] interpreted the fractional integral in terms of the fractal Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The author considered the evolution of the state of the physical system 퐽(푡) = 푡 ∫ 0 퐾(푡, 휏) 푓 (푡)푑휏 where the memoryfunction 퐾(푡, 휏) 푓 (푡) accountsfor the loss of some states of th system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the fractional index of integration equals the fractal dimension of the Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Podlubly [183] and Podlubny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [185] suggested the geometrical interpreation of the left-sided (equation (1)) and right-sided (equation (2)) RL integrals and of the RL (equations (3) - (4)) and the Caputo (equation (5)) derivatives, as well as of the Riesz potential that is the sum of the left-sided and right-sided RL fractional integrals 푅훼 푏 푓 (푥) = 1 Γ(훼) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 푥 ∫ 푎 (푥 − 푡) 훼−1 푓 (푡)푑푡 + 푏 ∫ 푥 (푡 − 푥) 훼−1 푓 (푡)푑푡 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb and of the Feller potential Φ훼 푓 (푥) = 푐퐽 훼 푎+ 푓 (푥) + 푑퐽 훼 푏− 푓 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The geometric interpretation by Podlubny is based on additing the third dimension 푔푥(푡) = 1 Γ(훼 + 1) [푥 훼 + (푥 − 푡) 훼] 13 to the pair (t, f (x)) and considering the three-dimensional line (푡, 푔푥(푡), 푓 (푡)) as the top edge of the "fence" that gives shadow on the wall in the (g,f) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Tarasov [57] proposed the ”informatic” (”computer science”) interpretation of the RL and the Caputo derivatives of the non-integer orders using the reconstructions from the infinite sequence of the derivatives of the integer orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Such reconstructions atre based on the Kotel’nikov theorem (also known as the sampling theorem) proved by Vladimir Kotel’nikov in 1933 and also by Claude Shannon 1949: under the certain restrictive conditions, function 푓 (푡) can be restored from its samples 푓 [푛] = 푓 (푛푇) according to the Whittaker-Shannon interpolation formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The author stressed that infinity of the sequences of the integer derivatives plays a fundamental role in represen- tation of the fractional derivatives that describe nonlocality and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Gómez-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [70] analysed the Caputo differentiation using the RC circuit for which the fractional version of the Ohm’s law and Kirchhoff’s law are written as 푣(푡) = 1 휎1−훾 푑훾푞 푑푡훾 , 푅 푑푞 푑푡 + 1 퐶 푞(푡) = 푣(푡) where 푞 is the electric charge, 푣 is the voltage, 푅 is the resistance of the conductor, 퐶 is the capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The parameter 휎 is introduced in order to be consistent with the dimensionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' it characterizes the fractional structures (the components that show the intermediate behaviour between conservative (capacitor) and dissipative (resistor) [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The authors derived the fractional differential equation for the RC circuit 푑훾 푑푡훾 + 1 휏훾 푞(푡) = 퐶 휏훾 푣(푡), 휏훾 = 푅퐶 휎1−훾 where 휏훾 is the time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Gómez-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' claimed that the differentiation is related to the memory effects that reflect the intrinsic dissipation in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Sierociuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [204] used the RC network to model the fractional order diffusion based on the analogy between the heat and electrical conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The authors showed that the equations for the capacitor and for the resistor in the transmission line could be used to get the diffusion equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the loosing of heat was represented by the additional resistors connected parallel to capacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Carpinteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [29] considered the mechanical interpretation of the Marchaud fractional derivative using the body springs connecting the non-adjacent points of the body with the stiffness decaying with the distance between the material points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='13 Local Fractional Derivatives The fractional derivatives are nonlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Several researches introduced the local variants [259] that are useful for study of the pointwise behaviour of the fractal and multifractal functions that describe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', the stress and deformation patterns in materials exhibiting the fractal-like microstructure [29] or the velocity field of turbulent fluid [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Kolwankar & Gangal [127, 128, 129, 126] defined the derivative via the RL deriva- tive as 픇푞 푓 (푦) = lim 푥→푦 퐷푞( 푓 (푥) − 푓 (푦)) (푥 − 푦)푞 if the limit exists and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 14 The local fractional Taylor formula is written as [242] 푓 (푥) = 푛 � 푖=0 푓 (푛) (푦) Γ(1 + 푛) (푥 − 푦)푛 픇훼 Γ(푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' + 훼) (푥 − 푦) 훼 + 푅훼(푥, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [249, 266] used similar definition 픇(푘) 푓 (휏) = lim 휏→휏0 푓 (휏) − 푓 (휏0) 휏푘 − 휏푘 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [39] proposed the local derivatives based on the integrals of the difference-quotient (IDQ) or the singular of difference-quotient (SIDQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' For example, the right SIDQ local derivative is 픇훼 푓 (푦) = 1 Γ(1 − 훼) lim ℎ→0+ 1 ∫ 0 (1 − 푡)−훼 푓 (푡ℎ + 푦) − 푓 (푦) ℎ훼 푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The local fractional derivative is essentially the fractal derivative [36, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' In contrast to the purely analytical approach of the fractional calculus, the fractal calculus follows the physical-geometric approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' to avoid confusion it is suggested to call the latter the scaled calculus [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractal ("Hausdorff") derivative on the time fractal is defined as [92] 휕 푓 휕푡휎 = lim 푡퐵→푡퐴 푓 (푡퐵) − 푓 (푡퐴) (푡퐵) 휎 − (푡퐴) 휎 where 휎 is the fractal dimension of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' A more general definition is formulated as [37, 10] 휕 휏 푓 휕푡휎 = lim 푡퐵→푡퐴 푓 휏(푡퐵) − 푓 휏(푡퐴) (푡퐵) 휎 − (푡퐴) 휎 Since the fractal derivative is the local operator, the numerical solution of the fractal derivative equations can be performed by the standard numerical techniques for the integer-order differential equations [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The similar properties have the so called "conformable" fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 "Conformable" Fractional Derivative Most fractional derivatives do not have the desirable properties [2, 118, 117]: the derivative of a constant is not zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' they do not obey the product rule 퐷 훼( 푓 푔) = 푓 퐷 훼(푔) + 푔퐷 훼 푓 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' they do not obey the quotient rule 퐷 훼( 푓 /푔) = (푔퐷 훼( 푓 ) − 푓 퐷 훼(푔))/푔2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' they do not obey the chain rule 퐷 훼( 푓 푔) = 푓 훼(푔(푡)푔훼(푡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' they do not obey in general 퐷 훼퐷훽 푓 = 퐷 훼+훽 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [118] and Katugampola [117, 116] suggested the so called "conformable" limit based [7] derivatives 퐷 훼 푓 (푡) = lim 휖 →0 푓 (푡 + 휖푡1−훼) − 푓 (푡) 휖 , 0 < 훼 < 1, 15 and 퐷 훼 푓 (푡) = lim 휖 →0 푓 (푡푒휖 푡−훼) − 푓 (푡) 휖 , 0 < 훼 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Since the conformable derivative is the extension of the classical derivative defi- nition, this derivative obeys the product rule, the quotient rule, the linearity property, the zero derivative for the constant and are valid for some extensions of the classical calculus such as the Rolle’s Theorem or Mean Value Theorem [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 2 Tempered Fractional Calculus Sabzikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [195] suggested a variant of the fractional calculus where power laws are tempered by the exponential factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The random walks model with the exponentially tempered power law jumps converges to a tempered stable motion [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' This tempered fractional diffusion is useful in the geophysical [157, 263] and financial [30] problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The authors considered two intervals for the parameter 훼: 0 < 훼 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The tempered fractional derivative 휕 훼,휆 푥 is defined as the function with the Fourier transform [(휆 + 푖푘) 훼 − 휆훼] ˆ푓 (푘) that in real space is written as 휕 훼,휆 푥 푓 (푥) = 훼 Γ(1 − 훼) ∞ ∫ 0 ( 푓 (푥) − 푓 (푥 − 푦))푒−휆푦푦−훼−1푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The negative tempered fractional derivative 휕 훼,휆 −푥 is defined as the function with the Fourier transform [(휆 − 푖푘) 훼 − 휆훼] ˆ푓 (푘) that in real space is written as 휕 훼,휆 −푥 푓 (푥) = 훼 Γ(1 − 훼) ∞ ∫ 0 ( 푓 (푥) − 푓 (푥 + 푦))푒−휆푦푦−훼−1푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 1 < 훼 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The tempered fractional derivative 휕 훼,휆 푥 is defined as the function with the Fourier transform [(휆 + 푖푘) 훼 − 휆훼 − 푖푘훼휆훼−1] ˆ푓 (푘) that in real space is 휕 훼,휆 푥 푓 (푥) = 훼(1 − 훼) Γ(2 − 훼) ∞ ∫ 0 ( 푓 (푥 − 푦) − 푓 (푥) + 푦 푓 ′(푥))푒−휆푦푦−훼−1푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The negative tempered fractional derivative 휕 훼,휆 푥 is defined as the function with the Fourier transform [(휆 − 푖푘) 훼 − 휆훼 + 푖푘훼휆훼−1] ˆ푓 (푘) that in real space is 휕 훼,휆 −푥 푓 (푥) = 훼(1 − 훼) Γ(2 − 훼) ∞ ∫ 0 ( 푓 (푥 + 푦) − 푓 (푥) − 푦 푓 ′(푥))푒−휆푦푦−훼−1푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 16 Sabzikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' introduced the positive tempered integral as ℑ훼,휆 + 푓 (푥) = 1 Γ(훼) 푥 ∫ −∞ 푓 (푢)(푥 − 푢) 훼−1푒−휆(푥−푢)푑푢 and the negative tempered integral as ℑ훼,휆 − 푓 (푥) = 1 Γ(훼) 푥 ∫ −∞ 푓 (푢)(푢 − 푥) 훼−1푒−휆(푢−푥)푑푢 called the RL tempered integrals since for 휆 = 0 they reduce to the usual RL integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The authors defined the RL tempered fractional derivatives D 훼,휆 ± as functions with the Fourier transform (휆 ± 푖푘) 훼 ˆ푓 (푘) that can be expressed D 훼,휆 ± 푓 (푥) = � 휕 훼,휆 ±푥 푓 (푥) + 휆훼 푓 (푥) 0 < 훼 < 1 휕 훼,휆 ±푥 푓 (푥) + 휆훼 푓 (푥) ± 훼휆훼−1 푓 ′(푥) 1 < 훼 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Evidently, integration and differentiation are the inverse operators: D 훼,휆 ± ℑ훼,휆 ± 푓 (푥) = 푓 (푥), ℑ훼,휆 ± D 훼,휆 ± 푓 (푥) = 푓 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The integration and differentiation operators have the semigroup property ℑ훼,휆 ± ℑ훽,휆 ± 푓 = ℑ훼+훽,휆 ± 푓 , D 훼,휆 ± D훽,휆 ± 푓 = D 훼+훽,휆 ± 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3 Fractional Differential Equations Generally, the fractal media could not be considered as continuous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The use of the non-integer dimensional spaces [174] is necessary to describe a fractal medium by the continuous models [216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional differential equations [53, 167, 121, 50] are non-local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' could incorporate the effects of the memory and the spatial correlations) and could be formulated in the distinct but mathematically equivalent forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Mainardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [155] compared the fractional extensions of the standard Cauchy problem 휕푢(푥, 푡) 휕푡 = 휕2푢(푥, 푡) 휕푥2 , 푥 ∈ R, t ∈ R+ 0, u(x, 0+) = u0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (11) The fundamental solution (or Green function) of (11), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the solution subjected to the initial condition 푢0(푥) = 훿(푥), is the Gaussian probability density function 푢(푥, 푡) = 1 2√휋 푡−1/2푒−푥2/(4푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Green function has the scaling property 푢(푥, 푡) = 푡1/2푈(푥/푡1/2), 푈(푥) is the reduced Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 17 The Cauchy problem (11) is equivalent to the integro-differential equation 푢(푥, 푡) = 푢0(푥) + 푡 ∫ 0 � 휕2푢(푥, 휏) 휕푥2 � 푑휏 where the initial condition is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional diffusion equation could be written with the use of the RL derivative 퐷1−훽 (훽 is the real number 0 < 훽 < 1) 휕푢(푥,푡) 휕푡 = 퐷1−훽 휕2푢(푥, 푡) 휕푥2 (12) or the Caputo derivative 퐷훽 ★ 퐷훽 ★푢(푥, 푡) = 휕2푢(푥, 푡) 휕푥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (13) The equations (12) and (13) are equivalent to the equation based on the use of the RL fractional integral of the order 훽 푢(푥, 푡) = 푢0(푥) + 퐽훽 � 휕2푢(푥, 휏) 휕푥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (14) Note that the equation (12) could be obtained by differentiating (14), the equation (14) can be derived by the fractional integration of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The equation (12) was studied by Metzler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [158] and by Saichev & Zaslavsky [196], the equation (14) by Gorenflo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [80, 79] and by Mainardi [146, 147], the integrodifferentialequation (14) by Schneider & Wyss [200] using the Mellin transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Mainardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [155] search for the fundamental solution of the equation (13) by applying the sequence of the Fourier F {푣(푥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 푘} = ˆ푣(푘) = ∞ ∫ −∞ 푒푖푘푥푣(푥)푑푥, 푘 ∈ R and the Laplace L{푤(푡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 푠} = ˜푤(푠) = ∞ ∫ 0 푒−푠푡푤(푡)푑푡, 푠 ∈ C transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Thus the Green function in the Fourier-Laplace domain is determined by ˆ˜푢(푘, 푠) = 푠훽−1 푠훽 + 푘2 , 0 < 훽 ≤ 1, R(푠) > 0, 푘 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (15) There are two strategies to determine the Green function in the space-time domain 푢(푥, 푡) related to the order in performing inversions in the expression (15) [155]: 18 1) Invert the Fourier transform to get ˜푢(푥, 푠) and then invert the Laplace transform [146, 147] or 2) invert the Laplace transform to get ˆ푢(푘, 푡) and then invert the Fourier transform [81, 151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Nieto [168] studied the linear fractional differential equation with the spatial RL derivative for initial or periodic boundary conditions and derived the maximumprinciple using the properties of the Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Compte [42] and West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [235] studied the equation for the hyperdiffusion (Lévy-flight diffusion) 휕푃 휕푡 = 퐷(−Δ) 훾 2 where the fractional 푛-dimensional Laplace operator (−Δ) 훾 2 is defined by its Fourier transform with respect to the spatial variable [53] F [(−Δ) 훾 2 푔(푥)] = |휔|훾F [푔(푥)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Luchko [144] derived the maximum principle fortheinitial-boundary-valueproblem for the time-fractional diffusion equation with Caputo derivative over the open bounded domain 퐺 × (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='푇), 퐺 ⊂ 푅푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The equation could subjected to the complex transformation [137, 94, 138] 푠 = 푥푆/Γ(1 + 훼) to convert to a partial differential equation2 For example, the heat conduc- tion equation (훼 is the fractal dimension of the fractal medium) 휕푇 휕푡 = 휕 훼 휕푥 훼 � 휆 휕 훼푇 휕푥 훼 � is converted into the equation 휕푇 휕푡 = 휕 휕푠 � 휆 휕푇 휕푠 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' That could be further transformed by introduction of the Boltzmann variable [140] 휒 = 푠/√푡 = 푥 훼/√푡Γ(1 + 훼) into the ordinary differential equation 푑 푑휒 � 휆 푑푇 푑휒 � + 휒 2 푑푇 푑휒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' For the general fractional differential equation in the Jumarie’s modification of the RL derivatives 푓 (푢, 푢훼 푡 , 푢훽 푥, 푢훾 푦, 푢휆 푧, 푢2훼 푡 , 푢2훽 푥 , 푢2훾 푦 , 푢2휆 푧 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' ) = 0 He & Li [95] suggested the following transforms 푠 = 푞푡훼 Γ(1 + 훼) , 푋 = 푝푥훽 Γ(1 + 훽) , 푌 = 푘푦훾 Γ(1 + 훾) , 푍 = 푙푧휆 Γ(1 + 휆) 2Such transformation is possible in the multidimensional case if the variables 푡, 푥, 푦, 푧 obey the following constraint [94] (푞, 푝, 푘, 푙 are constants) 휉 = 푞푡 훼 Γ(1 + 훼) + 푝푥훽 Γ(1 + 훽) + 푘푦훾 Γ(1 + 훾) + 푙푧휆 Γ(1 + 휆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 19 thus converting the fractional derivatives into classical derivatives 휕 훼푢 휕푡훼 = 푞 휕푢 휕푠 , 휕훽푢 휕푥훽 = 푝 휕푢 휕푋 , 휕훾푢 휕푦훾 = 푘 휕푢 휕푌 , 휕휆푢 휕푧휆 = 푙 휕푢 휕푍 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Babusci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [14] discussed relations between the differential equations and the theories of the pseudo-operators [220, 202] and the generalized integral transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Distributed order differential equations The distributed order differential equations (DODE) are a special class of the fractional differential equations [165, 253, 33, 34, 35, 125, 24, 77, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Chechkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [32] discussed the natural and the modifies forms of DODEs and noted that the latter in com- bination with the continuity equation and the retarded linear response equation for the flux exhibiting memory of the processes at the previous times admits a thermodynamic interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' DODEs are used to describe the accelerating subdiffusion, decelerating superdiffusion or transformation of the anomalous behaviour at the short times into the normal behaviour at the long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' For example, Metzler & Klafter [159] consid- ered the DODE for the description of the ultraslow diffusion with the logarithmic time dependence �푥2(푡)� ∝ log푘 푡 including the so called Sinai diffusion (푘 = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The concept of the distributed order differentiation is close to the variable order fractional operators that are useful for the study of the viscoelasticity, the reaction kinetics of proteins, the electrorheological fluids, the damage modelling [71, 41, 197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' There are two approaches to the formulation of the distributed order differential equations: 1) direct — a new variable does not assigned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 2) Independent variable approach — the order is considered as a function of some independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Mainardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [155] studied the fractional diffusion equation of distributed order 1 ∫ 0 푏(훽)[퐷훽푢(푥, 푡)]푑훽 = 휕2푢(푥, 푡) 휕푥2 , 푏(훽) ≥ 0, 1 ∫ 0 푏(훽)푑훽 = 1 (16) with 푥 ∈ R, 푡 ≥ 0 and the initial condition 푢(푥, 0+) = 훿(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The weight function 푏(훽) is called the order-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The authors used the Fourier and Laplace transforms to get the fundamental solution similar to a single-order case (15) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 ∫ 0 푏(훽)푠훽푑훽 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ˆ˜푢(푘, 푠) − 1 ∫ 0 푏(훽)푠훽−1푑훽 = −푘2 ˆ˜푢(푘, 푠) and ˆ˜푢(푘, 푠) = 퐵(푠)/푠 퐵(푠) + 푘2 , 푘 ∈ R, B(s) = 1 ∫ 0 b(훽)s훽d훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (17) In the case of small 푘 the equation (17) can be approximated as ˆ˜푢(푘, 푠) = 1 푠 � 1 − 푘2 퐵(푠) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' � 20 and the second moment is written as ˜휇2(푠) = − 휕2 휕푘2 ˆ˜푢(푘 = 0, 푠) = 2 퐵(푠) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (18) The special case of DODEs are the double-order fractional equations [32] 푏(훽) = 푏1훿(훽−훽1)+푏2훿(훽−훽2), 0 < 훽1 < 훽2 ≤ 1, 훽1 > 0, 훽2 > 0, 훽1+훽2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Asymptotic behaviour of 휇2(푡) follows from (18) for cases of the slow diffu- sion (the power-law growth, 푏(훽) = 푏1훿(훽 − 훽1) + 푏2훿(훽 − 훽2)) where ˜휇2(푠) = 2/(푏1푠훽1+1 + 푏2푠훽2+1) and the ultra-slow diffusion (the logarithmic growth, 푏(훽) = 1) with ˜휇2(푠) = 2ln 푠/푠(푠 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The distributed order equations allow to describe the more complex media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The time-fractional diffusion equation of the distributed order (16) is potentially more flexible to represent the local phenomena while the space-fractional diffusion equation of the distributive order is more suited to represent the variations in space [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 Special Functions There are special functions related to the differential equation similar to the classical case (such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=', the Bessel and the cylindrical functions, the classical orthogonal polynomials, Airy functions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=') [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The most important functions in the fractional calculus are the Mittag-Leffler function [86], the H-functions [98, 156, 124], the Wright functions [154, 152], the generalized Lommel-Write functions [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Mittag- Leffler function is even called the "Queen"-function of the fractional calculus [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Mittag-Leffler Functions The eigenfunction of the RL derivatives are the solutions of the equation [99] 퐷 훼 0+[ 푓 (푥)] = 휆 푓 (푥) where 휆 is the eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The eigenfunctions are 푓 (푥) = 푥1−훼퐸훼,훼(휆푥 훼) where 퐸훼,훽 = ∞ � 푘=0 푥퐾 Γ(훼푘 + 훽) (19) is the generalized Mittag-Leffler function (also called the Wiman’s function [203]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The more general eigenvalue equation for derivatives of the orders 훼 and 훽 is 퐷 훼,훽 0+ [ 푓 (푥)] = 휆 푓 (푥) The solution is [99] 푓 (푥) = 푥 (1−훽) (1−훼)퐸훼,훼+훽(휆푥 훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The special case is the equation 퐷 훼,1 0+ [ 푓 (푥)] = 휆 푓 (푥) with eigenfunction 푓 (푥) = 퐸훼(휆푥 훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The one-parameter Mittag-Lefler function is the particular case of (19) for 훽 = 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 21 Evidently [50], 퐸0,1(푥) = ∞ � 푘=0 1 Γ(1) = ∞ � 푘=0 푥푘 = 1 1 − 푥 , 퐸1(푥) = ∞ � 푘=0 푥푘 푘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' = exp(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' There are other special cases such as [86] 퐸2(−푥2) = 퐸2,1(−푥2) = cos(푥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 퐸2(푥2) = 퐸2,1(푥2) = cosh(푥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' for 푥 > 0 퐸1/2(푥1/2) = 퐸1/2(푥1/2) = (1 + 푒푟 푓 (푥)) exp(푥2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' for 푥 ∈ C and 푟 ∈ N 퐸1,푟 = 1 푥푟−1 � exp(푥) − 푟−2 � 푘=0 푥푘 푘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 퐸3(푥) = 1/2[푒푥1/3 + 2푒−1/2푥1/3 cos( √ 3/2푥1/3)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 퐸4(푥) = 1/2[cos(푥1/4) + cosh(푥1/4)] where 푒푟 푓 (푥) = 2/√휋 ∫ 푥 0 exp(−푡2)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Mittag-Leffler function 퐸1 satisfies the functional relation [50, 86] 퐸1(푥 − 푦) = 퐸1(푥)/퐸1(푦) and the relation between two Mittag-Leffler functions with different parameters 퐸푛1,푛2 (푥) = 푥퐸푛1,푛1+푛2 (푥) + 1/Γ(푛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Note that the frequently used relation 퐸훼(푎(푡 + 푠) 훼) = 퐸훼(푎푡훼)퐸훼(푎푠훼), 푡, 푠 ≥ 1 is valid only if 훼 = 0 or 훼 = 1 [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Asymptotic expansions and integral representations of the Mittag-Leffler functions could be found in the papers [74, 71, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Prabhakar [186] suggested the extension 퐸 훾 훼,훽(푥) = ∞ � 푛=0 (훾)푛 Γ(훼푛 + 훽) 푥푛 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푅푒(훼) > 0, 푅푒(훽) > 0 (훾)푛 is the Pochhammer symbol [203] (훾)0 = 1, (훾)푛 = 훾(훾 + 1)(훾 + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (훾 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The extension to the multi-index Mittag-Leffler functions [123, 124] 퐸( 1 휌푖 ),(휇푖) (푥) = ∞ � 푘=0 푥푘 Γ(휇1 + 푘/휌1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Γ(휇푚 + 푘/휌푚) is performed by replacing of the indices 훼 = 1/휌 and 훽 = 휇 by two sets of multi-indices 훼 → (1/휌1, 1/휌2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 1/휌푚) and 훽 → (휇1, 휇2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 휇푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' There are a couple of related functions [166] Barret’s function 푈(푥, 휆) = ∞ � 푘=1 휆푘−1푥푘훼푖 Γ(푘훼 − 푖 + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Rabotnov’s (fractional exponential) function [189, 21] E훼(훽, 푥) = 푥 훼 ∞ � 푛=0 훽푛푥푛(훼+1) Γ((푛 + 1)(1 + 훼)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 H Functions The H-function of order (푚, 푛, 푝, 푞) ∈ N4 is defined via the Mellin-Barnes type contour integral [53, 156] 퐻푚,푛 푝,푞 (푧) = 1 2휋푖 ∫ L H 푚,푛 푝,푞 푧푠푑푠 where 푧푠 = 푒푥푝[푠(푙푛|푧| + 푖푎푟푔푧)], H 푚,푛 푝,푞 = 퐴(푠)퐵(푠) 퐶(푠)퐷(푠) , 퐴(푠) = 푚 � 푗=1 Γ(푏 푗 − 훽 푗푠), 퐵(푠) = 푛 � 푗=1 Γ(1 − 푎 푗 + 훼푗푠), 퐶(푠) = 푞 � 푗=푚+1 Γ(1 − 푏 푗 + 훽 푗푠), 퐷(푠) = 푝 � 푗=푛+1 Γ(푎 푗 − 훼푗푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Here 푚, 푛, 푝, 푞 are integers satisfying 0 ≤ 푛 ≤ 푝, 1 ≤ 푚 ≤ 푞, 푚2 + 푛2 ≠ 0, 푎 푗( 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푝), 푏 푗( 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푞) are complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The integration contour L could be chosen in different ways: L = L−푖∞,푖∞ chosen to go from −푖∞ to 푖∞ leaving to the right all poles of P(퐴) of the functions Γ in 퐴(푠) and to the left all poles of P(퐵) of the functions Γ in 퐵(푠);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' L = L푖∞ is a loop beginning and ending at +∞ and encircling in the negative direction all the poles of P(퐴);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' L = L−푖∞ is a loop beginning and ending at −∞ and encircling in the negative direction all the poles of P퐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='3 Wright Functions The Write function is defined by the series representation that is convergentin the whole 푧-complex plane [152, 72, 73, 120] 푊휆,휇(푧) = ∞ � 푛=0 푧푛 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='Γ(휆푛 + 휇) , 휆 > −1, 휇 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The integral representation of the Write function is written as 푊휆,휇(푧) = 1 2휋푖 ∫ 퐻 푎 푒휎+푧휎−휆 푑휎 휎휇 where 퐻푎 is the Hankel path (a loop that starts from −∞ along the lower side of the negative real axis, encircles the circular area the origin with radius 휖 → 0 in the positive sense, and ends at −∞ along the upper side of the negative real axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' There are Write-type auxiliary functions 퐹휈(푧) = 푊−휈,0(푧), 푀휈(푧) = 푊−휈,1−휈(푧), where 0 < 휈 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' these functions are related 퐹휈(푧) = 휈푧푀휈(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 23 The series representations of the auxiliary functions are 퐹휈(푧) = ∞ � 푛=1 (−푧)푛 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='Γ(−휈푛) = 1 휋 ∞ � 푛=1 (−푧)푛−1 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Γ(휈푛 + 1) sin(휋휈푛) and 푀휈(푧) = 퐹휈(푧) = ∞ � 푛=0 (−푧)푛 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='Γ[−휈푛 + (1 − 휈))] = 1 휋 ∞ � 푛=1 (−푧)푛−1 (푛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='Γ(휈푛) sin(휋휈푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 4 Solution of Fractional Differential Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='1 Analytical Methods Numerous approximate analytical methods are known: the Adomian decomposition method (ADM) [131];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the combined Laplace-Adomian method (CLAM) [232];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the variational iteration method (VIM) [87, 237, 59, 239, 244]3 and its local (LVIM) [248, 244, 236, 246] and fractional (using the fractional order Lagrange multipliers) [119, 262] variants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the homotopy perturbation method (HPM)[1, 241, 251, 205, 190, 233] and its modification [238] and local fractional variant (LFHPM) [247];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the differential transformation method [109, 6, 66];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the heat-balance integral method (HBIM)[101, 103, 102];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the fractional complex transform method (FCTM) [245, 137, 136, 93, 209];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the local fractional Fourier series method (FSM) [250, 261];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the modified simple equation method [106, 252];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the method of images (limited to special spatial symmetries);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the Mellin integral transform method [145];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the local fractional decomposition method (LFDM) [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the fractional sub-equation method [260, 83];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3VIM includes three steps to determine the variational iteration formula: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' establishing the correction functional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' identifying the Lagrange multipliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' determining the initial iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The second step is the crucial one [236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 24 the Sumudu transform methods [231, 45] and its variant — the local fractional homotopy perturbation Sumudu transform mehod [265];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the theta-method [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the Picard succesive approximation method (PSAM) [243, 240];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the local Laplace transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Frequently analytical methods are variants of perturbation methods [226]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' For example, He [88, 89, 91] based his method to solve the general equation 퐴(푢)− 푓 (푟) = 0 with the general differential operator 퐴 divided into linear 퐿 and nonlinear 푁 parts 퐿(푢) + 푁(푢) − 푓 (푟) = 0 on the approach of Liao [139] (who used the two-parameter family of equations) by considering the one-parameter family (1− 푝)퐿(푢) + 푝푁(푢) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' He constructed the homotopy 푣(푟, 푝) : Ω × [0, 1] → 푅 that satisfies 퐻(푣, 푝) = (1 − 푝)[퐿(푣) − 퐿(푣0)] + 푝[퐴(푣) − 푓 (푟)] = 0 where the homotopy parameter 푝 ∈ [0, 1], 푣0 is the initial approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Evidently, 퐻(푣, 0) = 퐿(푣) − 퐿(푣0) = 0 and 퐻(푣, 1) = [퐴(푣) − 푓 (푟)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' In topology, 퐿(푣) − 퐿(푣0) is called deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The homotopy parameter 푝 is considered as a small parameter and the solution is written as a series 푣 = 푣0 + 푝푣1 + 푝2푣2 + 푝3푣3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' and when 푝 → 1 푢 = lim 푝→1 푣 = 푣0 + 푣1 + 푣2 + 푣3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The Adomian decomposition method (ADM) [3, 54] does not use linearization, perturbation or the Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The accuracy of the approximate analytical solutions can be verified by the direct substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The initial value is written as 퐿푢 + 푅푢 + 푁푢 = 푔 where 퐿 is the linear operator to be inverted, 푅 is the linear remainder operator and 푁 is the nonlinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Thus 퐿−1퐿푢 = 푢 − Φ, Φ incorporates the initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The solution and nonlinear term are decomposed into series 푢 = ∞ � 푛=0 푢푛, 푁푢 = ∞ � 푛=0 퐴푛 where 퐴푛 are the Adomian polynomials for 푁푢 = 푓 (푢) are [54] 퐴푛 = 1 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 휕푛 휕휆푛 푓 � ∞ � 푘=0 푢푘휆푘 � , 푛 = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Finally ∞ � 푛=0 푢푛 = Φ + 퐿−1푔 − 퐿−1 � 푅 ∞ � 푛=0 푢푛 + ∞ � 푛=0 퐴푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 25 The nonlinear term 푁푢(푥, 푡) can be also decomposed [238] as 푁푢 = ∞ � 푛=0 푝푛퐻푛 where He’s polynomials are [171, 68] 퐻푛(푢0, 푢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푢푛) = 1 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 휕푛 휕푝푛 � 푁 � 푛 � 푖=0 푝푖푢푖 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The fractional sub-equation method includes several steps [83]: Transformation of the nonlinear fractional equation in two variables 푥 and 푡 퐷 훼 푡 푢, 퐷 훼 푥 푢, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' ) = 0, 0 < 훼 ≤ 1, 퐷 훼 푡 푢 and 퐷 훼 푥 푢 are Jumarie modification of the RL derivtives, using the travelling wave transformation 푢(푥, 푡) = 푢(휉), 휉 = 푥 + 푐푡, where 푐 is a constant to be determined, to the equation 푃(푢, 푐푢′, 푢′, 푐퐷 훼 휉푢, 퐷 훼 휉푢, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' (20) The solution of the equation (20) is assumed to have the form 푢(휉) = −1 � 푖=−푛 푎푖휙푖 + 푎0 + 푛 � 푖=1 푎푖휙푖, where 푎푖(푖 = −푛, −푛 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푛 − 1, 푛) are constants to be determined, 휙 = 휙(휉) are functions that satisfy the following Riccati equation 퐷 훼 휉 휙(휉) = 휎휙2(휉), 휎 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Formulation of a set of overdetermined nonlinear algebraic equations for 푐 and 푎푖(푖 = −푛, −푛 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' , 푛 − 1, 푛) [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='2 Numerical Methods Diethelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [115] listed the requirement to the numerical methods that should be convergent, consistent of some reasonable order ℎ푝, stable, reasonably inexpensive to run, reasonably easy to program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Numerous methods are used in practice: finite difference, finite elements, radial basis functions, spectral methods, meshfree methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The numerical methods for the fractional differential equations usually are constructed by the modification of the meth- ods for the ordinary differential equations but require significantly more computation time and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The approximation of the fractional derivative needs the computation of the convolution integral that requires to sample and multiply the behaviour of two functions over the whole of the interval of integration leading to the operation count of 푂(푛2) where 푛 is number of sampling points [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' The reduction of the computational efforts is related to the fading memory property of the fractional derivatives that allows to restrict the integration interval — using the 26 short memory principle [46, 49] (also fixed memory principle [63] and logarithmic memory principle [85]), and using adaptive time stepping and basis selection [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Numerous methods are used to solve the fractional differential equations in practice: the finite difference [135, 199] (both the explicit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Euler [172] and the implicit [163, 67], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the Crank-Nicolson [211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 212] or the alternating direction implicit [257,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 258] schemes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' compact schemes [229,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 230,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 52]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the finite elements [104,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 267,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 192] (including least squres FEM [61],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Galerkin FEM [206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 108],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' discontinuous Galerkin FEM [164]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the spectral methods [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 58,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 255],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' the meshfree methods [55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 201] (including the radial basis functions methods that exploit cubic 휙 = 푟3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Gaussian 휙 = 푒푥푝(−푟2/푐2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' multiquadrics 휙 = √ 푐2 + 푟2 or inverse multiquadrics 휙 = 1/ √ 푐2 + 푟2 functions [234,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' 9]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Legendre wavelet collocation method [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' Bahuguna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [15], Hanert [84], Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfP_ap/content/2301.00037v1.pdf'} +page_content=' [48] and Deng & Li [47], Ford & Simpson [63], Ford & Connolly [62], Momani et 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a/6NE4T4oBgHgl3EQfBwtK/content/tmp_files/2301.04854v1.pdf.txt b/6NE4T4oBgHgl3EQfBwtK/content/tmp_files/2301.04854v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d2589728c4b3ca38830deca796d5392b1e4f9cb --- /dev/null +++ b/6NE4T4oBgHgl3EQfBwtK/content/tmp_files/2301.04854v1.pdf.txt @@ -0,0 +1,1328 @@ +Mixed-state Entanglement for AdS Born-Infeld Theory +Peng Liu 1,∗ Zhe Yang 1,† Chao Niu 1,‡ Cheng-Yong Zhang 1,§ and Jian-Pin Wu 2,3¶ +1 Department of Physics and Siyuan Laboratory, +Jinan University, Guangzhou 510632, China +2 Center for Gravitation and Cosmology, +College of Physical Science and Technology, +Yangzhou University, Yangzhou 225009, China +3 School of Aeronautics and Astronautics, +Shanghai Jiao Tong University, Shanghai 200240, China +Abstract +We study the mixed-state entanglement for AdS Born-Infeld (BI) theory. +We calculate the +mixed-state entanglement and investigate the relationship between it and the system parameters. +We find that the holographic entanglement entropy (HEE) and mutual information (MI) exhibit +monotonically increasing and decreasing behavior with BI factor b. However, the entanglement +wedge cross-section (EWCS) exhibits a very rich set of phenomena about system parameters. +EWCS always increases with b when b is small and then monotonically decreases with b. These +behaviors suggest that increasing the BI factor, which is essentially enhancing the coupling be- +tween the background geometry and the transport properties can always enhance the EWCS. The +coupling between the entanglement and the transport behaviors has also been studied in condensed +matter theories and is important to construct a stable quantum circuit. We also provide analytical +understanding of the above phenomenon. +∗Electronic address: phylp@email.jnu.edu.cn +†Electronic address: yzar55@stu2021.jnu.edu.cn +‡Electronic address: niuchaophy@gmail.com +§Electronic address: zhangcy@email.jnu.edu.cn +¶jianpinwu@yzu.edu.cn, corresponding author +1 +arXiv:2301.04854v1 [hep-th] 12 Jan 2023 + +Contents +I. Introduction +2 +II. Holographic Born-Infeld Theory And Information-Related Quantities +4 +A. The AdS Born-Infeld Model +4 +B. Holographic information-related quantities +7 +C. Computations of holographic geometric quantities +8 +1. The minimum surface +9 +2. The EWCS +10 +III. The Holographic Entanglement Entropy And The Holographic Mutual +Information +11 +IV. The Holographic Entanglement Wedge Cross-Section +14 +V. Discussion +18 +Acknowledgments +19 +References +19 +I. +INTRODUCTION +Quantum entanglement is the most distinguishing characteristic between quantum and +classical systems. Holographic gravity, condensed matter theory, quantum information, and +other areas have recently overlapped with each other on quantum entanglement. Numerous +quantum entanglement measurements have been discovered to be capable of diagnosing the +quantum phase transition of strong correlation systems and the topological quantum phase +transitions, as well as playing a key role in the emergence of spacetime [1–8]. +There are numerous types of quantum entanglement measurements, including entangle- +ment entropy (EE), mutual information (MI), R´enyi entanglement entropy, and negativity. +Among these quantum entanglement measurements, EE is commonly considered a useful +measure of pure state entanglement. However, EE is not applicable to measure the more +common mixed-state entanglement. To measure mixed-state entanglement, numerous new +2 + +entanglement measurements, such as the entanglement of purification, non-negativity, and +the entanglement of formation, have been proposed [9, 10]. On the other hand, calculating +the mixed-state entanglement measures is extremely difficult. +Gauge/gravity duality is a powerful tool for analyzing strongly correlated systems because +it connects entanglement-related physical quantities to geometric objects in dual gravity sys- +tems. In the dual gravity system, the holographic entanglement entropy (HEE) connects +the EE of a subregion on the boundary with the area of the minimum surface [5]. HEE +has been demonstrated to be able to detect quantum phase transitions and thermodynamic +phase transitions [11–15]. Recently, the R´enyi entropy was proposed to be proportional to +the minimal area of cosmic branes [16]. Moreover, the butterfly effect that reflects the dy- +namic properties of quantum systems, has been extensively studied in holographic theories +[17–26]. In addition, holographic duality of quantum complexity, a new information-related +quantity from the EE, was also proposed [27–33]. More recently, the EWCS was associated +with the area of the minimum cross-section of the entanglement wedge [34, 35]. The geomet- +ric prescription of EWCS provides a novel and powerful tool for studying the mixed-state +entanglement in holographic theories. +Among all the models in holographic theories, the Born-Infeld (BI) model is a special +class of models for nonlinear electromagnetic field theories. It was first proposed to eliminate +the divergent self-energy of Maxwell theory. Later, it was found that the BI theory can be +naturally derived from the string theory under the low-energy approximation. The BI model +under the holographic theories can be dual to the quantum chromodynamics (QCD) systems +[36, 37], and some condensed matter systems with novel transport behaviors, such as the +quantum liquid [38], the Mott-insulator [39], and the novel magneto-resistance phenomenon +[40, 41], which is consistent with the experimental phenomenon in [42, 43]. Various prop- +erties of the BI model, such as its thermodynamic properties, transport properties [44], the +complexity [45], have been extensively investigated. However, the question of how exactly +the BI factor b, which embodies the nonlinearity of this nonlinear electromagnetic field the- +ory, affects the properties of the system, especially the mixed-state entanglement properties, +remains to be answered. +This paper focuses on the effect of the BI factor on two measures of mixed-state en- +tanglement - MI and EWCS. When b → 0, the background geometry is AdS-Schwarzschild +solution, and the entanglement property of the system is decoupled from the transport prop- +3 + +erty of the system; while for non-zero b, the transport behaviors can affect the entanglement +property. Therefore, we interpret b as the degree of correlation between the entanglement +and transport properties of the metric when b increases from zero. Remind also that the +coupling between the transport properties and the entanglement is also an important topic +in condensed matter field theory, and is crucial for the construction of a stable quantum cir- +cuit [46–48]. For b → ∞, the system goes to the AdS-RN black brane system with a linear +Maxwell field. Therefore, the range b ∈ (0, ∞) represents the process that the Maxwell field +turns on and converges to a linear Maxwell field case. Our main goal is to explore how BI +factor b affects the MI and EWCS. +We organize this paper as follows: we introduce the holographic BI model in Sec. II A, +entanglement measures (HEE, MI, EWCS) and their holographic duality in Sec. II B. We +discuss the properties of HEE, MI (III) and EWCS (IV) systematically. Finally, we summa- +rize in Sec. V. +II. +HOLOGRAPHIC BORN-INFELD THEORY AND INFORMATION-RELATED +QUANTITIES +First, we review the holographic BI model. Following that, we review the concepts of the +HEE, MI, and EWCS with their holographic dual. Then, we elaborate upon our algorithms +proposed to calculate minimum surfaces and minimum cross-sections. +A. +The AdS Born-Infeld Model +The action of the 4-dimensional holographic BI model is, +S = +� +d4x√−g +� +R − 3Λ +16πG + +b2 +4πG +� +1 − +� +1 + 2F +b2 +�� +. +(1) +The parameter b is the BI factor, and Λ = − 3 +l2 with l the AdS radius. The solution of the +BI theory is, +ds2 = −f(r)dt2 + +1 +f(r)dr2 + r2hijdxidxj, +(2) +with +f(r) = r2 +l2 − 2M +r ++ +4Q22F1 +� +1 +4, 1 +2; 5 +4; − Q2 +b2r4 +� +3r2 ++ 2b2r2 +3 +� +1 − +� +Q2 +b2r4 + 1 +� +, +(3) +4 + +Q is the electric charge and M is the mass of the black brane. For l2 < 0 and l2 > 0 the +system is asymptotically dS and AdS, respectively. Here, we fix l2 = 1 for concreteness. +For k = 1, 0, −1 the hij denotes a sphere, a Ricci flat surface, and a hyperbolic surface, +respectively. Here, we focus on the planar case, i.e., k = 0. +At the horizon r = rh we have f(rh) = 0, and hence we arrive at the ADM mass +M = +4l2Q2 2F1 +� +1 +4, 1 +2; 5 +4; − Q2 +b2r4 +h +� +− 2b2l2r4 +h +� +Q2 +b2r4 +h + 1 + 2b2l2r4 +h + 3r4 +h +6l2rh +. +(4) +The Hawking temperature is, +T = rh +4π +� +3 − 2b2 +�� +Q2 +b2r4 +h ++ 1 − 1 +�� +. +(5) +The planar case is always thermodynamically stable [49]. Therefore, in this BI black brane +system, there is no thermodynamic phase transition. +The system is invariant under the rescaling, +(t, 1/r, x, y) → α(t, 1/r, x, y), Q → Q/α2, T → T/α, rh → αrh. +Other parameters such as b, β are all dimensionless. Therefore, we can fix rh = 1. Here, +we adopt √Q as the scaling unit, consequently, we need to divide physical quantity with +scaling dimension [n] by Qn/2. +For numerical convenience, we transform r into z ≡ rh/r such that the semi-infinite +domain r ∈ (rh, ∞) becomes a finite domain z ∈ [0, 1]. Then the metric becomes, +ds2 = 1 +z2 +� +−hdt2 + r2 +hdz2 +h ++ r2 +hdx2 + r2 +hdy2 +� +, +(6) +with +h(z) ≡4 +3Q2z3 +� +z 2F1 +�1 +4, 1 +2; 5 +4; −Q2z4 +b2 +� +− 2F1 +�1 +4, 1 +2; 5 +4; −Q2 +b2 +�� +− 2 +3b2 +� +z3 +� +1 − +� +Q2 +b2 + 1 +� ++ +� +Q2z4 +b2 ++ 1 − 1 +� +− z3 + 1. +(7) +And the dimensionless Hawking temperature becomes, +T = +b2 +� +2 − 2 +� +Q2 +b2 + 1 +� ++ 3 +4π√Q +. +(8) +5 + +FIG. 1: The contour plot of the Hawking temperature in the plane (b, rh), where the temperature +is only positive in the shaded region. +From the dimensionless Hawking temperature (8) we can find that, +lim +Q→0 T → ∞, +lim +Q→ +√ +12b2+9 +2b +T → 0. +(9) +Also, we can find that, +∂QT = 2b +� +b − +� +b2 + Q2 +� +− 3 +� +Q2 +b2 + 1 − 2Q2 < 0. +(10) +Therefore, the quantity Q is restricted to the range [0, +√ +12b2+9 +2b +] and that the temperature +T decreases as Q increases. This system is described by three variables (T, b , rh), with +only two of them being independent. We have also observed that for any given value of +b, the temperature T always increases with increasing rh, thus, the value of rh is uniquely +determined by a given temperature T. This can be seen in the Fig. 1. Therefore, we can +simplify the system to a two-parameter system (b, T). +When the parameter b → ∞, the background solution of our system converges to the +AdS-RN solution, and when b → 0, it becomes the AdS-Schwarzschild solution. When b is +zero, there is an electromagnetic field present, but the background solution is still the AdS- +Schwarzschild solution. This means that the entanglement-related physical quantities are +not affected by the conductivity of the system. However, as b increases, the electromagnetic +fields starts to affect the background solution, and thus has an impact on the entanglement +6 + +5 +4 +0.99 +0.88 +0.77 +3 +0.66 +0.55 +0.44 +2 +0.33 +0.22 +0.11 +0 +1 +0 +0 +1 +2 +3 +4 +5 +bstructure of the system. Therefore, we refer to increasing b from zero to infinity as the +process of turning on the coupling between the background and the conductivity, and finally +resulting in an AdS-RN system. +It is worth noting that the relationship between conductivity and entanglement-related +quantities is of great importance in condensed matter theories. Recent experiments have +shown that entanglement between quantum dots can persist despite the influence of surface +plasmon polariton (SPPs) transmission [47, 48, 50]. These findings are crucial for the devel- +opment of stable quantum circuits. Additionally, it has been found that at specific values +of the inter-dot distance d or detuning δ, the two-quantum-dot system can be in a highly +entangled state and be separate from the transmission of SPPs [46]. However, when d or δ +deviate from these values, the entanglement of quantum dots becomes highly correlated with +the transmission of SPPs. This suggests that decoupling of entanglement and transport can +exist in real physical systems and can be characterized by certain parameters. +Next, we will focus on how the entanglement-related physical quantities change as we +vary the parameter b. +B. +Holographic information-related quantities +Entanglement is a fundamental and intriguing aspect of quantum mechanics. One way +to quantify entanglement is through entanglement entropy (EE), which measures the degree +of entanglement between a subset of a system and the rest of the system. Specifically, the +entanglement entropy SA between subsets A and B of a system A ∪ B is defined as the von +Neumann entropy in terms of the reduced density matrix ρA. +SA(|ψ⟩) = −Tr [ρA log ρA] , +ρA = TrB (|ψ⟩⟨ψ|) . +(11) +It is easy to find that SA = SB for pure states [51]. Holographic duality theory relates the +holographic entanglement entropy (HEE) to the area of the minimum surface in dual gravity +systems [5] (see the left plot of Fig. 2). +EE is often used to measure the degree of entanglement in pure states, but it is not as +effective in measuring mixed state entanglement. For example, even when subsystems A and +B are not entangled, they can still have non-zero EE in a system composed of direct product +of the density matrices of ρA and ρB. This is because EE takes into account both quantum +7 + +entanglement and classical correlation, so it does not always provide a accurate measure +of the entanglement. As a result, other measures for mixed-state entanglement have been +proposed in the literature [9, 10]. The most direct mixed-state entanglement measure is MI. +For the subsystem A ∪ C separated by B, the mutual information (MI) is defined as: +I (A, B) := S (A) + S (B) − S (A ∪ B) , +(12) +This measures the mixed-state entanglement between A and B. It can be easily verified +that I (A, B) = 0 when ρAB = ρA ⊗ ρB, therefore MI have the property that direct product +states have zero entanglement. However, MI is not a perfect measure of mixed-state entan- +glement, as it is closely related to EE, and it’s properties are sometimes dominated by EE +or thermal entropy in certain situations. This indicates that other measures of mixed-state +entanglement should be used. +The entanglement wedge cross-section (EWCS) has been associated with the duality of +certain mixed-state entanglement measures, such as entanglement of purification, logarith- +mic negativity, and reflect entropy [53–55]. Takayanagi proposed that EWCS EW (ρAB) is +the area of the minimum cross-section ΣAB in connected entanglement wedge [34], i.e. (see +the right plot in Fig. 2), +EW (ρAB) = min +ΣAB +�Area (ΣAB) +4GN +� +. +(13) +It is important to note that if the entanglement wedge is disconnected, meaning the minimum +cross-section does not exist, the EWCS will be zero, it corresponds to cases with vanish- +ing MI. Additionally, the EWCS also satisfies some important inequalities as its quantum +information counterparts [34, 56] +Next, we present the algorithm for obtaining the minimum surfaces and EWCS. +C. +Computations of holographic geometric quantities +We examine the EWCS of an infinite strip with a homogeneous background for numerical +simplicity. For a generic homogeneous background +ds2 = gttdt2 + gzzdz2 + gxxdx2 + gyydy2, +(14) +where z = 0 represents the boundary of the asymptotic AdS spacetime. The left plot in +Fig. 3 is a visual representation of the minimum surface for an infinite strip along the y- +8 + +x +y +z +x +y +z +FIG. 2: The left plot: The minimum surface for a given width w. The right plot: The minimum +cross-section (green surface) of the entanglement wedge. +axis. Since the background is homogeneous, all metric components gµν only depend on the +coordinate z. +1. +The minimum surface +The minimum surface near the AdS boundary is perpendicular to the boundary, making +the spatial direction x an unsuitable parameter for finding the minimum surface. Ref. [58] +adopted the angle θ with tan θ = z/x, as the parameter for the minimum surface (see Fig. +3). Using this method, we can parametrize a surface as (x(θ), z(θ)) with area A given by +A = 2 +� π/2 +0 +� +x′(θ)2gxxgyy + z′(θ)2gyygzzdθ. +(15) +The resultant equations of motion read, +x′(θ)z′(θ)2 +� g′ +xx +2gxx ++ g′ +yy +gyy +− g′ +zz +2gzz +� ++ x′(θ)3 � +gyyg′ +xx + gxxg′ +yy +� +2gxxgzz ++ x′′(θ)z′(θ) − x′(θ)z′′(θ) = 0, +z(θ) − tan(θ)x(θ) = 0. +(16) +where g′ +## ≡ g′ +##(z). The boundary conditions are, +z(0) = 0, +x(0) = w, +z′(π/2) = 0, +x(π/2) = 0, +(17) +where w is the width of the strip. +9 + +2. +The EWCS +Given a biparty subsystem with minimum surfaces C1(θ1), C2(θ2), we solve the minimum +surface Cp1,p2 connecting p1 ∈ C1 and p2 ∈ C2. We parametrize Cp1,p2 with z, then the area +of Cp1,p2 reads, +A = +� +Cp1,p2 +� +gxxgyyx′(z)2 + gxxgzzdz. +(18) +The resultant equation of motion becomes, +x′(z)3 +� gxxg′ +yy +2gyygzz ++ g′ +xx +2gzz +� ++ x′(z) +�g′ +xx +gxx ++ g′ +yy +2gyy +− g′ +zz +2gzz +� ++ x′′(z) = 0, +(19) +with boundary conditions, +x(z(θ1)) = x(θ1), +x(z(θ2)) = x(θ2). +(20) +To obtain the EWCS, we need to locate the global minimum of the minimum surfaces +connecting C1(θ1), C2(θ2), i.e., the minimum cross-section. +Finding the minimum cross-section is a challenging task as it involves searching through +a two-dimensional parameter space (θ1, θ2). +However, it can be noted that the globally +minimum cross-section must be perpendicular to the minimum surfaces at the point of +intersection. This observation serves as a local constraint, which can greatly speed up the +search process. We demonstrate the methods of solving the EWCS in Fig. 3. For numerical +stability, it is better to implement the perpendicular conditions with normalized vectors as, +Q1(θ1, θ2) ≡ +gab +� ∂ +∂z +�a � +∂ +∂θ1 +�b +� +gcd +� ∂ +∂z +�c � ∂ +∂z +�d +� +gmn +� +∂ +∂θ1 +�m � +∂ +∂θ1 +�n +�������� +p1 += 0, +Q2(θ1, θ2) ≡ +gab +� ∂ +∂z +�a � +∂ +∂θ2 +�b +� +gcd +� ∂ +∂z +�c � ∂ +∂z +�d +� +gmn +� +∂ +∂θ2 +�m � +∂ +∂θ2 +�n +�������� +p2 += 0. +(21) +Note that Q1 and Q2 are both functions of the θ1 and θ2. Now, the search of the EWCS is +equivalent to finding the minimum surface ending at (θ1, θ2) where (21) is satisfied. +To determine the correct EWCS, we first select an initial seed (θ1, θ2) and use the Newton +iterative method to obtain feedback (δθ1, δθ2). By repeating this process, we can find the +minimum cross-section, which is the EWCS. It is crucial to carefully choose the initial +10 + +-1.5 +-1.0 +-0.5 +0.5 +1.0 +1.5 +x +0.2 +0.4 +0.6 +0.8 +1.0 +z +p1 +p2 +θ1 +θ2 +C1(θ1) +C2(θ2) +FIG. 3: +The demonstration of the EWCS. The p1 and p2 are the intersection points of the +minimum surface connecting those two minimum surfaces. The solid blue curve (parametrized +with θ1) and solid orange curve (parametrized with θ2) are minimum surfaces. +The thick red +curve is the minimum surface connecting p1 and p2. The blue arrows at the p1 and p2 are the +tangent vector +� ∂ +∂z +�a��� +p1 and +� ∂ +∂z +�a��� +p2 along the Cp1,p2, while the purple arrows are the tangent +vectors +� +∂ +∂θ1 +�a��� +p1 and +� +∂ +∂θ2 +�a��� +p2 along C1, C2, respectively. The dark dashed horizontal line is the +horizon. +values of (θ1, , θ2) for the iterations to converge. The numerical reliability is ensured by +the convergence of results when using different initial values or increasing the density of +discretization. For more technical details, refer to reference [57]. +Using the techniques outlined above, we will now examine mixed-state entanglement +measures for the BI model. Additionally, we will examine the correlation between the BI +factor b and information-related quantities. +III. +THE HOLOGRAPHIC ENTANGLEMENT ENTROPY AND THE HOLO- +GRAPHIC MUTUAL INFORMATION +We begin by examining the relationship between HEE, system parameters b and T. As +shown in Fig. 4, HEE, represented by S, increases monotonically with both b and T, but +their rate of increase is different. Initially, S increases slowly with T and its growth rate +with T becomes more pronounced as T increases. On the other hand, S increases quickly +with b at first and then slows down as b decreases. Next, we explain the behavior of S with +11 + +b=0.0001000 b=0.008791 +b=0.01765 +b=0.02868 +b=0.03912 +b=0.05289 +0.05 +0.10 +0.15 +0.20 +0.25 +T +-1.80 +-1.75 +-1.70 +-1.65 +-1.60 +S +w=0.8 +T=0.00100 T=0.09967 T=0.1389 +T=0.1614 +T=0.1856 +T=0.2110 +0.1 +0.2 +0.3 +0.4 +b +-1.80 +-1.75 +-1.70 +-1.65 +-1.60 +-1.55 +S +w=0.8 +FIG. 4: HEE vs T and b at width w = 0.8, respectively. +b and T, respectively. +When the horizon radius of the black brane increases, the minimum surface tends to be +closer to the horizon of the black brane, which makes the thermodynamic entropy dominate +the behavior of the HEE. Therefore, the growth of HEE with T as well as b, can be un- +derstood from the relation between rh and T or b. According to (5) we can deduce that rh +increases with increasing temperature and b, this can be seen by taking the derivative of rh +with respect to T and b. The results are, +∂Trh = +r4 +h +� +Q2 +b2r4 +h + 1 +r4 +h +� +2b2 +�� +Q2 +b2r4 +h + 1 − 1 +� ++ 3 +� +Q2 +b2r4 +h + 1 +� ++ 2Q2, +∂brh = +2rh +� +Q2 − 2b2r4 +h +�� +Q2 +b2r4 +h + 1 − 1 +�� +b +� +r4 +h +� +2b2 +�� +Q2 +b2r4 +h + 1 − 1 +� ++ 3 +� +Q2 +b2r4 +h + 1 +� ++ 2Q2 +�. +(22) +From the above equation, it is clear that ∂Trh is always positive, indicating that rh increases +as T increases. However, ∂brh can be positive or negative, depending on the specific pa- +rameter range. Further examination shows that ∂brh is always greater than zero when rh is +relatively large. This means that rh increases with b when rh is large, or when the minimum +surface is closer to the horizon of the black brane. +When b is relatively large, the system is approximately the AdS-RN system. The ar- +gument presented in [59] can be applied to prove that ∂TS > 0. Furthermore, for small +subregions, it can be inferred from the equations in [59] that ∂TS is close to 0, which ex- +plains the flat behavior of S along T for small temperatures. +After studying HEE, we proceed to investigate the behavior of MI with T and b. In the BI +model, the configurations for MI and EWCS are subsystems composed of a and b separated +12 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +b +T +I(3, 0.10, 2) +5.85 +6.63 +7.41 +8.19 +8.97 +9.75 +10.53 +11.31 +12.09 +12.87 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +b +T +I(3, 0.25, 2) +0.42 +0.84 +1.26 +1.68 +2.10 +2.52 +2.94 +3.36 +3.78 +4.20 +FIG. 5: +MI as a function of b and T for different configurations. +b=0.0001000 b=0.04554 +b=0.09392 +b=0.1591 +b=0.2526 +b=0.4000 +0.1 +0.2 +0.3 +0.4 +0.5 +T +0.5 +1.0 +1.5 +I +(a, p, c) = (0.5, 0.2, 0.35) +T=0.00100 T=0.1182 T=0.1614 +T=0.1856 +T=0.2110 T=0.2372 +0.1 +0.2 +0.3 +0.4 +b +1.30 +1.35 +1.40 +1.45 +1.50 +1.55 +I +(a, p, c) = (0.5, 0.2, 0.35) +FIG. 6: MI as a function of b and T for different configurations. +by region p. As seen in Fig. 5, MI decreases with increasing temperature and b. This is in +contrast to the behavior of HEE. Moreover, it is worth noting that MI can decrease to zero, +which is an indication of a disentanglement phase transition. We have also plotted the MI +for smaller configurations (see Fig. 6), and the qualitative phenomena remain the same. +As the subsystem c and the separation p change, the system undergoes a disentangling +phase transition, at which point the entanglement of two subsystems a and c vanishes. The +critical value of subsystem cc and separation pc are shown in Fig. 7. The left plot of Fig. 7 +shows that the critical value of subsystem cc increases with b and T; however, the right plot +of Fig. 7 shows that the critical value of the separation pc decreases with b and T. This is +as expected since increasing the temperature or b will tends to destroy the entanglement, +13 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +b +T +cc (a=0.6, p=0.3) +0.4234 +0.4408 +0.4582 +0.4756 +0.4930 +0.5104 +0.5278 +0.5452 +0.5626 +0.5800 +0.0 +0.1 +0.2 +0.3 +0.4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +b +T +pc (a=0.6, c=0.3) +0.1988 +0.2030 +0.2072 +0.2114 +0.2156 +0.2198 +0.2240 +0.2282 +0.2324 +0.2366 +FIG. 7: Critical configurations of cc and pc. +resulting in a larger subregion cc or a smaller separation pc. +Next, we explore the mixed-state entanglement through the EWCS. +IV. +THE HOLOGRAPHIC ENTANGLEMENT WEDGE CROSS-SECTION +In Fig. +8, we present the minimum surfaces and the corresponding minimum cross- +sections. It can be observed that the minimum surface is flatter when the temperature is +lower. This is due to the fact that the coordinate z is related to the horizon radius rh, and at +lower temperatures, a small rh will rescale z to zrh, resulting in a flatter minimum surface. +This makes it challenging to obtain precise enough solutions for the minimum surface since +the flat case is more singular in the θ coordinate. To overcome this issue, we redefine the +angle as z = ηx tan(θ), where η is a number related to the temperature. Only with this +technique, we can achieve precise enough solutions. +We show the EWCS vs b in Fig. 9, from which we can find that the EWCS can show very +delicate behaviors. The EWCS increases with b at first in a very narrow range of b, however, +it starts to decrease with b when b is relatively large and monotonically decreases with b. +This is in sharp contrast to the behavior of the HEE and MI, that only shows monotonical +behaviors (see Fig. 4 and Fig. 5). In addition, the EW changes slower with b than that with +T. The typical change is of order 10−4 and 10−3, respectively. This delicate behavior can +be captured precisely because the precision of our numerical methods can be up to 10−7. +Notice that the background is an AdS-Schwarzschild solution when b is 0, meanwhile, its +14 + +-���� +-���� +���� +���� +� +���� +���� +���� +���� +���� +���� +� +�=������ +�=������ +�=������ +�=������ +�=������ +FIG. 8: The illustration of EWCS. At the same configuration (a, p, c) = (0.1, 0.05, 0.06925) we +see that the minimum surface becomes flatter when decreasing the temperature. Meanwhile, the +minimum cross-section always ends at the point near the tops of the inner minimum surface, while +ends at the point away from the tops of the outer minimum surface. +0.0 +0.1 +0.2 +0.3 +0.4b +18.360 +18.365 +18.370 +18.375 +Ew +T=0.1565 T=0.2492 T=0.2537 +T=0.2654 T=0.2758 T=0.2971 +0.05 +0.10 +0.15 +0.20b +-0.02 +0.02 +0.04 +∂bEw +FIG. 9: EWCS vs T. This plot is obtained at (a, p, c) = (0.1, 0.05, 0.06925). When b is relatively +large, the EW converges to certain fixed values. For T = 0.2971 it can first increase, and later +decreases, and after that increases with b. Therefore, for very small b the EWCS increases with b, +irrespective of the values of the T and the configurations. +electromagnetic field is non-zero. At this point, the charge transport behavior of the system +is significantly different from that of the genuine AdS-Schwarzschild system. +Moreover, +since its geometry is still AdS-Schwarzschild, the entanglement-related geometric quantities +will be decoupled from the charge transport. +As b gradually increases, the background +geometry will receive back reactions from the Maxwell field. At this time, the entanglement- +related geometry starts to couple with the charge transports. Therefore, b can play a role +in measuring the relationship between entanglement and transport when b is small. As we +15 + +T=0.1211 +T=0.1364 +T=0.1461 +T=0.1569 +T=0.1687 +0.0 +0.1 +0.2 +0.3 +0.4b +4.864 +4.866 +4.868 +4.870 +4.872 +4.874 +4.876 +Ew +FIG. 10: The EWCS vs b for a larger configuration (a, p, c) = (0.5, 0.2, 0.3875). +have pointed out, the EWCS increases with b when b is very small, i.e., when the coupling +has just occurred. And when b increases further, the EWCS gradually shows a decreasing +behavior. Notice that simpler geometric quantities such as HEE, and MI only show a very flat +monotonic behavior. This indicates that EWCS, as a mixed-state entanglement, captures +very different properties from HEE and MI. +To understand the above behavior more clearly, we implement the following analytical +treatments. For small values of b, we can expand the expression of the EW (18) integral +with respect to b as, +EW = +� +Σ +� +� 1 +z2 +� +dx2 + +dz2 +(1 − z3) + bdz2 � +Γ +� 1 +4 +� ++ 8Γ +� 5 +4 +�� � +dz2 + dx2 (1 − z3) +�−1/2 +2 +� +3πT (1 − z3) (z2 + z + 1) Γ +� 1 +4 +� ++ O(b2) +� +� , +(23) +where the second term shows us that +dEW +db +> 0 for small values of b. This explains the +ubiquitous existence of the monotonically increasing behavior of EW vs b for small values of +b. From the holographic dual picture, it means that when the Maxwell field starts to turn +on from the BI case, the EW is increased. However, when further increasing b we find that +EW reaches local maximums and starts to decrease. When b is large, it can be expected that +the background system approaches the AdS-RN, a fixed background geometry. Therefore, +the EW will starts to converge to some fixed value. +Next, we show the EWCS in larger configurations in Fig. 10. As seen in Fig. 10, the +non-monotonicity of EW with b becomes more pronounced as the width of the configuration +increases. This means that the non-monotonicity exists over a wider interval. The reason +for this is that when the width is relatively small, the minimum surface and the minimal +16 + +b=0.1355 +b=0.1753 +b=0.2492 +b=0.2758 +b=0.3084 +0.20 +0.25 +0.30 +0.35 +0.40T +18.32 +18.33 +18.34 +18.35 +18.36 +18.37 +18.38 +Ew +b=0.0100 +b=0.0744 +b=0.1389 +b=0.2033 +b=0.2678 +0.20 +0.21 +0.22 +0.23 +0.24 +0.25 +0.26 +0.27T +4.81 +4.82 +4.83 +4.84 +4.85 +4.86 +Ew +FIG. 11: EWCS vs T. The left plot is obtained at (a, p, c) = (0.1, 0.05, 0.06925); while the right +plot is obtained for a larger configuration (a, p, c) = (0.5, 0.2, 0.3875). +cross-section are only slightly different from the properties of AdS. However, as the width +increases, they deviate more significantly from AdS. +Next, we examine the behavior of EWCS with temperature. When the configuration is +relatively small in BI systems, EWCS decreases monotonically with temperature, as shown +in the left plot of Fig. 11. It is worth noting that the non-monotonic behavior of EWCS at +extremely small temperatures has been studied in [59] for AdS-RN systems. Additionally, +we illustrate the behavior of EWCS with temperature for larger configurations in the right +plot of Fig. 11, which also shows that EWCS decreases monotonically with temperature. +Although the monotonic decreasing behaviors are similar, the EWCS curves for small con- +figurations differ from those for large configurations. By comparing the two plots in Fig. +11, crossovers of the EWCS curves with temperature can be observed in the larger configu- +ration, which reflects the non-monotonic behavior of EWCS with b. These findings suggest +that the behavior of EWCS is generally monotonically decreasing with temperature, and +this behavior is consistent with that of MI. +In order to more clearly demonstrate the relationship between the EWCS and variables +b and T, a contour plot of EWCS as a function of b and T is presented in Figure 12. This +plot illustrates the non-monotonic nature of EWCS with respect to b and the monotonic +decrease of EWCS as T increases. +17 + +FIG. 12: The EWCS vs b for a larger configuration (a, p, c) = (0.5, 0.2, 0.3875). +V. +DISCUSSION +In this paper, we study the behavior of HEE, MI, and the mixed-state entanglement +measure EWCS in the BI model. Our results shows that HEE increases monotonically with +both b and T, while MI decreases monotonically with both b and T. +Interestingly, the +behavior of EWCS with respect to b shows a non-monotonic trend. Specifically, when b is +small, EWCS increases with b, but it begins to decrease as b increases further. In contrast, +EWCS exhibits a consistent monotonically decreasing trend with T. +Moreover, we provide analytical explanations for the non-monotonic behavior of EWCS +with respect to b. Note that when b is small, b serves as a measure of the coupling between +the entanglement-related quantities and the charge transport of the system. Based on this +observation, we conjecture that increasing the coupling between the entanglement-related +quantities and the transport properties can enhance the EWCS of the system. This cou- +pling between transport behaviors and entanglement is also a topic of significant interest in +condensed matter theory, as seen in previous studies on nanowires [46], plasmonics [47, 50], +and plasmons [48]. +18 + +Ew at (a,p,c)=(0.5,0.2,0.3875) +0.40 +0.35 +5.066 +4.998 +0.30 +4.930 +4.862 +T +4.794 +0.25 +4.726 +4.658 +4.590 +0.20 +0.15 E +0.0 +0.1 +0.2 +0.3 +0.4 +bNext, we point out the issues that deserve further investigation. To begin, we can exam- +ine other BI-like theories, such as the BI theory with massive gravity, the BI theory with +Axions, and so on, to see if the non-monotonic behavior observed in this paper is general. +Furthermore, we can examine the effect of more general nonlinear EM field theories on the +entanglement-related physical quantities of the system, such as the more general nonlinear +EM fields [39, 60]. We are working on these directions. +Acknowledgments +Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work. +Zhe Yang would like to express appreciation to Feng-Ying Deng. This work is supported by +the Natural Science Foundation of China under Grant No. 11805083, 11905083, 12005077, +12147209, the Science and Technology Planning Project of Guangzhou (202201010655) and +Guangdong Basic and Applied Basic Research Foundation (2021A1515012374). J.-P.W. is +also supported by Top Talent Support Program from Yangzhou University. +[1] A. Osterloh, L. 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D 98 (2018) no.2, 026021 [arXiv:1711.03298 [hep-th]]. +23 + diff --git a/6NE4T4oBgHgl3EQfBwtK/content/tmp_files/load_file.txt b/6NE4T4oBgHgl3EQfBwtK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d9d2bb6e11a0273443a95e846bbf8d99a89020e --- /dev/null +++ b/6NE4T4oBgHgl3EQfBwtK/content/tmp_files/load_file.txt @@ -0,0 +1,1080 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf,len=1079 +page_content='Mixed-state Entanglement for AdS Born-Infeld Theory Peng Liu 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='∗ Zhe Yang 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='† Chao Niu 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='‡ Cheng-Yong Zhang 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='§ and Jian-Pin Wu 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3¶ 1 Department of Physics and Siyuan Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Jinan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Guangzhou 510632,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' China 2 Center for Gravitation and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Yangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Yangzhou 225009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' China 3 School of Aeronautics and Astronautics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' China Abstract We study the mixed-state entanglement for AdS Born-Infeld (BI) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We calculate the mixed-state entanglement and investigate the relationship between it and the system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We find that the holographic entanglement entropy (HEE) and mutual information (MI) exhibit monotonically increasing and decreasing behavior with BI factor b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, the entanglement wedge cross-section (EWCS) exhibits a very rich set of phenomena about system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' EWCS always increases with b when b is small and then monotonically decreases with b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' These behaviors suggest that increasing the BI factor, which is essentially enhancing the coupling be- tween the background geometry and the transport properties can always enhance the EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The coupling between the entanglement and the transport behaviors has also been studied in condensed matter theories and is important to construct a stable quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We also provide analytical understanding of the above phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' ∗Electronic address: phylp@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='cn †Electronic address: yzar55@stu2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='cn ‡Electronic address: niuchaophy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='com §Electronic address: zhangcy@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='cn ¶jianpinwu@yzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='cn, corresponding author 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='04854v1 [hep-th] 12 Jan 2023 Contents I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Holographic Born-Infeld Theory And Information-Related Quantities 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The AdS Born-Infeld Model 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Holographic information-related quantities 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Computations of holographic geometric quantities 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The minimum surface 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The EWCS 10 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The Holographic Entanglement Entropy And The Holographic Mutual Information 11 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The Holographic Entanglement Wedge Cross-Section 14 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Discussion 18 Acknowledgments 19 References 19 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' INTRODUCTION Quantum entanglement is the most distinguishing characteristic between quantum and classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Holographic gravity, condensed matter theory, quantum information, and other areas have recently overlapped with each other on quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Numerous quantum entanglement measurements have been discovered to be capable of diagnosing the quantum phase transition of strong correlation systems and the topological quantum phase transitions, as well as playing a key role in the emergence of spacetime [1–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' There are numerous types of quantum entanglement measurements, including entangle- ment entropy (EE), mutual information (MI), R´enyi entanglement entropy, and negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Among these quantum entanglement measurements, EE is commonly considered a useful measure of pure state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, EE is not applicable to measure the more common mixed-state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' To measure mixed-state entanglement, numerous new 2 entanglement measurements, such as the entanglement of purification, non-negativity, and the entanglement of formation, have been proposed [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' On the other hand, calculating the mixed-state entanglement measures is extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Gauge/gravity duality is a powerful tool for analyzing strongly correlated systems because it connects entanglement-related physical quantities to geometric objects in dual gravity sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' In the dual gravity system, the holographic entanglement entropy (HEE) connects the EE of a subregion on the boundary with the area of the minimum surface [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' HEE has been demonstrated to be able to detect quantum phase transitions and thermodynamic phase transitions [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Recently, the R´enyi entropy was proposed to be proportional to the minimal area of cosmic branes [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Moreover, the butterfly effect that reflects the dy- namic properties of quantum systems, has been extensively studied in holographic theories [17–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' In addition, holographic duality of quantum complexity, a new information-related quantity from the EE, was also proposed [27–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' More recently, the EWCS was associated with the area of the minimum cross-section of the entanglement wedge [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The geomet- ric prescription of EWCS provides a novel and powerful tool for studying the mixed-state entanglement in holographic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Among all the models in holographic theories, the Born-Infeld (BI) model is a special class of models for nonlinear electromagnetic field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' It was first proposed to eliminate the divergent self-energy of Maxwell theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Later, it was found that the BI theory can be naturally derived from the string theory under the low-energy approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The BI model under the holographic theories can be dual to the quantum chromodynamics (QCD) systems [36, 37], and some condensed matter systems with novel transport behaviors, such as the quantum liquid [38], the Mott-insulator [39], and the novel magneto-resistance phenomenon [40, 41], which is consistent with the experimental phenomenon in [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Various prop- erties of the BI model, such as its thermodynamic properties, transport properties [44], the complexity [45], have been extensively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, the question of how exactly the BI factor b, which embodies the nonlinearity of this nonlinear electromagnetic field the- ory, affects the properties of the system, especially the mixed-state entanglement properties, remains to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This paper focuses on the effect of the BI factor on two measures of mixed-state en- tanglement - MI and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When b → 0, the background geometry is AdS-Schwarzschild solution, and the entanglement property of the system is decoupled from the transport prop- 3 erty of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' while for non-zero b, the transport behaviors can affect the entanglement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, we interpret b as the degree of correlation between the entanglement and transport properties of the metric when b increases from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Remind also that the coupling between the transport properties and the entanglement is also an important topic in condensed matter field theory, and is crucial for the construction of a stable quantum cir- cuit [46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For b → ∞, the system goes to the AdS-RN black brane system with a linear Maxwell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, the range b ∈ (0, ∞) represents the process that the Maxwell field turns on and converges to a linear Maxwell field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Our main goal is to explore how BI factor b affects the MI and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We organize this paper as follows: we introduce the holographic BI model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' II A, entanglement measures (HEE, MI, EWCS) and their holographic duality in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We discuss the properties of HEE, MI (III) and EWCS (IV) systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Finally, we summa- rize in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' HOLOGRAPHIC BORN-INFELD THEORY AND INFORMATION-RELATED QUANTITIES First, we review the holographic BI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Following that, we review the concepts of the HEE, MI, and EWCS with their holographic dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Then, we elaborate upon our algorithms proposed to calculate minimum surfaces and minimum cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The AdS Born-Infeld Model The action of the 4-dimensional holographic BI model is, S = � d4x√−g � R − 3Λ 16πG + b2 4πG � 1 − � 1 + 2F b2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (1) The parameter b is the BI factor, and Λ = − 3 l2 with l the AdS radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The solution of the BI theory is, ds2 = −f(r)dt2 + 1 f(r)dr2 + r2hijdxidxj, (2) with f(r) = r2 l2 − 2M r + 4Q22F1 � 1 4, 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' − Q2 b2r4 � 3r2 + 2b2r2 3 � 1 − � Q2 b2r4 + 1 � , (3) 4 Q is the electric charge and M is the mass of the black brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For l2 < 0 and l2 > 0 the system is asymptotically dS and AdS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Here, we fix l2 = 1 for concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For k = 1, 0, −1 the hij denotes a sphere, a Ricci flat surface, and a hyperbolic surface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Here, we focus on the planar case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=', k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' At the horizon r = rh we have f(rh) = 0, and hence we arrive at the ADM mass M = 4l2Q2 2F1 � 1 4, 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' − Q2 b2r4 h � − 2b2l2r4 h � Q2 b2r4 h + 1 + 2b2l2r4 h + 3r4 h 6l2rh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (4) The Hawking temperature is, T = rh 4π � 3 − 2b2 �� Q2 b2r4 h + 1 − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (5) The planar case is always thermodynamically stable [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, in this BI black brane system, there is no thermodynamic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The system is invariant under the rescaling, (t, 1/r, x, y) → α(t, 1/r, x, y), Q → Q/α2, T → T/α, rh → αrh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Other parameters such as b, β are all dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, we can fix rh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Here, we adopt √Q as the scaling unit, consequently, we need to divide physical quantity with scaling dimension [n] by Qn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For numerical convenience, we transform r into z ≡ rh/r such that the semi-infinite domain r ∈ (rh, ∞) becomes a finite domain z ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Then the metric becomes, ds2 = 1 z2 � −hdt2 + r2 hdz2 h + r2 hdx2 + r2 hdy2 � , (6) with h(z) ≡4 3Q2z3 � z 2F1 �1 4, 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' −Q2z4 b2 � − 2F1 �1 4, 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' −Q2 b2 �� − 2 3b2 � z3 � 1 − � Q2 b2 + 1 � + � Q2z4 b2 + 1 − 1 � − z3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (7) And the dimensionless Hawking temperature becomes, T = b2 � 2 − 2 � Q2 b2 + 1 � + 3 4π√Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (8) 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 1: The contour plot of the Hawking temperature in the plane (b, rh), where the temperature is only positive in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' From the dimensionless Hawking temperature (8) we can find that, lim Q→0 T → ∞, lim Q→ √ 12b2+9 2b T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (9) Also, we can find that, ∂QT = 2b � b − � b2 + Q2 � − 3 � Q2 b2 + 1 − 2Q2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (10) Therefore, the quantity Q is restricted to the range [0, √ 12b2+9 2b ] and that the temperature T decreases as Q increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This system is described by three variables (T, b , rh), with only two of them being independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We have also observed that for any given value of b, the temperature T always increases with increasing rh, thus, the value of rh is uniquely determined by a given temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This can be seen in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, we can simplify the system to a two-parameter system (b, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When the parameter b → ∞, the background solution of our system converges to the AdS-RN solution, and when b → 0, it becomes the AdS-Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When b is zero, there is an electromagnetic field present, but the background solution is still the AdS- Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This means that the entanglement-related physical quantities are not affected by the conductivity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, as b increases, the electromagnetic fields starts to affect the background solution, and thus has an impact on the entanglement 6 5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='77 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='44 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='11 0 1 0 0 1 2 3 4 5 bstructure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, we refer to increasing b from zero to infinity as the process of turning on the coupling between the background and the conductivity, and finally resulting in an AdS-RN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' It is worth noting that the relationship between conductivity and entanglement-related quantities is of great importance in condensed matter theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Recent experiments have shown that entanglement between quantum dots can persist despite the influence of surface plasmon polariton (SPPs) transmission [47, 48, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' These findings are crucial for the devel- opment of stable quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Additionally, it has been found that at specific values of the inter-dot distance d or detuning δ, the two-quantum-dot system can be in a highly entangled state and be separate from the transmission of SPPs [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, when d or δ deviate from these values, the entanglement of quantum dots becomes highly correlated with the transmission of SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This suggests that decoupling of entanglement and transport can exist in real physical systems and can be characterized by certain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Next, we will focus on how the entanglement-related physical quantities change as we vary the parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Holographic information-related quantities Entanglement is a fundamental and intriguing aspect of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' One way to quantify entanglement is through entanglement entropy (EE), which measures the degree of entanglement between a subset of a system and the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Specifically, the entanglement entropy SA between subsets A and B of a system A ∪ B is defined as the von Neumann entropy in terms of the reduced density matrix ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' SA(|ψ⟩) = −Tr [ρA log ρA] , ρA = TrB (|ψ⟩⟨ψ|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (11) It is easy to find that SA = SB for pure states [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Holographic duality theory relates the holographic entanglement entropy (HEE) to the area of the minimum surface in dual gravity systems [5] (see the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' EE is often used to measure the degree of entanglement in pure states, but it is not as effective in measuring mixed state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For example, even when subsystems A and B are not entangled, they can still have non-zero EE in a system composed of direct product of the density matrices of ρA and ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This is because EE takes into account both quantum 7 entanglement and classical correlation, so it does not always provide a accurate measure of the entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As a result, other measures for mixed-state entanglement have been proposed in the literature [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The most direct mixed-state entanglement measure is MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For the subsystem A ∪ C separated by B, the mutual information (MI) is defined as: I (A, B) := S (A) + S (B) − S (A ∪ B) , (12) This measures the mixed-state entanglement between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' It can be easily verified that I (A, B) = 0 when ρAB = ρA ⊗ ρB, therefore MI have the property that direct product states have zero entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, MI is not a perfect measure of mixed-state entan- glement, as it is closely related to EE, and it’s properties are sometimes dominated by EE or thermal entropy in certain situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This indicates that other measures of mixed-state entanglement should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The entanglement wedge cross-section (EWCS) has been associated with the duality of certain mixed-state entanglement measures, such as entanglement of purification, logarith- mic negativity, and reflect entropy [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Takayanagi proposed that EWCS EW (ρAB) is the area of the minimum cross-section ΣAB in connected entanglement wedge [34], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (see the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 2), EW (ρAB) = min ΣAB �Area (ΣAB) 4GN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (13) It is important to note that if the entanglement wedge is disconnected, meaning the minimum cross-section does not exist, the EWCS will be zero, it corresponds to cases with vanish- ing MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Additionally, the EWCS also satisfies some important inequalities as its quantum information counterparts [34, 56] Next, we present the algorithm for obtaining the minimum surfaces and EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Computations of holographic geometric quantities We examine the EWCS of an infinite strip with a homogeneous background for numerical simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For a generic homogeneous background ds2 = gttdt2 + gzzdz2 + gxxdx2 + gyydy2, (14) where z = 0 represents the boundary of the asymptotic AdS spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 3 is a visual representation of the minimum surface for an infinite strip along the y- 8 x y z x y z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 2: The left plot: The minimum surface for a given width w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The right plot: The minimum cross-section (green surface) of the entanglement wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Since the background is homogeneous, all metric components gµν only depend on the coordinate z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The minimum surface The minimum surface near the AdS boundary is perpendicular to the boundary, making the spatial direction x an unsuitable parameter for finding the minimum surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' [58] adopted the angle θ with tan θ = z/x, as the parameter for the minimum surface (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Using this method, we can parametrize a surface as (x(θ), z(θ)) with area A given by A = 2 � π/2 0 � x′(θ)2gxxgyy + z′(θ)2gyygzzdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (15) The resultant equations of motion read, x′(θ)z′(θ)2 � g′ xx 2gxx + g′ yy gyy − g′ zz 2gzz � + x′(θ)3 � gyyg′ xx + gxxg′ yy � 2gxxgzz + x′′(θ)z′(θ) − x′(θ)z′′(θ) = 0, z(θ) − tan(θ)x(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (16) where g′ ## ≡ g′ ##(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The boundary conditions are, z(0) = 0, x(0) = w, z′(π/2) = 0, x(π/2) = 0, (17) where w is the width of the strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The EWCS Given a biparty subsystem with minimum surfaces C1(θ1), C2(θ2), we solve the minimum surface Cp1,p2 connecting p1 ∈ C1 and p2 ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We parametrize Cp1,p2 with z, then the area of Cp1,p2 reads, A = � Cp1,p2 � gxxgyyx′(z)2 + gxxgzzdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (18) The resultant equation of motion becomes, x′(z)3 � gxxg′ yy 2gyygzz + g′ xx 2gzz � + x′(z) �g′ xx gxx + g′ yy 2gyy − g′ zz 2gzz � + x′′(z) = 0, (19) with boundary conditions, x(z(θ1)) = x(θ1), x(z(θ2)) = x(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (20) To obtain the EWCS, we need to locate the global minimum of the minimum surfaces connecting C1(θ1), C2(θ2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=', the minimum cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Finding the minimum cross-section is a challenging task as it involves searching through a two-dimensional parameter space (θ1, θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, it can be noted that the globally minimum cross-section must be perpendicular to the minimum surfaces at the point of intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This observation serves as a local constraint, which can greatly speed up the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We demonstrate the methods of solving the EWCS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For numerical stability, it is better to implement the perpendicular conditions with normalized vectors as, Q1(θ1, θ2) ≡ gab � ∂ ∂z �a � ∂ ∂θ1 �b � gcd � ∂ ∂z �c � ∂ ∂z �d � gmn � ∂ ∂θ1 �m � ∂ ∂θ1 �n �������� p1 = 0, Q2(θ1, θ2) ≡ gab � ∂ ∂z �a � ∂ ∂θ2 �b � gcd � ∂ ∂z �c � ∂ ∂z �d � gmn � ∂ ∂θ2 �m � ∂ ∂θ2 �n �������� p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (21) Note that Q1 and Q2 are both functions of the θ1 and θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Now, the search of the EWCS is equivalent to finding the minimum surface ending at (θ1, θ2) where (21) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' To determine the correct EWCS, we first select an initial seed (θ1, θ2) and use the Newton iterative method to obtain feedback (δθ1, δθ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' By repeating this process, we can find the minimum cross-section, which is the EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' It is crucial to carefully choose the initial 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 z p1 p2 θ1 θ2 C1(θ1) C2(θ2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 3: The demonstration of the EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The p1 and p2 are the intersection points of the minimum surface connecting those two minimum surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The solid blue curve (parametrized with θ1) and solid orange curve (parametrized with θ2) are minimum surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The thick red curve is the minimum surface connecting p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The blue arrows at the p1 and p2 are the tangent vector � ∂ ∂z �a��� p1 and � ∂ ∂z �a��� p2 along the Cp1,p2, while the purple arrows are the tangent vectors � ∂ ∂θ1 �a��� p1 and � ∂ ∂θ2 �a��� p2 along C1, C2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The dark dashed horizontal line is the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' values of (θ1, , θ2) for the iterations to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The numerical reliability is ensured by the convergence of results when using different initial values or increasing the density of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For more technical details, refer to reference [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Using the techniques outlined above, we will now examine mixed-state entanglement measures for the BI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Additionally, we will examine the correlation between the BI factor b and information-related quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' THE HOLOGRAPHIC ENTANGLEMENT ENTROPY AND THE HOLO- GRAPHIC MUTUAL INFORMATION We begin by examining the relationship between HEE, system parameters b and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 4, HEE, represented by S, increases monotonically with both b and T, but their rate of increase is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Initially, S increases slowly with T and its growth rate with T becomes more pronounced as T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' On the other hand, S increases quickly with b at first and then slows down as b decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Next, we explain the behavior of S with 11 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0001000 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='008791 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='01765 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='02868 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='03912 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='05289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='25 T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='60 S w=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='00100 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='09967 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1389 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1614 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1856 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='55 S w=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 4: HEE vs T and b at width w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' b and T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When the horizon radius of the black brane increases, the minimum surface tends to be closer to the horizon of the black brane, which makes the thermodynamic entropy dominate the behavior of the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, the growth of HEE with T as well as b, can be un- derstood from the relation between rh and T or b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' According to (5) we can deduce that rh increases with increasing temperature and b, this can be seen by taking the derivative of rh with respect to T and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The results are, ∂Trh = r4 h � Q2 b2r4 h + 1 r4 h � 2b2 �� Q2 b2r4 h + 1 − 1 � + 3 � Q2 b2r4 h + 1 � + 2Q2, ∂brh = 2rh � Q2 − 2b2r4 h �� Q2 b2r4 h + 1 − 1 �� b � r4 h � 2b2 �� Q2 b2r4 h + 1 − 1 � + 3 � Q2 b2r4 h + 1 � + 2Q2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' (22) From the above equation, it is clear that ∂Trh is always positive, indicating that rh increases as T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, ∂brh can be positive or negative, depending on the specific pa- rameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Further examination shows that ∂brh is always greater than zero when rh is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This means that rh increases with b when rh is large, or when the minimum surface is closer to the horizon of the black brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When b is relatively large, the system is approximately the AdS-RN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The ar- gument presented in [59] can be applied to prove that ∂TS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Furthermore, for small subregions, it can be inferred from the equations in [59] that ∂TS is close to 0, which ex- plains the flat behavior of S along T for small temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' After studying HEE, we proceed to investigate the behavior of MI with T and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' In the BI model, the configurations for MI and EWCS are subsystems composed of a and b separated 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 b T I(3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='10, 2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='97 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='31 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='09 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 b T I(3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='25, 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5: MI as a function of b and T for different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0001000 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='04554 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='09392 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1591 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2526 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5 I (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='35) T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='00100 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1182 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1614 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1856 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2110 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='55 I (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='35) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 6: MI as a function of b and T for different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' by region p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5, MI decreases with increasing temperature and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This is in contrast to the behavior of HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Moreover, it is worth noting that MI can decrease to zero, which is an indication of a disentanglement phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We have also plotted the MI for smaller configurations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 6), and the qualitative phenomena remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As the subsystem c and the separation p change, the system undergoes a disentangling phase transition, at which point the entanglement of two subsystems a and c vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The critical value of subsystem cc and separation pc are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 7 shows that the critical value of subsystem cc increases with b and T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' however, the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 7 shows that the critical value of the separation pc decreases with b and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This is as expected since increasing the temperature or b will tends to destroy the entanglement, 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2366 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 7: Critical configurations of cc and pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' resulting in a larger subregion cc or a smaller separation pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Next, we explore the mixed-state entanglement through the EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' THE HOLOGRAPHIC ENTANGLEMENT WEDGE CROSS-SECTION In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 8, we present the minimum surfaces and the corresponding minimum cross- sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' It can be observed that the minimum surface is flatter when the temperature is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This is due to the fact that the coordinate z is related to the horizon radius rh, and at lower temperatures, a small rh will rescale z to zrh, resulting in a flatter minimum surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This makes it challenging to obtain precise enough solutions for the minimum surface since the flat case is more singular in the θ coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' To overcome this issue, we redefine the angle as z = ηx tan(θ), where η is a number related to the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Only with this technique, we can achieve precise enough solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We show the EWCS vs b in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 9, from which we can find that the EWCS can show very delicate behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The EWCS increases with b at first in a very narrow range of b, however, it starts to decrease with b when b is relatively large and monotonically decreases with b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This is in sharp contrast to the behavior of the HEE and MI, that only shows monotonical behaviors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' In addition, the EW changes slower with b than that with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The typical change is of order 10−4 and 10−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This delicate behavior can be captured precisely because the precision of our numerical methods can be up to 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Notice that the background is an AdS-Schwarzschild solution when b is 0, meanwhile, its 14 ���� ���� ���� ���� � ���� ���� ���� ���� ���� ���� � �=������ �=������ �=������ �=������ �=������ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 8: The illustration of EWCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' At the same configuration (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='06925) we see that the minimum surface becomes flatter when decreasing the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Meanwhile, the minimum cross-section always ends at the point near the tops of the inner minimum surface, while ends at the point away from the tops of the outer minimum surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4b 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='360 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='365 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='370 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='375 Ew T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1565 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2492 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2537 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2654 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2758 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='20b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='04 ∂bEw FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 9: EWCS vs T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This plot is obtained at (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='06925).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When b is relatively large, the EW converges to certain fixed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2971 it can first increase, and later decreases, and after that increases with b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, for very small b the EWCS increases with b, irrespective of the values of the T and the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' electromagnetic field is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' At this point, the charge transport behavior of the system is significantly different from that of the genuine AdS-Schwarzschild system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Moreover, since its geometry is still AdS-Schwarzschild, the entanglement-related geometric quantities will be decoupled from the charge transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As b gradually increases, the background geometry will receive back reactions from the Maxwell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' At this time, the entanglement- related geometry starts to couple with the charge transports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, b can play a role in measuring the relationship between entanglement and transport when b is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As we 15 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1211 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1364 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1461 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1569 T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='864 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='866 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='868 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='870 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='872 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='874 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='876 Ew FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 10: The EWCS vs b for a larger configuration (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3875).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' have pointed out, the EWCS increases with b when b is very small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=', when the coupling has just occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' And when b increases further, the EWCS gradually shows a decreasing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Notice that simpler geometric quantities such as HEE, and MI only show a very flat monotonic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This indicates that EWCS, as a mixed-state entanglement, captures very different properties from HEE and MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' To understand the above behavior more clearly, we implement the following analytical treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' For small values of b, we can expand the expression of the EW (18) integral with respect to b as, EW = � Σ � � 1 z2 � dx2 + dz2 (1 − z3) + bdz2 � Γ � 1 4 � + 8Γ � 5 4 �� � dz2 + dx2 (1 − z3) �−1/2 2 � 3πT (1 − z3) (z2 + z + 1) Γ � 1 4 � + O(b2) � � , (23) where the second term shows us that dEW db > 0 for small values of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This explains the ubiquitous existence of the monotonically increasing behavior of EW vs b for small values of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' From the holographic dual picture, it means that when the Maxwell field starts to turn on from the BI case, the EW is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, when further increasing b we find that EW reaches local maximums and starts to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When b is large, it can be expected that the background system approaches the AdS-RN, a fixed background geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Therefore, the EW will starts to converge to some fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Next, we show the EWCS in larger configurations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 10, the non-monotonicity of EW with b becomes more pronounced as the width of the configuration increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This means that the non-monotonicity exists over a wider interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The reason for this is that when the width is relatively small, the minimum surface and the minimal 16 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1355 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1753 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2492 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2758 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='40T 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='32 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='33 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='34 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='35 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='36 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='37 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='38 Ew b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0100 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0744 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1389 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2033 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='27T 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='83 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='84 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='85 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='86 Ew FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 11: EWCS vs T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' The left plot is obtained at (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='06925);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' while the right plot is obtained for a larger configuration (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3875).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' cross-section are only slightly different from the properties of AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' However, as the width increases, they deviate more significantly from AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Next, we examine the behavior of EWCS with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' When the configuration is relatively small in BI systems, EWCS decreases monotonically with temperature, as shown in the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' It is worth noting that the non-monotonic behavior of EWCS at extremely small temperatures has been studied in [59] for AdS-RN systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Additionally, we illustrate the behavior of EWCS with temperature for larger configurations in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 11, which also shows that EWCS decreases monotonically with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Although the monotonic decreasing behaviors are similar, the EWCS curves for small con- figurations differ from those for large configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' By comparing the two plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 11, crossovers of the EWCS curves with temperature can be observed in the larger configu- ration, which reflects the non-monotonic behavior of EWCS with b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' These findings suggest that the behavior of EWCS is generally monotonically decreasing with temperature, and this behavior is consistent with that of MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' In order to more clearly demonstrate the relationship between the EWCS and variables b and T, a contour plot of EWCS as a function of b and T is presented in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This plot illustrates the non-monotonic nature of EWCS with respect to b and the monotonic decrease of EWCS as T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 12: The EWCS vs b for a larger configuration (a, p, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3875).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' DISCUSSION In this paper, we study the behavior of HEE, MI, and the mixed-state entanglement measure EWCS in the BI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Our results shows that HEE increases monotonically with both b and T, while MI decreases monotonically with both b and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Interestingly, the behavior of EWCS with respect to b shows a non-monotonic trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Specifically, when b is small, EWCS increases with b, but it begins to decrease as b increases further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' In contrast, EWCS exhibits a consistent monotonically decreasing trend with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Moreover, we provide analytical explanations for the non-monotonic behavior of EWCS with respect to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Note that when b is small, b serves as a measure of the coupling between the entanglement-related quantities and the charge transport of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Based on this observation, we conjecture that increasing the coupling between the entanglement-related quantities and the transport properties can enhance the EWCS of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This cou- pling between transport behaviors and entanglement is also a topic of significant interest in condensed matter theory, as seen in previous studies on nanowires [46], plasmonics [47, 50], and plasmons [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 18 Ew at (a,p,c)=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3875) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='066 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='930 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='862 T 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='726 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='658 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='15 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='4 bNext, we point out the issues that deserve further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' To begin, we can exam- ine other BI-like theories, such as the BI theory with massive gravity, the BI theory with Axions, and so on, to see if the non-monotonic behavior observed in this paper is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Furthermore, we can examine the effect of more general nonlinear EM field theories on the entanglement-related physical quantities of the system, such as the more general nonlinear EM fields [39, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' We are working on these directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Acknowledgments Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Zhe Yang would like to express appreciation to Feng-Ying Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' This work is supported by the Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' 11805083, 11905083, 12005077, 12147209, the Science and Technology Planning Project of Guangzhou (202201010655) and Guangdong Basic and Applied Basic Research Foundation (2021A1515012374).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' is also supported by Top Talent Support Program from Yangzhou University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Osterloh, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Amico, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Falci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Fazio, “Scaling of Entanglement close to a Quantum Phase Transitions” Nature 416, 608 (2002) [arXiv:0202029 [quant-ph]] [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQfBwtK/content/2301.04854v1.pdf'} +page_content=' Amico, R.' metadata={'source': 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b/7dFLT4oBgHgl3EQfAS4p/content/tmp_files/2301.11965v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..626d9c6ee8ae61ec0930e6118f25aee5a2fab023 --- /dev/null +++ b/7dFLT4oBgHgl3EQfAS4p/content/tmp_files/2301.11965v1.pdf.txt @@ -0,0 +1,1773 @@ +The persistent homology of genealogical networks +Zachary M. Boyda,∗, Nick Callorb, Taylor Gledhillc, Abigail Jenkinsd, Robert Snellmane, Benjamin Webbf, +Raelynn Wonnacottg +aDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, zach boyd@byu.edu +bDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, n.b.callor@gmail.com +cDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, gledhilltaylor2@gmail.com +dDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, jenkins.abby@gmail.com +eDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, snellman@mathematics.byu.edu +fDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, bwebb@mathematics.byu.edu +gDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, raelynnwo@gmail.com +Abstract +Genealogical networks (i.e. family trees) are of growing interest, with the largest known data sets now +including well over one billion individuals. Interest in family history also supports an 8.5 billion dollar +industry whose size is projected to double within 7 years [FutureWise report HC-1137]. Yet little mathemat- +ical attention has been paid to the complex network properties of genealogical networks, especially at large +scales. +The structure of genealogical networks is of particular interest due to the practice of forming unions, e.g. +marriages, that are typically well outside one’s immediate family. In most other networks, including other +social networks, no equivalent restriction exists on the distance at which relationships form. To study the +effect this has on genealogical networks we use persistent homology to identify and compare the structure +of 101 genealogical and 31 other social networks. Specifically, we introduce the notion of a network’s +persistence curve, which encodes the network’s set of persistence intervals. We find that the persistence +curves of genealogical networks have a distinct structure when compared to other social networks. This +difference in structure also extends to subnetworks of genealogical and social networks suggesting that, even +with incomplete data, persistent homology can be used to meaningfully analyze genealogical networks. Here +we also describe how concepts from genealogical networks, such as common ancestor cycles, are represented +using persistent homology. We expect that persistent homology tools will become increasingly important in +genealogical exploration as popular interest in ancestry research continues to expand. +Keywords: persistent homology, genealogical networks, social networks, persistence curves, bottleneck +distance +1. Introduction +The study of genealogical networks, that is networks relating parents with children and spouses with +each other through successive generations is of rapidly growing interest, both because of genealogy’s pop- +ular appeal and its applications in genetics [1], sociology [2], population sciences [3], and economics [4]. +Growing data availability of rich, temporally resolved data is also driving interest in genealogy. For example, +∗Corresponding Author +arXiv:2301.11965v1 [q-bio.MN] 27 Jan 2023 + +FamilySearch has constructed a human family tree with over 1.40 billion individuals, based on 2.21 billion +sources, including 4.78 billion images (https://www.familysearch.org/en/newsroom/company-facts). Popu- +larization of DNA testing services and increasing availability of audio sources, geographic tags, occupation +metadata, and migration records combine to make genealogical networks some of the largest, most richly +featured, geospatially embedded temporal networks in existence. Examples of relevant academic studies +include methods for automatically constructing networks from documents [5, 6], analyzing marriage pat- +terns [4], structured population modeling, branching processes [7], and biconnected components [2, 8]. Of +particular interest to us are works that study distance to recent common ancestors, both theoretically and via +simulation (e.g. [9, 3]). A growing body of literature also uses genealogical networks for genetic inference, +as in [1]. +Related to these genealogical endeavors, a major goal of network science is to describe the structure +of such real-world networks. In this paper, we consider persistent homology as a tool to both analyze and +explore the structure of genealogical networks. Persistent homology, roughly speaking, is a method of +representing voids or gaps in the structure of a network, that distinguishes how significant these voids are to +the overall network structure. Persistent homology can be used to compare these voids across two networks +without requiring a correspondence between the individual vertices or edges, or even requiring the networks +to be the same size. The basic idea involves “filling in” the network with simplices (points, edges, triangles, +tetrahedra, etc.) and keeping track of how the network changes as we do so (see Section 3 for details). +Some similar applications of persistent homology in the study of networks include [26], [28], [30], [27]. +The collaboration networks studied in [26] are similar to the social networks that we use for comparison +in this paper, though our focus is primarily on distinguishing these from genealogical networks. Both [28] +and [30] apply persistent homology techniques to general randomized networks of various forms. It is also +possible to vary the technique for generating a topological object from a network, as in [27] where three +methods are compared. We also recommend [29] and [36] as good overviews of the general methods of +applying persistent homology. +For this paper, our method of constructing a topological representative for each network follows the same +general pattern as the work cited above. However, we also acknowledge the wide variety of alternatives for +encoding such information. [32] and [33] encode their information as point-clouds rather than graphs. A +higher-dimensional version of persistent homology is presented in [34], which may permit the inclusion +of time-varying networks. Finally, the formulation in [35] may allow for better analysis of corrupted or +too-large datasets. +We also wish to bring attention to four particular applications that demonstrate the versatility of persistent +homology. In each of these applications, persistent homology has been used to identify structural voids in +data and then to associate these voids to recognizable features in the underlying networks. It is the latter +use that we wish to emphasize. Robins et al. [10] have shown that voids found using persistent homology +correspond to percolating spheres in a porous material. In [11], structural voids arise when several groups +of neurons are strongly connected sequentially, but out-of-sequence pairs are only weakly connected. In +these neurological networks, persistent homology provides a way to identify and classify these different +sequences as well as quantify the strength of these connections. The application in [31] provides a method +for extending traditional genetic analysis tools to a parameterized family of datasets by constructing an +appropriate topological object. Lastly, [12] shows that structural voids or gaps can also represent much +more abstract concepts. In this case persistent voids are shown to correspond to the atonality in music +compositions. +Intuitively, the voids or gaps in genealogical networks should be quite different when compared with +2 + +other networks, such as social networks, since unions1 (such as marriages) in genealogical networks typically +form at specific distances, rather than through other mechanisms e.g. triadic closure. That is, distances +between individuals who form unions are typically not too small or too large (see Section 2). In contrast, +in other social networks, new connections can form at any distance but are often quite small [13]. This +difference in network growth between genealogical and other social networks causes differences in network +topology that are reflected in the network’s persistent homology. Thus persistent homology is a useful +descriptive tool for exploring and modeling the structure of genealogical networks. +Here, we propose a new method for representing persistent homology, which we call a persistence curve +(see Section 4). The persistence curves of many genealogical networks are very similar to each other, +and importantly the persistence curves of subsets of genealogical networks, that is, sampled genealogical +networks, are also similar to the persistence curves of unsampled genealogical networks (see Section 6). +To give our study of genealogical networks context we also study the persistent homology of social net- +works. We find that the same result holds for the social networks we consider, in that the persistence curves +of social networks show a common pattern and the persistence curves for social and sampled social networks +are similar (see Section 6). We confirm our analysis using another tool for comparing persistent homologies, +the bottleneck distance, which is also capable of detecting and differentiating the distinct homology patterns +between genealogical and other social networks. +In summary, we make the following contributions: +• Introduce the notion of a persistence curve and introduce the use of this together with the bottleneck +distance as a tool for the analysis of general networks. +• Report the distinct persistent homology structure of genealogical networks using both persistence +curves and the bottleneck distance. +• Link this structure to genealogically relevant concepts. +• Similarly, report the distinct persistence homology structure of social networks and compare this to +the structure of genealogical networks. +• Report evidence that persistent homology methods work well even in the presence of incomplete data. +This is particularly relevant given that genealogical data is often, if not necessarily, incomplete. +Throughout the paper, examples from family networks are contrasted with other social networks to highlight +the unique features of genealogical networks from a persistent homology point of view. +The paper is organized as follows. In Section 2 we describe both genealogical and social networks. In +Section 3 we define the persistent homology of a network and introduce the notion of persistence curves. In +Section 4 we define the bottleneck distance and show how both this distance and persistence curves can be +used to compare networks. In Section 5 we describe the genealogical and social data sets we use in our study +and give our experimental results in Section 6. Section 6 also includes a discussion of how certain structural +features of social and genealogical networks are represented using persistent homology. In Section 7 we +summarize our results and conclude with a discussion regarding the use of persistent homology as a tool for +analyzing general network structure and recovering network features. Throughout we give examples of each +of the concepts we introduce. +1In order to be inclusive of various relevant relationships in this paper, we use the word “union” to describe not only legal marriages +and common law marriages but also some others, including any relationship that produced children. +3 + +2. Background: Genealogical and Social Networks +We represent genealogical networks with a graph G = (V, E), where V = {1, 2, . . . , n} are the individ- +uals within the network, and E are the (genealogical) relationships. These relationships consist of both +parent-child edges and spouse (or more generally union) edges. For the sake of simplicity, these edges are +considered to be undirected. We note that the structure of a genealogical network is often thought of as +Out[1858]= +Tikopia Genealogical Network +Residence Hall Social Network +Figure 1: Left: The largest connected component of the Tikopia genealogical network consisting of 288 individuals from the island of +Tikopia in Polynesia from the year 1930 to 2008, is shown [14]. Parent-child edges are shown in blue and union edges are shown in +red. Right: The largest connected component of the Residence Hall social network consisting of 217 individuals and their friendships +from the Australian National University campus is shown [15]. +being “tree-like”, since genealogical networks are often constructed from an individual, their parents, their +grandparents, and so on, ignoring union edges. The result is a tree, i.e. a connected acyclic graph, if we +create only a few generations of the family. However, full genealogical networks are not trees due to the +presence, for example, of triangles consisting of two parents and a child (with the two parent-child edges +and one union edge). Because of the frequency of such cycles and the fact that they are the smallest possible +cycles, we refer to them as trivial cycles. The other typical familial cycle, or cycle found within a family +consisting of two parents and some number of children, is a cycle of length four consisting of two parents +and two children. +Although familial cycles are ubiquitous in genealogical networks, they are not the only cycles that can +form. Going far enough through an individual’s ancestors, it is often possible to find a nearest common +ancestor, i.e., a common ancestor of one’s father and mother. If such an ancestor exists (and it usually does +exist), then the genealogical network has a nontrivial cycle. We refer to this as a common ancestor cycle, +which consists of only parent-child edges. Other nontrivial cycles are possible in genealogical networks via +unions. For instance, a “double cousins” relationship occurs when two siblings from one family form unions +with two siblings from another family. The result is a union cycle, or a cycle that contains only union edges +and the parent-child edges connecting siblings. In genealogical networks, union and parent-child edges can +combine in any number of ways to create complex non-tree structures (see Figure 1 left). +A feature that is particular to genealogical networks is that union edges typically form at specific dis- +tances within these networks. Here the distance d(i, j) between i and j is the shortest path distance between +these individuals if such a path exists. Otherwise, it is infinite. In a genealogical network we refer to the +distance between two individuals before they form a union as the couple’s distance to union. For cultural, +genetic, and other reasons these distance are typically not small, i.e. usually larger than four. Consequently, +4 + +0 +5 +10 +15 +20 +25 +30 +0.00 +0.05 +0.10 +0.15 +Distance to Union +Fraction of Unions at Distance +Figure 2: The histogram representing the finite “distance to union” distances is shown where data is collected from 104 genealogical +networks from kinsources.net. The height of each bar represents the fraction of unions that form at a specific distance. +genealogical networks do not typically have small nonfamilial cycles and often have large extended cycles. +This is illustrated in Figure 2 where distance to union data is collected from 104 publicly available genealog- +ical networks given in Table 2 in the Appendix. Here familial cycles are omitted and the height of each bar +represents the fraction of unions that form at a specific distance. Noticeably, few unions form at distances +less than five with the large majority of distance falling between 5 and 10. +The observation that genealogical networks have large extended cycles is illustrated in Figure 3. Shown +left in orange is the distribution of cycle lengths of the San Marino genealogical network, a network of the +population of the Republic of San Marino from the 15th to the end of the 19th century [14]. In this network, +which consists of 28,586 individuals, there are 7,146 familial cycles of length three and 8,636 familial cycles +of length four. These are omitted in the figure so we can observe the lengths of the cycles forming a basis +of nonfamilial cycles in the network. For the sake of contrast, in blue is the distribution of cycle lengths +in a basis of the cycles found in the Deezer Europe social network, consisting of 28,281 individuals. Here, +similar to genealogical networks, a social network is represented by a graph G = (V, E) where the vertices V +also represent individuals. The difference is that in a social network the edges represent some type of social +interaction(s). The Deezer network is an online music streaming platform whose social network represents +individuals in Europe who use the platform where edges represent mutual user-follower relationships. +Noticeably, the San Marino network has relatively few nonfamilial basis cycles under length ten but +quite a few cycles with lengths greater than thirty. In contrast, the Deezer social network has a much tighter +distribution of basis cycles ranging from roughly five to fifteen in length. +To understand the extent to which these cycle distributions are related to the local structure of the associ- +ated networks we compare these to the cycle distribution of the associated configuration models of these two +networks, respectively. The configuration model is a model for generating random networks with a given +degree sequence [16]. Taking the degree sequences from both the San Marino genealogical and Deezer so- +cial network, we create ten versions of these networks each with the same degree sequences. The result of +averaging the basis cycle length distributions of these versions of the San Marino and Deezer networks is +shown in Figure 3 (center and right in red and green, respectively). While the cycle distribution for the San +Marino network is quite different from what the configuration model produces, the Deezer social network +is quite similar to the distribution predicted by its configuration model. This suggests that much of the cy- +cle structure in the Deezer social network is dominated by local interactions, whereas the cycles in the San +Marino genealogical network are affected by nonlocal mechanisms that form the network. This includes, +5 + +Out[51]= +0 +10 +20 +30 +40 +0.00 +0.05 +0.10 +0.15 +0.20 +0 +10 +20 +30 +40 +0.00 +0.05 +0.10 +0.15 +0.20 +0 +5 +10 +15 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +SM and DE Cycle Lengths +SM Configuration Model +DE Configuration Model +Figure 3: Left: Shown in orange is the distribution of the lengths of the cycles forming a basis of the nonfamilial cycle lengths in the +San Marino (SM) genealogical network. The analogous distribution of cycle lengths is shown in blue for all cycles in the Deezer Europe +(DE) social network. Center: Shown in orange is again the basis cycle length distribution of the San Marino genealogical network. In +red is the distribution of the basis cycle lengths averaged over ten realizations of the (loopy, multi-edged) configuration model on the +San Marino network. Since the configuration model generates graphs with the same degree distribution as the SM network, this panel +indicates that SM’s longer cycles do not arise simply from the degree distribution. Right: Shown in blue is again the basis cycle length +distribution of the Deezer social network. In green is the distribution of the basis cycle lengths averaged over ten realizations of the +configuration model on the Deezer social network. For this social network, the cycle length distribution can be mostly explained by the +degree distribution alone. +presumably, the nonlocal distance to union phenomena described above. +The relations we see in Figure 3 between the cycle length distribution for the San Marino genealogical +network and the Deezer social network are typical of the genealogical and social networks we consider in +Section 5. This suggests that cycle length distribution is a feature that can be used to distinguish genealog- +ical from social networks. Specifically, when we consider two networks with a similar number of cycles, +genealogical networks have a much wider distribution of cycle lengths than social networks. However, the +method used to calculate the cycle length distribution in Figure 3 does not provide any further insight into +this phenomenon. This limitation motivates us to apply tools from persistent homology which provides ways +to describe and measure the relation between any two network cycles. The additional structure that can be +obtained by these methods allow us to further distinguish the structure of genealogical and social networks +(see Section 6.1) and to relate the structural differences demonstrated in Figure 3 to mechanisms that produce +genealogical and social networks, respectively (see Section 6.3). +3. Persistent Homology of Networks +Persistent homology provides a method for studying cycles in a network. For the purposes of this paper, +a brief explanation of persistent homology will be given from the context of simplicial homology. For a +more in-depth treatment of simplicial homology, see Chapter 2.1 of [17]. For those readers who are either +familiar with the basics of persistent homology or who wish to skip the following technical discussion it is +possible to proceed to Section 5 where we discuss the social and genealogical networks we analyze. +For a network given by a graph G = (V, E) we define the distance matrix D(G) = [di j] to have entries +di j = d(i, j), which is the length of the shortest path between individual i and j. For each value δ that +appears in the distance matrix D(G), we form a simplicial complex Gδ as follows. The set of 0-simplices +is equivalent to the set of vertices of G, where each 0-simplex is identified with a single vertex. Since the +distinction between 0-simplices and vertices is purely formal, we will use the terms 0-simplex and vertex +interchangeably, and the 0-simplices will be indexed the same way as the vertices. The set of 1-simplices Eδ +corresponds to the set of edges {i, j} such that d(i, j) ≤ δ, where the edge {i, j} is identified with the 1-simplex +6 + +formed by i and j. Again the distinction here is unnecessary for our present discussion, so we will use the +same notation for 1-simplices and edges. However, the simplicial complex Gδ may also contain objects that +do not have equivalent representatives in the graph G, namely the n-simplices for n ≥ 2. For each integer +n ≥ 2, the set of n-simplices in Gδ consists of all n-simplices [a0 +a1 +. . . +an] such that d(ai, aj) ≤ δ for +0 ≤ i < j ≤ n. That is, Gδ includes an n-simplex σ if each vertex listed in σ is within δ of every vertex listed +in σ. +In order to simplify our remaining definitions, we extend our definition of Gδ to include all non-negative +integers. For i ≥ 0, let δi be the greatest entry of D(G) such that δi ≤ i. Let Gi = Gδi. This definition together +with our construction of Gδ ensures the following three important properties are true for all Gi. +1. For i < j, Gi is a subcomplex of G j, i.e. every simplex of Gi is a simplex of G j. +2. For i ≥ 1, there exists a subcomplex of Gi that can be identified with the original graph G. +3. Since G is finite, let M = maxij d(i, j), then, for all i ≥ M, Gi = GM. +(a) G0 +(b) G1 = G +(c) G2 +(d) G3 +Figure 4: The hexagonal network G = G1 in Example 3.1 is filled in as i increases from 0 to 3. This produces the simplicial complexes +G0,G1,G2,G3 shown left to right. +Example 3.1. (Hexagonal Network) Consider the hexagonal network G = (V, E) with six vertices, forming +a single cycle, shown in Figure 4(b). This network has the distance matrix +D(G) = +������������������������� +0 +1 +2 +3 +2 +1 +1 +0 +1 +2 +3 +2 +2 +1 +0 +1 +2 +3 +3 +2 +1 +0 +1 +2 +2 +3 +2 +1 +0 +1 +1 +2 +3 +2 +1 +0 +������������������������� +. +For the values i = 0, 1, 2, 3, we form four simplicial complexes, G0, G1, G2, and G3 where we let Gi = +(Vi, Ei). For i = 0, E0 is empty. Thus, G0 consists of six vertices. For i = 1 the set E1 contains the six +edges that form the network’s single cycle, so G1 = G. This graph has no trivial cycles (i.e., triangles), so +G1 contains no simplices of dimension greater than 1 (i.e., no n-simplices for n > 1). For i = 2 the set E2 +gains six additional edges. We also now have eight trivial cycles. Each of these cycles is the boundary of +a 2-simplex, so G2 contains these eight 2-simplices as well. However, no subset of these 2-simplices forms +the boundary of a 3-simplex, so G2 has no simplices of dimension greater than 2. For i = 3 the set E3 +contains all possible edges between the vertices of G, so all possible trivial cycles are present. Additionally, +all possible 2-simplices, and hence all possible n-simplices, are also present in G3. In particular, G3 is a +6-simplex with its boundary. Since M = 3 is the largest value we see in the distance matrix, then Gi = G3 +for i ∈ Z, i > 3. +7 + +The persistent homology of the network G measures how the homology of Gi changes as i increases. If +certain features can be identified across multiple values of i, we say they persist. Intuitively, features that +arise from the actual network structure should persist for many values of i, while features that arise because +of measurement error, ‘noise’, should only appear sporadically. The Stability Theorem (the Main Theorem +of [18]) states that if the error in measuring a network is bounded by some constant C, then the persistent +homology of the true network and the persistent homology of the noisy network will differ by at most C. We +will make this statement more precise in Section 4.1. +Here we give a formal definition of persistent homology in terms of simplicial homology, which we will +immediately follow this with equivalent definitions in the context of networks. We use Hp(Gi) to denote the +dimension-p simplicial homology of the simplicial complex Gi with coefficients in Z2, as Hp(X) is a vector +space of Z2. +Definition 1. (pth Persistent Homology) For a graph G, and integers i, j with 0 ≤ i ≤ j, let the function +φi, j : Hp(Gi) → Hp(G j) be the linear map induced by the inclusion Gi → G j. The pth persistent homology +of G, PHp(G) is the pair ({Hp(Gi)}i≥0, {φi,j}0≤i 0, H0(Gi) � Z2, which is equivalent to the vector space over Z2 with basis +{1}. For any v ∈ V, since i = 0 is the first time we see v, we call this the birth of v. At i = 1, since we have +removed all vertices except 1 from the basis, we say this is the death of those five 0-simplices. Since 1 will +always be in the basis for Gi, the death of 1 is said to be ∞. +Definition 4. (Representing Persistent Homology: Dimension 1) Let G = (V, E) be a network with one +connected component. For each i ≥ 0, we can identify the basis of H1(Gi) with a set Ci of cycles in Gi. The +Fundamental Theorem of Persistent Homology allows us to choose these cycles so that if σ is a cycle in Ci, +then exactly one of the following is true for any integer j ≥ 0: +1. σ does not exist in G j, in which case j < i, +2. σ is trivial or null-homotopic in G j, in which case i < j, +3. σ is a cycle in C j. +Thus, we will refer to the cycles in � +i≥0 Ci as the representatives of PH1(G). (Again, PH1(G) is actually +much larger than this. These are actually representatives of equivalence classes that form a basis for PH1(G) +as a vector space.) +We note that C0 is always empty, since there are no edges in G0. Furthermore, rank(H1(Gi)) = |Ci| for +all i ≥ 0. Because of the construction of the Gi all representatives of PH1(G) will be present in G1. One +can think of the representatives of PH1(G) as representing “large” cycles. More specifically, if a cycle σ is +contained in � +s≤i≤t Ci, then it must have a diameter of at least t and at least one pair of consecutive vertices +distance s apart. +Example 3.4. We now consider PH1(G) for the hexagonal network G in Figure 4(b). In both Figure 4(a) +and 4(b) we see that G0 has no cycles, G1 has exactly one cycle, and that the cycle in G1 is non-trivial. +In Figures 5(a) and 5(b), we have indicated some of the cycles in G2, namely the cycles 1,2,3,1; 3,4,5,3; +1,5,6,1; and 1,3,5,1 in Figure 5(a) and the cycle 1,2,3,5,1 in Figure 5(b). In fact, Figure 5(c) shows us that +G2 is an octahedron and therefore every cycle in G2 is either trivial or null-homotopic. Finally, G3 contains +even more cycles than G2, such as 1,3,6,1; but these are all null-homotopic since G3 also contains every +possible 2-simplex for six vertices. Therefore, PH1(G) has only one representative, the cycle 1,2,3,4,5,6,1; +which appears in G1, so we say that t = 1 is the birth of the cycle. The cycle is null-homotopic in G2, so +t = 2 is the death of the cycle. +We now turn our attention to PH2(G), but in order to represent PH2(G) we need to introduce some new +structure for the induced graphs. A triangle [a +b +c] in Gi is a set of three vertices, a, b, and c, that form a +trivial cycle in Gi. That is, the edges {a, b}, {b, c}, and {a, c} are all present in Gi. A closed surface in Gi is a +set of distinct triangles so that for each [a +b +c] in the set there is exactly one other triangle [a +b +d] also +in the set. A closed surface in Gi is trivial if the corresponding set of 2-simplices is null-homotopic in Gi. +That is, the closed surface is “filled in” by some collection of 3-simplices in Gi. For example, the octahedron +in Figure 5(c) is a non-trivial closed surface in G2 because there are no 3-simplices in G2. In G3, however, +we add edges between vertices at distance 3. In turn, we gain several 3-simplices, including [1 +2 +3 +6], +[1 +3 +5 +6], [3 +4 +5 +6], and [2 +3 +4 +6]. Figure 5(d) shows three of these 3-simplices to demon- +strate how the closed surface from G2 is filled in by all four. +9 + +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +6 +(a) G2 trivial cycles +(b) G2 null-homotopic cycle +(c) G2 sphere +(d) G3 select 3-simplices +Figure 5: A visual depiction of simplices and cycles present in G2. Left: Four trivial cycles filled by individual 2-simplices: [1 +2 +3], +[3 +4 +5], [1 +5 +6], and [1 +3 +5]. Center Left: A non-trivial, but null-homotopic cycle, 1, 2, 3, 5, 1 filled in by two 2-simplices +[1 +2 +3] and [1 +3 +5]. Center Right: All eight 2-simplices represented as the faces of a regular octahedron. Right: The closed +surface of G2 is filled in by four 3-simplices [1 +2 +3 +6], [1 +3 +5 +6](notshown), [3 +4 +5 +6], [2 +3 +4 +6]. +Definition 5. (Representing Persistent Homology: Dimension 2) Let G = (V, E) be a network with one +connected component. For each i ≥ 0, we can identify the basis for H2(Gi) with a set S i of non-trivial closed +surfaces in Gi. The Fundamental Theorem of Persistent Homology allows us to choose these representatives +so that if σ is a closed surface in S i, then exactly one of the following is true for any integer j ≥ 0 +1. σ does not exist in G j, in which case j < i, +2. σ is trivial in G j, in which case i < j, +3. σ is a cycle in S j. +Thus we will refer to the closed surfaces in � +i≥0 S i as the representatives of PH2(G). +The geometric intuition for PH2(G) is similar to that of PH1(G) in identifying large ‘voids’ in G. If +σ ∈ � +s≤i≤t S i, then σ is a closed surface with diameter at least t. The value of s is harder to describe, but is +related to the density of vertices. +Example 3.5. We now consider PH2(G) for the hexagonal graph G in Example 3.1. Recall from Example +3.4 that G0 and G1 have no trivial cycles, and therefore contain no closed surfaces. We can see in Figure +5 that G2 has exactly one closed surface and it must be non-trivial, since there are no 3-simplices. Finally, +G3 has many closed surfaces, but because it contains every possible 3-simplex on six vertices, these are all +trivial. Therefore, PH2(G) has only one representative, the octahedral closed surface in G2. This surface +first appears in G2, so t = 2 is its birth, and the surface is filled by a solid in G3, so t = 3 is its death. +Definition 6. (Persistence Intervals) Recall that the birth of a representative σ ∈ PHp(G) (vertex, cycle, +or closed surface) of the persistent homology of a network G is the smallest integer i so that σ ∈ Gi, and +the death of σ is the largest integer j so that σ ∈ G j−1 and σ is trivial in Gk for k ≥ j, if such an integer +exists. The persistence interval for σ is [a, b), where a and b are the birth and death of σ, respectively. +This represents the set of all parameter values i for which the equivalence class corresponding to σ is a +non-trivial element of Hp(Gi). The persistence of σ is b − a. +Example 3.6. We now finish our consideration of the persistent homology of G from Figure 4(b). Recall +from Example 3.3 that PH0(G) has six representatives. These all have birth t = 0. Five of these have a death +of t = 1, and one of these has a death of ∞. Therefore the persistence intervals for PH0(G) are [0, 1) × 5 and +[0, ∞) × 1. +10 + +From Example 3.4, we know PH1(G) has one representative, with birth t = 1 and death t = 2. Therefore +the corresponding persistence interval is [1, 2). Note that the diameter of the cycle is 3 and every pair of +consecutive vertices is distance 1 apart. This follows the idea mentioned earlier that the representatives of +PH1(G) indicate ‘large’ cycles. Specifically, the diameter of σ is at least the death of σ, and the birth of σ +is the maximum distance between consecutive vertices. +From Example 3.5, PH2(G) has one representative, with birth t = 2 and death t = 3. Therefore, the +persistence interval for that element is [2, 3). Note that the diameter of the corresponding set of vertices is 3 +in G. This also follows the idea mentioned earlier that PH2(G) identifies large ‘voids’ in G. Specifically, the +death of σ is a lower bound on the diameter of σ. +Given the representatives chosen in Definitions 3, 4, 5, and 6, we have the following three observations +regarding the persistent homology of a finite, undirected, unweighted graph G: +(i) If G has n vertices, then PH0(G) will have exactly n persistence intervals, with exactly one [0, ∞) interval +for each connected component and the rest will be [0, 1) intervals. +(ii) In dimension 1, PH1(G) describes the number and sizes of the non-trivial cycles in the original network. +The persistence intervals will all be of the form [1, b) for some integer b > 1. The value of b is related to +the diameter of the corresponding cycle. In the networks we have studied, we note that a persistence interval +[1, b) in PH1(G) corresponds to a simple cycle with between 3b − 2 and 3b vertices, inclusive. +(iii) In dimension 2, the voids we detect in PH2(G) tell us about the nontrivial intersections of cycles. Such +intersections are hard to visualize but, roughly speaking, a representative in PH2(G) can only form if several +large cycles intersect each other pairwise. +4. Comparing Networks using Persistent Homology +In this section we demonstrate how methods based on persistent homology can be used to compare +different networks. The two methods we introduce in this paper are based on using (a) the bottleneck +distance and (b) the persistence curves of a given set of networks. Both (a) and (b) rely on first computing +persistence intervals then analyzing the differences in these intervals. +The two networks we consider throughout this section to demonstrate these methods are the Tikopia ge- +nealogical network from Figure 1 (left) and the hexagonal network from Figure 4. The persistence intervals +for these networks are given in Table 1, respectively. +Dimension +Interval Type and Persistence +Tikopia +Hexagon +Dimension 0 +[0, ∞) × 8, [0, 1) × 286 +[0, ∞) × 1, [0, 1) × 1 +Dimension 1 +[1, 2) × 16, [1, 3) × 19, [1, 4) × 5, [1, 5) × 3, +[1, 6) × 2, [1, 7) × 1 +[1, 2) × 1 +Dimension 2 +[2, 3) × 4, [3, 4) × 11, [4, 5) × 12, [5, 6) × 4, +[6, 7) × 5, [7, 8) × 1, [8, 9) × 1 +[2, 3) × 1 +Table 1: The persistence intervals of the Tikopia genealogical network and the hexagon network are shown. Here the notation [a, b) × k +indicates that the network has k persistence intervals [a, b). The corresponding persistence diagrams are shown in Figure 6 and the +corresponding persistence curve for the Tikopia network is shown in Figure 7. +11 + +4.1. Persistence Diagrams and Bottleneck Distance +One common way to represent persistence intervals is to plot them as points in R × (R ∪ {∞}), which +is typically referred to as a persistence diagram. While this method of visualizing a network’s persistent +homology does not indicate how often a given persistence interval occurs, it does provide information on +what kind of persistence intervals occur for a given network. +Definition 7. (Persistence Diagrams) Let PHp(G) be the pth persistent homology of a network G. The +persistence diagram for PHp(G) is a multiset of points in R × (R ∪ {∞}) defined as follows. +• For each σ ∈ PHp(G) with persistence interval [a, b), we include one copy of the point (a, b). +• For each c ∈ R, we include infinitely many copies of the point (c, c). +Note that we include the points (a, a) to represent features in G that are considered trivial in PHp(G), +such as cycles consisting of exactly three vertices. This inclusion is necessary for us to define a meaningful +metric on the space of persistence diagrams. The metric we use here is called the bottleneck distance. +Definition 8. (Bottleneck Distance) Let S 1 and S 2 be persistence diagrams for two graphs G and H, re- +spectively. Let η range over the set of bijections from S 1 to S 2. Then the bottleneck distance between S 1 and +S 2 is +dB(S 1, S 2) = inf +η sup +x∈S 1 +∥x − η(x)∥∞. +The Fundamental Theorem of Persistent Homology (introduced in [19], explained well in [36] and [29]) +ensures that if two graphs are isomorphic, the corresponding persistence diagrams will be equal, and thus the +bottleneck distance will be 0. However, it is possible for non-isomorphic graphs to have identical persistence +diagrams. +Example 4.1. (Bottleneck Distance Between the Tikopia and Hexagonal Networks) Notice that the per- +sistence intervals for the Tikopia genealogical network (see Table 1) include, as a subset, the persistence +intervals from the hexagonal network we considered in Example 3.6. We can form a bijection between the +persistence diagrams of the Tikopia and hexagonal network by identifying the non-trivial intervals from the +hexagonal network with those of the Tikopia network. We then map any additional intervals from the Tikopia +network of the form [a, b) to the trivial interval [ a+b +2 , a+b +2 ). (The perceptive reader may notice that this is not +clearly a bijection, but there is a standard technique from set theory for modifying it to be bijective.) +This mapping is shown in Figure 6 (right). Here, [1, 7) is mapped to [4, 4). As this pair of points is +further apart than any other pair in this bijection, the bottleneck distance for the two networks is at most +three, since we take an infimum over all possible bijections. Conversely, there is no interval in the hexagonal +persistence diagram that is closer to [1, 7) than 3, so the bottleneck distance is at least three. Thus, the +bottleneck distance for these two persistence diagrams is exactly 3. +Suppose that two networks, each of which is connected, admit isometric embeddings in Rn. The Stability +Theorem [18] guarantees that if the Hausdorff distance between the embeddings is δ, then the bottleneck +distance for the corresponding persistence diagrams is at most δ. For example, if the PH1 persistence +diagrams differ by δ, then any attempt to pair up cycles in the networks must include at least one pair of +cycles for any isometric embedding that are δ apart in that embedding. In Section 6.1 we apply this idea to +a large collection of genealogical and social networks. +12 + +Hexagonal Network PD +Tikopia Network PD +Bottleneck Bijection +Figure 6: Left: The persistence diagram of the hexagonal network in Figure 4(b) is shown. Center: The persistence diagram of +the Tikopia genealogical network in Figure 1 (left) is shown. Right: A bottleneck bijection between the persistence intervals of the +hexagonal and Tikopia family network is shown. Orange lines show which points are matched to points of the form (a, a) where a ∈ R. +4.2. Persistence Curves +For the network data we consider, persistence diagrams obfuscate a key difference that we consider +important: the number of persistence intervals. For a simple example of this, consider networks of the form +V = {1, 2, . . . , n} with edges of the form {i, i + 1} for 1 ≤ i < n. For n ≥ 2, any network of this type will have +persistence intervals [0, 1) × (n − 1) and [0, ∞) × 1. However, when plotting the persistence diagram we will +only ‘see’ two points: (0, 1) and (0, ∞). +To address this limitation, we introduce the notion of a persistence curve as a new way to visualize +the persistent homology of a network (see Definition 9). The difference between the persistence curve and +the persistence diagram of a network is that the persistence curve also includes the number of intervals of +a particular type. To create a persistence curve we first compute a network’s persistence intervals, then +sort the intervals of a given dimension by their persistence into a bar graph. For instance, in dimension +1 the Tikopia genealogical network has thirteen [1, 2) intervals, nineteen [1, 3) intervals, etc. which are +sequentially stacked as shown in Figure 7 (left) to create what we will call a barcode. To create the associated +persistence curve we connect the endpoints of each subsequent bar as shown in Figure 7 (right). +In dimension-one, the birth times of our intervals will all start at 1, as the networks we consider are +unweighted, undirected, and connected. This means that in this dimension the resulting bar graph is also a +plot of the death times for each interval. For higher-dimensions, which have varied birth times, we also plot +the lengths of the intervals but for simplicity we start at 1 as in dimension-one. +A formal definition of a network’s persistence curves is the following. +Definition 9. (Persistence Curves) Let G = (V, E) be a network with nonempty vertex and edge sets. Let +{[a j, bj)} be the set of all persistence intervals for each σj ∈ PHn(G) where j ∈ N. For all n ∈ N the +persistence curve PHn(G) is the linear interpolation of the set of points {(bj − (aj − 1), j)} where b j−1 − +(aj−1 − 1) ≤ bj − (aj − 1). +Visualizing persistence intervals as a curve allows us to compare the persistent homology of different +networks in a similar fashion to persistence diagrams while retaining different information. In particular, we +can see how many intervals there are of a given persistence, whereas the persistence diagram only indicates +the presence of such an interval. In what follows we will typically plot the persistence curves of multiple +networks on the same axes to indicate what differences exist in the persistent homology of different networks +(cf. Section 6). +13 + +6-Cycle Persistence Diagram +Tikopia Persistence Diagram +Superimposed Persistence Diagram +1 +8- +8 +8 +6 +4 +eath +: +(a, a) +a, a) +2 +(e 'e) +Dimension 0 +Dimension 0 +Dimension 1 +Dimension 1 +Matched Points +Dimension 2 +Dimension 2 +Tikopia Network +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +1 +Birth +Birth +BirthTikopia Network Barcode +Tikopia Network Persistence Curve +Figure 7: Left: The barcode of the Tikopia genealogical network in dimension 1 is shown. The individual bars are formed from the +persistence intervals given in Table 1. Right: The associated persistence curve for the Tikopia network in Figure 1 is shown. +5. Data +The data we consider in this paper is of two types; genealogical network data and other social network +data. The genealogical networks we consider are drawn from ninety-seven genealogical networks found +in[14], which range in size from n = 17 to 5, 016 individuals. The social network data we use is taken from +twenty-seven different social networks obtained from [20, 21, 22, 23]. These range in size from n = 16 to +2, 539 individuals. (See Table 2 in the Appendix for a full description of this data set.) +Although many larger genealogical and social network data sets are available we are limited by both the +temporal and spacial complexity of the algorithm used to compute persistence intervals. The program we +used, called Ripser (from the python package Ripser) [24], has a computational and spacial complexity of +O((n + m)3) where n is the number of individuals and m is the number of edges in a network. The number +n+m is the number of simplicies in the network. In the genealogical networks we consider there are between +n + m = 41 to 15, 735 simplicies and in the social networks we consider between n + m = 41 to 19, 056 +simplices. +To understand how a network’s persistence intervals are effected by the completeness or incompleteness +of data we also consider subnetworks sampled from a few, much larger, genealogical and social networks. +These sampled networks are created by randomly selecting an individual with a single neighbor, i.e. a +vertex of degree 1, then performing a breadth-first-search starting with this individual to find the η closest +individuals in the network to this individual. Because of the spatial and computational limitations of Ripser +we choose 600 ≤ η ≤ 3, 000 to ensure we can compute the persistence intervals of these sampled networks. +In total we sampled from four different genealogical networks and four different social networks. These are +the Advogat, LastFM Asia, Deezer HU and Deezer RO social networks and the genealogical networks 96–99 +shown in Table 2, respectively. We sampled from each of these networks five times each to create a total of +20 sampled genealogical networks and 20 sampled social networks. The reason we begin our breadth-first +search with a vertex of degree 1 is to ensure that our sampled networks have vertices both on the boundary +and the interior of the original network we sampled to better mimic the structure of the original genealogical +and social networks. +Apart from the (i) genealogical and social networks we consider and (ii) sampled versions of these +networks, we also consider what we refer to as (iii) atypical genealogical networks. There are a number of +14 + +FamilyDimension1 +40 +30 +Interval Index +20 +10 +0 +1 +2 +m +4 +-5 +6 +1 +7 +Intervals (Birth and Death Time)FamilyDimension1 +40 +Interval Index +20 +10 +2 +3 +4 +5 +6 +7 +Intervals(BirthandDeathTime)genealogical networks that appear to be created with no attempt to represent all or even a fraction of the +familial relationships. For example, the US Presidents network, cited as Atyp. Gen. Network 2 in Table 2, +follows the shortest genealogical path between presidents leaving out extraneous relationships. We consider +a number these atypical genealogical networks, which form a contrast to the more standard genealogical +networks we consider especially in terms of their peristent homology. A description of each of the (i) +genealogical, social, (ii) sampled genealogical, sampled social, and (iii) atypical genealogical networks we +consider is given at the end of the Appendix. +Figure 8: PCA projections of the bottleneck distances between networks are shown. Left: The bottleneck distance between each of +the twenty sampled genealogical and sampled social networks is shown. Center: The bottleneck distances are shown between the +genealogical, social, and atypical genealogical networks we consider. Right: The bottleneck distances in the center panel are shown for +only the genealogical and social networks we consider. +6. Results +Here we compare genealogical and other social networks using the (a) bottleneck distance and the (b) +persistence curves defined in Section 4 (see Definitions 8 and 9, respectively). For those who have skipped +Sections 3 and 4, the bottleneck distance gives us a distance between two networks based on the differences +in their persistent homology. Persistence curves give us a way of visualizing this difference but in greater +detail (cf. Figure 7). +6.1. Network Comparison using Bottleneck Distance +Here we compute the bottleneck distance between every pair from the social and genealogical networks +we consider. To visualize these results we use principal component analysis to identify the two components +that account for the most variance and then plot this data in R2 (see Figure 8). +From each part of Figure 8 we can see that genealogical networks are generally separated from social net- +works and form clusters that are easily distinguished. For the sampled networks (shown left), we can easily +separate genealogical and social networks, and we can identify at least two distinct subclasses of genealogi- +cal networks. However, the bottleneck distance does an inferior job separating the non-sampled genealogical +and social networks (shown center and right). The exception are the atypical genealogical networks, whose +persistence intervals differ significantly enough from all of the other networks to be distinguishable as a third +class of networks (shown center). +15 + +Sampled Genealogical and Social Networks +Genealogical, Social, and atypical Network +Genealogical and Social Networks +Genealogical +24 +Genealogical +Genealogical +Social +Social +8 +Social +Atypical +15 +6 +-2 +- +1 +21 +31 +t +- +15 +2Figure 9: Comparison of persistence curves for full networks vs sampled networks, grouped by dimension and type of network. Upper +Row: Sampling social networks typically stretches the persistence curve in only one axis without affecting the other axis. Lower Row: +Sampling genealogical networks typically shrink the persistence curve in both axes. Overall the average slope for social networks tends +to increase when sampled, while genealogical networks experience a decrease in average slope. +6.2. Comparison of Genealogical and Social Networks using Persistence Curves +Persistence curves give us a new alternative way of comparing networks. The advantage of using these +curves compared to the bottleneck distance is that these curves give us a more detailed picture of how +the number of persistence intervals varies from network to network. This allows us to better differentiate +the structure of genealogical networks from social networks as well as observe the structure common to +genealogical networks and those common to social networks, respectively. +In Figure 9 the persistence curves for the unsampled genealogical and unsampled social networks are +shown in blue and red, respectively. The atypical genealogical networks are shown in green. The social +networks have persistence curves that are quite vertical in both dimension 1 and dimension 2. For dimension +1, this indicates that most cycles in a social network are close to being trivial; either because they have a +relatively small circumference or because they can be decomposed into a union of cycles with small circum- +ferences. In particular, most of the social networks have a maximum death time of three (see Definition 2), +which corresponds to having a basis of cycles whose maximal circumference is at most nine. In other words, +any cycle of circumference ten or more decomposes as the union of smaller cycles. For dimension 2, the +steepness of the persistence curves indicate the presence of many distinct, yet similar, paths between certain +pairs of vertices. +In contrast, the genealogical networks have persistence curves that have a much more horizontal profile +16 + +Social vs. Sampled Sccial Networks: Dimension 1 +Social vs. Sampled Sccial Networks: Dimension 2 +O- +Social +50D0 +Social +Sampled Social +Sampled Social +ofIntervals +3100 +31D0 +2400 +aqnn +2400 +8DO +0 - +0 +2D +25 +3.D +3.5 +4.D +2D +25 +3.D +3.5 +4.D +Genealogical vs. Sampled Genealaogical Networks: Dimensian 1 +Genealogical vs. Sampled Genealagical Networks: Dimensian 2 +160.0 +Atypical +3500 +Atypical +1400 +Genealogical +310 +Genealogical +Number of Intervaba +1200 +Sampled Genealogical +of Intervals +Sampled Genealogical +2500 +24D0 +aqnn +1500 +1400 +240 +500 +0 - +0- +25 +5.D +SL +14.0 +12.5 +15.0 +17.5 +25 +5.D +14.0 +12.5 +15.0 +17.5indicating that most cycles are quite long and there are fewer ‘alternate paths’ between pairs of vertices. +In the extreme, the atypical genealogical networks are nearly flat in dimension 1, which reflects the fact +that these atypical networks were intentionally constructed to have very few cycles. In dimension 2, the +atypical networks show a similar slope to most of the typical genealogical networks, but the size of the +alternative paths in these networks are much larger. This is likely due to the high number of individuals +who were added only to link distant individuals, e.g. presidents. In a typical genealogical network, the +additional relationships between such individuals would allow large cycles to decompose but in the atypical +genealogical networks this in not the case. +Figure 10: Upper Row: Comparison of persistence curves for full networks by type. Lower Row: Comparison of persistence curves for +sampled networks by type, excluding atypical genealogical networks. In each dimension, the average slope for genealogical networks +is typically lower than the average slope for a social network. The atypical genealogical networks have the lowest average slope and +much greater total length. The behavior for average slopes is more pronounced for sampled networks than for full networks. +In Figure 10, we see the persistence curves for the sampled genealogical and sampled social networks +shown in blue and red, respectively. The atypical genealogical networks are shown in green. Again the social +networks have persistence curves that are quite vertical in both dimensions, although these curves are not as +tall as in the case of unsampled social networks. This indicates that as a social network is sampled it retains a +similar proportion of close-to-trivial cycles, but may lose many of the alternative paths between vertices that +appear in dimension 2. By contrast, for genealogical networks the persistence curves indicate the complete +loss of very large cycles in conjunction with a proportional loss of close-to-trivial cycles. In dimension +2, genealogical networks experience a more severe loss of alternative paths than the social networks. As a +result, though sampling shrinks the scale of the persistence curves for social and genealogical networks, they +remain visually distinct. +17 + +Genealogical vs. Sccial: Dimension 1 +Genealogical vs. Sccial Networks: Dimension 2 +O +Atypical +5000 +Atypical +Genealogical +Genealogical +Social +Number of Intervals +Social +40 +3100 +2400 +24D0 +0 +25 +5.D +7.5 +14.0 +12.5 +15.0 +17.5 +25 +5.D +7.5 +14.0 +012.515.017.5 +Sampled Geneakogical vs. Sampled Social Networks: Dimensian 1 +Sampled Geneakogical vs. Sampled Social Networks: Dimensian 2 +24D0 +Sampled Genealogical +24D0 +Sampled Genealogical +Sampled Social +Sampled Social +of Intervals +of Intervaba +1500 +1500 +1400 +1400 +500 +0 +2D +25 +3.0 +3.5 +4.0 +4.5 +5.D +2 +3 +6As in the bottleneck distance plots, genealogical and social networks appear to cluster together in that +they have similar types of persistence curve. In fact, this is true whether or not the networks are sampled or +unsampled. This suggests that even with incomplete data social network and genealogical networks have a +distinguishable persistent homology, at least at the scales we consider. +It is worth mentioning that, while the bottleneck distance plots show us to an extent how different ge- +nealogical and social networks are the persistence curves show us what are differences are. The distance +plots in Figure 8 do have the advantage of simplicity, however, and could presumably be used to more +quickly identify differences in networks that are not as apparent as those we find between genealogical and +social networks. +6.3. Connections +It is also possible to use persistent homology to study properties of a network, such as the number of +connected components, the typical size of cycles, or even “missing links” in the data. For genealogical +and social networks, we can convert these mathematical concepts into more familiar ideas such as family +groups or common ancestors. This also allows us to make conjectures about the persistent homology for +such networks by converting standard assumptions about families or social networks into the language of +persistence. +In dimension 0, the number of connected components determines the number of [0, ∞) intervals, and the +total number of distinct vertices is the number of [0, ∞) intervals plus the number of [0, 1) intervals. In the +context of a genealogical network, each connected component represents a family group that is not related +to the other family groups by any known connection. Thus, if a given family network is indeed a single +“family” of relatives, there should be exactly one [0, ∞) interval. In our Tikopia example we have eight +[0, ∞) intervals each of which correspond to exactly one connected component of this genealogical network. +(Note that Figure 1 (left) shows only the largest of these components). In this example, most of the the other +‘family groups’ are actually individuals with no relation edges in the network. +In social networks, the connected components create what could be referred to as friend groups. Unlike +genealogical networks, there are usually few restrictions on which edges form in a social network. As +such, we do not have a conjecture about the number of [0, ∞) intervals in this setting in general. However, +sampling any network as described in Section 5 will result in a new network with a single [0, ∞) interval. +Moving to dimension 1, persistence intervals in this dimension describe the way that each connected +component is internally structured. In sufficiently large genealogical networks, we will see three kinds of +features that we call common ancestors, union cycles, and hybrid cycles. A common ancestor cycle occurs +when two descendants of an individual form a union or have a child together. We use the term union cycle to +refer to situations where a cycle is formed through union edges and edges connecting two siblings. The final +type of cycle of note, the hybrid cycles, are those formed by any other combination of parent-child edges +and union edges, which includes everything that is not a strict common ancestor or union cycle. These three +types of cycles are illustrated in Figure 11, where marriage edges are indicated by red edges and parent- +child edges are indicated by blue edges. We show a common ancestor in Figure 11(a). Figure 11(b) is an +example of a union cycle in which two siblings in one family form unions with two siblings in another, +where only a single parent in each family is shown. In Figure 11(c) we give an example of a θ-cycle, which +is the union of a common ancestor cycle and two overlapping hybrid cycles. This example comes from +siblings of one family marrying cousins from another family. These cycles can be any length theoretically, +but cultural norms affect the typical size and number of each type of cycle differently. Recording practices +and incomplete data also limit whether these cycles appear in a given dataset. Thus having a description +of these cycles together with an understanding of the culture may help identify errors in the recorded data. +18 + +Conversely, understanding the distribution of cycles in high fidelity datasets can help identify the underlying +cultural norms and help extrapolate where individuals are missing in incomplete data sets. +(a) Common Ancestor Cycle +(b) Union Cycle +(c) θ-Cycle +Figure 11: Left: A common ancestor cycle. The top most vertex is a common ancestor of the lowest vertex. The horizontal red line is a +marriage, all other lines are parent-children edges. Center: A union cycle, specifically the double cousin situation described in Section +2. The left-most and right-most vertices are parents of their neighboring vertices. The two horizontal red lines are marriage edges. +Right: A θ-cycle formed by a common ancestor cycle with two overlapping hybrid cycles. +Since many cultures avoid marrying close relatives, common ancestor cycles tend to have a fairly large +circumference. In the Tikopia network (see Figure 1) we see persistence intervals with death values as high +as 7 corresponding to cycles with a circumference of at least 21 individuals, which appear to be common +ancestor cycles. This partially explains why persistence curves are so flat: there are relatively few minimal +common ancestor cycles in a network, but they have very high persistence. More precisely, if the distance to +union (the total number of individuals in a common ancestor cycle) is n, then the persistence of that cycle is +⌊n/3⌋. However, the representatives of persistent homology only include a basis for these cycles, instead of +including every possible distinct cycle. In particular, a large common ancestor cycle will decompose into the +union of two hybrid cycles if the hybrid cycles are each shorter than the common ancestor cycle, as shown +in Figure 11(c). Persistent homology will reflect the size of the two smaller cycles instead of the larger +common ancestor cycle. We note that it is possible to identify the actual cycles chosen for our basis, but the +software we used does not provide that information and size of the networks prohibits us from identifying +the cycles manually. +In social networks, we see that highly persistence cycles are quite rare. In order to have a cycle of +persistence 3, for instance, we need a loop with circumference 9 or higher with no shorter paths between any +two vertices in the loops. It may be that phenomena like the small-world effect or, more colloquially, six- +degrees of freedom limit the maximal persistence of social networks. We see this reflected in our example +data sets with a maximum persistence of 3 for all but one of the social networks. +7. Conclusion +In this paper, we explore the persistent homology structure of genealogical networks, motivated by the +observation that family links tend to form in a fixed range of intermediate distances, which makes genealog- +ical networks homologically distinct from most other social networks. We also introduce the notion of a +persistence curve, which can be used to summarize and compare the persistent homology structure of any +19 + +network. We also relate specific genealogical structures, such as the common ancestor cycle, to homology +objects. +We find that, in the presence of incomplete data homology analysis is still genealogically useful. We +note missing data due to recording practices and incomplete data (a ubiquitous feature of real genealogical +networks), limits the kind of cycles that appear in a given dataset. Thus having a description of these cycles +together with an understanding of the culture may help identify errors in the recorded data. Conversely, +understanding the distribution of cycles in high fidelity datasets can help identify the underlying cultural +norms and help extrapolate where individuals are missing in incomplete data sets. +There are several interesting directions in which this work could be expanded. For example, our work +has made it clear that there is a real need to analyze the persistent homology of large networks, with at least +tens of thousands of nodes, since family formation generally takes place at these scales. The Ripser library +we relied on was not able to reach these scales. Additionally, we are very interested in creating random +graph models which reflect the actual homology of human family networks—a first attempt at this by our +group has been fairly successful at the scale of hundreds of nodes [25]. More broadly, there is a need to +model the ground truth human family network. All the extant data sources represent biased, limited, and +noisy subnetworks, while the true interest of the genealogical community is in the ground truth network. +Tools for signal denoising, image inpainting, and graph extrapolation, for example, could be useful in this +context. Finally, an important aspect of genealogical networks is the relationship between various support- +ing documents/metadata and the links that are discoverable through them. For example, one can consider +optimal document collection strategies with a limited budget or document collection that is fair in terms of +capturing minority information, which is often underrepresented. +8. Declarations +8.1. Availability of data and materials +Links to the datasets generated and/or analysed during the current study can be found in Table 2. Code to +replicate and extend this work can be found at https://github.com/AbigailJ32/The-persistent-homology-of- +genealogical-networks. +8.2. Competing interests +The authors declare that they have no competing interests. +8.3. Funding +ZB, BW, and AJ, were supported by a BYU CPMS CHIRP grant. ZB was additionally supported by NFS +award #2137511 and Army Research Office grant #W911NF-18-1-0244, and the James S. McDonnell Foun- +dation 21st Century Science Initiative—Complex Systems Scholar Award grant #2200203. BW was addi- +tionally supported by the Simons Foundation grant #714015. The views and conclusions contained in this +document are those of the authors and should not be interpreted as representing the official policies, either +expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is autho- +rized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation +herein. +8.4. Authors’ contributions +Designed the experiments: ZB, NC, BW, RW. Performed the experiments: RF, RW. Wrote the paper: ZB, +NC, TG, AJ, RS, BW, RW. All authors read and approved the final manuscript. +20 + +8.5. Acknowledgements +We acknowledge helpful conversations with Joseph Price and the FamilySearch Engineering Research team. +We also acknowledge Kolton Baldwin for helping to improve our code and simulations. +9. Appendix +Here we indicate both the genealogical and social networks used in our persistent homology computa- +tions (see Section 6). We distinguish the datasets by network type: Friendship/Acquaintance, Social Media, +Collaboration/Business, Disease Transmission, Information Sharing, Genealogical, and Atypical Genealog- +ical networks. We also provide the network name, number of vertices and edges in the network, and a +citation where the network can be found. Also, a special thanks to Kolton Baldwin for help with numerical +simulations on this paper. +Table 2: Social and Genealogical Network Data Sets. +Network Data +Network Type & Name +Vertices +Edges +Citation +Social Networks +Friendship & +Aquaintance +Dolphins +62 +159 +http://www-personal.umich.edu/∼mejn/netdata/ +Zachary Karate Club +34 +78 +http://vlado.fmf.uni-lj.si/pub/networks/data/ucinet/ucidata.htm#zachary +Residence Hall +217 +2672 +http://konect.cc/networks/moreno oz/ +Highland Tribes +16 +58 +http://konect.cc/networks/ucidata-gama/ +Seventh Graders +29 +376 +http://konect.cc/networks/moreno seventh/ +Physicians +241 +1098 +http://konect.cc/networks/moreno innovation/ +Highschool +70 +366 +http://konect.cc/networks/moreno highschool/ +Dutch College +32 +354 +http://konect.cc/networks/moreno vdb/ +Sampson’s monastery +25 +322 +http://vladowiki.fmf.uni-lj.si/doku.php?id=pajek:data:esna3:sampson +Adolescent health +2539 +12969 +http://konect.cc/networks/moreno health/ +Hamsterster friends +2952 +12534 +http://konect.cc/networks/petster-hamster-friend/ +Social Network 1 +32 +220 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as1.net +Social Network 2 +32 +191 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as2.net +Social Network 5 +32 +90 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as5.net +Social Network 7 +32 +61 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as7.net +Social Network 8 +32 +79 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as8.net +Social Network 9 +32 +58 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as9.net +Social Media +Firm Hi-Tech +33 +124.5 +https://networkrepository.com/soc-firm-hi-tech.php +Wiki-Vote +889 +2.9K +https://networkrepository.com/soc-wiki-Vote.php +FB-PAGES-FOOD +620 +2.1K +https://networkrepository.com/fb-pages-food.php +Advogato +6541 +51127 +http://konect.cc/networks/advogato/ +LastFM Asia +7624 +27806 +https://snap.stanford.edu/data/feather-lastfm-social.html +Deezer HU +47538 +222887 +https://snap.stanford.edu/data/gemsec-Deezer.html +Deezer RO +41773 +125826 +https://snap.stanford.edu/data/gemsec-Deezer.html +Collaboration & +Business +Social Network 4 +32 +218 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as4.net +Social Network 6 +32 +103 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as6.net +Social Network 11 +32 +83 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as11.net +Social Network 12 +32 +65 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as12.net +Disease Transmission +Taro Exchange +22 +78 +http://konect.cc/networks/moreno taro/ +21 + +Network Name & Type +Vertices +Edges +Citation +Information Sharing +Social Network 3 +32 +119 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as3.net +Social Network 10 +32 +80 +http://vlado.fmf.uni-lj.si/pub/networks/doc/ECPR/assign.1/as10.net +Genealogical Networks +Genealogical Network 1 +310 +322 +https://www.kinsources.net/kidarep/dataset-209-mowanjum-kalumburu.xhtml +Genealogical Network 2 +303 +537 +https://www.kinsources.net/kidarep/dataset-2-mbuti-village-1957-af03.xhtml +Genealogical Network 3 +371 +718 +https://www.kinsources.net/kidarep/dataset-58-ojibwa-1930-nd07.xhtml +Genealogical Network 4 +795 +1387 +https://www.kinsources.net/kidarep/dataset-150-achuar-pastaza.xhtml +Genealogical Network 5 +636 +1151 +https://www.kinsources.net/kidarep/dataset-92-chenchu-1940-as02.xhtml +Genealogical Network 6 +782 +1366 +https://www.kinsources.net/kidarep/dataset-28-trio-1960s.xhtml +Genealogical Network 7 +128 +202 +https://www.kinsources.net/kidarep/dataset-23-shoshone-1880-nd11.xhtml +Genealogical Network 8 +439 +626 +https://www.kinsources.net/kidarep/dataset-70-genesis.xhtml +Genealogical Network 9 +244 +481 +https://www.kinsources.net/kidarep/dataset-66-waimiri-atroari.xhtml +Genealogical Network 10 +410 +746 +https://www.kinsources.net/kidarep/dataset-240-kodiak.xhtml +Genealogical Network 11 +337 +572 +https://www.kinsources.net/kidarep/dataset-51-wilcania.xhtml +Genealogical Network 12 +216 +378 +https://www.kinsources.net/kidarep/dataset-22-ainu-1880-as01.xhtml +Genealogical Network 13 +77 +134 +https://www.kinsources.net/kidarep/dataset-69-slavey-1911-nd12.xhtml +Genealogical Network 14 +815 +1582 +https://www.kinsources.net/kidarep/dataset-7-pakaa-nova.xhtml +Genealogical Network 15 +20 +28 +https://www.kinsources.net/kidarep/dataset-38-wanindiljaugwa-1948-au06.xhtml +Genealogical Network 16 +219 +371 +https://www.kinsources.net/kidarep/dataset-171-suya.xhtml +Genealogical Network 17 +17 +24 +https://www.kinsources.net/kidarep/dataset-31-family.xhtml +Genealogical Network 18 +168 +221 +https://www.kinsources.net/kidarep/dataset-14-labrador-inuit-1776-nu02.xhtml +Genealogical Network 19 +64 +109 +https://www.kinsources.net/kidarep/dataset-91-takamiut-1927-64-nu03.xhtml +Genealogical Network 20 +1423 +3211 +https://www.kinsources.net/kidarep/dataset-258-todas.xhtml +Genealogical Network 21 +645 +1097 +https://www.kinsources.net/kidarep/dataset-65-igluligmiut-1961-nu07.xhtml +Genealogical Network 22 +4463 +8416 +https://www.kinsources.net/kidarep/dataset-115-charlevoix.xhtml +Genealogical Network 23 +48 +86 +https://www.kinsources.net/kidarep/dataset-41-vedda-1905-as04.xhtml +Genealogical Network 24 +104 +172 +https://www.kinsources.net/kidarep/dataset-71-igluligmiut-1960-61-nu08.xhtml +Genealogical Network 25 +1263 +2021 +https://www.kinsources.net/kidarep/dataset-223-samburu.xhtml +Genealogical Network 26 +80 +132 +https://www.kinsources.net/kidarep/dataset-10-apache-1932-nd01.xhtml +Genealogical Network 27 +1269 +2395 +https://www.kinsources.net/kidarep/dataset-24-ayd-nl-yoruk-2005.xhtml +Genealogical Network 28 +299 +532 +https://www.kinsources.net/kidarep/dataset-13-tory.xhtml +Genealogical Network 29 +19 +30 +https://www.kinsources.net/kidarep/dataset-21-ngatatjara-1966-au04.xhtml +Genealogical Network 30 +399 +592 +https://www.kinsources.net/kidarep/dataset-204-dogon-konsogu-donyu.xhtml +Genealogical Network 31 +377 +712 +https://www.kinsources.net/kidarep/dataset-49-alyawarra-1971-au01.xhtml +Genealogical Network 32 +1263 +2021 +https://www.kinsources.net/kidarep/dataset-223-samburu.xhtml +Genealogical Network 33 +118 +192 +https://www.kinsources.net/kidarep/dataset-39-eyak-1890.xhtml +Genealogical Network 34 +98 +161 +https://www.kinsources.net/kidarep/dataset-75-nunamiut-1885-nu11.xhtml +Genealogical Network 35 +479 +830 +https://www.kinsources.net/kidarep/dataset-19-ojibwa-1949-nd08.xhtml +Genealogical Network 36 +1695 +3206 +https://www.kinsources.net/kidarep/dataset-103-tikuna-arara.xhtml +Genealogical Network 37 +256 +441 +https://github.com/AbigailJ32/The-persistent-homology-of-genealogical-networks +Genealogical Network 38 +798 +1416 +https://www.kinsources.net/kidarep/dataset-229-nucoorilma-tingha.xhtml +Genealogical Network 39 +738 +1212 +https://www.kinsources.net/kidarep/dataset-32-yaraldi.xhtml +Genealogical Network 40 +525 +855 +https://github.com/AbigailJ32/The-persistent-homology-of-genealogical-networks +Genealogical Network 41 +619 +1224 +https://www.kinsources.net/kidarep/dataset-251-nunivak.xhtml +Genealogical Network 42 +3008 +6074 +https://www.kinsources.net/kidarep/dataset-80-torshan.xhtml +Genealogical Network 43 +278 +464 +https://www.kinsources.net/kidarep/dataset-62-dogrib-1911-25-59-nd04.xhtml +Genealogical Network 44 +105 +172 +https://www.kinsources.net/kidarep/dataset-5-konkama-1931-44-51-eu02.xhtml +Genealogical Network 45 +240 +395 +https://www.kinsources.net/kidarep/dataset-158-tikar.xhtml +Genealogical Network 46 +4178 +7351 +https://www.kinsources.net/kidarep/dataset-45-obidos.xhtml +Genealogical Network 47 +216 +286 +https://www.kinsources.net/kidarep/dataset-254-port-keats.xhtml +Genealogical Network 48 +147 +242 +https://www.kinsources.net/kidarep/dataset-78-pul-eliya-1954-simpler-version.xhtml +Genealogical Network 49 +277 +516 +https://www.kinsources.net/kidarep/dataset-213-sarmi.xhtml +Genealogical Network 50 +330 +622 +https://www.kinsources.net/kidarep/dataset-73-parakana.xhtml +Genealogical Network 51 +35 +53 +https://www.kinsources.net/kidarep/dataset-81-gundangborn-1948-au02.xhtml +Genealogical Network 52 +48 +76 +https://www.kinsources.net/kidarep/dataset-84-hare-1956-nd05.xhtml +22 + +Network Name & Type +Vertices +Edges +Citation +Genealogical Network 53 +105 +245 +https://www.kinsources.net/kidarep/dataset-87-arara.xhtml +Genealogical Network 54 +116 +220 +https://www.kinsources.net/kidarep/dataset-89-nunamiut-1960-nu13.xhtml +Genealogical Network 55 +116 +176 +https://www.kinsources.net/kidarep/dataset-226-jie.xhtml +Genealogical Network 56 +657 +1166 +https://www.kinsources.net/kidarep/dataset-27-nyungar.xhtml +Genealogical Network 57 +659 +1288 +https://www.kinsources.net/kidarep/dataset-3-anuta-1972.xhtmlj +Genealogical Network 58 +112 +182 +https://www.kinsources.net/kidarep/dataset-15-oodnadatta.xhtml +Genealogical Network 59 +218 +353 +https://www.kinsources.net/kidarep/dataset-17-lainiovouma-1952-eu03.xhtml +Genealogical Network 60 +90 +119 +https://www.kinsources.net/kidarep/dataset-12-miwuyt-1967-au03.xhtml +Genealogical Network 61 +289 +477 +https://www.kinsources.net/kidarep/dataset-9-konkama-1951-eu01.xhtml +Genealogical Network 62 +1463 +1969 +https://www.kinsources.net/kidarep/dataset-306-nobles-ile-de-france-1000-1440.xhtml +Genealogical Network 63 +4109 +6517 +https://www.kinsources.net/kidarep/dataset-287-duu-rea.xhtml +Genealogical Network 64 +29 +48 +https://www.kinsources.net/kidarep/dataset-46-hatfields-and-mccoys.xhtml +Genealogical Network 65 +40 +59 +https://www.kinsources.net/kidarep/dataset-33-angmagsalik-1884-nu01.xhtml +Genealogical Network 66 +294 +441 +https://www.kinsources.net/kidarep/dataset-18-tikopia-1930.xhtml +Genealogical Network 67 +502 +786 +https://www.kinsources.net/kidarep/dataset-34-netsilik-1922-nu09.xhtml +Genealogical Network 68 +83 +126 +https://www.kinsources.net/kidarep/dataset-8-semang-1924-50-as03.xhtml +Genealogical Network 69 +95 +157 +https://www.kinsources.net/kidarep/dataset-4-shoshone-1860-nd10.xhtml +Genealogical Network 70 +2588 +5651 +https://www.kinsources.net/kidarep/dataset-61-kelkummer.xhtml +Genealogical Network 71 +88 +144 +https://www.kinsources.net/kidarep/dataset-77-apache-1935-nd02.xhtml +Genealogical Network 72 +1513 +2217 +https://www.kinsources.net/kidarep/dataset-90-omaha-1880.xhtml +Genealogical Network 73 +3014 +5454 +https://www.kinsources.net/kidarep/dataset-128-ammonni.xhtml +Genealogical Network 74 +139 +201 +https://www.kinsources.net/kidarep/dataset-79-paiute-1880-nd09.xhtml +Genealogical Network 75 +5016 +10719 +https://www.kinsources.net/kidarep/dataset-249-baruya.xhtml +Genealogical Network 76 +125 +202 +https://www.kinsources.net/kidarep/dataset-242-tlingit.xhtml +Genealogical Network 77 +272 +445 +https://www.kinsources.net/kidarep/dataset-36-copper-1922-nu10.xhtml +Genealogical Network 78 +378 +609 +https://www.kinsources.net/kidarep/dataset-52-apache-1936-nd03.xhtml +Genealogical Network 79 +926 +1951 +https://www.kinsources.net/kidarep/dataset-68-surui.xhtml +Genealogical Network 80 +706 +1177 +https://www.kinsources.net/kidarep/dataset-60-mbuti-forest-1957-af02.xhtml +Genealogical Network 81 +435 +672 +https://www.kinsources.net/kidarep/dataset-64-melombo.xhtml +Genealogical Network 82 +128 +114 +https://www.kinsources.net/kidarep/dataset-164-kaingang.xhtml +Genealogical Network 83 +169 +275 +https://www.kinsources.net/kidarep/dataset-11-top-of-the-mountain.xhtml +Genealogical Network 84 +178 +274 +https://www.kinsources.net/kidarep/dataset-37-igluligmiut-1921-nu05.xhtml +Genealogical Network 85 +87 +111 +https://www.kinsources.net/kidarep/dataset-216-tiwi.xhtml +Genealogical Network 86 +2049 +4159 +https://www.kinsources.net/kidarep/dataset-35-chuukese-1947-1940.xhtml +Genealogical Network 87 +868 +980 +https://www.kinsources.net/kidarep/dataset-20-saudi-royal-genealogy.xhtml +Genealogical Network 88 +2821 +5079 +https://www.kinsources.net/kidarep/dataset-30-manus-1929.xhtml +Genealogical Network 89 +454 +980 +https://www.kinsources.net/kidarep/dataset-74-arawete.xhtml +Genealogical Network 90 +304 +472 +https://www.kinsources.net/kidarep/dataset-42-nunamiut-tareumiut-1900-nu12.xhtml +Genealogical Network 91 +367 +671 +https://www.kinsources.net/kidarep/dataset-48-wanindiljaugwa-1941-au05.xhtml +Genealogical Network 92 +3151 +4289 +https://www.kinsources.net/kidarep/dataset-54-feistritz-am-gael-1990.xhtml +Genealogical Network 93 +2975 +5107 +https://www.kinsources.net/kidarep/dataset-159-cocama-cocamilla.xhtml +Genealogical Network 94 +585 +1249 +https://www.kinsources.net/kidarep/dataset-44-torres-strait.xhtml +Genealogical Network 95 +334 +530 +https://www.kinsources.net/kidarep/dataset-6-igluligmiut-1949-nu06.xhtml +Genealogical Network 96 +9595 +14988 +https://www.kinsources.net/kidarep/dataset-93-sainte-catherine.xhtml +Genealogical Network 97 +28586 +51446 +https://www.kinsources.net/kidarep/dataset-76-san-marino.xhtml +Genealogical Network 98 +18645 +32439 +https://www.kinsources.net/kidarep/dataset-307-bwa-slam-biogsurvey.xhtml +Genealogical Network 99 +8809 +15643 +https://www.kinsources.net/kidarep/dataset-194-kel-owey.xhtml +Atypical Genealogical +Networks +Atyp. 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Tillmann, et al., A roadmap for the computation of persistent homology, +EPJ Data Science (2017) +26 + diff --git a/7dFLT4oBgHgl3EQfAS4p/content/tmp_files/load_file.txt b/7dFLT4oBgHgl3EQfAS4p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..101b5b1dcaa0696b3b3ecee86067476a3ce72c1b --- /dev/null +++ b/7dFLT4oBgHgl3EQfAS4p/content/tmp_files/load_file.txt @@ -0,0 +1,1358 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf,len=1357 +page_content='The persistent homology of genealogical networks Zachary M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Boyda,∗, Nick Callorb, Taylor Gledhillc, Abigail Jenkinsd, Robert Snellmane, Benjamin Webbf, Raelynn Wonnacottg aDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, zach boyd@byu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='edu bDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='callor@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='com cDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, gledhilltaylor2@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='com dDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, jenkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='abby@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='com eDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, snellman@mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='byu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='edu fDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, bwebb@mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='byu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='edu gDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, raelynnwo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='com Abstract Genealogical networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' family trees) are of growing interest, with the largest known data sets now including well over one billion individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Interest in family history also supports an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='5 billion dollar industry whose size is projected to double within 7 years [FutureWise report HC-1137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Yet little mathemat- ical attention has been paid to the complex network properties of genealogical networks, especially at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The structure of genealogical networks is of particular interest due to the practice of forming unions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' marriages, that are typically well outside one’s immediate family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In most other networks, including other social networks, no equivalent restriction exists on the distance at which relationships form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' To study the effect this has on genealogical networks we use persistent homology to identify and compare the structure of 101 genealogical and 31 other social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Specifically, we introduce the notion of a network’s persistence curve, which encodes the network’s set of persistence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We find that the persistence curves of genealogical networks have a distinct structure when compared to other social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This difference in structure also extends to subnetworks of genealogical and social networks suggesting that, even with incomplete data, persistent homology can be used to meaningfully analyze genealogical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Here we also describe how concepts from genealogical networks, such as common ancestor cycles, are represented using persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We expect that persistent homology tools will become increasingly important in genealogical exploration as popular interest in ancestry research continues to expand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Keywords: persistent homology, genealogical networks, social networks, persistence curves, bottleneck distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Introduction The study of genealogical networks, that is networks relating parents with children and spouses with each other through successive generations is of rapidly growing interest, both because of genealogy’s pop- ular appeal and its applications in genetics [1], sociology [2], population sciences [3], and economics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Growing data availability of rich, temporally resolved data is also driving interest in genealogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For example, ∗Corresponding Author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='11965v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='MN] 27 Jan 2023 FamilySearch has constructed a human family tree with over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='40 billion individuals, based on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='21 billion sources, including 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='78 billion images (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='familysearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='org/en/newsroom/company-facts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Popu- larization of DNA testing services and increasing availability of audio sources, geographic tags, occupation metadata, and migration records combine to make genealogical networks some of the largest, most richly featured, geospatially embedded temporal networks in existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Examples of relevant academic studies include methods for automatically constructing networks from documents [5, 6], analyzing marriage pat- terns [4], structured population modeling, branching processes [7], and biconnected components [2, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Of particular interest to us are works that study distance to recent common ancestors, both theoretically and via simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' [9, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' A growing body of literature also uses genealogical networks for genetic inference, as in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Related to these genealogical endeavors, a major goal of network science is to describe the structure of such real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In this paper, we consider persistent homology as a tool to both analyze and explore the structure of genealogical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Persistent homology, roughly speaking, is a method of representing voids or gaps in the structure of a network, that distinguishes how significant these voids are to the overall network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Persistent homology can be used to compare these voids across two networks without requiring a correspondence between the individual vertices or edges, or even requiring the networks to be the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The basic idea involves “filling in” the network with simplices (points, edges, triangles, tetrahedra, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=') and keeping track of how the network changes as we do so (see Section 3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Some similar applications of persistent homology in the study of networks include [26], [28], [30], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The collaboration networks studied in [26] are similar to the social networks that we use for comparison in this paper, though our focus is primarily on distinguishing these from genealogical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Both [28] and [30] apply persistent homology techniques to general randomized networks of various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' It is also possible to vary the technique for generating a topological object from a network, as in [27] where three methods are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We also recommend [29] and [36] as good overviews of the general methods of applying persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For this paper, our method of constructing a topological representative for each network follows the same general pattern as the work cited above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' However, we also acknowledge the wide variety of alternatives for encoding such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' [32] and [33] encode their information as point-clouds rather than graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' A higher-dimensional version of persistent homology is presented in [34], which may permit the inclusion of time-varying networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Finally, the formulation in [35] may allow for better analysis of corrupted or too-large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We also wish to bring attention to four particular applications that demonstrate the versatility of persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In each of these applications, persistent homology has been used to identify structural voids in data and then to associate these voids to recognizable features in the underlying networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' It is the latter use that we wish to emphasize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' [10] have shown that voids found using persistent homology correspond to percolating spheres in a porous material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In [11], structural voids arise when several groups of neurons are strongly connected sequentially, but out-of-sequence pairs are only weakly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In these neurological networks, persistent homology provides a way to identify and classify these different sequences as well as quantify the strength of these connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The application in [31] provides a method for extending traditional genetic analysis tools to a parameterized family of datasets by constructing an appropriate topological object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Lastly, [12] shows that structural voids or gaps can also represent much more abstract concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In this case persistent voids are shown to correspond to the atonality in music compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Intuitively, the voids or gaps in genealogical networks should be quite different when compared with 2 other networks, such as social networks, since unions1 (such as marriages) in genealogical networks typically form at specific distances, rather than through other mechanisms e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' triadic closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' That is, distances between individuals who form unions are typically not too small or too large (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In contrast, in other social networks, new connections can form at any distance but are often quite small [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This difference in network growth between genealogical and other social networks causes differences in network topology that are reflected in the network’s persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Thus persistent homology is a useful descriptive tool for exploring and modeling the structure of genealogical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Here, we propose a new method for representing persistent homology, which we call a persistence curve (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The persistence curves of many genealogical networks are very similar to each other, and importantly the persistence curves of subsets of genealogical networks, that is, sampled genealogical networks, are also similar to the persistence curves of unsampled genealogical networks (see Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' To give our study of genealogical networks context we also study the persistent homology of social net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We find that the same result holds for the social networks we consider, in that the persistence curves of social networks show a common pattern and the persistence curves for social and sampled social networks are similar (see Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We confirm our analysis using another tool for comparing persistent homologies, the bottleneck distance, which is also capable of detecting and differentiating the distinct homology patterns between genealogical and other social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In summary, we make the following contributions: Introduce the notion of a persistence curve and introduce the use of this together with the bottleneck distance as a tool for the analysis of general networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Report the distinct persistent homology structure of genealogical networks using both persistence curves and the bottleneck distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Link this structure to genealogically relevant concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Similarly, report the distinct persistence homology structure of social networks and compare this to the structure of genealogical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Report evidence that persistent homology methods work well even in the presence of incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This is particularly relevant given that genealogical data is often, if not necessarily, incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Throughout the paper, examples from family networks are contrasted with other social networks to highlight the unique features of genealogical networks from a persistent homology point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In Section 2 we describe both genealogical and social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In Section 3 we define the persistent homology of a network and introduce the notion of persistence curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In Section 4 we define the bottleneck distance and show how both this distance and persistence curves can be used to compare networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In Section 5 we describe the genealogical and social data sets we use in our study and give our experimental results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Section 6 also includes a discussion of how certain structural features of social and genealogical networks are represented using persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In Section 7 we summarize our results and conclude with a discussion regarding the use of persistent homology as a tool for analyzing general network structure and recovering network features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Throughout we give examples of each of the concepts we introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 1In order to be inclusive of various relevant relationships in this paper, we use the word “union” to describe not only legal marriages and common law marriages but also some others, including any relationship that produced children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Background: Genealogical and Social Networks We represent genealogical networks with a graph G = (V, E), where V = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' , n} are the individ- uals within the network, and E are the (genealogical) relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' These relationships consist of both parent-child edges and spouse (or more generally union) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For the sake of simplicity, these edges are considered to be undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We note that the structure of a genealogical network is often thought of as Out[1858]= Tikopia Genealogical Network Residence Hall Social Network Figure 1: Left: The largest connected component of the Tikopia genealogical network consisting of 288 individuals from the island of Tikopia in Polynesia from the year 1930 to 2008, is shown [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Parent-child edges are shown in blue and union edges are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Right: The largest connected component of the Residence Hall social network consisting of 217 individuals and their friendships from the Australian National University campus is shown [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' being “tree-like”, since genealogical networks are often constructed from an individual, their parents, their grandparents, and so on, ignoring union edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The result is a tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' a connected acyclic graph, if we create only a few generations of the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' However, full genealogical networks are not trees due to the presence, for example, of triangles consisting of two parents and a child (with the two parent-child edges and one union edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Because of the frequency of such cycles and the fact that they are the smallest possible cycles, we refer to them as trivial cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The other typical familial cycle, or cycle found within a family consisting of two parents and some number of children, is a cycle of length four consisting of two parents and two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Although familial cycles are ubiquitous in genealogical networks, they are not the only cycles that can form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Going far enough through an individual’s ancestors, it is often possible to find a nearest common ancestor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=', a common ancestor of one’s father and mother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' If such an ancestor exists (and it usually does exist), then the genealogical network has a nontrivial cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We refer to this as a common ancestor cycle, which consists of only parent-child edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Other nontrivial cycles are possible in genealogical networks via unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For instance, a “double cousins” relationship occurs when two siblings from one family form unions with two siblings from another family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The result is a union cycle, or a cycle that contains only union edges and the parent-child edges connecting siblings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In genealogical networks, union and parent-child edges can combine in any number of ways to create complex non-tree structures (see Figure 1 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' A feature that is particular to genealogical networks is that union edges typically form at specific dis- tances within these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Here the distance d(i, j) between i and j is the shortest path distance between these individuals if such a path exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Otherwise, it is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In a genealogical network we refer to the distance between two individuals before they form a union as the couple’s distance to union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For cultural, genetic, and other reasons these distance are typically not small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' usually larger than four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Consequently, 4 0 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='15 Distance to Union Fraction of Unions at Distance Figure 2: The histogram representing the finite “distance to union” distances is shown where data is collected from 104 genealogical networks from kinsources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The height of each bar represents the fraction of unions that form at a specific distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' genealogical networks do not typically have small nonfamilial cycles and often have large extended cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This is illustrated in Figure 2 where distance to union data is collected from 104 publicly available genealog- ical networks given in Table 2 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Here familial cycles are omitted and the height of each bar represents the fraction of unions that form at a specific distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Noticeably, few unions form at distances less than five with the large majority of distance falling between 5 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The observation that genealogical networks have large extended cycles is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Shown left in orange is the distribution of cycle lengths of the San Marino genealogical network, a network of the population of the Republic of San Marino from the 15th to the end of the 19th century [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In this network, which consists of 28,586 individuals, there are 7,146 familial cycles of length three and 8,636 familial cycles of length four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' These are omitted in the figure so we can observe the lengths of the cycles forming a basis of nonfamilial cycles in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For the sake of contrast, in blue is the distribution of cycle lengths in a basis of the cycles found in the Deezer Europe social network, consisting of 28,281 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Here, similar to genealogical networks, a social network is represented by a graph G = (V, E) where the vertices V also represent individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The difference is that in a social network the edges represent some type of social interaction(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The Deezer network is an online music streaming platform whose social network represents individuals in Europe who use the platform where edges represent mutual user-follower relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Noticeably, the San Marino network has relatively few nonfamilial basis cycles under length ten but quite a few cycles with lengths greater than thirty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In contrast, the Deezer social network has a much tighter distribution of basis cycles ranging from roughly five to fifteen in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' To understand the extent to which these cycle distributions are related to the local structure of the associ- ated networks we compare these to the cycle distribution of the associated configuration models of these two networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The configuration model is a model for generating random networks with a given degree sequence [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Taking the degree sequences from both the San Marino genealogical and Deezer so- cial network, we create ten versions of these networks each with the same degree sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The result of averaging the basis cycle length distributions of these versions of the San Marino and Deezer networks is shown in Figure 3 (center and right in red and green, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' While the cycle distribution for the San Marino network is quite different from what the configuration model produces, the Deezer social network is quite similar to the distribution predicted by its configuration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This suggests that much of the cy- cle structure in the Deezer social network is dominated by local interactions, whereas the cycles in the San Marino genealogical network are affected by nonlocal mechanisms that form the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This includes, 5 Out[51]= 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='20 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='20 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='35 SM and DE Cycle Lengths SM Configuration Model DE Configuration Model Figure 3: Left: Shown in orange is the distribution of the lengths of the cycles forming a basis of the nonfamilial cycle lengths in the San Marino (SM) genealogical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The analogous distribution of cycle lengths is shown in blue for all cycles in the Deezer Europe (DE) social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Center: Shown in orange is again the basis cycle length distribution of the San Marino genealogical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In red is the distribution of the basis cycle lengths averaged over ten realizations of the (loopy, multi-edged) configuration model on the San Marino network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Since the configuration model generates graphs with the same degree distribution as the SM network, this panel indicates that SM’s longer cycles do not arise simply from the degree distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Right: Shown in blue is again the basis cycle length distribution of the Deezer social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In green is the distribution of the basis cycle lengths averaged over ten realizations of the configuration model on the Deezer social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For this social network, the cycle length distribution can be mostly explained by the degree distribution alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' presumably, the nonlocal distance to union phenomena described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The relations we see in Figure 3 between the cycle length distribution for the San Marino genealogical network and the Deezer social network are typical of the genealogical and social networks we consider in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This suggests that cycle length distribution is a feature that can be used to distinguish genealog- ical from social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Specifically, when we consider two networks with a similar number of cycles, genealogical networks have a much wider distribution of cycle lengths than social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' However, the method used to calculate the cycle length distribution in Figure 3 does not provide any further insight into this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This limitation motivates us to apply tools from persistent homology which provides ways to describe and measure the relation between any two network cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The additional structure that can be obtained by these methods allow us to further distinguish the structure of genealogical and social networks (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='1) and to relate the structural differences demonstrated in Figure 3 to mechanisms that produce genealogical and social networks, respectively (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Persistent Homology of Networks Persistent homology provides a method for studying cycles in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For the purposes of this paper, a brief explanation of persistent homology will be given from the context of simplicial homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For a more in-depth treatment of simplicial homology, see Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='1 of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For those readers who are either familiar with the basics of persistent homology or who wish to skip the following technical discussion it is possible to proceed to Section 5 where we discuss the social and genealogical networks we analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For a network given by a graph G = (V, E) we define the distance matrix D(G) = [di j] to have entries di j = d(i, j), which is the length of the shortest path between individual i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For each value δ that appears in the distance matrix D(G), we form a simplicial complex Gδ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The set of 0-simplices is equivalent to the set of vertices of G, where each 0-simplex is identified with a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Since the distinction between 0-simplices and vertices is purely formal, we will use the terms 0-simplex and vertex interchangeably, and the 0-simplices will be indexed the same way as the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The set of 1-simplices Eδ corresponds to the set of edges {i, j} such that d(i, j) ≤ δ, where the edge {i, j} is identified with the 1-simplex 6 formed by i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Again the distinction here is unnecessary for our present discussion, so we will use the same notation for 1-simplices and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' However, the simplicial complex Gδ may also contain objects that do not have equivalent representatives in the graph G, namely the n-simplices for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For each integer n ≥ 2, the set of n-simplices in Gδ consists of all n-simplices [a0 a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' an] such that d(ai, aj) ≤ δ for 0 ≤ i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' That is, Gδ includes an n-simplex σ if each vertex listed in σ is within δ of every vertex listed in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In order to simplify our remaining definitions, we extend our definition of Gδ to include all non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i ≥ 0, let δi be the greatest entry of D(G) such that δi ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Let Gi = Gδi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This definition together with our construction of Gδ ensures the following three important properties are true for all Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i < j, Gi is a subcomplex of G j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' every simplex of Gi is a simplex of G j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i ≥ 1, there exists a subcomplex of Gi that can be identified with the original graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Since G is finite, let M = maxij d(i, j), then, for all i ≥ M, Gi = GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' (a) G0 (b) G1 = G (c) G2 (d) G3 Figure 4: The hexagonal network G = G1 in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='1 is filled in as i increases from 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This produces the simplicial complexes G0,G1,G2,G3 shown left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' (Hexagonal Network) Consider the hexagonal network G = (V, E) with six vertices, forming a single cycle, shown in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This network has the distance matrix D(G) = ������������������������� 0 1 2 3 2 1 1 0 1 2 3 2 2 1 0 1 2 3 3 2 1 0 1 2 2 3 2 1 0 1 1 2 3 2 1 0 ������������������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For the values i = 0, 1, 2, 3, we form four simplicial complexes, G0, G1, G2, and G3 where we let Gi = (Vi, Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i = 0, E0 is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Thus, G0 consists of six vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i = 1 the set E1 contains the six edges that form the network’s single cycle, so G1 = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' This graph has no trivial cycles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=', triangles), so G1 contains no simplices of dimension greater than 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=', no n-simplices for n > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i = 2 the set E2 gains six additional edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We also now have eight trivial cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Each of these cycles is the boundary of a 2-simplex, so G2 contains these eight 2-simplices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' However, no subset of these 2-simplices forms the boundary of a 3-simplex, so G2 has no simplices of dimension greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' For i = 3 the set E3 contains all possible edges between the vertices of G, so all possible trivial cycles are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Additionally, all possible 2-simplices, and hence all possible n-simplices, are also present in G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' In particular, G3 is a 6-simplex with its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Since M = 3 is the largest value we see in the distance matrix, then Gi = G3 for i ∈ Z, i > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' 7 The persistent homology of the network G measures how the homology of Gi changes as i increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' If certain features can be identified across multiple values of i, we say they persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Intuitively, features that arise from the actual network structure should persist for many values of i, while features that arise because of measurement error, ‘noise’, should only appear sporadically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The Stability Theorem (the Main Theorem of [18]) states that if the error in measuring a network is bounded by some constant C, then the persistent homology of the true network and the persistent homology of the noisy network will differ by at most C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We will make this statement more precise in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Here we give a formal definition of persistent homology in terms of simplicial homology, which we will immediately follow this with equivalent definitions in the context of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' We use Hp(Gi) to denote the dimension-p simplicial homology of the simplicial complex Gi with coefficients in Z2, as Hp(X) is a vector space of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' (pth Persistent Homology) For a graph G, and integers i, j with 0 ≤ i ≤ j, let the function φi, j : Hp(Gi) → Hp(G j) be the linear map induced by the inclusion Gi → G j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dFLT4oBgHgl3EQfAS4p/content/2301.11965v1.pdf'} +page_content=' The pth persistent homology of G, PHp(G) is the pair ({Hp(Gi)}i≥0, {φi,j}0≤i0} is the set of activities directly-follow a in L at least +once and σ ∈ A∗. Directly-follow activities sorting is sortDFA(a, L) = σ such +that {b ∈ σ} = A and |σ| = |A| and ∀1≤i 0, the algorithm +provides a performance guarantee of 1 − ϵ. +- Empirical analysis. +We study the empirical performance of the proposed local-improvement +algorithm against baseline methods on both synthetic and real-world networks. Overall, we observe +that the empirical approximation ratio of the proposed algorithm is much higher than 1/2, which is +our theoretical guarantee. +3 + +2 Related Work +Integration in public housing. Issues regarding segregation and the need for enhancing integration +have been documented extensively in the social science literature (e.g., [11, 24, 21, 26]). In partic- +ular, many works on segregation in social networks (e.g., [16, 18]) stem from the pioneering models +proposed by Schelling [31], where agents move between vertices to improve their utility values. While +Schelling’s framework allows the study of agent dynamics, Benabbou et al. [4] study integration in +public housing allocation from a planning perspective. In particular, they formulate the setting as a +weighted matching problem where the set of available houses is partitioned into blocks, and agents +are assigned (by some central agency) to blocks to maximize a utility measure while satisfying some +diversity constraints. They establish the NP-hardness of the problem and present an approxima- +tion algorithm based on a result of Stamoulis [34]. A number of other studies have also addressed +integration in the context of public housing from a social science perspective (e.g., [28, 19, 22, 17]). +The problem formulations and the algorithmic techniques used in Benabbou et al. [4] and in our +work are significantly different. First, Benabbou et al. [4] examine a weighted matching problem. +Their model does not use any network structure for the residences, whereas our work approaches +the problem from a graph theoretic standpoint, with the underlying network playing an important +role in the formulation. Further, the integration index studied in our work is defined w.r.t graph +structures, whereas the measure used in [4] is based on constraints on the ethnicity quotas for blocks. +More importantly, the goal of our work is to find an assignment that maximizes the integration level, +whereas the goal in [4] is to maximize the overall utility of agents under a diversity constraint. +Integration indices. Various indices to measure the level of integration in a population are surveyed +in [24]. +However, most of those indices cannot be naturally extended to a network setting. +The +integration index IoA considered in our work was proposed by Agarwal et al. [1]2 in the context of the +Schelling Game on networks, where agents can change locations to increase their utilities. Agarwal +et al. explore several properties (e.g., the integration price of anarchy/stability) of the index from a +game theoretic perspective. Further, they show that finding an assignment for which all agents are +integrated (i.e., each agent has at least one neighbor of a different type) is NP-hard [1]. +Approximation algorithms. Our approximation algorithm for general IM-IoA is based on a local- +improvement scheme. A well-known problem for which a local-improvement algorithm provides an +approximation guarantee of 1/2 is the unweighted MaxCut problem [25]. We note that the analysis +used to establish the performance guarantees of the local-improvement methods for MaxCut and IM- +IoA are substantially different. In particular, MaxCut has no cardinality constraints, and the objective +is defined w.r.t edges. In contrast, IM-IoA requires that a specified number of vertices be assigned +2In Agarwal et al. [1], the index is called “degree of integration”. In our work, the term “degree” is used to denote the +degree of a vertex. We use the term “index of integration” to denote the index proposed in [1]. +4 + +to type-1 agents, and the objective is defined w.r.t vertices. One can also formulate IM-IoA as a +non-monotone submodular function maximization problem. Since such a formulation requires a strict +equality constraint (involving type-1 agents), the best known performance guarantee under the general +non-monotone submodular maximization framework with such a constraint is 0.356 [7]. +3 Problem Definition +We study the problem of assigning vertices in a graph to a group of agents, such that the integration +level of the resulting layout of agents in the graph is maximized. We begin with some definitions and +then provide a formal definition of the maximization problem. +Graphs and agents. Let G = (V,E) be an undirected graph, where V is a set of vertices representing +vacant residences, and E is a set of edges representing the neighborhood relationship between resi- +dences. Let A be the set of agents to be assigned to V. We consider a setting where the set of agents +is divided into two demographic subgroups. Formally, A is partitioned into two subsets A1 and A2; +we refer to agents in Ai as type i agents, i = 1,2. Let k = ∣A1∣ denote the number of type-1 agents, so +n − k is the number of type-2 agents. Without loss of generality, we let k ≤ n/2, and we refer to A1 as +the minority subgroup. Lastly, we assume that ∣V∣ = ∣A∣; that is, the number of vertices is the same as +the number of agents. +Assignment. An assignment is a mapping from vertices to agents. To simplify proofs, we use an +equivalent definition where an assignment is a mapping from vertices to agent types. In particular, +an assignment P ∶ V → {1,2} is a function that assigns an agent type to each vertex in V, such that +k vertices are assigned type-1 and n − k vertices are assigned type-2. In such an assignment, a type-i +vertex is occupied by a type-i agent, i = 1,2. We remark that the above definition of an assignment is +mathematically equivalent to defining an assignment to be a mapping from V to A. +The index of integration. We consider the integration index proposed in [1] and apply it to our +context. +▷ Definition 3.1 (Index of agent-integration (IoA) [1]). Given an assignment P, an agent +x ∈ A is integrated if x has at least one neighbor in G whose type is different from that of x. Let +A′ be the set of integrated agents under P. The index of agent-integration of P is then defined as +the number of integrated agents in A: +IoA(P) = ∣A′∣ +(1) +Equivalently, a vertex u ∈ V is integrated under P if the agent assigned to u is integrated. Thus, +we may also view the index as IoA(P) = ∣V′∣ where V′ is the set of integrated vertices under P. We +remark that these two definitions of IoA are mathematically equivalent. +5 + +The optimization problem. +We define the problem Integration Maximization-Index of +Agent Integration (IM-IoA). +▷ Definition 3.2 (IM-IoA). Given a graph G = (V,E), a set A of agents with k type-1 and n − k +type-2 agents, find an assignment P such that IoA(P) is maximized. +4 Approximation for General Graphs +IM-IoA is NP-hard, as established in [1]. In this section, we present a local-improvement algorithm +for IM-IoA and show that the algorithm achieves a factor 1/2 approximation for general graphs. +For convenience in presenting the proofs, we consider an assignment from the perspective of vertices +rather than that of the agents. As stated earlier, these two definitions are equivalent. +The algorithm. We start from a random assignment P. In each iteration of the algorithm, we find +(if possible) a pair of type-1 and type-2 vertices such that swapping their types strictly increases the +objective. In particular, let u be a type-1 vertex, and v be a type-2 vertex. We swap the types of u +and v (i.e., u becomes type-2 and v becomes type-1) if and only if the resulting new assignment P′ +has a strictly higher IoA; that is, IoA(P) < IoA(P′). The algorithm terminates when no such swap +can be made. The pseudocode is given in Algorithm (1). +Algorithm 1: Local-Improvement-IoA +Input +: A graph G = (V,E), k, where k ≤ ∣V∣/2 +Output: An assignment P +1 P ← a random assignment & Updated ← True +2 while Updated do +3 +Updated ← False +4 +for x ∈ V1(P) do +5 +for y ∈ V2(P) do +6 +P′ ← the assignment where P′(x) = P(y) and P′(y) = P(x) +7 +if IoA(P′) > IoA(P) then +8 +P = P′, Updated ← True & break +9 return P +4.1 Analysis of the algorithm +Given a problem instance of IM-IoA, let P be a saturated assignment3 returned by Algorithm (1). +Let P∗ be an optimal assignment that achieves the maximum objective, denoted by OPT. We assume +that P ≠ P∗. In this section, we show that IoA(P) ≥ 1/2 ⋅ IoA(P∗) = 1/2 ⋅ OPT, thereby establishing a +3An assignment is saturated if no pairwise swap of types between a type-1 and a type-2 vertices can increase the +objective. +6 + +1/2 approximation. Due to the page limit, we sketch the proof here; the full proof appears in the +appendix. +Given an assignment P, which is a mapping from vertices to agent types, we call a vertex v a +type-1 (or type-2) vertex if P(v) = 1 (or P(v) = 2). Let V1(P) and V2(P) denote the set of type-1 +and type-2 vertices under P. Let VU +1(P) and VU +2(P) denote the set of uncovered4 type-1 and type-2 +vertices under P. For each vertex u, let N U +u(P) denote the set of neighbors of u that are uncovered +under P, and let Γu(P) denote the set of different-type neighbors of u that are uniquely covered by +u, i.e., Γu(P) is the set of vertices v such that (i) v is a neighbor of u, (ii) the type of v is different +from the type of u, and (iii) v has no other neighbor whose type is the same as u’s type. +▷ Observation 4.1. The index IoA(P) = n − ∣VU +1(P)∣ − ∣VU +2(P)∣. +We now consider the following mutually exclusive and collectively exhaustive cases of VU +1(P) and +VU +2(P) under the saturated assignment P. We start with a simple case where all the type-2 vertices +under P are integrated. +Case 1: VU +2(P) = ∅. +Under this case, all vertices in V2(P) are integrated which gives +IoA(P) ≥ ∣VU +2(P)∣ = n − k ≥ 1 +2 ⋅ n ≥ 1 +2 ⋅ OPT +(2) +The above case trivially implies that the algorithm provides a 1/2 approximation. We now look at the +remaining case where VU +2(P) ≠ ∅. +Case 2: VU +2(P) ≠ ∅. +Under this case, there exists at least one vertex in V2(P) that is not integrated. We first show that +VU +1(P) and VU +2(P) cannot both be non-empty. +▷ Lemma 4.2. For a saturated assignment P, if VU +2(P) ≠ ∅, then VU +1(P) = ∅. +Proof. (Sketch) Let y ∈ VU +2(P) be a vertex of type-2 that is not integrated (i.e., all neighbors of y are +of type-2). For contradiction, suppose VU +1(P) ≠ ∅. Now let x ∈ VU +1(P) be an non-integrated vertex of +type-1 whose neighbors are all of type-1. Let P′ denote the assignment where we switch the types +between x and y, that is, P′(x) = P(y) = 2, P′(y) = P(x) = 1, while the types of all other vertices +remain unchanged. One can verify that IoA(P′) ≥ IoA(P) + 2, that is, switching the types of x and +y increases the index IoA by at least 2. This implies the existence of an improvement move from P, +which contradicts the fact that P is saturated. It follows that VU +1(P) = ∅. +∎ +4Under an assignment, a vertex is “covered” if it is integrated and “uncovered” otherwise. +7 + +Lemma (4.2) implies that under case 2 (i.e., VU +2(P) ≠ ∅), we have VU +1(P) = ∅. We now consider +the following two mutually exclusive and collectively exhaustive subcases under Case 2 and show that +the approximation factor under each subcase is 1/2. +Subcase 2.1: VU +2(P) ≠ ∅, and Γx(P) ≠ ∅, +∀x ∈ V1(P), that is, for each type-1 vertex x ∈ V1(P), +there is at least one type-2 neighbor of x that is uniquely covered (“made integrated”) by x. +Suppose P ≠ P∗, that is, for some vertices x ∈ V, P(x) ≠ P∗(x). Let ˜V2−1 = {v ∈ V +∶ P(v) = +2,P∗(v) = 1} be the set of vertices that are type-2 under P, but are type-1 under P∗. Analogously, +let ˜V1−2 = {v ∈ V ∶ P(v) = 1,P∗(v) = 2} be the set of vertices of type-1 under P, but are of type-2 +under P∗. Observe that ∣˜V2−1∣ = ∣˜V1−2∣. We may view P∗ as the result of a transformation from P +under pairwise swaps of types between ˜V2−1 and ˜V1−2. An example is given in Figure (2). We present +a key lemma that bounds the difference between the objective values of P and P∗. +▷ Lemma 4.3 (Subcase 2.1). Let P be a saturated assignment under subcase 2.1, and let P∗ be an +optimal assignment. We have +IoA(P∗) − IoA(P) ≤ +∑ +y∈˜V2−1∖VU +2(P) +∣(N(y) ∩ VU +2(P)∣ +(3) ++ +∑ +y∈˜V2−1∩VU +2(P) +(∣(N(y) ∩ VU +2(P)∣ + 1). +Proof. (Sketch) Since P is saturated, Lemma (4.2) implies that all type-1 vertices under P are in- +tegrated. Thus, the difference IoA(P∗) − IoA(P) is at most the number of type-2 vertices that are +integrated under P∗ but are not integrated under P. +Let f ∶ ˜V1−2 → ˜V2−1 be an arbitrary bijective mapping. We may regard P∗ as a result of the +transformation from P via pairwise swaps of types between vertices specified by f (i.e., the type of +x ∈ ˜V1−2 is swapped with the type of f(x) ∈ ˜V2−1). Observe that only vertices in VU +2(P) that are +adjacent to ˜V2−1 (or within ˜V2−1) under P can be newly integrated under P∗ after swapping ˜V1−2 +with ˜V2−1 (by the definition of VU +2(P), vertices in ˜V1−2 have no neighbors in VU +2(P).). It follows that +for each vertex y ∈ ˜V2−1, at most ∣(N(y) ∩ VU +2(P)∣ of its neighbors can become newly integrated after +transforming from P to P∗. Further, if also y ∈ ˜V2−1 ∩ VU +2(P), y itself could also be newly integrated +after the swap. We then have +IoA(P∗) − IoA(P) ≤ ∣ +⋃ +y∈˜V2−1 +N(y) ∩ VU +2(P)∣ + ∣˜V2−1 ∩ VU +2(P)∣ +≤ +∑ +y∈˜V2−1∖VU +2(P) +∣(N(y) ∩ VU +2(P)∣ +(4) ++ +∑ +y∈˜V2−1∩VU +2(P) +(∣(N(y) ∩ VU +2(P)∣ + 1) +8 + +where the last inequality follows from the union bound. +∎ +Figure 2: Two assignments P and P∗ where type-1 and type-2 vertices are highlighted in blue and +red, respectively. In this case, ˜V2−1 = {x3,x4} and ˜V1−2 = {x1,x2}. We may then transform +P into P∗ by swapping types between the pair (x1,x3) and between (x2,x4). Note that this +example is only to demonstrate how ˜V2−1 and ˜V1−2 are defined, as P cannot be a saturated +assignment returned by the algorithm. +We now proceed to show that the difference between IoA and IoA established in Lemma (4.3) is +at most IoA(P), thereby establishing IoA(P) ≥ 1 +2 ⋅ IoA(P∗). Recall that for each vertex x ∈ V, Γx(P) +is the set of neighbors of x whose types are different from x, and are uniquely covered by x under P. +By the definition of Subcase 2.1, Γx(P) is not empty for all x ∈ V1(P). We first argue that for any +y ∈ VU +2(P) and any x ∈ V1(P), we have ∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣. +▷ Lemma 4.4 (Subcase 2.1). Given a saturated assignment P, for any y ∈ VU +2(P) and any x ∈ V1(P), +we have +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣. +Proof. (Sketch) Given that y is not integrated under P, x and y cannot be adjacent. Since P is a +saturated assignment, if the types of x and y are to be swapped, the number of newly integrated +vertices would be at most the number of newly non-integrated vertices. Further, one can verify that +the number of vertices that are newly integrated is at least ∣N(y) ∩ VU +2(P)∣ + 1, and the number of +vertices that are newly non-integrated is at most ∣Γx(P)∣ + 1. Since P is saturated, it follows that +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣. This concludes the proof. +∎ +We now establish the next Lemma, which bounds the size of N(y) ∩ VU +2(P) for y ∈ V2(P) ∖ VU +2(P) +and x ∈ V1(P). +▷ Lemma 4.5 (Subcase 2.1). Given a saturated assignment P, for any y ∈ V2(P) ∖ VU +2(P) and any +x ∈ V1(P), we have +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ + 1 +Proof. (Sketch) We partition V2(P) ∖ VU +2(P) into two subsets B and C, as follows. Subset B is the set +of integrated type-2 vertices whose neighbors are all integrated under P, i.e., B = {y ∈ V2(P)∖VU +2(P) ∶ +9 + +PN(y) ∩ VU +2(P) = ∅}. Subset C, the complement of B, is the set of integrated type-2 vertices with at +least one non-integrated neighbor under P, i.e., C = {y ∈ V2(P) ∖ VU +2(P) ∶ N(y) ∩ VU +2(P) ≠ ∅}. The +lemma clearly holds if y ∈ B. Further, we show that for the case when y ∈ C, no type-1 neighbors of +y is uniquely covered by y under P (i.e., Γy(P) = ∅). Further, suppose y ∈ C, consider an objective +non-increasing move from P where we swap the types between x and y. If y is a neighbor of x under +P, one can verify that the the maximum loss is ∣Γx(P)∣ and the minimum gain is ∣N(y) ∩ VU +2(P)∣. +Thus +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ +(5) +On the other hand, if y is not a neighbor of x under P, one can verify that the maximum loss is +∣Γx(P)∣ + 1 and the minimum gain is ∣N(y) ∩ VU +2(P)∣. Thus +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ + 1 +(6) +This concludes the proof. +∎ +We are now ready to establish IoA(P) ≥ 1 +2 ⋅ IoA(P∗) under Subcase 2.1. +▷ Lemma 4.6 (Subcase 2.1). Suppose VU +2(P) ≠ ∅ and Γx(P) ≠ ∅,∀x ∈ V1(P), we have IoA(P) ≥ +1 +2 ⋅ IoA(P∗) where P∗ is an optimal assignment that gives the maximum objective. +Proof. (Sketch) Note that ˜V2−1 is a subset of V2(P). Further, Observe that Γx(P) are disjoint for +different vertices x ∈ V1(P). Now, by Lemma (4.3) to and (4.5), We have +IoA(P∗) − IoA(P) +≤ +⎛ +⎜ +⎝ +∑ +y∈˜V2−1 +∣Γf−1(y)(P)∣ +⎞ +⎟ +⎠ ++ ∣˜V2−1∣ +≤ ∣V2(P) ∖ VU +2(P)∣ + ∣V1(P)∣ +(7) +≤ IoA(P) +where Inequality (7) follows from ∣˜V2−1∣ = ∣˜V1−2∣ ≤ ∣V1(P)∣ and (∑y∈˜V2−1 ∣Γf−1(y)(P)∣) ≤ ∣V2(P) ∖ +VU +2(P)∣. +∎ +We now have shown that if VU +2(P) ≠ ∅ and Γx(P) ≠ ∅,∀x ∈ V1(P), the algorithm gives a 1/2 +approximation. We proceed to the final subcase. +Subcase 2.2: VU +2(P) ≠ ∅, and Γx(P) = ∅,∃x ∈ V1(P), that is, there exists at least one type-1 vertex +10 + +x ∈ V1(P) such that for each type-2 neighbor y of x, y is adjacent to at least one type-1 vertex other +than x. +▷ Lemma 4.7 (Subcase 2.2). Under subcase 2.2, for each non-integrated type-2 vertex y ∈ VU +2(P), all +type-2 neighbors of y are integrated (i.e., N(y) ∩ VU +2(P) = ∅) under P. That is, the vertices in VU +2(P) +form an independent set of G. +Proof. (Sketch) Given such a x ∈ V1(P) defined in Subcase 2.2, for contradiction, suppose there exists +a non-integrated type-2 vertex y ∈ VU +2(P) such that at least one type-2 neighbor, denoted by y′ ∈ N(y), +of y is not integrated under P (note that all neighbors of y are of type-2 since y is not integrated). +Now consider a new assignment P′ where we switch the types between x and y. One can verify that +IoA(P′) ≥ IoA(P)+1, that is, after the switch, the index IoA would increase by at least 1. This implies +the existence of an improvement move from P, which contradicts P being a saturated assignment. +Thus, no such a non-integrated type-2 vertex y′ of y can exist. +∎ +Observe that IoA(P) = (n−∣VU +2(P)∣). With Lemma (4.7) in place, we now argue that the size of VU +2(P) +cannot be too large. +▷ Lemma 4.8 (Subcase 2.2). Under Subcase 2.2, ∣VU +2(P)∣ ≤ n +2 +Proof. (Sketch) Let Y ∶= {y ∈ V2(P) ∖ VU +2(P) ∶ N(y) ∩ VU +2(P) ≠ ∅} be the set of type-2 integrated +vertices whose has at least one non-integrated type-2 neighbor. +We first note that Γy(P) (if not +empty) are mutually disjoint for different y ∈ Y. It follows that IoA(P) ≥ ∣Y∣ + ∑y∈Y ∣Γy(P)∣. Suppose +we switch the types between such a vertex x and a vertex y ∈ Y, and let P′ denote the resulting new +assignment. One can verify that the maximum loss of objective after the swap is ∣Γy(P)∣ + 1, whereas +the minimum gain is ∣N(y) ∩ VU +2(P)∣. Since P is a saturated assignment returned by the algorithm, +we must have IoA(P) ≥ IoA(P′). Therefore, ∣N(y) ∩ VU +2(P)∣ ≤ ∣Γy(P)∣ + 1, ∀y ∈ Y. Overall, we have +that +∣VU +2(P)∣ = ∣ ⋃ +y∈Y +N(y) ∩ VU +2(P)∣ +(8) +≤ ∣V1(P)∣ + ∣Y∣ +(9) +≤ ∣V1(P)∣ + ∣V2(P) ∖ VU +2(P)∣ +(10) += n − ∣VU +2(P)∣ +(11) +It immediately follows that ∣VU +2(P)∣ ≤ n +2 . +∎ +Lastly, Since IoA(P) = n − ∣VU +2(P)∣, by Lemma (4.8), we have IoA(P) = n − ∣VU +2(P)∣ ≥ 1 +2 ⋅ n ≥ +1 +2 ⋅ IoA(P∗), thereby establishing a 1/2 approximation for Subcase 2.2. Overall, we have shown that a +11 + +saturated assignment P returned by Algorithm (1) gives a 1/2-approximation for IM-IoA. Thus: +▷ Theorem 4.9. Algorithm (1) gives a 1 +2-approximation for IM-IoA. +Analysis is tight. We present a class of problem instances where the approximation ratio of the +solution produced by Algorithm (1) can be arbitrarily close to 1/2. Therefore, the ratio 1/2 in the +statement of Theorem (4.9) cannot be improved, so our analysis is tight. The proof appears in the +Appendix. +▷ Proposition 4.10. For every ϵ > 0, there exists a problem instance of IM-IoA for which there is +a saturated assignment P such that IoA(P) ≤ (1 +2 + ϵ) ⋅ OPT. +5 Subgroups With Similar Sizes +In this section, we study the problem instances when the number of type-1 agents is a constant fraction +of the total number of agents, that is, k = α⋅n for some constant 0 ≤ α ≤ 1/2. We refer to this problem +as αn-IM-IoA. For example, α = 1/2 represents the bisection constraint. We first show that αn-IM-IoA +remains computationally intractable. See the Appendix for the proof. +▷ Theorem 5.1. The problem αn-IM-IoA is NP-hard. +5.1 A semidefinite programming approach +We now present an approximation algorithm for αn-IM-IoA based on semidefinite programming (SDP) +relaxation [14]. The overall scheme is inspired by the work of Frieze and Jerrum [12] on the Max- +Bisection problem. Given a graph G = (V,E), each vertex i ∈ V has a binary variable xi ∈ {−1,1} +such that xi = −1 if i is of type-1, and xi = 1 if i is of type-2. +First, we observe that a valid +quadratic program (QP) is (see Appendix for the proof): maximize∑i∈V maxj∈N (i) {1−xixj +2 +} s.t. +∑i 0, based on the technique introduced in [2], we present +a polynomial time approximation scheme that achieves a (1 − ϵ) approximation for IM-IoA. +PTAS Outline. Let q = 2 ⋅ ⌈1/ϵ⌉. We start with a plane embedding of G, which partitions the set of +vertices into ℓ layers for some integer ℓ ≤ n. Let Vi be the set of vertices in the ith layer, i = 1,...,ℓ. For +each r = 1,...,q, observe that we may partition the vertex set into t + 1 subsets, where t = ⌈(ℓ − r)/q⌉, +such that the (i) the first subset W(1,r) consists of the first r layers, (ii) the last subset W(t+1,r) +consists of the last ((l − r) mod q) layers, and (iii) each ith subset W(i,r) in the middle contains +q layers in sequential order. Let Wr = {W(1,r),...,W(t+1,r)} be such a partition. Let G(i,r) be the +subgraph induced on W(i,r), i = 1.,,,t + 1. It is known that each G(i,r) is a q-outerplanar graph with +treewidth O(q) [5], which is bounded. Let Gr = ⋃i G(i,r). By Theorem (6.1), we can solve the problem +optimally on each Gr, r = 1,...,q, in polynomial time. The algorithm then returns the solution with +the largest objective over all r = 1,...,q. Using the fact that q is fixed, one can verify that the running +time of the overall scheme is polynomial in n. +15 + +▷ Theorem 6.2. The PTAS algorithm gives a factor (1−ϵ) approximation on planar graphs for any +fixed ϵ > 0. +Proof. (Sketch) Let q = 2 ⋅ ⌈1/ϵ⌉. We show that the algorithm gives a 1 − 2/q ≥ 1 − ϵ approximation. +Let P∗ be an assignment of agents on G that gives the maximum number of integrated agents. Fix +an integer r in the range [1 .. q], and let Wr = {W(1,r),...,W(t+1,r)} be a partition of the vertex set as +described above. Let Pr be an assignment on Gr that is obtained from the proposed algorithm. We +now look at the assignments Pr and P∗, restricted to vertices in Wr. Specifically, let P(i,r) and P∗ +(i,r) +be the assignment of agents restricted to the subset W(i,r) under Pr and P∗, respectively. Further, let +IoA(P(i,r)) be the number of integrated agents in G(i,r) under Pr, and IoA(P∗ +(i,r)) be the number of +integrated agents in G(i,r) under P∗. +Define ∆r = IoA(P∗) − ∑t+1 +i=1 IoA(P∗ +(i,r)). Integrated vertices that are left uncounted can only exist +on the two adjacent layers between each pair of subgraphs G(i,r) and G(i+1,r), i = 1,...t. Let V∗ be the +set of integrated vertices under P∗. We then have, ∆r ≤ ∑t +j=0 (V∗ ∩ Vj⋅q+r) + (V∗ ∩ Vj⋅q+r+1). It follows +that minr=1,...,q{∆r} ≤ 2 +q ⋅ IoA(P∗). One can then verify that IoA(Pr∗) ≥ (1 − 2 +q) ⋅ IoA(P∗) where r∗ = +arg minr=1,...,q{∆r}. Lastly, let ˆP be an assignment returned by the algorithm, ˆP = arg maxr IoA(Pr). +It follows that IoA( ˆP) ≥ (1 − 2 +q) ⋅ IoA(P∗). +∎ +7 Experimental Evaluation +We evaluate the empirical performance of the proposed local improvement algorithm for IM-IoA under +several scenarios. Our results demonstrate the high effectiveness of the algorithm on both synthetic +and real-world networks. +7.1 Experimental setup +Networks. We selected networks based on their sizes and application domain, as shown in Table (1). +Specifically, Gnp and Power-law are synthetic networks generated using the Erd˝os-R`enyi [10] and +Barab´asi-Albert [3] models, respectively. City is a synthetic network of a residential area in a U.S. +city; here, vertices are houses, and any pair of houses within 100 yards are considered as neighbors. +Arena and Google+ are mined social networks obtained from a public repository [20]. +Algorithms. We evaluate the performance of Local-Improvement algorithm using the following base- +lines: (1) Greedy: Initially, all vertices are occupied by type-2 agents; then iteratively k of these are +replaced by type-1 agents in a greedy manner. Specifically, in each iteration, a replacement that causes +the largest increase in the objective value is chosen. (2) Random: a random subset of k vertices are +chosen for type-1 agents, and the remaining vertices are assigned to type-2 agents. +16 + +Network +Type +n +m +Max deg +Gnp +Random +1,000 +4,975 +36 +Power-law +Random +1,000 +5,015 +355 +City +Residential +7,444 +238,802 +165 +Arena +Social +10,680 +24,316 +205 +Google+ +Social +23,613 +39,182 +2,761 +Table 1: List of networks +Evaluation metrics. We use two metrics to quantify the performance of algorithms: (i) the integra- +tion ratio µ = obj/n (i.e., the fraction of integrated agents) and (ii) the empirical approximation ratio +γ = obj/OPT where OPT is the optimal value. The value OPT is computed by solving an integer +linear program (ILP) using Gurobi [27]. +Machine and reproducibility. Experiments were performed on an Intel Xeon(R) Linux machine +with 64GB of RAM. The source code and selected datasets are in the Technical Appendix. +7.2 Experimental results +We present an overview of the results under the following experimental scenarios. +Empirical ratio across networks. +We first study the empirical approximation ratio γ of the +algorithms on different networks. For the three large networks, namely City, Arena and Google+, the +ILP solver didn’t terminate even though it was run for 24 hours. Therefore, we restricted our focus to +smaller subgraphs of these networks. For each subgraph, we fixed the number k of minority agents to +be 10% of n, where n is the number of vertices in the network. The empirical ratio for each algorithm +is then averaged over 100 repetitions. +Representative results for the empirical ratio are shown in Fig. (3). Overall, we observe that the +effectiveness of Local-Improvement and Greedy are close to the optimal value, with Local-Improvement +outperforming Greedy by a small margin. Specifically, the empirical ratio of Local-Improvement is +greater than 0.85 on all tested instances. As one would expect, the empirical ratio of Random is much +lower than its counterparts. Overall, we note that the empirical ratio of Local-Improvement is much +higher than its theoretical guarantee of 1/2. Recall from Section (4) that there are instances where +Local-Improvement produces solutions that are of 1/2 of the optimal value. Our experimental findings +indicate such worst-case instances did not occur in these experiments. We also note that empirically +Greedy is comparable to Local Improvement. However, no known performance guarantee for Greedy +has been established. In contrast, as shown in Section (4), Local Improvement provides a guarantee of +1/2. +17 + +Gnp +Power-law +City* +Arena* +Google+* +Network +0.00 +0.25 +0.50 +0.75 +1.00 +Approximation Ratio γ +Local-Improvement +Greedy +Random +Figure 3: The empirical approximation ratio γ for algorithms. The number of vertices and edges (n, +m) for each subgraph are as follows. City*: (1607,50112), Arena*: (1981,9132), Google+*: +(2000, 5042). +0.05 +0.10 +0.15 +0.20 +0.25 +The fraction of minority agents +0.00 +0.25 +0.50 +0.75 +1.00 +Integration Ratio µ +(a) Gnp network +0.05 +0.10 +0.15 +0.20 +0.25 +The fraction of minority agents +0.00 +0.25 +0.50 +0.75 +1.00 +Local-Improvement +Greedy +Random +(b) City network +Figure 4: The change of the fraction of integrated agents as the fraction of minority agents increases. +The networks are Gnp and City shown in Table (1). +Variations on the number of minority agents. Next, we study the integration ratio µ (i.e., the +fraction of integrated agents) obtained by the algorithms under the scenario where the fraction of +minority agents (k) increases from 0.01 to 0.25. The representative results for Gnp and City networks +are shown in Fig. (4). +Overall, we observe that as the fraction of minority agents increases, the +integration ratio µ grows monotonically for all algorithms. Similar results are observed for all the +chosen networks. Despite the monotonicity observed in the experiments, we remark that the objective +value that an algorithm can obtain is general non-monotone as k increases. (A simple example is a +star where the objective is maximized for k = 1 when the type-1 agent is placed at the center. It is +easy to verify that as k increases, the optimal objective decreases.) +Change of objective as local improvement proceeds. +Lastly, we study the increase in the +objective value as the number of swaps used in Local-Improvement is increased. Results are shown in +Fig. (5) for gnp networks with 1000 nodes and average degrees varying from 10 to 30. Overall, we +observe a linear relationship between the objective value and the number of swaps. +18 + +0 +25 +50 +75 +100 +125 +150 +175 +200 +Number of Swaps +0.6 +0.7 +0.8 +0.9 +1.0 +Integration Ratio µ +Avg deg: 10 +Avg deg: 15 +Avg deg: 20 +Avg deg: 25 +Avg deg: 30 +Figure 5: The change in the number of integrated agents as Local-Improvement proceeds. The under- +lying gnp networks have 1,000 vertices; the average degree varies from 10 to 30. +8 Conclusions +We considered an optimization problem that arises in the context of placing agents on a network to +maximize the integration level. Since the general problem is NP-hard, we presented approximation +algorithms with provable performance guarantees for several versions of the problem. +Our work +suggests several directions for further research. First, it is of interest to investigate approximation +algorithms with better performance guarantees for the general problem. One possible approach is to +consider local improvement algorithms that instead of swapping just one pair of vertices to increase +the number of integrated vertices, swap up to j pairs, for some fixed j ≥ 2 in each iteration. One can +also study the problem under network-based extensions of other integration indices proposed in the +social science literature [24]. Another direction is the scenario where the total number of agents is less +than the number of nodes (so that some nodes remain unoccupied by agents). In addition, one can +also study the variant where there are agents of three or more types, and the notion of integration is +defined by requiring the neighborhood of an agent to include a certain number of agents of the other +types. Overall, this topic offers a variety of interesting new problems for future research. +19 + +4 Appendix: Additional Materials for Section 4 +Notation +Definition +P +An assignment return by the algorithm +P∗ +An optimal assignment +Vi(P) +The set of type-i vertices under P +VU +i (P) +The set of uncovered type-i vertices under P +N U +v (P) +The set of neighbors of v ∈ V that are uncovered under P +Γv(P) +The set of different-type neighbors of v that are uniquely covered by v +Type-i vertex (under P) +A vertex occupied by a type-i agent +An uncovered vertex (under P) +A vertex that is not integrated +Table 2: A notation table +Agarwal et al. [1] establish that IM-IoA is NP-hard5. +We now further study its solvability. +For +convenience in presenting the proofs, we define an assignment from the perspective of vertices of the +underlying graph, rather than the perspective of the agents. We remark that the two definitions are +equivalent. +Assignment. An assignment P ∶ V → A is a function that assigns an agent type in {1,2} to each +vertex (location) in V, such that k vertices are assigned type-1 and n − k vertices are assigned type-2. +Given an assignment P, we call a vertex v a type-1 (or type-2) vertex if P(v) = 1 (or P(v) = 2). Let +V1(P) and V2(P) denote the set of type-1 and type-2 vertices under P. Let VU +1(P) and VU +2(P) denote +the set of uncovered type-1 and type-2 vertices under P. For each vertex u, let N U +u(P) denote the set +of neighbors of u that are uncovered under P, and let Γu(P) denote the set of different-type neighbors +of u that are uniquely covered by u, i.e., Γu(P) is the set of vertices v such that (i) v is a neighbor +of u, (ii) the type of v is different from the type of u, and (iii) v has no other neighbors whose types +are the same as u’s type. +4.1 The algorithm +In this section, we present a local-search scheme that achieves a 2-approximation for IM-IoA. Without +lose of generality, we assume (i) k ≤ n − k, that is, type-1 is the minority type, and (ii) the site graph +G is connected (we may examine each connected component independently if G is not connected). +The algorithm. The algorithm is based on an incremental-improvement scheme. We start from a +5The work by Agarwal et al. [1] did not attempt to address the hardness of IM-IoA, as IoA is not the main result in +that paper. +20 + +random assignment P. In each iteration of the algorithm, we find a pair (if such a pair exists) of type-1 +and type-2 vertices such that swapping their types strictly increases the objective. In particular, let +u and v be a type-1 and type-2 vertices, respectively. We swap the types of u and v (i.e., u becomes +type-2 and v becomes type-1) if and only if the resulting new assignment P′ incurs a strictly higher +IoA, that is, IoA(P) < IoA(P′). The algorithm terminates when no such improvement move can be +made. The pseudocode is given below. +4.2 Analysis of the algorithm +We now investigate the performance of Algorithm (1). Let P be a saturated assignment6 returned by +Algorithm (1). All the analyses are given under P unless specified otherwise. Recall that VU +1(P) ⊆ V is +the set of type-1 vertices are not integrated under P. That is, for each vertex x ∈ VU +1(P), all neighbors +of x under P are also of type-1. Similarly, let VU +2(P) ⊆ V be the set of type-2 vertices who are not +integrated under P. An example of such sets are given in Figure (6). +Figure 6: An assignment where type-1 and type-2 vertices are highlighted in blue and red, respectively. +In this example, VU +1(P) = {x} and VU +2(P) = {y}. Note that this example is only to demon- +strate how the two sets are defined, as later we show that if both VU +1(P) ≠ ∅ and VU +2(P) ≠ ∅, +this assignment cannot be a saturated assignment. +▷ Observation 4.1. The index IoA(P) = n − ∣VU +1(P)∣ − ∣VU +2(P)∣. +We now consider the following mutually exclusive and collectively exhaustive cases of VU +1(P) and +VU +2(P) under the saturated assignment P. We start with a simple warm-up case where all the type-2 +vertices under P are integrated. +Case 1: VU +2(P) = ∅. +6An assignment is saturated if no pairwise swap of types between a type-1 and a type-2 vertices can increase the +objective. +21 + +Under this case, all vertices in V2(P) are integrated which gives +IoA(P) ≥ ∣VU +2(P)∣ = n − k ≥ 1 +2 ⋅ n +(15) +The above case trivially implies a 2-approximation of the algorithm. We now look at the remaining +case where VU +2(P) ≠ ∅. +Case 2: VU +2(P) ≠ ∅. +Under this case, there exists at least one vertex in V2(P) that is not integrated. We now study the +approximation ratio. +▷ Lemma 4.2. For a saturated assignment P, if VU +2(P) ≠ ∅, then VU +1(P) = ∅. +Proof. Let y ∈ VU +2(P) be a vertex of type-2 that is not integrated (i.e., all neighbors of y are of +type-2). For contradiction, suppose VU +1(P) ≠ ∅.. Now let x ∈ VU +1(P) be an non-integrated vertex of +type-1 whose neighbors are all of type-1. Let P′ denote the assignment where we switch the types +between x and y, that is, P′(x) = P(y) = 2, P′(y) = P(x) = 1, while the types of all other vertices +remain unchanged. +▷ Claim 4.2.1. IoA(P′) ≥ IoA(P) + 2, that is, switching the types of x and y increases the index +IoA by at least 2. +We now establish the above claim. First observe that after the switch, only the integration status of +vertices in {x,y}∪N(x)∪N(y) can change, where N(x) and N(y) are neighbors of x and y. Given +that all neighbors of x are of type-1 under P, and y is of type-2, switching P(x) with P(y) can only +increase the number of integrated neighbors in N(x). Similarly, switching P(y) with P(x) can only +increase the number of integrated neighbors in N(y). Further, note that x (who was not integrated +in P) will be integrated after the switch, as N(x) consists of (only) vertices of type-1. By the same +argument, y (who was again not integrated in P) will be integrated after the switch. It follows that +after the switch, the index IoA would increase by at least 2, that is, IoA(P′) ≥ IoA(P) + 2. This +conclude the claim. One may check Figure (6) for a visualization. +The claim above implies the existence of an improvement move from P, which contradicts P +being a saturated assignment. It follows that no such an x ∈ VU +1(P) exists and thus VU +1(P) = ∅. +∎ +Lemma 4.2 immediately implies that under case 2 (i.e., VU +2(P) ≠ ∅), we must have VU +1(P) = ∅. +IoA(P) ≥ ∣V1∣ = k +(16) +22 + +We now argue for a stronger approximation ratio of 2. Consider the following two mutually exclusive +and collectively exhaustive subcases under Case 2. Recall that for each vertex x, Γx(P) is the set +of different-type neighbors of x that are uniquely covered (i.e. “made integrated”) by x under P. +Formally, if x ∈ V1(P), then Γx(P) = {y ∈ V2(P) ∪ N(x) ∶ y ∉ ⋃x′∈V1(P)∖{x} N(x′)} +Subcase 2.1: VU +2(P) ≠ ∅, and +Γx(P) ≠ ∅, ∀x ∈ V1(P) +that is, for each type-1 vertex x ∈ V1(P), there is at least one type-2 neighbors y of x that is +uniquely covered (i.e. “made integrated“) by x. +Recall that P is a saturated assignment returned by the algorithm. By Lemma (4.2), we know that +all vertices in V1(P) are integrated under P. Thus, the total number of integrated vertices under P +equals ∣V1(P)∣ = k plus the number of vertices in V2(P) that are adjacent to vertices in V1(P) (It +immediately follows that IoA(P) ≥ 2 ⋅ k +n). Let P∗ by an optimal assignment that gives the maximum +number of integrated vertices. We now argue that IoA(P) ≥ 1 +2 ⋅ IoA(P∗). +Suppose P ≠ P∗, that is, for some vertices x ∈ V, P(x) ≠ P∗(x). Let ˜V2−1 = {v ∈ V +∶ P(v) = +2,P∗(v) = 1} be the set of vertices that are type-2 under P, but are type-1 under P∗. Analogously, +let ˜V1−2 = {v ∈ V ∶ P(v) = 1,P∗(v) = 2} be the set of vertices of type-1 under P, but are of type-2 +under P∗. Observe that ∣˜V2−1∣ = ∣˜V1−2∣. We may view P∗ as the result of a transformation from P +under pairwise swaps of types between ˜V2−1 and ˜V1−2. An example is given in Figure (7). We present +a key lemma that bounds the difference in the objective value between P and P∗. +Figure 7: Two assignments P and P∗ where type-1 and type-2 vertices are highlighted in blue and +red, respectively. In this case, ˜V2−1 = {x3,x4} and ˜V1−2 = {x1,x2}. We may then transform +P into P∗ by swapping types between the pair (x1,x3) and between (x2,x4). Note that this +example is only to demonstrate how ˜V2−1 and ˜V1−2 are defined, as P cannot be a saturated +assignment returned by the algorithm. +▷ Lemma 4.3 (Subcase 2.1). Let P be a saturated assignment that satisfies subcase 2.1, and let +23 + +PP∗ be an optimal assignment. We have +IoA(P∗) − IoA(P) ≤ +∑ +y∈˜V2−1∖VU +2(P) +∣(N(y) ∩ VU +2(P)∣ + +∑ +y∈˜V2−1∩VU +2(P) +(∣(N(y) ∩ VU +2(P)∣ + 1) +(17) +Proof. Since P is saturated, Lemma (4.2) implies that all type-1 vertices under P are integrated. +Thus, the difference IoA(P∗)−IoA(P) is at most the number of type-2 vertices that are integrated +under P∗ but are not integrated under P. +Let f ∶ ˜V1−2 → ˜V2−1 be an arbitrary bijective mapping. We may regard P∗ as a result of the +transformation from P via pairwise swaps of types between vertices specified by f (i.e., the type of +x ∈ ˜V1−2 is swapped with the type of f(x) ∈ ˜V2−1). Observe that only vertices in VU +2(P) that are +adjacent to ˜V2−1 (or within ˜V2−1) under P can be newly integrated under P∗ after swapping ˜V1−2 +with ˜V2−1 (by the definition of VU +2(P), vertices in ˜V1−2 have no neighbors in VU +2(P).). It follows +that for each vertex y ∈ ˜V2−1, at most ∣(N(y)∩VU +2(P)∣ of its neighbors can become newly integrated +after transforming from P to P∗. Further, if also y ∈ ˜V2−1 ∩ VU +2(P), y itself could also be newly +integrated after the swap. We then have +IoA(P∗) − IoA(P) ≤ ∣ +⋃ +y∈˜V2−1 +N(y) ∩ VU +2(P)∣ + ∣˜V2−1 ∩ VU +2(P)∣ +(18) +≤ +∑ +y∈˜V2−1∖VU +2(P) +∣(N(y) ∩ VU +2(P)∣ + +∑ +y∈˜V2−1∩VU +2(P) +(∣(N(y) ∩ VU +2(P)∣ + 1) +(19) +where the last inequality follows from the union bound. This completes the proof. +∎ +We note that the bound derived in Lemma (4.3) is not tight for many problem instances. Never- +theless, later we will see that such a bound is enough for our purpose of showing a 1 +2 approximation. +Further, we note that there indeed exist a class of problem instances where this bound is exact. +Lemma (4.3) bounds the maximum difference between IoA(P∗) and IoA(P), which is +∑ +y∈˜V2−1∖VU +2(P) +∣(N(y) ∩ VU +2(P)∣ + +∑ +y∈˜V2−1∩VU +2(P) +(∣(N(y) ∩ VU +2(P)∣ + 1) +We now proceed to show that the above difference is at most IoA(P), thereby establishing IoA(P) ≥ +1 +2 ⋅ IoA(P∗). All the discussion below are under P unless stated otherwise. Recall that for each vertex +x ∈ V, Γx(P) is the set of neighbors of x whose types are different from x, and are uniquely covered +by x under P. By the definition of Subcase 2.1, Γx(P) is not empty for all x ∈ V1(P). We first argue +that for any y ∈ VU +2(P) and any x ∈ V1(P), we have ∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣. +24 + +▷ Lemma 4.4 (Subcase 2.1). Given a saturated assignment P, for any y ∈ VU +2(P) and any x ∈ +V1(P), we have +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ +Proof. Given that y is not integrated under P, x and y cannot be adjacent. Since P is a saturated +assignment, if the types of x and y are to be swapped, the number of newly integrated vertices +would be at most the number of newly non-integrated vertices. We now examine the integration +status of vertices in the closed neighborhood of x and y under P after such a swap: +For y and its neighbors +⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ +All vertices in N(y) ∩ VU +2(P) become newly integrated +All vertices in N(y) ∩ (A2 ∖ VU +2(P)) remain integrated +The vertex y itself becomes newly integrated +⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭ +For x and its neighbors +⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ +All vertices in N(x) ∖ Γx(P) remain integrated +Some vertices in Γx(P) may become newly non-integrated +The vertex x itself may become newly non-integrated +⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭ +Overall, the number of vertices that are newly integrated is at least ∣N(y)∩VU +2(P)∣+1, and the +number of vertices that are newly non-integrated is at most ∣Γx(P)∣ + 1. Since P is saturated, it +follows that: +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ +(20) +This concludes the proof. +∎ +We now show that for any y ∈ V2(P)∖VU +2(P) and any x ∈ V1(P), we have ∣N(y)∩VU +2(P)∣ ≤ ∣Γx(P)∣+1. +▷ Lemma 4.5 (Subcase 2.1). Given a saturated assignment P, for any y ∈ V2(P) ∖ VU +2(P) and +any x ∈ V1(P), we have +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ + 1 +Proof. We partition V2(P) ∖ VU +2(P) into two subsets B and C, as follows. Subset B is the set of +integrated type-2 vertices whose neighbors are all integrated under P, i.e., +B = {y ∈ V2(P) ∖ VU +2(P) ∶ N(y) ∩ VU +2(P) = ∅} +Subset C, the complement of B, is the set of integrated type-2 vertices with at least one non- +integrated neighbor under P, i.e., C = {y ∈ V2(P) ∖ VU +2(P) ∶ N(y) ∩ VU +2(P) ≠ ∅}. The Lemma +25 + +clearly holds if y ∈ B since then ∣N(y) ∩ VU +2(P)∣ = 0. We now present a key claim for the case when +y ∈ C: +▷ Claim 4.5.1. For all vertices y ∈ C, no type-1 neighbors of y is uniquely covered by y under P +(i.e., Γy(P) = ∅). +For contradiction, suppose there exists a type-1 neighbors x ∈ N(y) ∩ V1(P) of y such that x is +not adjacent to any other type-2 vertices under P. Then by the definition of subcase 2.1 (i.e., each +type-1 vertex uniquely covers at least one type-2 vertex), x is the only type-1 neighbor of y. One +then can easily verify that exchanging the types between x and y strictly increase the objective of +P, contradicting the fact that P is saturated. This conclude the proof of Claim (4.5.1). +We continue to assume that y ∈ C and consider an objective non-increasing move from P where +we swap the types between x and y. If y is a neighbor of x under P, then by Claim (4.5.1), one +can verify that the the maximum loss is ∣Γx(P)∣ and the minimum gain is ∣N(y) ∩ VU +2(P)∣. Thus +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ +(21) +On the other hand, if y is not a neighbor of x under P, one can verify that the maximum loss +is ∣Γx(P)∣ + 1 and the minimum gain is ∣N(y) ∩ VU +2(P)∣. Thus +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γx(P)∣ + 1 +(22) +This concludes the proof. +∎ +We are now ready to establish IoA(P) ≥ 1 +2 ⋅ IoA(P∗) under Subcase 2.1. +▷ Lemma 4.6 (Subcase 2.1). Suppose VU +2(P) ≠ ∅ and Γx(P) ≠ ∅,∀x ∈ V1(P), we have +IoA(P) ≥ 1 +2 ⋅ IoA(P∗) +where P∗ is an optimal assignment that gives the maximum objective. +Proof. Note that ˜V2−1 is a subset of V2(P). Further, Observe that Γx(P) are disjoint for different +26 + +vertices x ∈ V1(P). Now, by Lemma (4.3) to and (4.5), We have +IoA(P∗) − IoA(P) +≤ +∑ +y∈˜V2−1∖VU +2(P) +∣(N(y) ∩ VU +2(P)∣ + +∑ +y∈˜V2−1∩VU +2(P) +(∣(N(y) ∩ VU +2(P)∣ + 1) +(Lemma (4.3)) +≤ +∑ +y∈˜V2−1∖VU +2(P) +(∣Γf−1(y)(P)∣ + 1) + +∑ +y∈˜V2−1∩VU +2(P) +(∣Γf−1(y)(P)∣ + 1) +(Lemma (4.4) & (4.5)) += +⎛ +⎜ +⎝ +∑ +y∈˜V2−1 +∣Γf−1(y)(P)∣ +⎞ +⎟ +⎠ ++ ∣˜V2−1∣ +≤ ∣V2(P) ∖ VU +2(P)∣ + ∣V1(P)∣ +(23) +≤ IoA(P) +where Inequality (23) follows from ∣˜V2−1∣ = ∣˜V1−2∣ ≤ ∣V1(P)∣ and (∑y∈˜V2−1 ∣Γf−1(y)(P)∣) ≤ ∣V2(P) ∖ +VU +2(P)∣. This concludes the proof. +∎ +We now have shown that if VU +2(P) ≠ ∅ and Γx(P) ≠ ∅,∀x ∈ V1(P), the algorithm gives a 2 approxi- +mation. We proceed to the last subcase. +Subcase 2.2: VU +2(P) ≠ ∅, and +Γx(P) = ∅, +∃x ∈ V1(P) +that is, there exists at least one type-1 vertex x ∈ V1(P) such that for each type-2 neighbor y of x, +y is adjacent to at least one type-1 vertex other than x. +▷ Lemma 4.7 (Subcase 2.2). Under subcase 2.2, for each non-integrated type-2 vertex y ∈ VU +2(P), +all type-2 neighbors of y are integrated (i.e., N(y) ∩ VU +2(P) = ∅) under P. +Proof. Given such a x ∈ V1(P) defined in Subcase 2.2, for contradiction, suppose there exists a non- +integrated type-2 vertex y ∈ VU +2(P) such that at least one type-2 neighbor, denoted by y′ ∈ N(y), +of y is not integrated under P (note that all neighbors of y are of type-2 since y is not integrated). +Now consider a new assignment P′ where we switch the types between x and y. +▷ Claim 4.7.1. We have IoA(P′) ≥ IoA(P) + 1, that is, after the switch, the index IoA would +increase by at least 1. +We now establish the claim. Similar to Lemma (4.2), only the integration status of vertices in +{x,y} ∪ N(x) ∪ N(y) can change. We first consider the integration states of vertices in N(x). Let +IoA(N(x),P) denote the number of integrated vertices in the neighborhood of x under P, and let +∆IoA(N(x),P) = IoA(N(x),P′) − IoA(N(x),P) denote the change in the integrated vertices in +27 + +the neighborhood of x after the switch. Let N1(x,P) and N2(x,P) denote the set of type-1 and +type-2 neighbors of x under P, respectively. +Under Subcase 2.2, each vertex N2(x,P) is adjacent to at least one type-1 vertex in additional +to x, thus, all vertices in N2(x,P) remain integrated after we swap types between y and x. Further, +it is easy to see that the swap cannot decrease the number of integrated vertices in N1(x,P). It +follows that +∆IoA(N(x),P) ≥ 0 +Now consider the integration states of vertices in N(y). First observe that N1(y,P) = ∅ since +y ∈ VU +2(P) is not integrated. +Also, swapping the types between y and x will not decrease the +number of integrated vertices in N2(y,P). In fact, since there exists a vertex y′ ∈ N2(y,P) who is +not integrated under P, the swap makes y′ integrated (as x and y′ are of different types). It follows +that +∆IoA(N(y),P) ∶= IoA(N(y),P′) − IoA(N(y),P) ≥ 1 +Lastly, we consider the integration states of x and y. In particular, x is integrated under P, +and after the swap, it might become non-integrated. On the other hand, y is not integrated under +P, it must become newly integrated after the swap. Nevertheless, the net increase of the number +of integrated vertices in {x,y} is at least 0 when we change from P to P′. Overall, it follows that +IoA(P′) − IoA(P) = ∆IoA(N(x),P) + ∆IoA(N(y),P) + ∆IoA({x,y},P) ≥ 1 +(24) +This concludes the claim. Note that the claim implies the existence of an improvement move +from P, which contradicts P being a saturated assignment returned by Algorithm (1). Thus, no +such a non-integrated type-2 vertex y′ of y can exist, that is, for each y ∈ VU +2(P), all (type-2) +neighbors of y are integrated. This concludes the proof. +∎ +Lemma (4.7) implies that under Subcase 2.2, the vertices in VU +2(P) form an independent set of G, as +stated in the corollary below. +▷ Corollary 4.7.1 (Subcase 2.2). Under Subcase 2.2, vertices in VU +2(P) form an independent set +of G. +Observe that IoA(P) = (n − ∣VU +2(P)∣). With Lemma (4.7) in place, we now argue the size of VU +2(P) +cannot be too large. +28 + +▷ Lemma 4.8 (Subcase 2.2). Under Subcase 2.2, we have +∣VU +2(P)∣ ≤ n +2 +Proof. Let +Y ∶= {y ∈ V2(P) ∖ VU +2(P) ∶ N(y) ∩ VU +2(P) ≠ ∅} +be the set of type-2 integrated vertices whose has at least one non-integrated type-2 neighbor. Recall +that Γy(P) is the set of type-1 neighbors of y who are uniquely covered by y under P. We first +note that Γy(P) (if not empty) are mutually disjoint for different y ∈ Y. It follows that +IoA(P) ≥ ∣Y∣ + ∑ +y∈Y +∣Γy(P)∣ +(25) +Now revisit the definition of subcase 2.2. In particular there exists a type-1 vertex x ∈ V1(P) such +that each type-2 neighbor of x is also covered by (i.e., adjacent to) at least one other type-1 vertex. +Suppose we switch the types between such a vertex x and a vertex y ∈ Y, and let P′ denote the +resulting new assignment. Observe the following in P′ +▷ Claim 4.8.1. All vertices in N(x) remains integrated in P′. +This holds since these neighbors are either of (i) type-1 which are now adjacent to x of type-2 +in P′, or of (ii) type-2 which are adjacent to at least one other type-1 vertex. +▷ Claim 4.8.2. All vertices in N(y) ∩ VU +2(P) become newly integrated in P′, and all vertices in +Γy(P) may become newly non-integrated in P′. The integration status of all other vertices in N(y) +remain unchanged from P to P′. +One can easily verify the above claim based on the fact that y is of type-1 under P′. Lastly, +note that in y remains integrated in P′ since y is of type-1 in P′ and has at least one type-2 +neighbor. +On the other hand, x (who was integrated in P) might not be integrated in P′. +It +follows that the maximum loss of objective after the swap is ∣Γy(P)∣ + 1, where as the minimum +gain is ∣N(y)∩VU +2(P)∣. Since P is a saturated assignment returned by the algorithm, we must have +IoA(P) ≥ IoA(P′). It follows that +∣N(y) ∩ VU +2(P)∣ ≤ ∣Γy(P)∣ + 1, ∀y ∈ Y +(26) +29 + +Lastly, by Corollary (4.7.1), vertices in VU +2(P) form an independent set of G. Thus, +∣VU +2(P)∣ = ∣ ⋃ +y∈Y +N(y) ∩ VU +2(P)∣ +(27) +Overall, we have that +∣VU +2(P)∣ = ∣ ⋃ +y∈Y +N(y) ∩ VU +2(P)∣ +(Corollary (4.7.1) +(28) +≤ ∑ +y∈Y +∣N(y) ∩ VU +2(P)∣ +(Union bound) +(29) +≤ ∑ +y∈Y +(∣Γy(P)∣ + 1) +(Eq. (26)) +(30) +≤ ∣V1(P)∣ + ∣Y∣ +(Γy(P) are mutually disjoint) +(31) +≤ ∣V1(P)∣ + ∣V2(P) ∖ VU +2(P)∣ +(By Y ⊆ V2(P) ∖ VU +2(P)) +(32) += n − ∣VU +2(P)∣ +(33) +It immediately follows that +∣VU +2(P)∣ ≤ n +2 +(34) +This concludes the proof. +∎ +Lastly, Since IoA(P) = n − ∣VU +2(P)∣, by Lemma (4.8), we have IoA(P) = n − ∣VU +2(P)∣ ≥ 1 +2 ⋅ n ≥ 1 +2 ⋅ +IoA(P∗), thereby establishing a 1/2 approximation for Subcase 2.2. Overall, we have shown that a +saturated assignment P returned by Algorithm (1) gives a 2-approximation for IM-IoA. The Theorem +immediately follows. +▷ Theorem 4.9. Algorithm (1) gives a 1 +2-approximation for IM-IoA. +Analysis is tight +We now present a class of problem instances where the approximation ratio of the solution produced by +Algorithm (1) can be arbitrarily close to 1/2. Therefore, the ratio 1/2 in the statement of Theorem (4.9) +cannot be improved, so our analysis is tight. +▷ Proposition 4.10. For every ϵ > 0, there exists a problem instance of IM-IoA for which there +is a saturated assignment P such that IoA(P) ≤ (1 +2 + ϵ) ⋅ OPT. +Proof. Recall that k is the number of type-1 vertices. We first present the construction of the graph +G = (V,E). Let W1 be a set of k vertices that form a clique. For each v ∈ W1, we introduce a set Uv +30 + +... +... +... +... +... +... + -clique +Figure 8: A pictorial example of a problem instance where Algorithm (1) gives an assignment whose +approximation ratio is 1 +2. +of k vertices outside the clique that are adjacent v. Let W2 = ⋃v∈W1 Uv denote the union of these +sets. All vertices in W2 are also adjacent to a new vertex w, and we further make this vertex w +adjacent to exactly one vertex in W1. Lastly, we added a total of k stars, each of which consists of +k −1 vertices (i.e., each star has a center vertex and k −2 leaf vertices). We then connect the center +of each star to one vertex in a unique Uv. This completes the construction. An example is given in +Figure (8). +Now consider an assignment P where all vertices in W1 are of type-1 (recall that k = ∣W1∣), and +the rest of vertices are of type-2. One can verify that such an assignment is saturated (and thus +could be returned by Algorithm (1)). On the other hand, an assignment P∗ that gives a strictly +higher objective is where we assign (i) type-1 to one vertex in W1, (i) assign w to type-1, and (iii) +the centers of k − 2 stars (with any two stars being left out) are assigned with type-1. The rest of +vertices are of type-2. +One can verify that IoA(P) = k2+k+1, and IoA(P∗) = 2k2−2k+4. The ratio IoA(P)/IoA(P∗) = +1/2 as k goes to infinity. Since IoA(P∗) ≤ OPT where OPT is the optimal objective of a problem +instance, the claim follows. +∎ +5 Appendix : Additional Material for Section 5 +We study the problem instances when the number of type-1 agents is a constant fraction of the total +number of agents, that is, k = α ⋅ n for some constant 0 ≤ α ≤ 1/2. +We refer to this problem as +αn-IM-IoA. For example, α = 1/2 implies the bisection constraint. +31 + +5.1 Intractability remains +We first show that αn-IM-IoA problem remains intractable. +▷ Theorem 5.1. The problem αn-IM-IoA is NP-hard. +Proof. We present a reduction from the general IM-IoA problem to αn-IM-IoA where α = 1 +2. Let +Π1 = ⟨G,A,n⟩ be an instance of IM-IoA, A = {A1,A2}, where k = ∣A1∣ is the number of type-1 agents +that needs to be assigned, and n = ∣V(G)∣ is the total number of agents. The decision question asks +whether there exists an assignment of agent-types for Π1 such that all the n vertices are integrated. +This question is known to be NP-hard [1]. +An instance, Π2 = ⟨G′,A′,2n⟩, A′ = {A′ +1,A′ +2}, of the bisection version of IM-IoA consists of the +following components. To from the graph G′, the first component G1 a copy of G. Let G2 be a +graph formed by first creating a simple path I with k − 1 vertices, then for each vertex v on the +path, we introduce a new vertex (not on the path) that is uniquely adjacent to v. Overall, G2 has +2k − 2 vertices. An example of G2 is shown in Figure (9). Let G3 be a star graph with n − 2k + 2 +vertices (i.e., one center with n−2k +1 leaf vertices). Lastly, the final graph G′ consists of the three +aforementioned connected components: G1, G2, and G3. One can verify that G′ has 2n vertices (and +thus the number of agents is 2n). We set the number of type-1 agents ∣A′ +1∣ = n, corresponding to +the bisection constraint ∣A′ +1∣ = 1/2 ⋅ ∣A′∣. +We now argue that Π1 admits an assignment where all n vertices are integrated if and only if +Π2 has an assignment where all 2n vertices are integrated. +(⇒) Suppose Π1 has an assignment P on G such that all vertices are integrated. We now present +an assignment P′ for G′ such that all vertices are integrated in Π2. Specifically, we discuss how +types are assigned on G1, G2, and G3. The assignment of agent-type on G1 is the same as that +of on G under P. Next, for G2, we set all the k −1 vertices on the path I to type-1 (i.e., taken +by type-1 agents), and the rest of k − 1 vertices are of type-2. Lastly, for G3, all the n − 2k + 1 +leaf vertices are of type-1, and the center vertex is of type-2. The completes the construction +of P′. One can verify that the total number of type-1 vertices is k + (k − 1) + (n − 2k + 1) = n, +and further, all the 2n vertices are integrated under P′. +(⇐) Suppose Π2 has an assignment P′ on G′ such that all vertices are integrated. We show that +there exists an assignment P on G such that all vertices are integrated in Π1. Consider the +assignment P′ restricted to G2 and G3. We first observe that any assignment that makes all +vertices in G2 integrated must has exactly k −1 type-1 vertices in G2. As for G3, there are two +possible assignments that makes all vertices integrated: either (i) having one type-2 vertex +at the center and the leaf vertices are of type-1 or (ii) vise versa. Note that under the first +32 + +assignment, the total number of type-1 vertices placed on G2 and G3 is (k−1)+(n−2k+1) = n−k. +Since the total number of type-1 vertices is n, there are exactly k type-1 vertices in G1. Thus, +the assignment P is obtained by restricting P′ to G1. On the other hand, under the second type +of assignment in G3, the total number of type-1 vertices placed in G2 and G3 is (k −1)+1 = k. +That is, there are n − k type-1 vertices in G1 under P′. Then P is obtained by flipping the +types of vertices (i.e., type-1 changes to type-2, vise versa) assigned in G1 under P′. +This concludes the proof. +∎ +... +... + vertices +Figure 9: An example of graph G2 +5.2 A semidefinite programming approach +Remark. Our approximation results are given in terms of the expected approximation ratio γ. One +can obtain a w.h.p. bound by (i) running the algorithm K rounds, and (ii) output the best solution. +In particular, one can verify that for each round, the probability of producing an approximation factor +(1−ϵ)⋅γ is at least 1 −(1−γ)/(1−(1−ϵ)⋅γ) for arbitrarily small constant ϵ > 0. One can then choose +a large enough K to obtain a high probability bound, while K remains a polynomial of n. +We now present an approximation algorithm based on a semidefinite programming (SDP) relax- +ation. Given a graph G = (V,E), each vertex i ∈ V has a binary variable xi ∈ {−1,1} such that xi = −1 +if i is of type-1, and xi = 1 if i is of type-2. To start with, a quadratic program (QP) of αn-IM-IoA +and its SDP relaxation can be formulated as follows: +QP ∶ +maximize +∑ +i∈V +max +j∈N (i) {1 − xixj +2 +} +s.t. +∑ +i 0, +α((1−ϵ)⋅2αGW − µ−µ2 +α−α2 ) +µ +is minimum at µ = +√ +α(1 − α)(1 − ϵ) ⋅ 2αGW . +Let γ = +√ +α(1 − α)(1 − ϵ) ⋅ 2αGW , we then have +f(ˆV +′ +1) ≥ +α ((1 − ϵ) ⋅ 2αGW − γ−γ2 +α−α2 ) +γ +⋅ OPTSDP +(57) +Lastly, since Pr[Z′ > (1 − ϵ) ⋅ 2αGW ] ≥ 1 − ϵ, +E[f(ˆV +′ +1)] ≥ +α ((1 − ϵ) ⋅ 2αGW − γ−γ2 +α−α2 ) +γ +⋅ (1 − ϵ) ⋅ OPTSDP +(58) +For small enough ϵ, say ϵ = 10−3, the approximation ratio is greater than 1/2 for α in range +[0.4,0.5]. For example, α = 0.45 gives a ratio of 0.5781, and α = 0.5 gives a ratio of 0.6492. +6 Appendix : Additional Material for Section 6 +In this section, we first show that IM-IoA can be solved in polynomial time on treewidth bounded +graphs. Based on this result, we further present a polynomial time approximation scheme (PTAS) for +the problem on planar graphs. +39 + +6.1 A dynamic programming algorithm for treewidth bounded graphs +The concept treewidth of a graph is first introduced in the seminal work by Robertson and Seymour [29]. +Many intractable problems have since enjoyed polynomial time algorithms when underlying graphs +have bounded treewidth. In this section, we present a dynamic programming algorithm that solves +IM-IoA in polynomial time (w.r.t. n) for the class of graphs that are treewidth bounded. +Dynamic programming setup. Given an instance of IM-IoA with graph G = (V,E) and the number +k of minorities, let T = (I,F) be a tree decomposition of G with a bounded treewidth σ. For each +Xi ∈ I, consider the set of bags in the subtree rooted at Xi in T , and let Yi be the set of all vertices +in these bags. Let G[Yi] denote the graph G induced on Yi. For each bag Xi, we define an array Hi +to keep track of the optimal objectives in G[Yi]. +A naive definition that fails. One immediate way is to define Hi(S,γ) to be the optimal objective +in G[Yi] such that (i) S ⊆ Xi are of type-1, Xi∖S are of type-2; (ii) there are a total of γ type-1 vertices +and ∣Yi∣−r type-2 vertices in G[Yi]. As a result, for each Xi, its corresponding Hi has O(2σ ⋅n) entries, +which is polynomial w.r.t. n since σ is bounded. Despite the simplicity of this definition, however, it +is unclear how to correctly update these arrays. For example, suppose Xi is of the type introduce, let +Xj be the child of Xi. Let v ∉ S be the vertices that is introduced to Xi. One might try to update +Hi(S,γ) by doing Hi(S,γ) = Hj(S,γ) + w(v,Xi), where w(v,Xi) is the number of newly integrated +vertices in Xi after v being introduced to the set. This formulation looks correct at the first glance +since v is not adjacent to any vertices in Yi other than those in Xi. Thus, it seems that the impact +this extra vertex v can cause is only restricted within Xi. However, we remark this far from true, and +that the above computation is not optimal. In particular, consider the example given in Fig (10). + +Figure 10: An example where the naive dp approach fails. Given an example graph G, the subgraph +on the left is G induced on Yj, where Xj = {x,y}. The subgraph on the right is G induced +on Yi, where Xj = {x,y,v}. The set S = {x,y}, and γ = 4. Optimal assignments that yields +Hj(S,γ) and Hi(S,γ) are given where blue vertices are of type-1, and red vertices are of +type-2. In particular, Hj(S,γ) = 6 and Hi(S,γ) = 9. Note that the naive dp approach +would set Hi(S,γ) to be 6 + 1 = 7 which is not optimal. +40 + +An alternative definition. +We introduce another dimension to the above definition of Hi. +In +particular, let Hi(S,S′,γ) be the optimal objective in G[Yi] such that +(i) Vertices in the subset S ⊆ Xi are of type-1, and vertices in Xi ∖ S are of type-2. +(ii) Vertices in S′ ⊆ Xi are to be treated integrated. +(iii) There is a total of γ type-1 vertices and ∣Yi∣ − γ type-2 vertices in G[Yi]. +The resulting Hi has O(4σ ⋅ n) entries. The algorithm then proceeds in a bottom-up fashion from the +leaves to the root in T . We now discuss how the array Hi should be updated for each bag Xi. +Update Scheme +(i) Leaf: For all γ = 0,...,min{∣Yi∣,k} (recall that k is the total number of type-1 agents) and for +all S ⊆ Xi s.t. ∣S∣ = γ, let Zi(S) be the set of integrated vertices in G[Xi] under the assignment +(S,Xi ∖ S). For all S′ ⊆ Xi, we have +Hi(S,S′,γ) = ∣S′ ∪ Zi(S)∣ +(59) +(ii) Introduce: Let Xj be the child of Xi, and let v be the vertex introduced to Xi (i.e., v ∈ Xi and +v ∉ Xj). For all γ = 0,...,min{∣Yi∣,k}, and for all S ⊆ Xi s.t. ∣S∣ ≤ γ, let Zi(S) be the set of +integrated vertices in G[Xi] under the assignment (S,Xi ∖ S). For all S′ ⊆ Xi: +- If v ∈ S, +Hi(S,S′,γ) = Hj(S ∖ {v},S′ ∪ Zi(S) ∖ {v},γ − 1) + 1(v) +(60) +- If v ∉ S, +Hi(S,S′,γ) = Hj(S,S′ ∪ Zi(S) ∖ {v},γ) + 1(v) +(61) +where 1(v) is an indicated variable that equals to 1 if and only if v is integrated in G[Xi] under +the assignment (S,Xi ∖ S). +(iii) Forget: Let Xj be the child of Xi, and let v be the vertex forgot by Xi (i.e., v ∉ Xi and v ∈ Xj). +For all γ = 0,...,min{∣Yi∣,k}, and for all S ⊆ Xi s.t. ∣S∣ ≤ γ, let Zi(S) be the set of integrated +vertices in G[Xi] under the assignment (S,Xi ∖ S). For all S′ ⊆ Xi: +Hi(S,S′,γ) = max{Hj(S,Zi(S) ∪ S′,γ),Hj(S ∪ {v},Zi(S) ∪ S′,γ)} +(62) +(iv) Join: Let Xj1 and Xj2 be the two children of Xi. +Note that Xj1 = Xj2 = Xi. +For all γ = +0,...,min{∣Yi∣,k}, and for all S ⊆ Xi s.t. ∣S∣ ≤ γ, let Zi(S) be the set of truly integrated vertices +in G[Xi] under the assignment (S,Xi ∖ S). +41 + +For all S′ ⊆ Xi, let Qi(S,S′) = Xi∖(S′ ∪ Zi(S)) be the set of vertices that are not truly integrated +in G[Xi] under the assignment (S,Xi∖S), and also should not be treated as integrated (i.e., ver- +tices in Qi(S,S′) are not in S′). We consider all subsets Qj1(S,S′) ⊆ Qi(S,S′) and Qj2(S,S′) ⊆ +Qi(S,S′), let ¯Qj1(S,S′) = Qi(S,S′) ∖ Qj1(S,S′) and ¯Qj2(S,S′) = Qi(S,S′) ∖ Qj2(S,S′). +Consider the solutions Hj1(S,S′ ∪Zi(S)∪ ¯Qj1(S,S′),γ1) and Hj2(S,S′ ∪Zi(S)∪ ¯Qj2(S,S′),γ2). +Let P[Qj1(S,S′),γ1] and P[Qj2(S,S′),γ2] be two corresponding assignments, restricted to Yj1 +and Yj2, that yield the objective Hj1(S,S′ ∪ Zi(S) ∪ ¯Qj1(S,S′),γ1) and Hj2(S,S′ ∪ Zi(S) ∪ +¯Qj2(S,S′),γ2), respectively. +Such assignments can be easily obtained during the bottom-up +process. Lastly, let W(P[Qj1(S,S′),γ1]) and W(P[Qj2(S,S′),γ2]) be the set of truly integrated +vertices in Yj1∖(S′ ∪ Zi(S)) and Yj2∖(S′ ∪ Zi(S)) under the assignments P[Qj1(S,S′),γ1] and +P[Qj2(S,S′),γ2], respectively. +Hi(S,S′,γ) is computed as follows +Hi(S,S′,γ) = max +D {∣W(P[Qj1(S,S′),γ1]) ∪ W(P[Qj2(S,S′),γ2])∣} + ∣S′ ∪ Zi(S)∣ +(63) +where D = {γ1,γ2,Qj1(S,S′),Qj2(S,S′) ∶ γ1+γ2−∣S∣ = γ,γ1 ≥ ∣S∣,γ2 ≥ ∣S∣,Qj1(S,S′) ⊆ Qi(S,S′), +Qj2(S,S′) ⊆ Qi(S,S′)} is the set of variables. +▷ Theorem 6.1. The problem IM-IoA can be solved optimally in polynomial time on tree-width +bounded graphs. +Proof. We first analyze the correctness of the update scheme. The optimality of the first three +update rules (i.e., leaf, introduce, forget) easily follows from induction. We further discuss the case +where Xi is of type join. In particular, we argue that the optimal objective Hi(S,S′,γ) has the +value shown in Equation (63). +Consider an optimal assignment P∗ +i on G[Yi] that achieves the optimal objective Hi(S,S′,γ). +Let γ∗ +1 and γ∗ +2 be the number of type-1 vertices in G[Yj1] and in G[Yj2], respectively, under P∗ +i . +Since Yj1 and Yj2 only share Xi as a common set, we have γ∗ +1 + γ∗ +2 − ∣S∣ = γ. We may consider P∗ +i +as a union of two assignments, P∗ +j1 and P∗ +j2, where P∗ +j1 and P∗ +j2 are P∗ +i restricted to G[Yj1] and in +G[Yj2], respectively. +Recall that Qi(S,S′) = Xi ∖ (S′ ∪ Zi(S)) is the set of vertices that are not truly integrated in +G[Xi] under the assignment (S,Xi ∖ S), and also should not be treated as integrated (i.e., vertices +in Qi(S,S′) are not in S′). Note that it is possible that some vertices in Qi(S,S′) are integrated +under P∗ +j1 and P∗ +j1. In particular, let Q∗ +j1(S,S′) ⊆ Qi(S,S′) and Q∗ +j2(S,S′) ⊆ Qi(S,S′) be the set +of vertices in Qi(S,S′) that are good under P∗ +j1 and P∗ +j2, respectively. +42 + +Observe that γ∗ +1,γ∗ +2,Q∗ +j1(S,S′),Q∗ +j2(S,S′) ∈ D. Let P[Q∗ +j1(S,S′),γ∗ +1] and P[Q∗ +j2(S,S′),γ∗ +2] be +two corresponding assignments returned by the proposed update scheme, restricted to Yj1 and Yj2, +that yield the objective Hj1(S,S′ ∪ Zi(S) ∪ ¯Q∗ +j1(S,S′),γ∗ +1) and Hj2(S,S′ ∪ Zi(S) ∪ ¯Q∗ +j2(S,S′),γ∗ +2), +respectively. In particular, we have +IoA(P[Q∗ +j1(S,S′),γ∗ +1]) = Hj1(S,S′ ∪ Zi(S) ∪ ¯Q∗ +j1(S,S′),γ∗ +1) +(64) += ∣S′ ∪ Zi(S) ∪ ¯Q∗ +j1(S,S′)∣ + ∣W(P[Qj1(S,S′),γ1]) ∖ Xj1∣ +(65) ++ ∣W(P[Qj1(S,S′),γ1]) ∩ Q∗ +j1(S,S′)∣ +(66) +Consider the particular instance (S,S′ ∪ Zi(S) ∪ ¯Q∗ +j1(S,S′),γ∗ +1) for G[Yj1]. Since P[Q∗ +j1(S,S′),γ∗ +1] +is an optimal solution and P∗ +j1 is a feasible solution of this instance, it follows that +∣W(P∗ +j1) ∖ Xj1∣ + ∣W(P∗ +j1) ∩ Q∗ +j1(S,S′)∣ = ∣W(P∗ +j1)∣ +(67) +≤ ∣W(P[Qj1(S,S′),γ1]) ∖ Xj1∣ +(68) ++ ∣W(P[Qj1(S,S′),γ1]) ∩ Q∗ +j1(S,S′)∣ +(69) +Similarly, we also have +∣W(P∗ +j2)∣ ≤ ∣W(P[Qj2(S,S′),γ2]) ∖ Xj2∣ + ∣W(P[Qj2(S,S′),γ2]) ∩ Q∗ +j2(S,S′)∣ +(70) +The objective of P∗ +i for the instance (S,S′,γ) on Xi is of the form: +IoA(P∗ +i ) = Hi(S,S′,γ) +(71) += ∣S′ ∪ Zi(S)∣ + ∣W(P∗ +j1) ∪ W(P∗ +j2)∣ +(72) += ∣S′ ∪ Zi(S)∣ + ∣W(P∗ +j1)∣ + ∣W(P∗ +j2)∣ − ∣Q∗ +j1(S,S′) ∩ Q∗ +j2(S,S′)∣ +(73) +Consider the placement Pi which is the union of P[Q∗ +j1(S,S′),γ∗ +1] and P[Q∗ +j2(S,S′),γ∗ +2]. Note that +Pi is a feasible solution to the problem instance (S,S′,γ) on Xi since γ∗ +1 +γ∗ +2 −∣S∣ = γ, and Yj1 only +overlaps with Yj1 on Xi. The objective of Pi for the instance (S,S′,γ) on Xi satisfies the following +43 + +inequality: +IoA(Pi) = ∣S′ ∪ Zi(S)∣ + ∣W(P[Qj1(S,S′),γ1]) ∪ W(P[Qj2(S,S′),γ2])∣ +(74) +≥ ∣S′ ∪ Zi(S)∣ +(75) ++ ∣W(P[Qj1(S,S′),γ1]) ∖ Xj1∣ + ∣W(P[Qj1(S,S′),γ1]) ∩ Q∗ +j1(S,S′)∣ +(76) ++ ∣W(P[Qj2(S,S′),γ2]) ∖ Xj2∣ + ∣W(P[Qj2(S,S′),γ2]) ∩ Q∗ +j2(S,S′)∣ +(77) +− ∣W(P[Qj1(S,S′),γ1]) ∩ Q∗ +j1(S,S′) ∩ W(P[Qj2(S,S′),γ2]) ∩ Q∗ +j2(S,S′)∣ +(78) +≥ IoA(P∗ +i ) +(79) +Lastly, since IoA(P∗ +i ) is the optimal, the above inequality implies equaliy, that is, +∣S′ ∪ Zi(S)∣ + ∣W(P[Qj1(S,S′),γ1]) ∪ W(P[Qj2(S,S′),γ2])∣ = IoA(P∗ +i ) +(80) +This concludes the proof of correctness. As for the running time, one can verify that for each bag +Xi, if Xi is of the type leaf, forget or introduce, we need time O(4σ ⋅ n) to update all entries in Hi, +where σ is the treewidth. On the other hand, if Xi is of the type join, we need time O(16σ ⋅ n3) to +update all entries in Hi. Overall, since the number of bags in the tree decomposition is polynomial +w.r.t n, and σ is bounded, the update scheme runs in polynomial time w.r.t. n. This concludes the +proof. +∎ +6.2 PTAS on planar graphs +One can easily verify that IM-IoA remains hard on planar graphs. +Given a planar graph G and +for any constant ϵ > 0, we present a polynomial time approximation scheme that achieves a (1 − ϵ) +approximation for the IM-IoA. First, based on the algorithm for treewidth bounded graphs, observe +that +▷ Observation 6.2. Given a graph G such that each connected component is tree-width bounded, +the problem IM-IoA can be solved in polynomial time on G. +PTAS algorithm. Let q = 2 ⋅ ⌈1/ϵ⌉. We start with a plane embedding of G which divides the set of +vertices into ℓ layers. Let Vi be the set of vertices in the ith layer, i = 1,...,ℓ. For each r = 1,...,q, +observe that we may partition the vertex set into t + 1 subsets, t = ⌈(ℓ − r)/q⌉, such that the (i) +the first subset W(1,r) consists of the first r layers, (ii) the last subset W(t+1,r) consists of the last +((l − r) mod q) layers, and (iii) each ith subset W(i,r) in the middle contains q layers in sequential +order. Let Wr = {W(1,r),...,W(t+1,r)} be such a partition. Let G(i,r) be the subgraph induced on +W(i,r), i = 1.,,,t + 1. It is known that each G(i,r) is an outerplanar graph with a bounded treewidth +44 + +O(q) [5]. Let Gr = ⋃i G(i,r). Then by Observation (6.2), we can solve the problem optimally on each +Gr, r = 1,...,q in polynomial time. The algorithm then returns the solution with the largest objective +over all r = 1,...,q. One can easily verify that the overall scheme runs in polynomial time w.r.t. n. +▷ Theorem 6.3. The algorithms gives a factor (1 − ϵ) approximation on planar graphs for any +fixed ϵ > 0. +Proof. Recall that k is the number of type-1 agents (and n−k is the number of type-2 agents). Let +q = 2 ⋅ ⌈1/ϵ⌉. We show that the algorithm gives a 1 − 2/q ≥ 1 − ϵ approximation. The case for q < 3 is +trivially true. +Let P∗ be an assignment of agents on G that gives the maximum number of integrated agents. +Fix a r = 1,...,q, let Wr = {W(1,r),...,W(t+1,r)} be a partition of the vertex set as described above. +Let Pr be an assignment on Gr that is obtained from the proposed algorithm. We now look at +the assignment Pr and P∗, restricted to vertices in Wr. Specifically, let P(i,r) and P∗ +(i,r) be the +assignment of agents restricted to the subset W(i,r) under Pr and P∗, respectively. Further, let +IoA(P(i,r)) be the number of integrated agents in G(i,r) under Pr, and IoA(P∗ +(i,r)) is the number of +integrated agents in G(i,r) under P∗. We first observe that +IoA(Pr) = +t+1 +∑ +i=1 +IoA(P(i,r)) +(81) +which is true since G(i,r)’s are disconnected. Further, by the fact that Pr is optimal on Gr, we have +t+1 +∑ +i=1 +IoA(P(i,r)) ≥ +t+1 +∑ +i=1 +IoA(P∗ +(i,r)) +(82) +Note that ∑t+1 +i=1 IoA(P∗ +(i,r)) could be less than IoA(P∗), which is the optimal objective on G. Let +t+1 +∑ +i=1 +IoA(P∗ +(i,r)) = IoA(P∗) − ∆r +where ∆r ≥ 0 is the difference. We note that the integrated vertices that are left uncounted can +only exist on the two adjacent layers between each pair of G(i,r) and G(i+1,r), i = 1,...t. Let V∗ be +the set of integrated vertices under P∗. We then have, +∆r ≤ +t +∑ +j=0 +(V∗ ∩ Vj⋅q+r) + (V∗ ∩ Vj⋅q+r+1) +(83) +Since the layers are a partition of the vertex set, and each layer gets counted exactly twice in the +45 + +above sum, We have, +q +∑ +r=1 +∆r = 2 ⋅ IoA(P∗) +(84) +It follows that +min +r=1,...,q{∆r} ≤ 2 +q ⋅ IoA(P∗) +(85) +Let r∗ = arg minr=1,...,q{∆r}. By equation (81) and (82), we have +IoA(Pr∗) ≥ (1 − 2 +q ) ⋅ IoA(P∗) +(86) +Lastly, let ˆP be the assignment returned by the algorithm, that is, ˆP = arg maxr IoA(Pr). By +Equations (81) to (86), we have +IoA( ˆP) ≥ (1 − 2 +q ) ⋅ IoA(P∗) +(87) +This concludes the proof. +∎ +46 + +References +[1] Aishwarya Agarwal, Edith Elkind, Jiarui Gan, and Alexandros Voudouris. +Swap stability in +schelling games on graphs. In Proceedings of the AAAI Conference on Artificial Intelligence, +number 02, pages 1758–1765, (online), 2020. AAAI Press. +[2] Brenda S Baker. Approximation algorithms for np-complete problems on planar graphs. Journal +of the ACM (JACM), 41(1):153–180, 1994. +[3] Albert-L´aszl´o Barab´asi and R´eka Albert. Emergence of scaling in random networks. science, +286(5439):509–512, 1999. +[4] Nawal Benabbou, Mithun Chakraborty, Xuan-Vinh Ho, Jakub Sliwinski, and Yair Zick. Diversity +constraints in public housing allocation. 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Nature human +behaviour, 4(11):1124–1134, 2020. +49 + diff --git a/A9E1T4oBgHgl3EQfDQPu/content/tmp_files/load_file.txt b/A9E1T4oBgHgl3EQfDQPu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e3c319d85908d6a1881f1977241fb3dc7515195 --- /dev/null +++ b/A9E1T4oBgHgl3EQfDQPu/content/tmp_files/load_file.txt @@ -0,0 +1,1558 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf,len=1557 +page_content='Assigning Agents to Increase Network-Based Neighborhood Diversity Zirou Qiu,1,2 Andrew Yuan,2 Chen Chen,2 Madhav V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Marathe,1,2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Ravi,2,3 Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Rosenkrantz,2,3 Richard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Stearns,2,3 Anil Vullikanti1,2 1 1Computer Science Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 2Biocomplexity Institute and Initiative, University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 3Computer Science Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', University at Albany – SUNY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Abstract Social segregation is a persistent problem in society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Despite of the existing strategic plans to advance diversity, we continue to witness spatial segregation of people by demo- graphic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Motivated by real-world applications, such as public-housing allocation for low-income individuals, we examine the problem of assigning a group of agents to ver- tices in a graph that represents spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Agents are of two types (subgroups) characterized by certain sensitive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The goal is to construct an assignment that maximizes the level of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Specifically, we quantify the diversity by the number of well-integrated agents, that is, the agents who have at least one neighbor of a different type in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given the intractable nature of this maximization problem, we focus on developing approximation algorithms with provable performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We first propose a local-improvement algorithm for general graphs with a constant factor 1/2 ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, for a special case where the sizes of two subgroups are similar, we present a semidefinite programming approach that yields an approximation factor better than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We also show that the problem can be solved efficiently when the underlying graph is treewidth-bounded, and then use this result to obtain a polynomial time approxi- mation scheme (PTAS) for the problem on planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Lastly, we conduct experiments to evaluate the performance of the proposed algorithms on realistic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='02876v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='DS] 7 Jan 2023 1 Introduction Various countries have public housing initiatives that offer low-income individuals secure and affordable residences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Housing options are typically allocated by government agencies and involve a process of assigning applicants on a waiting list to vacant residences [36, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given that these applicants often come from a variety of demographic groups, the spatial distribution of public housing partially shapes the demographic structure of local communities [32, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The promotion and cultivation of integrated communities is an objective of contemporary societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' It has been shown that integration can improve a country’s financial performance, reduce the disparity between demographic groups, and advance social prosperity in general [8, 23, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Conversely, segregated neighborhoods widen the socioeconomic divide in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' As noted by many social scientists, residential segregation remains a persistent problem that directly contributes to the uneven distribution of resources and limited life chances for some groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', [30, 35, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In this work, we look at the problem of promoting community integration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', diversity) in the context of housing assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Indeed, public housing programs often take diversity into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In Singapore, there are established policies to ensure that a certain ethnic quota must be satisfied for each project at the neighborhood level [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', cities like Chicago and New York place emphasis on the value of having integrated communities1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Nevertheless, formal computational methods for improving the level of integration in the housing assignment process have received limited attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Motivated by the above considerations, we investigate the scenario of public housing allocation from an algorithmic perspective and provide systematic approaches to design assignment strategies that enhance community integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Formally, we model a housing project as a network G = (V,E) where V is the set of vacant residences, and edges in E represent proximity between residences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We are also given a set A of agents representing the applicants to be assigned to residences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In this work, we examine the setting where agents are partitioned into two demographic subgroups: type-1 and type-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Without loss of generality, we assume that the number of type-1 agents does not exceed the number of type-2 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Due to this constraint, we sometimes use the phrase “minority agents” for type-1 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=') We also assume that the number of vacant residences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', ∣V∣) equals the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Our goal is to construct an assignment (bijective mapping) P of residences to agents that maximizes the the integration level of the layout of agents on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' To quantify the integration level of a given assignment P, we use the index of integration (IoA) metric proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' This index is defined as the number of integrated agents, that is, agents with at least one neighbor of a different type in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' An illustrative example is given in Fig (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We refer to 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='gov/content/dam/city/depts/dcd/Housing%20Programs/20733_37_5_Year_Plan_Report_ final_WEB_C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='pdf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' https://furmancenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='org/files/NYCHA_Diversity_Brief_Final-04-30-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='pdf 2 the above assignment problem as Integration Maximization - Index of Agent Integration (IM-IoA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We note that this problem could also arise in other settings where integration is preferred, such as dormitory assignments for freshmen in universities [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Figure 1: An example assignment of two type-1 agents (blue) and six type-2 agents (red) on a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Vertices with integrated agents are labeled by dashed circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The index of integration for this assignment (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', the number of integrated agents) is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The problem of maximizing IoA is shown to be NP-hard in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Nevertheless, the authors of [1] did not further address optimization questions for the problem, as their focus is on game theoretic properties for IoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In this work, we focus on developing approximation algorithms with provable performance guarantees for IM-IoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Our main contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Approximation for general instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We present a local-improvement algorithm that guar- antees a factor 1/2 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We further show that our analysis is tight by presenting an example that achieves this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' While it is possible to derive an approximation for the problem using a general result in [7], the resulting performance guarantee is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='356, which is weaker than our factor of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Improved approximation for special instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For the case when the number of type-1 agents is a constant fraction α of the total number of agents, 0 < α ≤ 1/2, we present a semidefinite programming (SDP) based randomized algorithm that yields approximation ratios in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='516,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='649] for α is in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='403,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For example, when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='45, the ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='578, and when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='5, the ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' A polynomial time approximation scheme for planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We present a dynamic pro- gramming based algorithm that solves IM-IoA in polynomial time on graphs with bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Using this result in conjunction with a technique due to Baker [2], we obtain a polynomial time ap- proximation scheme (PTAS) for the problem on planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For any fixed ϵ > 0, the algorithm provides a performance guarantee of 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Empirical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We study the empirical performance of the proposed local-improvement algorithm against baseline methods on both synthetic and real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Overall, we observe that the empirical approximation ratio of the proposed algorithm is much higher than 1/2, which is our theoretical guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 3 2 Related Work Integration in public housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Issues regarding segregation and the need for enhancing integration have been documented extensively in the social science literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', [11, 24, 21, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In partic- ular, many works on segregation in social networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', [16, 18]) stem from the pioneering models proposed by Schelling [31], where agents move between vertices to improve their utility values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' While Schelling’s framework allows the study of agent dynamics, Benabbou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' [4] study integration in public housing allocation from a planning perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In particular, they formulate the setting as a weighted matching problem where the set of available houses is partitioned into blocks, and agents are assigned (by some central agency) to blocks to maximize a utility measure while satisfying some diversity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' They establish the NP-hardness of the problem and present an approxima- tion algorithm based on a result of Stamoulis [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' A number of other studies have also addressed integration in the context of public housing from a social science perspective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', [28, 19, 22, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The problem formulations and the algorithmic techniques used in Benabbou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' [4] and in our work are significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' First, Benabbou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' [4] examine a weighted matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Their model does not use any network structure for the residences, whereas our work approaches the problem from a graph theoretic standpoint, with the underlying network playing an important role in the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, the integration index studied in our work is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='t graph structures, whereas the measure used in [4] is based on constraints on the ethnicity quotas for blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' More importantly, the goal of our work is to find an assignment that maximizes the integration level, whereas the goal in [4] is to maximize the overall utility of agents under a diversity constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Integration indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Various indices to measure the level of integration in a population are surveyed in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' However, most of those indices cannot be naturally extended to a network setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The integration index IoA considered in our work was proposed by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' [1]2 in the context of the Schelling Game on networks, where agents can change locations to increase their utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' explore several properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', the integration price of anarchy/stability) of the index from a game theoretic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, they show that finding an assignment for which all agents are integrated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', each agent has at least one neighbor of a different type) is NP-hard [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Our approximation algorithm for general IM-IoA is based on a local- improvement scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' A well-known problem for which a local-improvement algorithm provides an approximation guarantee of 1/2 is the unweighted MaxCut problem [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We note that the analysis used to establish the performance guarantees of the local-improvement methods for MaxCut and IM- IoA are substantially different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In particular, MaxCut has no cardinality constraints, and the objective is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='t edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In contrast, IM-IoA requires that a specified number of vertices be assigned 2In Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' [1], the index is called “degree of integration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In our work, the term “degree” is used to denote the degree of a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We use the term “index of integration” to denote the index proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 4 to type-1 agents, and the objective is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='t vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' One can also formulate IM-IoA as a non-monotone submodular function maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Since such a formulation requires a strict equality constraint (involving type-1 agents), the best known performance guarantee under the general non-monotone submodular maximization framework with such a constraint is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='356 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 3 Problem Definition We study the problem of assigning vertices in a graph to a group of agents, such that the integration level of the resulting layout of agents in the graph is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We begin with some definitions and then provide a formal definition of the maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Graphs and agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let G = (V,E) be an undirected graph, where V is a set of vertices representing vacant residences, and E is a set of edges representing the neighborhood relationship between resi- dences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let A be the set of agents to be assigned to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We consider a setting where the set of agents is divided into two demographic subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Formally, A is partitioned into two subsets A1 and A2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' we refer to agents in Ai as type i agents, i = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let k = ∣A1∣ denote the number of type-1 agents, so n − k is the number of type-2 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Without loss of generality, we let k ≤ n/2, and we refer to A1 as the minority subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Lastly, we assume that ∣V∣ = ∣A∣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' that is, the number of vertices is the same as the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' An assignment is a mapping from vertices to agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' To simplify proofs, we use an equivalent definition where an assignment is a mapping from vertices to agent types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In particular, an assignment P ∶ V → {1,2} is a function that assigns an agent type to each vertex in V, such that k vertices are assigned type-1 and n − k vertices are assigned type-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In such an assignment, a type-i vertex is occupied by a type-i agent, i = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We remark that the above definition of an assignment is mathematically equivalent to defining an assignment to be a mapping from V to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The index of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We consider the integration index proposed in [1] and apply it to our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1 (Index of agent-integration (IoA) [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given an assignment P, an agent x ∈ A is integrated if x has at least one neighbor in G whose type is different from that of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let A′ be the set of integrated agents under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The index of agent-integration of P is then defined as the number of integrated agents in A: IoA(P) = ∣A′∣ (1) Equivalently, a vertex u ∈ V is integrated under P if the agent assigned to u is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Thus, we may also view the index as IoA(P) = ∣V′∣ where V′ is the set of integrated vertices under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We remark that these two definitions of IoA are mathematically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 5 The optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We define the problem Integration Maximization-Index of Agent Integration (IM-IoA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2 (IM-IoA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given a graph G = (V,E), a set A of agents with k type-1 and n − k type-2 agents, find an assignment P such that IoA(P) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 4 Approximation for General Graphs IM-IoA is NP-hard, as established in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In this section, we present a local-improvement algorithm for IM-IoA and show that the algorithm achieves a factor 1/2 approximation for general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For convenience in presenting the proofs, we consider an assignment from the perspective of vertices rather than that of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' As stated earlier, these two definitions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We start from a random assignment P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In each iteration of the algorithm, we find (if possible) a pair of type-1 and type-2 vertices such that swapping their types strictly increases the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In particular, let u be a type-1 vertex, and v be a type-2 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We swap the types of u and v (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', u becomes type-2 and v becomes type-1) if and only if the resulting new assignment P′ has a strictly higher IoA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' that is, IoA(P) < IoA(P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The algorithm terminates when no such swap can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The pseudocode is given in Algorithm (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Algorithm 1: Local-Improvement-IoA Input : A graph G = (V,E), k, where k ≤ ∣V∣/2 Output: An assignment P 1 P ← a random assignment & Updated ← True 2 while Updated do 3 Updated ← False 4 for x ∈ V1(P) do 5 for y ∈ V2(P) do 6 P′ ← the assignment where P′(x) = P(y) and P′(y) = P(x) 7 if IoA(P′) > IoA(P) then 8 P = P′, Updated ← True & break 9 return P 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1 Analysis of the algorithm Given a problem instance of IM-IoA, let P be a saturated assignment3 returned by Algorithm (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let P∗ be an optimal assignment that achieves the maximum objective, denoted by OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We assume that P ≠ P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In this section, we show that IoA(P) ≥ 1/2 ⋅ IoA(P∗) = 1/2 ⋅ OPT, thereby establishing a 3An assignment is saturated if no pairwise swap of types between a type-1 and a type-2 vertices can increase the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 6 1/2 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Due to the page limit, we sketch the proof here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' the full proof appears in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given an assignment P, which is a mapping from vertices to agent types, we call a vertex v a type-1 (or type-2) vertex if P(v) = 1 (or P(v) = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let V1(P) and V2(P) denote the set of type-1 and type-2 vertices under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let VU 1(P) and VU 2(P) denote the set of uncovered4 type-1 and type-2 vertices under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For each vertex u, let N U u(P) denote the set of neighbors of u that are uncovered under P, and let Γu(P) denote the set of different-type neighbors of u that are uniquely covered by u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', Γu(P) is the set of vertices v such that (i) v is a neighbor of u, (ii) the type of v is different from the type of u, and (iii) v has no other neighbor whose type is the same as u’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The index IoA(P) = n − ∣VU 1(P)∣ − ∣VU 2(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We now consider the following mutually exclusive and collectively exhaustive cases of VU 1(P) and VU 2(P) under the saturated assignment P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We start with a simple case where all the type-2 vertices under P are integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Case 1: VU 2(P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Under this case, all vertices in V2(P) are integrated which gives IoA(P) ≥ ∣VU 2(P)∣ = n − k ≥ 1 2 ⋅ n ≥ 1 2 ⋅ OPT (2) The above case trivially implies that the algorithm provides a 1/2 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We now look at the remaining case where VU 2(P) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Case 2: VU 2(P) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Under this case, there exists at least one vertex in V2(P) that is not integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We first show that VU 1(P) and VU 2(P) cannot both be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For a saturated assignment P, if VU 2(P) ≠ ∅, then VU 1(P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) Let y ∈ VU 2(P) be a vertex of type-2 that is not integrated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', all neighbors of y are of type-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For contradiction, suppose VU 1(P) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Now let x ∈ VU 1(P) be an non-integrated vertex of type-1 whose neighbors are all of type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let P′ denote the assignment where we switch the types between x and y, that is, P′(x) = P(y) = 2, P′(y) = P(x) = 1, while the types of all other vertices remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' One can verify that IoA(P′) ≥ IoA(P) + 2, that is, switching the types of x and y increases the index IoA by at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' This implies the existence of an improvement move from P, which contradicts the fact that P is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' It follows that VU 1(P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ 4Under an assignment, a vertex is “covered” if it is integrated and “uncovered” otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 7 Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2) implies that under case 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', VU 2(P) ≠ ∅), we have VU 1(P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We now consider the following two mutually exclusive and collectively exhaustive subcases under Case 2 and show that the approximation factor under each subcase is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1: VU 2(P) ≠ ∅, and Γx(P) ≠ ∅, ∀x ∈ V1(P), that is, for each type-1 vertex x ∈ V1(P), there is at least one type-2 neighbor of x that is uniquely covered (“made integrated”) by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Suppose P ≠ P∗, that is, for some vertices x ∈ V, P(x) ≠ P∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let ˜V2−1 = {v ∈ V ∶ P(v) = 2,P∗(v) = 1} be the set of vertices that are type-2 under P, but are type-1 under P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Analogously, let ˜V1−2 = {v ∈ V ∶ P(v) = 1,P∗(v) = 2} be the set of vertices of type-1 under P, but are of type-2 under P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Observe that ∣˜V2−1∣ = ∣˜V1−2∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We may view P∗ as the result of a transformation from P under pairwise swaps of types between ˜V2−1 and ˜V1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' An example is given in Figure (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We present a key lemma that bounds the difference between the objective values of P and P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='3 (Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let P be a saturated assignment under subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1, and let P∗ be an optimal assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We have IoA(P∗) − IoA(P) ≤ ∑ y∈˜V2−1∖VU 2(P) ∣(N(y) ∩ VU 2(P)∣ (3) + ∑ y∈˜V2−1∩VU 2(P) (∣(N(y) ∩ VU 2(P)∣ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) Since P is saturated, Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2) implies that all type-1 vertices under P are in- tegrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Thus, the difference IoA(P∗) − IoA(P) is at most the number of type-2 vertices that are integrated under P∗ but are not integrated under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Let f ∶ ˜V1−2 → ˜V2−1 be an arbitrary bijective mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We may regard P∗ as a result of the transformation from P via pairwise swaps of types between vertices specified by f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', the type of x ∈ ˜V1−2 is swapped with the type of f(x) ∈ ˜V2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Observe that only vertices in VU 2(P) that are adjacent to ˜V2−1 (or within ˜V2−1) under P can be newly integrated under P∗ after swapping ˜V1−2 with ˜V2−1 (by the definition of VU 2(P), vertices in ˜V1−2 have no neighbors in VU 2(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' It follows that for each vertex y ∈ ˜V2−1, at most ∣(N(y) ∩ VU 2(P)∣ of its neighbors can become newly integrated after transforming from P to P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, if also y ∈ ˜V2−1 ∩ VU 2(P), y itself could also be newly integrated after the swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We then have IoA(P∗) − IoA(P) ≤ ∣ ⋃ y∈˜V2−1 N(y) ∩ VU 2(P)∣ + ∣˜V2−1 ∩ VU 2(P)∣ ≤ ∑ y∈˜V2−1∖VU 2(P) ∣(N(y) ∩ VU 2(P)∣ (4) + ∑ y∈˜V2−1∩VU 2(P) (∣(N(y) ∩ VU 2(P)∣ + 1) 8 where the last inequality follows from the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ Figure 2: Two assignments P and P∗ where type-1 and type-2 vertices are highlighted in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' In this case, ˜V2−1 = {x3,x4} and ˜V1−2 = {x1,x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We may then transform P into P∗ by swapping types between the pair (x1,x3) and between (x2,x4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Note that this example is only to demonstrate how ˜V2−1 and ˜V1−2 are defined, as P cannot be a saturated assignment returned by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We now proceed to show that the difference between IoA and IoA established in Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='3) is at most IoA(P), thereby establishing IoA(P) ≥ 1 2 ⋅ IoA(P∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Recall that for each vertex x ∈ V, Γx(P) is the set of neighbors of x whose types are different from x, and are uniquely covered by x under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' By the definition of Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1, Γx(P) is not empty for all x ∈ V1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We first argue that for any y ∈ VU 2(P) and any x ∈ V1(P), we have ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γx(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='4 (Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given a saturated assignment P, for any y ∈ VU 2(P) and any x ∈ V1(P), we have ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γx(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) Given that y is not integrated under P, x and y cannot be adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Since P is a saturated assignment, if the types of x and y are to be swapped, the number of newly integrated vertices would be at most the number of newly non-integrated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, one can verify that the number of vertices that are newly integrated is at least ∣N(y) ∩ VU 2(P)∣ + 1, and the number of vertices that are newly non-integrated is at most ∣Γx(P)∣ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Since P is saturated, it follows that ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γx(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ We now establish the next Lemma, which bounds the size of N(y) ∩ VU 2(P) for y ∈ V2(P) ∖ VU 2(P) and x ∈ V1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='5 (Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given a saturated assignment P, for any y ∈ V2(P) ∖ VU 2(P) and any x ∈ V1(P), we have ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γx(P)∣ + 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) We partition V2(P) ∖ VU 2(P) into two subsets B and C, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Subset B is the set of integrated type-2 vertices whose neighbors are all integrated under P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', B = {y ∈ V2(P)∖VU 2(P) ∶ 9 PN(y) ∩ VU 2(P) = ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Subset C, the complement of B, is the set of integrated type-2 vertices with at least one non-integrated neighbor under P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', C = {y ∈ V2(P) ∖ VU 2(P) ∶ N(y) ∩ VU 2(P) ≠ ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The lemma clearly holds if y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, we show that for the case when y ∈ C, no type-1 neighbors of y is uniquely covered by y under P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', Γy(P) = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, suppose y ∈ C, consider an objective non-increasing move from P where we swap the types between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' If y is a neighbor of x under P, one can verify that the the maximum loss is ∣Γx(P)∣ and the minimum gain is ∣N(y) ∩ VU 2(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Thus ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γx(P)∣ (5) On the other hand, if y is not a neighbor of x under P, one can verify that the maximum loss is ∣Γx(P)∣ + 1 and the minimum gain is ∣N(y) ∩ VU 2(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Thus ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γx(P)∣ + 1 (6) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ We are now ready to establish IoA(P) ≥ 1 2 ⋅ IoA(P∗) under Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='6 (Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Suppose VU 2(P) ≠ ∅ and Γx(P) ≠ ∅,∀x ∈ V1(P), we have IoA(P) ≥ 1 2 ⋅ IoA(P∗) where P∗ is an optimal assignment that gives the maximum objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) Note that ˜V2−1 is a subset of V2(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Further, Observe that Γx(P) are disjoint for different vertices x ∈ V1(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Now, by Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='3) to and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='5), We have IoA(P∗) − IoA(P) ≤ ⎛ ⎜ ⎝ ∑ y∈˜V2−1 ∣Γf−1(y)(P)∣ ⎞ ⎟ ⎠ + ∣˜V2−1∣ ≤ ∣V2(P) ∖ VU 2(P)∣ + ∣V1(P)∣ (7) ≤ IoA(P) where Inequality (7) follows from ∣˜V2−1∣ = ∣˜V1−2∣ ≤ ∣V1(P)∣ and (∑y∈˜V2−1 ∣Γf−1(y)(P)∣) ≤ ∣V2(P) ∖ VU 2(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ We now have shown that if VU 2(P) ≠ ∅ and Γx(P) ≠ ∅,∀x ∈ V1(P), the algorithm gives a 1/2 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We proceed to the final subcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2: VU 2(P) ≠ ∅, and Γx(P) = ∅,∃x ∈ V1(P), that is, there exists at least one type-1 vertex 10 x ∈ V1(P) such that for each type-2 neighbor y of x, y is adjacent to at least one type-1 vertex other than x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='7 (Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Under subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2, for each non-integrated type-2 vertex y ∈ VU 2(P), all type-2 neighbors of y are integrated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=', N(y) ∩ VU 2(P) = ∅) under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' That is, the vertices in VU 2(P) form an independent set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) Given such a x ∈ V1(P) defined in Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2, for contradiction, suppose there exists a non-integrated type-2 vertex y ∈ VU 2(P) such that at least one type-2 neighbor, denoted by y′ ∈ N(y), of y is not integrated under P (note that all neighbors of y are of type-2 since y is not integrated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Now consider a new assignment P′ where we switch the types between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' One can verify that IoA(P′) ≥ IoA(P)+1, that is, after the switch, the index IoA would increase by at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' This implies the existence of an improvement move from P, which contradicts P being a saturated assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Thus, no such a non-integrated type-2 vertex y′ of y can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ Observe that IoA(P) = (n−∣VU 2(P)∣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' With Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='7) in place, we now argue that the size of VU 2(P) cannot be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='8 (Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Under Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2, ∣VU 2(P)∣ ≤ n 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' (Sketch) Let Y ∶= {y ∈ V2(P) ∖ VU 2(P) ∶ N(y) ∩ VU 2(P) ≠ ∅} be the set of type-2 integrated vertices whose has at least one non-integrated type-2 neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We first note that Γy(P) (if not empty) are mutually disjoint for different y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' It follows that IoA(P) ≥ ∣Y∣ + ∑y∈Y ∣Γy(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Suppose we switch the types between such a vertex x and a vertex y ∈ Y, and let P′ denote the resulting new assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' One can verify that the maximum loss of objective after the swap is ∣Γy(P)∣ + 1, whereas the minimum gain is ∣N(y) ∩ VU 2(P)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Since P is a saturated assignment returned by the algorithm, we must have IoA(P) ≥ IoA(P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Therefore, ∣N(y) ∩ VU 2(P)∣ ≤ ∣Γy(P)∣ + 1, ∀y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Overall, we have that ∣VU 2(P)∣ = ∣ ⋃ y∈Y N(y) ∩ VU 2(P)∣ (8) ≤ ∣V1(P)∣ + ∣Y∣ (9) ≤ ∣V1(P)∣ + ∣V2(P) ∖ VU 2(P)∣ (10) = n − ∣VU 2(P)∣ (11) It immediately follows that ∣VU 2(P)∣ ≤ n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∎ Lastly, Since IoA(P) = n − ∣VU 2(P)∣, by Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='8), we have IoA(P) = n − ∣VU 2(P)∣ ≥ 1 2 ⋅ n ≥ 1 2 ⋅ IoA(P∗), thereby establishing a 1/2 approximation for Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Overall, we have shown that a 11 saturated assignment P returned by Algorithm (1) gives a 1/2-approximation for IM-IoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Thus: ▷ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Algorithm (1) gives a 1 2-approximation for IM-IoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Analysis is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We present a class of problem instances where the approximation ratio of the solution produced by Algorithm (1) can be arbitrarily close to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Therefore, the ratio 1/2 in the statement of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='9) cannot be improved, so our analysis is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The proof appears in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For every ϵ > 0, there exists a problem instance of IM-IoA for which there is a saturated assignment P such that IoA(P) ≤ (1 2 + ϵ) ⋅ OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 5 Subgroups With Similar Sizes In this section, we study the problem instances when the number of type-1 agents is a constant fraction of the total number of agents, that is, k = α⋅n for some constant 0 ≤ α ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We refer to this problem as αn-IM-IoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' For example, α = 1/2 represents the bisection constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' We first show that αn-IM-IoA remains computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' See the Appendix for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ▷ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The problem αn-IM-IoA is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='1 A semidefinite programming approach We now present an approximation algorithm for αn-IM-IoA based on semidefinite programming (SDP) relaxation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' The overall scheme is inspired by the work of Frieze and Jerrum [12] on the Max- Bisection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' Given a graph G = (V,E), each vertex i ∈ V has a binary variable xi ∈ {−1,1} such that xi = −1 if i is of type-1, and xi = 1 if i is of type-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' First, we observe that a valid quadratic program (QP) is (see Appendix for the proof): maximize∑i∈V maxj∈N (i) {1−xixj 2 } s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQfDQPu/content/2301.02876v1.pdf'} +page_content=' ∑i 0, is given by the following +presentation: +⟨x1, . . . , x2n, v | [x2i−1, x2i] = v, [xj, xk] = 0, 1 ≤ i ≤ n, (j, k) ̸= (2i − 1, 2i)⟩. +Corollary 3.3. Let H2n+1 be the Heisenberg Lie algebra of dimension 2n+1. Then +B0(H2n+1) = 0. +Proof. Let L2n be the generalized Heisenberg Lie algebra of rank n(2n − 1) and M +be its ideal generated by [x2r−1, x2r] − [x2s−1, x2s], 1 ≤ r < s ≤ n and [xt, xu], 1 ≤ +t < u ≤ 2n, (t, u) ̸= (2i − 1, 2i) for any i ≤ n. Then it is easy to see that L2n/M is +isomorphic to H2n+1. Since B0(L2n) = 0 by Theorem 3.2, it follows from Theorem +3.1 that +B0(L2n/M) ∼= +M ∩ L′ +2n +⟨K(L2n) ∩ M⟩. +Since for 1 ≤ r < s ≤ n, +[x2r−1 − x2s−1, x2r + x2s] = [x2r−1, x2r] − [x2s−1, x2s] + [x2r, x2s−1] + [x2r−1, x2s], +it follows that M∩L′ +2n = ⟨K(L2n)∩M⟩. Thus B0(L2n/M) = 0 so that B0(H2n+1) = +0. +□ +We now proceed to prove Theorem 1.5. +Proof of Theorem 1.5: Let d be a natural number greater than 4n and Ld be +the d-generated freest generalized Heisenberg Lie algebra generated by x1, x2, . . . xd. +Let M be the ideal generated by [x1, x2]+[x3, x4], [x5, x6]+[x7, x8], . . . [x4n−3, x4n−2]+ +[x4n−1, x4n]. Since B0(Ld) = 0, it follows from Theorem 3.1 that +dim B0(Ld/M) = dim +L′ +d ∩ M +⟨K(Ld) ∩ M⟩. +Next we prove that K(Ld) ∩ M = {0}. For this let l ∈ K(Ld) ∩ M. Since l ∈ M +there exists γ′ +ks such that +l = +n−1 +� +k=0 +γk+1 +� +[x4k+1, x4k+2] + [x4k+3, x4k+4] +� +. + +8 +P. K. RAI +Also there exist αi’s and βj’s such that +l = +� +d +� +i=1 +αixi, +d +� +j=1 +βjxj +� +because l ∈ K(Ld). Hence +n−1 +� +k=0 +γk+1 +� +[x4k+1, x4k+2] + [x4k+3, x4k+4] +� += +d−1 +� +i=1 +d +� +j=i+1 +(αiβj − αjβi)[xi, xj]. +It follows that +(3.1) +αiβj − αjβi = 0 when (i, j) ̸= (2k + 1, 2k + 2) for any k = 0, 1, . . . n − 1, +and +γk+1 = α4k+1β4k+2 − α4k+2β4k+1 = α4k+3β4k+4 − α4k+4β4k+3 +for any k = 0, 1, . . . n − 1. From Equation 3.1 we have +(3.2) +α4k+1β4k+3 − α4k+3β4k+1 = 0 +(3.3) +α4k+1β4k+4 − α4k+4β4k+1 = 0 +(3.4) +α4k+2β4k+3 − α4k+3β4k+2 = 0 +(3.5) +α4k+2β4k+4 − α4k+4β4k+2 = 0, +for k = 0, 1, . . .n − 1. +Suppose α4k+1 = β4k+1 = 0. Then γk+1 = 0. Assume then that α4k+1 = 0 but +β4k+1 ̸= 0. From Equations 3.2 and 3.3, α4k+3 = α4k+1 = 0. As a result, γk+1 = 0 +in this case as well. Thus we have shown that if α4k+1 = 0, then γk+1 = 0. Similarly, +if either of α4k+2, α4k+3, α4k+4, β4k+1, β4k+2, β4k+3, β4k+4 is zero, then γk+1 = 0. +Hence, we can now assume that neither of α4k+i, β4k+i is zero for i = 1, 2, 3, 4. +From Equations 3.2 and 3.3 we can deduce that β4k+3/β4k+4 = α4k+3/α4k+4. +Hence γk+1 = 0 so that l = 0. It follows that K(Ld) ∩ M = {0}. +Also, M ≤ L′ +d. +Hence dim B0(Ld/M) = dim(M) = n. By Corollary 1.2 +dim B(Ld/M) = n. Taking L to be Ld/M completes the proof. +Proof of Theorem 1.6: Let (θ, φ) be an isoclinism between L and M, i.e., θ : +L +Z(L) �→ +M +Z(M) and φ : γ2(L) �→ γ2(M) be isomorphisms and whenever θ(liZ(L)) = +miZ(M) for i = 1, 2, we have φ([l1, l2]) = [m1, m2]. Let ¯f ∈ B0(L) where f : +L × L �→ Ω be a cocycle. Define cf : M × M �→ Ω by cf(m1, m2) = f(l1, l2), where +l1 and l2 are given by θ−1(mi + Z(M)) = li + Z(L) for i = 1, 2. The rest of the +proof follows from the following lemmas: +Lemma 3.4. Let cf be the map defined above. Then +(i) cf is well defined. +(ii) cf is a 2 cocycle. +(iii) cf ∈ B0(M). + +BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS +9 +Proof. Since f is bilinear and f(k, l) = 0 whenever [k, l] = 0. +It follows that +f(l1 + z1, l2 + z2) = f(l1, l2) for every z1, z2 ∈ Z(L). This shows that cf is well- +defined. +To see that cf is a 2-cocycle, let m1, m2, m3 ∈ M and let θ−1(mi + Z(M)) = +li + Z(L) for i = 1, 2, 3. It is obvious that θ−1(m1 + m2 + Z(M)) = l1 + l2 + Z(L). +Therefore cf(m1 + m2, m3) = f(l1 + l2, l3) which equals f(l1, l3) + f(l2, l3) that +is equal to cf(m1, m3) + cf(m2, m3). Similarly cf(m1, m2 + m3) = cf(m1, m2) + +cf(m1, m3). Thus cf is bilinear. Also, it is easy to see that cf is alternating because +f is alternating. Next, For i, j, k ∈ {1, 2, 3} note that cf([mi, mj], mk) = f([li, lj], lk) +because +θ−1� +[mi, mj] + Z(M) +� += θ−1�� +mi + Z(M), mj + Z(M) +�� += +� +θ−1� +mi + Z(M) +� +, θ−1� +mj + Z(M) +�� += +� +li + Z(L), lj + Z(L) +� += [li, lj] + Z(L). +t follows that cf is a 2-cocycle, since f is a 2-cocycle. +To see that cf ∈ B0(M), suppose that [m1, m2] = 0. +But then [l1, l2] = 0 +because φ([l1, l2]) = [m1, m2]. Since f ∈ B0(L) it follows that f(l1, l2) = 0. Hence +cf(m1, m2) = 0. +Lemma 3.5. The map η : B0(L) �→ B0(M) defined by η(f) = cf is an isomor- +phism. +Proof. We begin by ensuring that the map is well-defined. To verify this consider +σ : L × L �→ Ω to be a coboundary. Then +cf+σ(m1, m2) = (f + σ)(l1, l2) = f(l1, l2) + σ(l1, l2) = cf(m1, m2) + cσ(m1, m2). +Thus we have, cf+σ = cf +cσ. Notice that cσ is a coboundary because σ is cobound- +ary. Therefore cf = cf+σ, i.e., η(f) = η(f + σ). This proves that η is well-defined. +In a similar fashion one can see that cf1+f2 = cf1 + cf2 and cαf1 = αcf1 for each +α ∈ Ω and each cocycles f1, f2 from L × L to Ω. So that η(f1 + f2) = η(f1) + η(f2) +and η(αf1) = αη(f1). Thus η is a linear map. +Finally, in order to see that η is a bijection, we define another map χ : B0(M) �→ +B0(L) in the same way as η is defined from B0(L) to B0(M). Then it is easy to see +that ηχ and χη both are identity maps and thus η is a bijection. This completes +the proof. +□ +□ +References +[1] P. Batten, Covers and multipliers of Lie algebras, Dissertation, North Carolina State Uni- +versity, 1993. 2, 3, 4, 5 +[2] F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad. +Nauk SSSR, Ser. Mat. 51 (1987), no. 3, article no. 688. 1 +[3] B. Kunyavski˘ı, The Bogomolov multiplier of finite simple groups, Cohomological and geo- +metric approaches to rationality problems, 209–217, Progr. Math., 282, Birkh¨auser Boston, +Inc., Boston, MA, 2010. 1 + +10 +P. K. RAI +[4] B. Kunyavski˘ı, Some New Parallels Between Groups and Lie Algebras, or What Can Be +Simpler than Multiplication Table?, EMS Newsl. 118 (2020), 5–13. 2, 3 +[5] P. Moravec, Unramified Brauer groups of finite and infinite groups, Am. J. Math. 134 (2012), +no. 6, 1679-1704. 1 +[6] Z. A. Rostami, M. Parvizi, P. Niroomand, The Bogomolov multiplier of Lie algebras, Hacet. +J. Math. Stat. 49 (2020), 1190- 1205. 1, 2, 6, 7 +[7] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), +no. 1, 71-84. 1 +(Pradeep K. Rai) Mahindra University, Hyderabad, Telangana,, India +Email address: raipradeepiitb@gmail.com + diff --git a/CtA0T4oBgHgl3EQfAf94/content/tmp_files/load_file.txt b/CtA0T4oBgHgl3EQfAf94/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f464fbd516c6f601983ed6b62002305284fd5f4 --- /dev/null +++ b/CtA0T4oBgHgl3EQfAf94/content/tmp_files/load_file.txt @@ -0,0 +1,391 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf,len=390 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='01963v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='RA] 5 Jan 2023 BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS PRADEEP K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' RAI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' In the work of Rostami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=', the Bogomolov multiplier of a Lie algebra L over a field Ω is defined as a particular factor of a subalgebra of the exterior product L ∧ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' If L is finite dimensional, we identify this object as a certain subgroup of the second cohomology group H2(L, Ω) by deriving a Hopf-Type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' As an application, we affirmatively answer two questions posed by Kunyavski˘ı regarding the invariance of the Bogomolov multiplier under isoclinism of Lie algebras and the existence of a family of Lie algebras with Bogomolov multipliers of unbounded dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Introduction The Bogomolov multiplier of a finite group G is a cohomological invariant of G that has been studied in connection with Noether’s problem, which asks whether the fixed subfield k(G) of the function field k(xg : g ∈ G) is purely transcendental over an algebraically closed field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The Bogomolov multiplier has played a key role in the discovery of counter examples to Noether’s problem over the complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Saltman [7] was the first to provide such counter examples, showing that if the unramified cohomology group H2 nr(k(G), C) is nonzero, then G has a negative solution to Noether’s problem over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Bogomolov later showed that H2 nr(k(G), C) is naturally isomorphic to the subgroup B0(G) of the second coho- mology group H2(G, C) consisting of classes that vanish when restricted to the abelian subgroups of G [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' This has led to the discovery of numerous other counter examples to Noether’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Kunyavski˘ı later gave the name “Bogomolov mul- tiplier” to B0(G) [3], and Moravec provided a homological description of B0(G) as a quotient of H2(G, C) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Following Moravec’s construction, Rostami et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' al [6] extended the notion of Bogomolov multiplier to Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' For the convenience of the reader we define it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a Lie algebra over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The exterior square of L is defined to be the Lie algebra L∧L generated by the symbols m ∧ n, where m, n ∈ L, subject to the following relations: (i) α(m ∧ n) = αm ∧ n = m ∧ αn, (ii) (m + m′) ∧ n = m ∧ n + m′ ∧ n, (iii) m ∧ (n + n′) = m ∧ n + m ∧ n′, (iv) [m, m′] ∧ n = m ∧ [m′, n] − m′ ∧ [m, n], (v) m ∧ [n, n′] = [n′, m] ∧ n − [n, m] ∧ n′, (vi) [(m ∧ n), (m′ ∧ n′)] = −[n, m] ∧ [m′, n′], (vii) m ∧ n = 0 whenever m = n, 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' 17B56, 14E08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Bogomolov multiplier, Lie algebras, Second cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' RAI for all α ∈ Ω, m, m′, n, n′ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' It is easy to see that K : L × L �→ [L, L] given by (m, n) �→ [m, n] induces a homomorphism ¯K : L∧L �→ [L, L], such that ¯K(m ∧ n) = [m, n], for all m, n ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' It is known that the kernel of ¯K is isomorphic to the Schur Multiplier H2(L, Ω) (defined below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' We denote it by M(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Define M0(L) to be the group ⟨m ∧ n | m, n ∈ L, [m, n] = 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The factor group M(L) M0(L) is defined to be the Bogomolov multiplier B(L) of the Lie algebra L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' In this article, we define a cohomological object B0(L) for a finite dimensional Lie algebra L over a field Ω, and show that it is isomorphic to the Bogomolov multiplier B(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Before that, we recall the definition of the Schur multiplier of a finite dimensional Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra and A be a trivial L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' A map f : L × L �→ A said to be a 2-cocycle if it is bilinear, alternating and satisfies f([x1, x2], x3) + f([x2, x3], x1) + f([x3, x1], x2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' And f is said to be a 2-coboundry if there exists a linear σ : L �→ A such that f(x1, x2) = −σ([x1, x2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The sets of 2-cocycles and 2-coboundries are denoted by Z2(L, A) and B2(L, A), respectively and form abelian groups with respect to usual addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The group Z2(L, A)/B2(L, A) is said to be the second cohomology group with coefficients in A, and is denoted by H2(L, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Schur multiplier of the Lie algebra L is defined as the abelian Lie algebra H2(L, Ω), considering Ω as a central L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' We are now ready to define B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' For a finite dimensional Lie algebra L over Ω, we define B0(L) as follows: B0(L) = {f ∈ H2(L, Ω) | f(x1, x2) = 0 whenever [x1, x2] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Batten [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='6] established the following Hopf Formula for the Schur multiplier of the Lie algebra L: H2(L, Ω) ∼= F ′ ∩ R [F, R] , where 1 �→ R �→ F �→ L �→ 1 is a free a presentation of L and F ′ is the derived subalgebra of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let K(L) denote the set {[x, y] | x, y ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' In the following theorem we derive a Hopf-type formula for B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra with a free presentation L ∼= F/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then B0(L) ∼= F ′∩R ⟨K(F )∩R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The following corollary follows from [6, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1] and the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then B(L) ∼= B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' As an application we answer a couple of questions of Kunyavski˘ı [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' He asked the following questions for finite dimensional Lie algebras L: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' [4, Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1] Can the dimension of B(L) be as large as possible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS 3 Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' [4, Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2] Is B(L) invariant under isoclinism of Lie algebras?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Two Lie algebras L and K are said to be isoclinic if there exist isomorphisms α : L/Z(L) �→ K/Z(K) and β : L′ �→ K′ such that the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' L Z(L) × L Z(L) φ −−−−→ L′ α×α \uf8e6\uf8e6� β \uf8e6\uf8e6� K Z(K) × K Z(K) θ −−−−→ K′ where φ � l1 + Z(L), l2 + Z(L) � = [l1, l2] for l1, l2 ∈ L and θ � k1 + Z(K), k2 + Z(K) � = [k1, k2] for k1, k2 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The pair (α, β) is called an isoclinism between L and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' In the following theorems we give an affirmative answer to Questions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let n ≥ 1 be a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' There exists a finite dimensional nilpotent Lie algebra L of nilpotency class 2 such that dimension of B(L) is greater than or equal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L and M be two isoclinic finite dimensional Lie Algebras over the field Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then B(L) ∼= B(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Hopf-Type Formula Consider a finite dimensional Lie algebra L over a field Ω, a central ideal H of L, and a trivial L-module A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The restriction map Res : Hom(L, A) → Hom(H, A) is defined as follows: for a homomorphism f : L → A, Res(f) is the restriction of f to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' There is also an inflation map Inf : Hom(L/H, A) → Hom(L, A) defined by sending a homomorphism α ∈ Hom(L/H, A) to the homomorphism α′ ∈ Hom(L, A) defined by α′(x, y) = α(β(x), β(y)) for all x, y ∈ L, where β : L → L/H is the natural group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Another inflation map Inf : H2(L/H, A) → H2(L, A) is defined similarly by sending [α] ∈ H2(L/H, A) to [α′] ∈ H2(L, A) where α′(x, y) = α(β(x), β(y)) for all x, y ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Next, we define a transgression map Tra : Hom(H, A) → H2(L/H, A) as follows: for a fixed section µ of β, define a map f : L/H ×L/H → L by f(x, y) = [µ(x), µ(y)]−µ([x, y]) for all x, y ∈ L/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Given a homomorphism χ ∈ Hom(H, A), we can verify that χf ∈ Z2(L/H, A) and that the cohomology class of χf does not depend on the choice of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The transgression map Tra is then defined as the map that sends a homomorphism χ to the cohomology class of χf We are now ready to quote some results required for our subsequent investiga- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The following 5-term exact sequence was established by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Batten in her Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Thesis [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1] Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a Lie algebra, H be central ideal of L, 1 �→ H �→ L �→ L/H �→ 1 be the natural exact sequence and A be a trivial L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then the 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' RAI induced sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1) 1 −→ Hom(L/H, A) Inf −−→ Hom(L, A) Res −−→ Hom(H, A) Tra −−→ H2(L/H, A) Inf −−→ H2(L, A) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a lie algebra and T be a subset of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' By ⟨T ⟩ we denote the subspace of L generated by T and by HomT (L, A) we denote the set of those homomor- phisms which maps T to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The following lemma is instrumental in our subsequent investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra over the field Ω and H be a central ideal of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then Tra(λ) ∈ B0(L/H) if, and only if λ ∈ HomT (H, Ω) where T = ⟨K(L) ∩ H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let µ : L/H → L be a section such that µ(0) = 0 and let Tra be defined using µ as Tra(λ) = [λf], where f(x, y) = [µ(x), µ(y)] − µ([x, y]) ∀ x, y ∈ L/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let λ ∈ HomT (H, Ω) and x, y ∈ L/H be such that [x, y] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then λf(x, y) = λ([µ(x), µ(y)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' But [µ(x), µ(y)] ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Therefore λf(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' This proves that [λf] ∈ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Thus Tra(λ) ∈ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Conversly, suppose that Tra(λ) ∈ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let l1, l2 ∈ L such that [l1, l2] ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Notice that [l1, l2] = [µ(l1+H), µ(l2+ H)] because H is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Also µ([l1 + H, l2 + H]) = µ([l1, l2] + H) = 0 because [l1, l2] ∈ H and µ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Therefore, λ([l1, l2] = λ([µ(l1 + H), µ(l2 + H)]) − λµ([l1 + H, l2 + H]) = λf(l1 + H, l2 + H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Since Tra(λ) = [λf] ∈ B0(L/H) and [l1 + H, l2 + H] = 0 in L/H we have that λf(l1 + H, l2 + H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Thus λ([l1, l2]) = 0 whenever [l1, l2] ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' This proves that λ ∈ HomT (H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra over the field Ω, H be central ideal of L, and T = ⟨K(L) ∩ H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then the induced sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2) 1 −→ Hom(L/H, Ω) Inf −−→ Hom(L, Ω) Res −−→ HomT (H, Ω) tra −−→ B0(L/H) inff −−→ B0(L) is exact, where tra and inff are the restrictions of Tra and Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The theorem follows from the exactness of the Sequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2 and the following straight forward observations: (i) Res(Hom(L, Ω) ≤ HomT (H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' (ii) Inf(B0(L/H) ≤ B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' □ The next theorem follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2 and [1, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra, L∗ be its cover with A ≤ L∗ satisfying the following three conditions (1) A ≤ Z(L∗) ∩ [L∗, L∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS 5 (2) A ∼= M(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' (3) L ∼= L∗/A, and T = ⟨K(L∗) ∩ A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then the map tra : HomT (A, Ω) �→ B0(L) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a finite dimensional Lie algebra and L∗ be its cover with A ≤ L∗ satisfying the following three conditions (1) A ≤ Z(L∗) ∩ [L∗, L∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' (2) A ∼= M(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' (3) L ∼= L∗/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then B0(L) ∼= A ⟨K(L∗)∩A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let T = ⟨K(L∗)∩A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Since A is an abelian lie algebra, homomorphisms are nothing but linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' It follows that HomT (A, Ω) ∼= Hom � A T , Ω � ∼= A T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' The result follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let H be a central ideal in L and T = ⟨K(L)∩H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then L′∩H ⟨K(L)∩H⟩ is isomorphic to the image of the map tra : HomT (H, Ω) �→ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' In particular, L′∩H ⟨K(L)∩H⟩ ∼= B0(L/H) if the map tra is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='3, we have the following exact sequence Hom(L, Ω) Res −−→ HomT (H, Ω) tra −−→ B0(G/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' It follows that HomT (H,Ω) Res(Hom(L,Ω)) is isomorphic to the image of tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Thus, to prove the theorem, we only need to show that L′ ∩ H ⟨K(L) ∩ H⟩ ∼= HomT (H, Ω) Res(Hom(L, Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Since H is abelian it follows that the natural restriction map Res1 : HomT (H, Ω) → HomT (L′ ∩ Z, Ω) is surjetive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Therefore HomT (H, Ω) ker Res1 ∼= HomT (L′ ∩ H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' If we consider the natural restriction map Res2 : Hom(H, Ω) �→ Hom(L′ ∩ H, Ω), it is straight forward to note that ker Res1 = ker Res2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let J be the subset of Hom(H, Ω) consisting of all χ which can be extended to a homomorphism L → Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Invoking the proof of [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2] we have J = ker Res2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' But it is obvious that J = Res(Hom(L, Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Hence, it follows that HomT (H, Ω) Res(Hom(L, Ω)) ∼= HomT (L′ ∩ H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' But HomT (L′ ∩ H, Ω) ∼= Hom �L′ ∩ H T , Ω � ∼= L′ ∩ H T because L′ ∩ H is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1: Let ¯R = R [F,R] and ¯F = F [F,R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then ¯R is a cen- tral ideal of ¯F and L ∼= ¯ F ¯ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' By [1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='4] the transgression map Tra : Hom( ¯R, Ω) �→ H2(L, Ω) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Therefore by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2 the map tra : Hom⟨K( ¯ F )∩ ¯R⟩( ¯R, Ω) �→ B0(L) is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' It therefore follows, from Theorem 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' RAI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='6, that B0(L) ∼= ¯ R∩[ ¯ F, ¯ F ] ⟨K( ¯ F )∩ ¯R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' But ¯R ∩ [ ¯F , ¯F] = F ′∩R [F,R] and ⟨K( ¯F) ∩ ¯R⟩ = ⟨K(F )∩R⟩ [F,R] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Hence B0(L) ∼= F ′∩R ⟨K(F )∩R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Applications The proof of the following proposition is exactly the same as the proof of [6, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let L be a Lie algebra with a free presentation L ∼= F/R, and M be an ideal of L, such that T = ker(F �→ L/M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then the sequence 0 �→ R ∩ ⟨K(F) ∩ T ⟩ ⟨K(F) ∩ R⟩ �→ B0(L) �→ B0(L/M) �→ M ∩ L′ ⟨K(L) ∩ M⟩ �→ 0, is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' A Lie algebra L is called generalized Heisenberg of rank n if L′ = Z(L) and dim L′ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' A freest generalized Heisenberg Lie algebra is a d-generated (minimally generated by d elements) generalized Heisenberg Lie algebra of rank 1 2d(d−1) for some d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' We shall denote it by Ld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Notice that Ld has the following presentation: ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' , xd, yij | [xi, xj] = yij, 1 ≤ i < j ≤ d, class 2⟩, and dim Ld = 1 2d(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let Ld be the freest generalized Heisenberg Lie algebra of rank 1 2d(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Then B0(Ld) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let f : Ld × Ld �→ Ω be a 2-cocycle such that ¯f ∈ B0(Ld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let B = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' , xd, [xi, xj] | 1 ≤ i < j ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Define a map µ : B �→ Ω as follows: µ(xi) = 0 for 1 ≤ i ≤ d, µ([xi, xj]) = −f(xi, xj) for 1 ≤ i < j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Since B is basis for Ld we can extend this map linearly to Ld and call it σ so that σ is a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Let x, y ∈ Ld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'} +page_content=' Since B is a basis we can write x and y as x = d � i=1 αixi + � 1≤i , +(3) +with (−)B = −1, if B is (a function of) an odd product of γa’s, otherwise (−)B = 1 [19], |ψoc > is +defined in Eq. (33). It is proven in [8] (App.I, Statement 3, 3.a, 3.b) that all the relations of Eq. (2) +remain valid also after the assumption of Eq. (31). +The “basis vectors” describing internal spaces of fermion and boson fields are chosen to be eigenstates +of all the Cartan subalgebra members. There are d +2 commuting operators of the Lorentz algebra in the +even dimensional spaces, Eq. (28), and d−1 +2 +in odd dimensional spaces, Eq. (29). +If Sab, a ̸= b, (or ˜Sab or Sab = Sab + ˜Sab) are members of the Cartan subalgebra group of the +Lorentz algebra in the internal space of fermion and boson fields, then it is not difficult to find the +eigenstate of each of the members just by taking into account relations of Eq. (2: Sab 1 +2(γa + ηaa +ik γb) = +k +2 +1 +2(γa+ ηaa +ik γb) and Sab 1 +2(1+ i +kγaγb) = k +2 +1 +2(1+ i +kγaγb), with k2 = ηaaηbb. The first eigenstate is nilpotent, +( 1 +2(γa + ηaa +ik γb))2 = 0 and the second eigenstate is projector ( 1 +2(1 + i +kγaγb))2 = 1 +2(1 + i +kγaγb). +Let us introduce the graphic notation, following Ref. [9, 18, 19]. +ab +(k): += +1 +2(γa + ηaa +ik γb) , +ab +[k]:= 1 +2(1 + i +kγaγb) , +ab +˜ +(k): += +1 +2(˜γa + ηaa +ik ˜γb) , +ab +˜[k]: 1 +2(1 + i +k ˜γa˜γb) , +( +ab +(k))† += +ab +(−k) , +( +ab +(k))2 = 0 , +( +ab +[k])† = +ab +[k] , +( +ab +[k])2 = +ab +[k] . +(4) +After taking into account Eq. (2) the relations follow +γa +ab +(k) += +ηaa +ab +[−k], +γb +ab +(k)= −ik +ab +[−k], +γa +ab +[k]= +ab +(−k), +γb +ab +[k]= −ikηaa +ab +(−k) , +˜γa +ab +(k) += +−iηaa +ab +[k], +˜γb +ab +(k)= −k +ab +[k], +˜γa +ab +[k]= +i +ab +(k), +˜γb +ab +[k]= −kηaa +ab +(k) , +(5) +More relations can be found in App. A. +2.1 +Properties of Clifford odd and Clifford even “basis vectors” in even +dimensional spaces +In each even dimensional space there are 2 +d +2 −1 members of the Clifford odd “basis vectors” appearing +2 +d +2−1 families, and the same number of 2 +d +2 −1 their Hermitian conjugated partners appearing in 2 +d +2 −1 +families. +4 + +There are two orthogonal groups of the Clifford even “basis vectors”. The members of each group +have their Hermitian conjugated partners within the same group. +Clifford odd “basis vectors” +We find the Clifford odd “basis vectors”, describing the internal space of fermion fields, as products +of odd numbers of nilpotents and the rest of projectors, if each nilpotent and each projector is the +eigenstate of one of the Cartan subalgebra members. +Let us call the Clifford odd ”basis vectors” ˆbm† +f , if this is the mth member of the family f. +Let us choose the first member ˆb1† +1 , if d = 2(2n + 1), as the product of nilpotents only. +d = 2(2n + 1) , +ˆb1† +1 = +03 +(+i) +12 +(+) +56 +(+) · · · +d−1 d +(+) , +ˆb2† +1 = +03 +[−i] +12 +[−] +56 +(+) · · · +d−1 d +(+) , +· · · +ˆb2 +d +2 −1† +1 += +03 +[−i] +12 +[−] +56 +(+) . . . +d−3 d−2 +[−] +d−1 d +[−] , +· · · . +(6) +In the case that d = 4n, n = 1, 2, .., the first member must have one projector. +d = 4n , +ˆb1† +1 = +03 +(+i) +12 +(+) +56 +(+) · · · +d−1 d +[+] , +· · · . +(7) +All the rest members of the same family, 2 +d +2 −1 − 1, follow by the application of all possible Sab on ˆb1† +1 , +while all the rest 2 +d +2 −1 − 1 families follow by the application of all possible ˜Sab on all the members of +the starting family. +The Hermitian conjugated partners (ˆbm† +f )† of the “basis vectors” ˆbm† +f +follow from these 2 +d +2 −1 × 2 +d +2 −1 +“basis vectors” by replacing each nilpotent +ab +(k) with +ab +(−k). +Choosing the vacuum state equal to +|ψoc >= +2 +d +2 −1 +� +f=1 +ˆbm +f ∗Aˆbm† +f +| 1 > , +(8) +for one of the members m, anyone, of the odd irreducible representation f, with | 1 >, which is the +vacuum without any structure — the identity — it follows that ˆbm +f |ψoc >= 0. +Each Clifford odd “basis vector” carries the family quantum number, and so does its Hermitian +conjugated partner. One correspondingly finds that the “basis vectors” and their Hermitian conjugated +partners fulfil the postulates for the second quantized fermion fields. +ˆbm +f ∗A|ψoc > += +0. |ψoc > , +ˆbm† +f +∗A|ψoc > += +|ψm +f > , +{ˆbm +f ,ˆbm′ +f‘ }∗A+|ψoc > += +0. |ψoc > , +{ˆbm† +f ,ˆbm′† +f‘ }∗A+|ψoc > += +0. |ψoc > , +{ˆbm +f ,ˆbm′† +f‘ }∗A+|ψoc > += +δmm′ +ff‘ |ψoc > , +(9) +5 + +where ∗A represents the algebraic multiplication of ˆbm† +f +and ˆbm′ +f′ among themselves and with the vacuum +state |ψoc > of Eq.(8). Eq. (9) follows by taking into account Eq. (2). +These “basis vectors” are not yet the representatives of the creation and annihilation operators: +They must be tensor, ∗T, products of the “basis vectors” and the basis in ordinary momentum or +coordinate space [8] 2. +Clifford even “basis vectors” +We can find the Clifford even “basis vectors” describing the internal space of the boson fields as +products of even numbers of nilpotents and the rest of projectors if each nilpotent and each projector +is the eigenstate of one of the Cartan subalgebra members. +Let us call the Clifford even “basis vectors” iAm† +f , i = I, II. There are namely two groups of Clifford +even basis vectors”. Each group has 2 +d +2 −1 × 2 +d +2 −1 members. +Let us choose the starting Clifford even “basis vector”, i=IA1† +1 , to be the product of projectors +ab +[k], +with k = i for S03, and k = 1 for the rest 2 +d +2−1 − 1 members of the Cartan subalgebra. +I ˆ +A1† +1 = +03 +[+i] +12 +[+] · · · +d−1 d +[+] . +(10) +The starting Clifford even “basis vector” of the second group i=IIA1† +1 can again be the product of pro- +jectors only, but in this case with +03 +[−i] instead of +03 +[+i] and for all the rest 2 +d +2 −1−1 members of the Cartan +subalgebra with k = +1. (This starting member can not be obtained from IA1† +1 by the application of +Sab’s or ˜Sab’s, since these operators always change the eigenvalues of two Cartan subalgebra members.) +II ˆ +A1† +1 = +03 +[−i] +12 +[+] · · · +d−1 d +[+] . +(11) +The rest of the members of each group follow from the starting member by the application of either +Sab’s or ˜Sab’s. +Since S01 transforms +03 +[+i] +12 +[+] into +03 +(−i) +12 +(−1), while ˜S01 transforms +03 +[+i] +12 +[+] into +03 +(+i) +ab +(+), we immedi- +ately see that the Clifford even “basis vector” have the Hermitian conjugated partners within the same +group of 2 +d +2 −1 × 2 +d +2 −1 members. +Clifford even “basis vectors” applying on Clifford odd “basis vectors. +Let us apply IA1† +1 , which is made of projectors +ab +[k] only, with k = i for S03, and k = 1 for the rest +members of the Cartan subalgebra, on ˆb1† +1 , which is the product of nilpotents only, with eigenvalue of +S03 equal k = i and of the rest of Cartan subalgebra members equal to k = 1. +Taking into account Eqs. (34, 35) one sees that this application, IA1† +1 ∗A ˆb1† +1 , leaves ˆb1† +1 unchanged. +When applying IA2† +1 , with the first two projectors transformed into two nilpotents, +03 +(−i) +12 +(−1), and all +the rest remain the same, we see that this application transforms ˆb1† +1 into ˆb2† +1 (= +03 +[−i] +12 +[−1] +56 +(+) +78 +(+) .... (all +the rest remains the same). The application of IA2† +1 on ˆb1† +1 obviously changes the eigenvalues of S03 and +of S12 of ˆb1† +1 for integer values, −i and −1, respectively. +2In even dimensional spaces with d = 4n, one proceeds as we did in d = 2(2n + 1) dimensional case after taking +into account the requirement that the odd number of nilpotents forms the anti-commuting “basis vectors” describing the +internal space of fermions: The starting “basis vector” ˆb1† +1 +must have one projector, while all the rest are nilpotents. +Sab’s then generate all the members of one family, while ˜Sab’s generate all the families. The “basis vectors” and their +Hermitian conjugated partners fulfil on the vacuum state, Eq. (33), the anti-commuting postulates of Eq. (9). +6 + +We conclude: The algebraic application, ∗A, of the Clifford even ”basis vectors” on the Clifford odd +”basis vectors”, describing the internal space of fermion fields, change their eigenvalues of the Cartan +subalgebra members for 0 or for integer values, ±i, or ±1, leading to +I ˆ +Am† +f‘ ∗A ˆbm′† +f +→ +� +ˆbm† +f +, +or zero . +(12) +Clifford even “basis vectors” applying on Clifford even “basis vectors” +It is not difficult to see, by taking into account Eqs. (34, 35), that the algebraic applications of +IAf† +1 ∗A IIAm′† +f‘ += 0 = IIAm′† +f‘ +∗A IAm† +f , for all (m, m′, f, f‘). +The algebraic application, ∗A, of iAm† +f ∗A iAm′† +f‘ +within each of the two groups give in general non zero +contribution, demonstrating the properties of the internal spaces of the gauge fields to the corresponding +fermion fields, the internal space of which are described by the Clifford odd “basis vectors”. +In each of the two groups, there are 2 +d +2 −1 members, which are products of projectors only. They are +self adjoint and have the eigenvalues of all the Cartan subalgebra members equal zero: Sab = Sab + ˜Sab. +All the rest iAm† +f +(there are 2 +d +2 −1 × (2 +d +2 −1 − 1) members) appear in pairs; Hermitian conjugated to +each other. Their mutual algebraic products form one of 2 +d +2 −1 self-adjoint members. +The algebraic multiplication of the Clifford even “basis vectors” on the Clifford even “basis vectors” +lead to +i ˆ +Am† +f +∗A +i ˆ +Am′† +f‘ +→ +� +i ˆ +Am† +f‘ , +or zero . i = (I, II) . +(13) +The reader can find in Ref. [7, 9] the Clifford odd and the Clifford even ”basis vectors” in the case +that the dimension of the space is d = (5 + 1), describing the internal space of fermion and boson fields, +respectively, illustrated by figures. +2.2 +Properties of the Clifford odd and Clifford even ”basis vectors” in odd +dimensional spaces +In this Subsect. 2.2 the Clifford odd and Clifford even “basis vectors” in odd dimensional spaces [12, 9] +are discussed. +While in even dimensional spaces the Clifford odd “basis vectors” fulfil the postulates for the second +quantized fermion fields, Eq. (9), and Clifford even ”basis vectors” have all the properties of the internal +spaces of their corresponding gauge fields, Eqs. (12, 13), the Clifford odd and even ”basis vectors” +have in odd dimensional spaces unusual properties resembling properties of the internal spaces of the +Faddeev-Popov ghosts, as we shall see in what follows. +Looking in d = (2n+1)dimensional cases, n = 1, 2, . . . , for the Clifford odd and Clifford even “basis +vectors” in 2n-dimensional part of space we find half of the “basis vectors” with properties presented +in Eqs. (6, 7, 10). In Eqs. (14, 15) they are presented on the left hand side. +The rest of the “basis vectors” follow applying S0 2n+1 on the obtained half of the Clifford odd and +the Clifford even “basis vectors”. Since S0 2n+1 are Clifford even operators; they do not change oddness +or evenness of the “basis vectors”. +One finds for the Clifford odd “basis vectors” correspondingly the additional 2 +d−1 +2 −1 members, ap- +pearing in 2 +d−1 +2 −1 families and the same number of their Hermitian conjugated partners on the right +7 + +hand side of Eq. (14). +d = +2(2n + 1) + 1 +ˆb1† +1 = +03 +(+i) +12 +(+) +56 +(+) · · · +d−2 d−1 +(+) +, +ˆb1† +2 +d−1 +2 +−1+1 = +03 +[−i] +12 +(+) +56 +(+) · · · +d−2 d−1 +(+) +γd , +ˆb2† +1 = +03 +[−i] +12 +[−] +56 +(+) · · · +d−2 d−1 +(+) +, +ˆb2† +2 +d−1 +2 +−1+1 = +03 +(+i) +12 +[−] +56 +(+) · · · +d−2 d−1 +(+) +γd , +· · · +· · · +ˆb2 +d−1 +2 +−1† +1 += +03 +[−i] +12 +[−] +56 +(+) . . . +d−2 d−1 +[−] +, +ˆb2 +d−1 +2 +−1† +2d−12−1+1 = +03 +(+i) +12 +[−] +56 +(+) . . . +d−2 d−1 +[−] +γd , +· · · +· · · . +(14) +The right handed half of “basis vectors” follows from the left handed “basis vectors” or from their +Hermitian conjugated partners by the application of S0d on the left handed part. The application of +˜S0d on the left handed part of the “basis vectors” generates the whole set of 2d−2 members of the Clifford +odd ”basis vectors” from the right hand side 3. +When applying on the Clifford even “basis vectors” appearing on the left hand side of Eq. (15) +the operators S0 2n+1 the additional two groups of 2 +d−1 +2 −1× 2 +d−1 +2 −1 “basis vectors” follow, presented in +Eq. (15) on the right hand side. +The 2d−2 Clifford odd objects presented on the right hand side of Eq. (14), and for the special +cases of Eqs. (23, 25), although they are the superposition of the Clifford odd products of γa’s, do not +manifest properties of “basis vectors” and their Hermitian conjugated partners, presented on the left +hand side of Eq. (14), and for the special cases of Eqs. ( 23, 25). +The eigenstates appearing on the right hand side of Eq. (14) can be divided into two groups which +are orthogonal to each other, having their Hermitian conjugated partners within the same group or are +self adjoint. Although they are Clifford odd objects they resemble the properties of the Clifford even +partners of the “basis vectors”, appearing on the left hand side of Eq. (15). +Let us see the application of the operators S0d and ˜S0d on the Clifford even “basis vectors” on the +even dimensional part of the d = 2(2n + 1) + 1 space. The Clifford even “basis vectors” must have an +even number of nilpotents, which means that in d = 2(2n + 1), we must have at least one projector. +To obtain all the Clifford even “basis vectors” we must apply on these starting Clifford even “basis +vectors”, presented in Eq. (15) on the left hand side, the operators S0d and ˜S0d. +d = +2(2n + 1) + 1 +IA1† +1 = +03 +(+i) +12 +(+) +56 +(+) · · · +d−2 d−1 +[+] +, +IA1† +2d−12−1+1 = +03 +[−i] +12 +(+) +56 +(+) · · · +d−2 d−1 +[+] +γd , +IA2† +1 = +03 +[−i] +12 +[−] +56 +(+) · · · +d−2 d−1 +[+] +, +IA2† +2d−12−1+1 = +03 +(+i) +12 +[−] +56 +(+) · · · +d−2 d−1 +[+] +γd , +· · · +· · · +IA2 +d−1 +2 +−1† +1 += +03 +[−i] +12 +[−] +56 +[−] . . . +d−2 d−1 +[+] +, +IA2 +d−1 +2 +−1† +2d−12−1+1 = +03 +(+i) +12 +[−] +56 +[−] . . . +d−2 d−1 +[+] +γd , +· · · +· · · . +(15) +The right hand side of Eq. (15), and for the special cases of the Clifford even part of Eqs. ( 23, +25), are the Cliffdord even “basis vectors” as there are their left handed partners. But they resemble +properties of the left handed “basis vectors”; presented in Eq. (14), and for the special cases of the +Clifford odd part of Eqs. ( 23, 25). These Clifford even objects can be arranged into 2 +d−1 +2 −1 members +3The application of S0d and ˜S0d on the left hand side part of the Hermitian conjugated group to the Clifford odd +”basis vectors” generate the same 2d−2 Clifford odd “basis vectors” as the S0 d and ˜S0 d when applying on the left hand +side “basis vectors”. Correspondingly we now have twice 2d−2 Clifford odd eigenstates of the d−1 +2 +Cartan subalgebra +members. +8 + +in 2 +d−1 +2 −1 families of “basis vectors” and into a separate group of their Hermitian conjugated partners. +However, they are the Clifford even “basis vectors”. +Let us point out that the Lorentz transformations in internal spaces of fermion and boson fields +transform the left hand sides of Eq. ((14) and of Eq. ((15) into the corresponding right hand sides and +opposite. +If we apply algebraically the Clifford even “basis vectors” appearing on the right hand side of Eq. (15) +on the Clifford odd “basis vectors” appearing on the right hand side of Eq. (14), we end up with the +Clifford odd “basis vector” appearing on the left hand side of Eq. (14), or on one of their Hermitian +conjugated partners. Or we obtain zero. +If we apply algebraically the Clifford even “basis vectors” appearing on the right hand side of Eq. (15) +on the Clifford odd “basis vectors” appearing on the left hand side of Eq. (14), we end up with the +Clifford odd “basis vectors” appearing on the right hand side of Eq. (14). +In the next section, we discuss concrete cases to make discussions more transparent. +Let us conclude this section with what we have learned: +a. In d = 2n + 1 dimensional spaces, n = 1, 2, . . . , there are two kinds of the Clifford odd “basis +vectors”: +a.i. The “basis vectors” are products of an odd number of nilpotents and the rest of the projectors. +These “basis vectors” appear in 2 +d−1 +2 −1 families, each family has 2 +d−1 +2 −1 members. They anti-commute, +fulfilling together with their Hermitian conjugated partners the postulates for the second quantized +fermion fields. Their Hermitian conjugated partners appear in a separate group. +a.ii. Applying on these Clifford odd “basis vectors” the operators S0d and ˜S0d the additional two times +2 +d−1 +2 −1× 2 +d−1 +2 −1 of the Clifford odd “basis vectors” follow. These Clifford odd “basis vectors” resemble +the properties of the Clifford even “basis vectors” from the case b.i. presented below; They form two +orthogonal groups. The members of each group have their Hermitian conjugated partners within the +same group, or they are self-adjoint. +b. In d = 2n + 1 dimensional spaces, n = 1, 2, . . . , there are two kinds of the Clifford even “basis +vectors”: +b.i. The “basis vectors” are products of even number of nilpotents and the rest of the projectors. These +“basis vectors” appear in two orthogonal groups with 2 +d−1 +2 −1×2 +d−1 +2 −1 members. Each group have their +Hermitian conjugated members within their own group, or they are self-adjoint. They commute, fulfill- +ing the postulates for the second quantized boson fields, the gauge fields of the corresponding fermion +fields of the case a.i.. +b.ii. Applying on these “basis vectors” the operators S0d and ˜S0d the additional two times 2 +d−1 +2 −1× +2 +d−1 +2 −1 Clifford even “basis vectors” follow. These Clifford even “basis vectors” resemble the properties +of the Clifford odd “basis vectors” of the case a.i.; They form two groups with 2 +d−1 +2 −1 members in +each of the 2 +d−1 +2 −1 families. Their Hermitian conjugated partners appear in a separate group. But they +commute. +c.i. When Clifford even “basis vectors” of the kind b.i. algebraically apply on the Clifford odd +“basis vectors” of the kind a.i. they transfer to the Clifford odd “basis vectors” the integer values of +the Cartan subalgebra members (±i, ±1 or 0) or they give zero. +c.ii. When Clifford even basis vectors” of the kind b.ii. algebraically apply on the Clifford odd “basis +vectors” of the kind a.ii. they transfer to the Clifford odd “basis vectors” the integer values of the +Cartan subalgebra members, (±i, ±1 or 0) or they give zero as in the case c.i.. +d.i. While the Clifford odd “basis vectors” in even dimensional spaces have well-defined handedness, +since the operator of handedness is the Clifford even operator, Eq. (26), the eigenvectors of the operator +9 + +of handedness in odd dimensional spaces are the superposition of the “basis vectors” of the kind a.i. +and of the kind a.ii.. +3 +“Basis vectors” in even, d = 2n for n = 1, 2, and odd, d = 2n+1 +for n = 1, 2, dimensional spaces +The internal spaces for fermion and boson fields in even and odd dimensional spaces for simple cases +are discussed: In Subsect. 3.1 for the choices d = (1+1), d = (3+1) and in Subsect. 3.2 for d = (0+1), +d = (2 + 1) and d = (4 + 1). This section is meant as an illustration of Sect. 2. +In Refs. [7, 9, 10, 8, 12, 11] the reader can find the definition of the “basis vectors” as the eigenstates +of the Cartan subalgebra of the Lorentz algebra in internal spaces of fermion and boson fields. “Basis +vectors” are written as superposition of the Clifford odd (for fermions) and the Clifford even (for bosons) +products of γa’s. “Basis vectors” for fermions have either left or right handedness, Γd (the handedness +is defined in Eq. (26)), and appear in families (the family quantum numbers are determined by ˜γa’s, +with ˜Sab = +i +4{˜γa, ˜γb}−). The Clifford odd “basis vectors” have their Hermitian conjugated partners +in a separate group. “Basis vectors” for bosons have no families and have their Hermitian conjugated +partners within the same group, Sect. 2. +The “basis vectors” in odd dimensional spaces differ in properties from the “basis vectors” in even +dimensional spaces, as we have concluded in the previous Sect. 2. +Half of the Clifford odd “basis vectors” have properties as in even dimensional spaces 4. +The +remaining half of the Clifford odd “basis vectors” gain properties of the Clifford even “basis vectors”. +Half of the Clifford even “basis vectors” have properties as in even dimensional spaces. The remaining +half of the Clifford even “basis vectors” gain properties of the Clifford odd “basis vectors”. Since the +operator of handedness is is the Clifford odd object (it is the product of odd number of γa’s), only the +superposition of the Clifford odd and the Clifford even “basis vectors” have a definite handedness 5. +3.1 +“Basis vectors” in even dimensional spaces: d = (1 + 1), (3 + 1) +To illustrate the differences in properties of the internal spaces of fermion and boson fields in even and +odd dimensional spaces, simple cases are discussed. The definition of nilpotents and projectors and the +relations among them can be found in Eq. (4) and App. A. +d = (1 + 1) +There are 4 (2d=2) “eigenvectors” of the Cartan subalgebra members, Eq. (28), S01 and S01 of the +Lorentz algebra Sab and Sab = S01 + ˜S01 (Sab = i +4{γa, γb}− ˜Sab = i +4{˜γa, ˜γb}−), representing one Clifford +odd “basis vector” ˆb1† +1 = +01 +(+i) (m=1), appearing in one family (f=1) and correspondingly one Hermitian +conjugated partner ˆb1 +1 = +01 +(−i) 6 and two Clifford even “basis vector” IA1† +1 = +01 +[+i] and IIA1† +1 = +01 +[−i], both +self-adjoint. +4The same choice of the Cartan subalgebra members is made in the case d = (2n + 1) and in the case of d = 2n. The +Lorentz transformations in the internal space of fermion and boson fields transform in Eqs. (14, 15) the left hand sides +into the right hand sides and opposite. +5Correspondingly the eigenvectors of the Cartan subalgebra members have both handednesses, Γ(2n+1) = ±1. +6It is our choice which one, +01 +(+i) or +01 +(−i), we choose as the “basis vector” ˆb1† +1 , and which one is its Hermitian conjugated +partner. The choice of the “basis vectors” determines the vacuum state |ψoc >, Eq. (8). For ˆb1† +1 = +01 +(+i), the vacuum state +is |ψoc >= +01 +[−i] (due to the requirement that ˆb1† +1 |ψoc > is nonzero, while ˆb1 +1|ψoc > is zero), which is the Clifford even object. +10 + +Correspondingly we have, after using Eqs. (2, 32), two Clifford odd and two Clifford even eigenvectors +of the Cartan subalgebra members +Clifford odd +ˆb1† +1 += +01 +(+i) , +ˆb1 +1 = +01 +(−i) , +Clifford even +IA1† +1 += +01 +[+i] , +IIA1† +1 = +01 +[−i] . +(16) +The two Clifford odd “basis vectors” are Hermitian conjugated to each other. The choice is made that +ˆb1† +1 is the “basis vector”, the second Clifford odd object is its Hermitian conjugated partner. Defining +the handedness as Γ(1+1) = γ0γ1, Eq. (26), it follows, using Eq. (30), that Γ(1+1) ˆb1† +1 = ˆb1† +1 . ˆb1† +1 is the +right handed “basis vector” 7. +Each of the two Clifford even “basis vectors” is self adjoint ((I,IIA1† +1 )† = I,IIA1† +1 ). +Let us notice, taking into account Eqs. (30, 34), that +{ˆb1 +1(≡ +01 +(−i)) ∗A ˆb1† +1 (≡ +01 +(+i))}|ψoc > += +IIA1† +1 (≡ +01 +[−i])|ψoc >= |ψoc > , +{ˆb1† +1 (≡ +01 +(+i)) ∗A ˆb1 +1(≡ +01 +(−i))}|ψoc > += +0 , +IA1† +1 (≡ +01 +[+i]) ∗A ˆb1† +1 (≡ +01 +(+i))|ψoc > += ˆb1† +1 (≡ +01 +(+i))|ψoc > , +IA1† +1 (≡ +01 +[+i])ˆb1 +1(≡ +01 +(−i))|ψoc > += +0 , +IA1† +1 ∗A +IIA1† +1 += +0 = IIA1† +1 ∗A +IA1† +1 . +(17) +The case with d = (3 + 1) is more informative: +d = (3 + 1) +In d = (3 + 1) there are 16 (2d=4) “eigenvectors” of the Cartan subalgebra members (S03, S12) and +(S03, S12) of the Lorentz algebras Sab and Sab , Eq. (28). +Half of them are the Clifford odd “basis vectors”, appearing in two families 2 +4 +2−1, f = (1, 2)), +each with two (2 +4 +2−1, m = (1, 2)), members, ˆbm† +f , and 2 +4 +2 −1× 2 +4 +2 −1 Hermitian conjugated partners ˆbm +f +appearing in a separate group (not reachable by Sab). +There are 2 +4 +2 −1 × 2 +4 +2−1 Clifford even ”basis vectors”, the members of the group IAm† +f , which are +Hermitian conjugated to each other or are self adjoint, all reachable by Sab from any starting ”basis +vector” IA1† +1 . And there is another group of 2 +4 +2 −1 × 2 +4 +2−1 Clifford even ”basis vectors”, they are the +members of IIAm† +f , again either Hermitian conjugated to each other or are self adjoint. All are reachable +from the starting vector IIA1† +1 by the application of Sab. +Choosing the right handed Clifford odd “basis vectors” as +f = 1 +f = 2 +˜S03 = i +2, ˜S12 = −1 +2 +˜S03 = − i +2, ˜S12 = 1 +2 +S03 +S12 +ˆb1† +1 = +03 +(+i) +12 +[+] +ˆb1† +2 = +03 +[+i] +12 +(+) +i +2 +1 +2 +ˆb2† +1 = +03 +[−i] +12 +(−) +ˆb2† +2 = +03 +(−i) +12 +[−] +− i +2 +−1 +2 , +(18) +7We could choose left handed “basis vectors” if choosing ˆb1† +1 = +01 +(−i), but the choice of handedness would remain only +one. +11 + +we find for their Hermitian conjugated partners +S03 = − i +2, S12 = 1 +2 +S03 = i +2, S12 = −1 +2 +˜S03 +˜S12 +ˆb1 +1 = +03 +(−i) +12 +[+] +ˆb1 +2 = +03 +[+i] +12 +(−) +− i +2 +−1 +2 +ˆb2 +1 = +03 +[−i] +12 +(+) +ˆb2 +2 = +03 +(+i) +12 +[−] +i +2 +1 +2 . +(19) +The vacuum state on which the Clifford odd ”basis vectors apply is equal to: |ψoc >= +1 +√ +2( +03 +[−i] +12 +[+] ++ +03 +[+i] +12 +[+]) 8. +Let us recognize that all the Clifford odd ”basis vectors” are orthogonal: +ˆbm† +f +∗A ˆbm′† +f′ += 0. +Let us present 2 +4 +2−1 × 2 +4 +2−1 Clifford even ”basis vectors”, the members of the group IAm† +f , which are +Hermitian conjugated to each other or are self adjoint 9 +S03 +S12 +S03 +S12 +IA1† +1 = +03 +[+i] +12 +[+] +0 +0 +, IA1† +2 = +03 +(+i) +12 +(+) +i +1 +IA2† +1 = +03 +(−i) +12 +(−) +−i +−1 +, IA2† +2 = +03 +[−i] +12 +[−] +0 +0 , +(20) +and 2 +4 +2−1 × 2 +4 +2 −1 Clifford even ”basis vectors”, the members of the group IIAm† +f , m = (1, 2), f = (1, 2), +which are again Hermitian conjugated to each other or are self adjoint +S03 +S12 +S03 +S12 +IIA1† +1 = +03 +[+i] +12 +[−] +0 +0 +, IIA1† +2 = +03 +(+i) +12 +(−) +i +−1 +IIA2† +1 = +03 +(−i) +12 +(+) +−i +1 +, IIA2† +2 = +03 +[−i] +12 +[+] +0 +0 . +(21) +The Clifford even “basis vectors” have no families. The two groups which are not reachable by Sab are +orthogonal. +IAm† +f +∗A +IIAm′† +f‘ += 0, +for any (m, m′, f, f‘) . +(22) +Even dimensional spaces have the properties of the fermion and boson second quantized fields. The +reader can find discussions about d = (5 + 1)- dimensional case in [9, 8] and the references therein. +3.2 +“Basis vectors” in odd dimensional spaces with d = (2 + 1), (4 + 1) +Half of the Clifford odd and Clifford even Clifford objects in 2n + 1-dimensional cases can be found by +treating the Clifford odd “basis vectors” and their Hermitian conjugated partners and the Clifford even +“basis vectors” in 2(2n + 1) (or 4n) dimensional part of space. The properties of these “basis vectors” +are presented in Eqs. (6, 7, 10, 11). +The rest of the “basis vectors” follow by the application of S0d on the “basis vectors” determining +the internal space of fermion and boson fields in 2(2n + 1) (or 4n) dimensional part of space. Since S0d +are the Clifford even operators, they do not change oddness or evenness of the “basis vectors” or their +8The case SO(1, 1) can be viewed as a subspace of the case SO(3, 1), recognizing the “basis vectors” +03 +(+i) +12 +[+] and +03 +(−) +12 +[−] with +03 +(+i) and +03 +(−i), respectively, as carrying two different handedness in d = (1 + 1), but each of them carries a +different “charge” S12. In the whole internal space, all the Clifford odd “basis vectors” have only one handedness. +9Let be repeated that Sab = Sab + ˜Sab [9]. +12 + +Hermitian conjugated partners. But they do change their properties: +i. +In even dimensional subspace, 2(2n + 1) of d = 2(2n + 1) + 1) (or 4n of d = 4n + 1) the +Clifford odd “basis vectors”, ˆbm† +f , have 2 +d−1 +2 −1 members, m, in 2 +d−1 +2 −1 families, f, and their Hermitian +conjugated partners appear in a separate group of 2 +d−1 +2 −1 members in 2 +d−1 +2 −1 families. The Clifford even +“basis vectors” appear in two mutually orthogonal groups, each with 2 +d−1 +2 −1× 2 +d−1 +2 −1 members. +ii. +The second part of “basis vectors” and their Hermitian conjugated partners, obtained from the +first part by the application of S0d with the same number of either the Clifford odd or of the Clifford +even objects as the first part, manifest: +The Clifford odd “basis vectors” appear in two mutually orthogonal groups, each with 2 +d−1 +2 −1× 2 +d−1 +2 −1 +members, self adjoint or with the Hermitian conjugated partners within the same group. The Clifford +even “basis vectors” appear in 2 +d−1 +2 −1 members, m, in 2 +d−1 +2 −1 families, f, and their Hermitian conju- +gated partners appear in a separate group of 2 +d−1 +2 −1 members in 2 +d−1 +2 −1 families. +iii. While ˆbm† +f +have in even dimensional spaces one handedness only (either right or left, depending +on the definition of handedness), in odd dimensional spaces, the operator of handedness is a Clifford odd +object — the product of an odd number of γa’s, Eq. (26), (still commuting with Sab) — transforming +the Clifford odd “basis vectors” into Clifford even “basis vectors” and opposite. Correspondingly are +the eigenvectors of the operator of handedness the superposition of the Clifford odd and the Clifford +even basis vectors”, offering in odd dimensional spaces the right and left handed eigenvectors of the +operator of handedness. +Let us illustrate the above mentioned properties of the “basis vectors” in odd dimensional spaces, +starting with the simplest case: +d=(2+1) +In d = (2 + 1) there are 8 (2d=3) “eigenvectors” of the Cartan subalgebra members (S01) and (S01) +of the Lorentz algebras Sab and Sab , Eq. (29). +Half of the Clifford odd and Clifford even “basis vectors” and their Hermitian conjugated partners +can be taken from Eq. (16), the rest half are obtained by the application of S02 or ˜S02 on the first half. +One obtains +d = +2 + 1 +Clifford odd +ˆb1† +1 = +01 +(+i) , +ˆb1† +2 = +01 +[−i] γ2 , +ˆb1 +1 = +01 +(−i) , +ˆb1 +2 = +01 +[+i] γ2 , +Clifford even +IA1† +1 = +01 +[+i] , +IA1† +2 = +01 +(−i) γ2 , +IIA1† +1 = +01 +[−i] , +IIA1† +2 = +01 +(+i) γ2 . +(23) +One clearly sees that the left hand side of the Clifford odd “basiss vectors” and the right hand side of +the Clifford even “basis vectors”, although the first are the Clifford odd objects and the later Clifford +even objects, have similar properties. +Like: +ˆb1 +1 ∗A ˆb1† +1 = IA1† +2 ∗A +IIA1† +2 = +01 +(−i) +01 +(+i)= +01 +[−i]= |ψoc > . +13 + +And the right hand side of the Clifford odd “basis vectors” contains two self adjoint orthogonal +“basis vectors” as the left hand side of the two Clifford even “basis vectors” does. +Let us find the eigenvectors of the operator of handedness Γ(2+1) = iγ0γ1γ2. Since it is the Clifford +odd object, its eigenvectors are the superposition of Clifford odd and Clifford even “basis vectors”. +Γ(2+1){ +01 +[−i] ±i +01 +[−i] γ2} = ∓{ +01 +[−i] ±i +01 +[−i] γ2} , +Γ(2+1){ +01 +(+i) ±i +01 +(+i) γ2} = ∓{ +01 +(+i) ±i +01 +(+i) γ2} , +Γ(2+1){ +01 +[+i] ±i +01 +[+i] γ2} = ±{ +01 +[+i] ±i +01 +[+i] γ2} , +Γ(2+1){ +01 +(−i) γ2 ± i +01 +(−i)} = ±{ +01 +(−i) γ2 ± i +01 +(−i)} . +(24) +d=(4+1) +In d = (4 + 1) there are 32 (2d=5) “eigenvectors” of the Cartan subalgebra members (S03, S12) and +(S03, S12) of the Lorentz algebras Sab and Sab, Eq. (29). +Half of the Clifford odd and Clifford even “basis vectors” and their Hermitian conjugated partners +can be taken from Eqs. (18, 19, 20, 21), the rest half follows by the application of S05 or ˜S05 on the +first half. +d = +4 + 1 +Clifford odd +ˆb1† +1 = +03 +(+i) +12 +[+] , ˆb1† +2 = +03 +[+i] +12 +(+) , +ˆb1† +3 = +03 +[−i] +12 +[+i] γ5 , ˆb1† +4 = +03 +(−i) +12 +(+) γ5 , +ˆb2† +1 = +03 +[−i] +12 +(−) , ˆb2† +2 = +03 +(−i) +12 +[−] , +ˆb2† +3 = +03 +(+i) +12 +(−) γ5 , ˆb2† +4 = +03 +[+i] +12 +[−] γ5 , +ˆb1 +1 = +03 +(−i) +12 +[+] , ˆb1 +2 = +03 +[+i] +12 +(−) , +ˆb1 +3 = +03 +[+i] +12 +[+] γ5 , ˆb1 +4 = +03 +(−i) +12 +(−) γ5 , +ˆb2 +1 = +03 +[−i] +12 +(+) , ˆb2 +2 = +03 +(+i) +12 +[−] , +ˆb2 +3 = +03 +(+i) +12 +(+) γ5 , ˆb2 +4 = +03 +[−i] +12 +[−] γ5 , +Clifford even +IA1† +1 = +03 +[+i] +12 +[+] , +IA1† +2 = +03 +(+i) +12 +(+) , +IA1 +3 = +03 +(−i) +12 +[+] γ5 , +IA1 +4 = +03 +[−i] +12 +(+) γ5 , +IA2† +1 = +03 +(−i) +12 +(−i) , +IA2† +2 = +03 +[−i] +12 +[−] , +IA2 +3 = +03 +[+i] +12 +(−) γ5 , +IA2 +4 = +03 +(+i) +12 +[−] γ5 , +IIA1† +1 = +03 +[−i] +12 +[+] , +IIA1† +2 = +03 +(−i) +12 +(+) , +IIA1† +3 = +03 +(+i) +12 +[+] γ5 , +IIA1† +4 = +03 +[+i] +12 +(+) γ5 , +IIA2† +1 = +03 +(+i) +12 +(−) , +IIA2† +2 = +03 +[+i] +12 +[−] , +IIA2† +3 = +03 +[−i] +12 +(−) γ5 , +IIA2† +4 = +03 +(−i) +12 +[−] γ5 . +(25) +One notices that the right hand side of the Clifford odd “basis vectors” appear in two mutually +orthogonal groups, each one with either self-adjoint members or with the Hermitian conjugated partners +within the same group. +The members of one group +ˆb1† +3 = +03 +[−i] +12 +[+i] γ5 , ˆb1† +4 = +03 +(−i) +12 +(+) γ5 , ˆb2† +3 = +03 +(+i) +12 +(−) γ5 , ˆb2† +4 = +03 +[+i] +12 +[−] γ5 +have the properties, except the commutativity (they are namely, the Clifford odd objects), as the group +of Clifford even objects +IIA1† +1 = +03 +[−i] +12 +[+] , IIA1† +2 = +03 +(−i) +12 +(+) , IIA2† +1 = +03 +(+i) +12 +(−) , IIA2† +2 = +03 +[+i] +12 +[−] . +14 + +The comparable properties also have the Clifford odd members of the group +ˆb1 +3 = +03 +[+i] +12 +[+] γ5 , ˆb1 +4 = +03 +(−i) +12 +(−) γ5 , ˆb2 +3 = +03 +(+i) +12 +(+) γ5 , ˆb2 +4 = +03 +[−i] +12 +[−] γ5 , +and the Clifford even members of the group +IA1† +1 = +03 +[+i] +12 +[+] , IA1† +2 = +03 +(+i) +12 +(+) , IA2† +1 = +03 +(−i) +12 +(−i) , IA2† +2 = +03 +[−i] +12 +[−] . +The members of both groups have Hermitian conjugated partners within the same group or are self- +adjoint. +On the other side, the members of the Clifford even group +IIA1† +3 = +03 +(+i) +12 +[+] γ5 , IIA1† +4 = +03 +[+i] +12 +(+) γ5 , IIA2† +3 = +03 +[−i] +12 +(−) γ5 , IIA2† +4 = +03 +(−i) +12 +[−] γ5 , +have their Hermitian conjugated partners in a separate group +IA1 +3 = +03 +(−i) +12 +[+] γ5 +IA1 +4 = +03 +[+i] +12 +(−) γ5 , IA2 +3 = +03 +[−i] +12 +(+) γ5 , IA2 +4 = +03 +(+i) +12 +[−] γ5 , +just like the Clifford odd objects on the left hand side +ˆb1† +1 = +03 +(+i) +12 +[+] , ˆb1† +2 = +03 +[+i] +12 +(+) , ˆb2† +1 = +03 +[−i] +12 +(−) , ˆb2† +2 = +03 +(−i) +12 +[−] , +which have their Hermitian conjugated partners in a separate group +ˆb1 +1 = +03 +(−i) +12 +[+] , ˆb1 +2 = +03 +[+i] +12 +(−) , ˆb2 +1 = +03 +[−i] +12 +(+) , ˆb2 +2 = +03 +(+i) +12 +[−] . +The “basis vectors” of the right hand side keep oddness if they are partners of the Clifford odd +“basis vectors” on left hand side, but demonstrate properties of the Clifford even objects on the left +hand side. +The “basis vectors” of the right hand side keep evenness if they are partners of the Clifford even +“basis vectors” on the left hand side, but demonstrate properties of the Clifford odd objects on the left +hand side. +After algebraically application of, for example, IIA1† +3 (= +03 +(+i) +12 +[+] γ5 on ˆb1† +4 = +03 +(−i) +12 +(+) γ5 we are left +with ˆb1† +2 = +03 +[+i] +12 +(+). +The eigenvectors of the operator of handedness in d = (4 + 1), Γ(4+1) = γ0γ1γ2γ3γ5, are the su- +perposition of the Clifford odd and Clifford even “basis vectors”, as for example: Γ(4+1)(ˆb1† +1 [= +03 +(+i) +12 +[+] +] ± IIA1† +3 [= +03 +(+i) +12 +[+] γ5]) = ∓((ˆb1† +1 ± IIA1† +3 ). +We can conclude that neither Clifford odd nor Clifford even “basis vectors”, have in odd dimensional +spaces the properties which they do demonstrate in even dimensional spaces: The properties which +empower the Clifford odd “basis vectors” to describe the internal space of fermion fields and the Clifford +even “basis vectors” to describe the internal space of the corresponding gauge fields: After enlarging the +“basis vectors” in a tensor product, ∗T, with the basis in ordinary space [9], the corresponding creation +and annihilation operators manifest the properties required by the postulates for the second quantized +either fermion or boson fields, respectively. +In odd dimensional spaces, half of the Clifford odd “basis vectors” demonstrate properties of the +Clifford even “basis vectors” and half of the Clifford even “basis vectors” demonstrate properties of the +Clifford odd “basis vectors”. Arbitrary Lorentz transformations transform the left hand sides into the +right sides and vice versa. +These are properties of the internal spaces of the ghost scalar fields, used in the quantum field theory +to make contributions of the Feynman diagrams finite. +15 + +4 +Discussion +This article discusses the properties of the internal spaces of fermion and boson fields in even and odd +dimensional spaces, if the internal spaces are described by the Clifford odd and even “basis vectors”, +which are the superposition of odd or even products of operators γa’s. “Basis vectors” are arranged +into algebraic products of nilpotents and projectors, which are eigenvectors of the Cartan subalgebra +of the Lorentz algebra Sab in the internal space of fermion and bosons fields. +The Clifford odd “basis vectors”, which are products of an odd number of nilpotents and the rest +of projectors, offer in even dimensional spaces the description of the internal space of fermion fields. +Each irreducible representation of the Lorentz algebra is equipped with the family quantum number +determined by the second kind of the Clifford operators ˜γa’s. The Clifford odd “basis vectors” anti- +commute. Their Hermitian conjugated partners appear in a different group. In a tensor product with +the basis in ordinary space, the “basis vectors” and their Hermitian conjugated partners form the +creation and annihilation operators which, applied on the vacuum state or on the Hilbert space ([8] +and the references therein), fulfil the anti-commutation relations postulated for the second quantized +fermion fields, offering therefore the explanation for the postulates. +In d = 2(2n + 1), n ≥ 7, the creation and annihilation operators, applying on the vacuum state, or +the Hilbert space, offer the description of all the properties of the observed quarks and leptons and +antiquarks and antileptons ([8] and the references therein) 10. +The Clifford even “basis vectors”, which are products of an even number of nilpotents and the rest of +projectors offer in even dimensional spaces the description of the internal space of boson fields, the gauge +fields of the corresponding fermion fields, described by the Clifford odd “basis vectors”. The Clifford +even “basis vectors” commute. They do not appear in families and have their Hermitian conjugated +partners in the same group or are self-adjoint. In a tensor product with the basis in ordinary space, the +Clifford even “basis vectors” form the creation and annihilation operators, which fulfil the commutation +relations postulated for the second quantized boson fields. In d = 2(2n + 1), n ≥ 7, these creation and +annihilation operators offer the description of all the properties of the observed gauge fields as well as +of Higgs’s scalar field, explaining also the Yukawa couplings. +This way of describing the internal space of boson fields with the Clifford even “basis vectors”, +although very promising, needs further studies to understand what new it can bring into understanding +of the second quantization of fermion and boson fields. In particular, it must be understood what new, +if anything, does bring the replacement in a simple starting action in d = 2(2n + 1), n ≥ 7, Eq. (1), of +vielbeins, f aα, and the two kinds of the spin connection fields, ωabα (the gauge fields of Sab) and ˜ωabα +(the gauge fields of ˜Sab) in the covariant derivative p0α +p0α = pα − 1 +2Sabωabα − 1 +2 +˜Sab˜ωabα , +with +p0α = pα − +� +mf +I ˆ +Am† +f +ICm +fα − +� +mf +II ˆ +Am† +f +ICm +fα . +The relations among I ˆ +Am† +f +ICm +fα and ωabα, and II ˆ +Am† +f +IICm +fα and ˜ωabα, not discussed directly in this +article [9], need additional study. +Not only that the description of the internal spaces of the fermion and boson fields with the Clifford +odd and Clifford even “basis vectors” in even dimensional spaces offers an explanation for the second +quantized postulates for fermion and boson fields, for all the assumptions of the standard model, and +for several so far observed phenomena, making several predictions, also the description of the internal +spaces of the fermion and boson fields in odd dimensional spaces seems meaningful for an explanation +10Quarks and leptons and antiquarks and antileptons appear in the same irreducible representation +16 + +for the ghosts, postulated by Fadeev and Popov [20]. introduced into gauge quantum field theories to +take care of the consistency of the path integral formulation of the quantum field theory. +Let us repeat what we have learned in this paper, Subsect. 2.2, Subsect. 3.2, about properties of the +Clifford even and the Clifford odd objects in odd dimensional spaces: +Neither Clifford odd nor Clifford even “basis vectors” have in odd dimensional spaces the properties +which they do demonstrate in even dimensional spaces, the properties which empower the Clifford odd +“basis vectors” to describe the internal space of fermion fields and the Clifford even “basis vectors” to +describe the internal space of the corresponding gauge fields. +In odd dimensional spaces, namely, half of the Clifford odd ”basis vectors”, although anticommuting, +demonstrate properties of the Clifford even “basis vectors” in even dimensional spaces and half of the +Clifford even “basis vectors”, although commuting, demonstrate properties of the Clifford odd “basis +vectors” in even dimensional spaces. These “basis vectors” obviously resemble properties of the internal +spaces of the ghost scalar fields, used in the quantum field theory to make contributions of the Feynman +diagrams finite 11. These are properties of the internal spaces of the ghost scalar fields used in the +quantum field theory to make contributions of the Feynman diagrams finite. +Also, properties of the Clifford odd and the Clifford even ”basis vectors” in odd dimensional spaces +need further study. +A +Some useful formulas +This appendix contains helpful relations needed in this paper. For more detailed explanations, and for +proofs, the reader is kindly asked to read [8] and the references therein. +The operator of handedness Γd is for fermions determined as follows. +Γ(d) = +� +a +(√ηaaγa) · +� +(i) +d +2 , +for d even , +(i) +d−1 +2 , +for d odd, +(26) +The Clifford objects γa’s and ˜γa’s fulfil the relations +{γa, γb}+ += +2ηab = {˜γa, ˜γb}+ , +{γa, ˜γb}+ += +0 , +(a, b) = (0, 1, 2, 3, 5, · · · , d) , +(γa)† += +ηaa γa , +(˜γa)† = ηaa ˜γa . +(27) +In the paper the signature ηaa = diag(1, −1, −1, . . . , −1) is used. +The choice of the Cartan subalgebra members is made for d even +S03, S12, S56, · · · , Sd−1 d , +S03, S12, S56, · · · , Sd−1 d , +˜S03, ˜S12, ˜S56, · · · , ˜Sd−1 d , +Sab = Sab + ˜Sab , +(28) +and for d odd +S03, S12, S56, · · · , Sd−2 d−1 , +S03, S12, S56, · · · , Sd−2 d−1 , +˜S03, ˜S12, ˜S56, · · · , ˜Sd−2 d−1 , +Sab = Sab + ˜Sab . +(29) +11Arbitrary Lorentz transformations in odd dimensional spaces transform the left hand sides of Eqs. (14, 15, 23, 25) +into the right sides and vice versa. +17 + +Nilpotents and projectors are defined as follows [1, 18, 19] +ab +(k): += +1 +2(γa + ηaa +ik γb) , +ab +[k]:= 1 +2(1 + i +kγaγb) , +(30) +with k2 = ηaaηbb. +One finds, taking Eq. (2) into account, and assuming +{˜γaB += +(−)B i Bγa} |ψoc > , +(31) +with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise (−)B = 1 [19], |ψoc > is +defined in Eq. (33), the eigenvalues of the Cartan subalgebra operators +Sab +ab +(k)= k +2 +ab +(k) , +˜Sab +ab +(k)= k +2 +ab +(k) , +Sab +ab +[k]= k +2 +ab +[k] , +˜Sab +ab +[k]= −k +2 +ab +[k] . +(32) +The vacuum state for the Clifford odd ”basis vectors”, |ψoc >, is defined as +|ψoc >= +2 +d +2 −1 +� +f=1 +ˆbm +f ∗Aˆbm† +f +| 1 > . +(33) +Taking into account Eq. (2) it follows +γa +ab +(k) += +ηaa +ab +[−k], +γb +ab +(k)= −ik +ab +[−k], +γa ab +[k]= +ab +(−k), +γb ab +[k]= −ikηaa +ab +(−k) , +˜γa +ab +(k) += +−iηaa ab +[k], +˜γb +ab +(k)= −k +ab +[k], +˜γa +ab +[k]= +i +ab +(k), +˜γb +ab +[k]= −kηaa +ab +(k) , +ab +(k) +† += +ηaa +ab +(−k) , +( +ab +(k))2 = 0 , +ab +(k) +ab +(−k)= ηaa ab +[k] , +ab +[k] +† += +ab +[k] , +( +ab +[k])2 = +ab +[k] , +ab +[k] +ab +[−k]= 0 , +ab +(k) +ab +[k] += +0 , +ab +[k] +ab +(k)= +ab +(k) , +ab +(k) +ab +[−k]= +ab +(k) , +ab +[k] +ab +(−k)= 0 , +ab +˜ +(k) +† += +ηaa +ab +˜ +(−k) , +( +ab +˜ +(k))2 = 0 , +ab +˜ +(k) +ab +˜ +(−k)= ηaa +ab +˜ +[k] , +ab +˜ +[k] +† += +ab +˜ +[k] , +( +ab +˜ +[k])2 = +ab +˜[k] , +ab +˜ +[k] +ab +˜ +[−k]= 0 , +ab +˜ +(k) +ab +˜[k] += +0 , +ab +˜ +[k] +ab +˜ +(k)= +ab +˜ +(k) , +ab +˜ +(k) +ab +˜ +[−k]= +ab +˜ +(k) , +ab +˜ +[k] +ab +˜ +(−k)= 0 . +(34) +One can further find +Sac +ab +(k) +cd +(k) += +− i +2ηaaηcc +ab +[−k] +cd +[−k] , +Sac ab +[k] +cd +[k]= i +2 +ab +(−k) +cd +(−k) , +Sac +ab +(k) +cd +[k] += +− i +2ηaa +ab +[−k] +cd +(−k) , +Sac ab +[k] +cd +(k)= i +2ηcc +ab +(−k) +cd +[−k] . +(35) +B +Acknowledgment +The author thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, +Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory by +offering the room and computer facilities and Matjaˇz Breskvar of Beyond Semiconductor for donations, +in particular for the annual workshops entitled ”What comes beyond the standard models”, and N.B. +Nielsen, L. Bonora and M. Blagojevic for fruitful discussions which have just started on this topic and +might hopefully continue. +18 + +References +[1] N. Mankoˇc Borˇstnik, ”Spinor and vector representations in four dimensional Grassmann space”, J. +of Math. Phys. 34 (1993) 3731-3745. +[2] N. Mankoˇc Borˇstnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett. B 292 (1992) +25-29. +[3] N. Mankoˇc Borˇstnik, ”Unification of spin and charges in Grassmann space?”, hep-th 9408002, +IJS.TP.94/22, Mod. Phys. Lett.A (10) No.7 (1995) 587-595. +[4] P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610. +[5] H.A. Bethe, R.W. Jackiw, ”Intermediate quantum mechanics”, New York : W.A. Benjamin, 1968. +[6] S. Weinberg, ”The quantum theory of fields”, Cambridge, Cambridge University Press, 2015. +[7] N. 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Mankoˇc Borˇstnik N S, ”The spin-charge-family theory is explaining the origin of families, of +the Higgs and the Yukawa couplings”, J. of Modern Phys. 4 (2013) 823 [arXiv:1312.1542]. +[17] N.S. Mankoˇc Borˇstnik, ”Clifford odd and even objects in even and odd dimensional spaces”, +Proceedings to the 25rd Workshop ”What comes beyond the standard models”, 6 - 12 of July, 2022, +Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, A. Kleppe, DMFA Zaloˇzniˇstvo, Ljubljana, December 2022, +[arXiv: ]. +[18] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep-th/0111257]. +[19] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, “How to generate families of spinors”, J. of Math. Phys. +44 4817 (2003) [arXiv:hep-th/0303224]. +[20] Faddeev, L. D.; Popov, V. (1967). ”Feynman diagrams for the Yang-Mills field”. Phys. Lett. B. 25 +(1): 29. Bibcode:1967PhLB...25...29F. doi:10.1016/0370-2693(67)90067-6. +20 + diff --git a/GdE3T4oBgHgl3EQfWQoN/content/tmp_files/load_file.txt b/GdE3T4oBgHgl3EQfWQoN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..44a07b3743c2fc634d8eb2ddcc91dbbe6e166061 --- /dev/null +++ b/GdE3T4oBgHgl3EQfWQoN/content/tmp_files/load_file.txt @@ -0,0 +1,738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf,len=737 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='04466v1 [hep-th] 7 Jan 2023 Clifford odd and even objects in even and odd dimensional spaces describing internal spaces of fermion and boson fields Norma Susana Mankoˇc Borˇstnik Department of Physics, University of Ljubljana SI-1000 Ljubljana, Slovenia, norma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='mankoc@fmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='uni-lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='si January 12, 2023 Abstract In a long series of works, it has been demonstrated, that the spin-charge-family theory offers the explanation for all in the standard model assumed properties of the second quantized fermion and boson fields, offering several predictions as well as explanations for several of the observed phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The theory assumes a simple starting action in even dimensional spaces with d ≥ (13 + 1) with massless fermions interacting with gravity only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The internal spaces of fermion and boson fields are described by the Clifford odd and even objects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' This contribution discusses the properties of the fermion and boson fields in odd dimensional spaces, d = (2n + 1), with the internal spaces of fermion and boson fields described again by the Clifford odd and even objects, respectively, pointing out that their properties differ essentially from the properties in even dimensional spaces, resembling the ghost needed when looking for final solutions with Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Keywords: Second quantization of fermion and boson fields with Clifford algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' beyond the standard model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Kaluza-Klein-like theories in higher dimensional spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Clifford algebra in odd dimensional spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ghosts in quantum field theories 1 introduction 30 years ago, I recognized that there are two kinds of Clifford algebra objects, γa’s and ˜γa’s [1, 2, 3], originating in the Grassmann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford and the Grassmann algebras can be used to describe the internal space of fermions in even dimensional spaces: The superposition of odd products of either γa’s or ˜γa’s, anti-commute, fulfilling on the vacuum states the anti-commutation relations [11] of the second quantization postulates for fermion fields [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The superposition of odd products of either γa’s or ˜γa’s, appear in irreducible representations [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Only one kind of fermions has been observed so far, appearing in several families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' If we use one of the two kinds of the Clifford algebra objects, say γa’s, to describe the internal space of fermions, and the second kind of the Clifford algebra objects, ˜γa’s, to describe the family quantum numbers of each of the irreducible representation determined by γa’s, we are left with one kind of fermions [16, 19], Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='3) of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 1 In any even dimensional space there are 2 d 2 −1 of the Clifford odd “basis vectors”, appearing in 2 d 2 −1 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They are the superposition of odd products γa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' All the members of any family are orthogonal to all the members of the same and all the other families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Their Hermitian conjugated partners appear in a separate group, again with 2 d 2 −1 members in 2 d 2−1 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford odd “basis vectors” have in even dimensional spaces only left or only right handedness, depending on the definition (Γ = �d a(√ηaaγa) · (i) d 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In any even dimensional space there are two groups of 2 d 2 −1× 2 d 2 −1 of the Clifford even “basis vectors”, which are the superposition of even products γa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The family quantum number has no meaning for the Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The members of one group are orthogonal to the members of another group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The members of any of the two groups of the Clifford even “basis vectors” have their Hermitian conjugated partners within the same group [12, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The superposition of even products of γa’s (or ˜γa’s), commute, fulfilling the commutation rela- tions [11] of the second quantization postulates for boson fields [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford even “basis vectors” have properties of the gauge fields of the corresponding Clifford odd “basis vectors”, what becomes transparent after the algebraic multiplication, ∗A, of the Clifford even “basis vectors” on the Clifford odd “basis vectors” and opposite, as well as of the Clifford even “basis vectors” among themselves [12, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Algebraic multiplication is distributive and associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The properties of the Clifford odd and the Clifford even “basis vectors” in even dimensional spaces is shortly overviewed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='1, showing that the Clifford odd “basis vectors”, applying on the appropriate vacuum states, manifest the postulates of the second quantized fermion fields, while the Clifford even “basis vectors” manifest the postulates for their gauge fields, the second quantized boson fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The properties of the fermion and boson fields in odd dimensional spaces differ drastically from the properties of the fermion and boson fields in even dimensional spaces: The Clifford odd “basis vectors” do not manifest the properties of the second quantized fermion fields in even dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Although anti-commuting, they instead manifest properties of the Clifford even “basis vectors” in even dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' And the Clifford even “basis vectors” do not manifest the properties of the second quantized boson fields in even dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Although commuting, they instead manifest properties of the Clifford odd “basis vectors” in even dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In addition, since the operator of handedness has in odd dimensional spaces the Clifford odd char- acter (Γ = �d a(√ηaaγa) · (i) d−1 2 ), it transforms the Clifford odd “basis vectors” into the Clifford even “basis vectors” [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The eigenstates of the operator of handedness are in odd dimensional spaces correspondingly the superposition of the Clifford odd and the Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The properties of the Clifford odd and the Clifford even ”basis vectors” in odd dimensional spaces are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In d = (13 + 1) dimensional space the Clifford odd “basis vectors”, if analysed from the point of view of the subgroups of the standard model groups, offer the description of the internal spaces of all the so far observed quarks and leptons and antiquarks and antileptons as assumed by the standard model before the electroweak phase transition, including in addition the right handed neutrinos and left handed antineutrions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Quarks and antiquarks and leptons and antileptons appear as sixty-four (64) members in two times four families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The corresponding Clifford even “basis vectors” offer the description of the internal spaces of the corresponding vector and scalar gauge fields [8, 12, 9, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The spin-charge-family theory, describing the internal spaces of fermion and boson fields by using the Clifford odd and even algebras in d = (13+1)-dimensional space, offers not only the explanation for the postulates of the second quantized fermion and boson fields, and the explanation for all the standard model assumptions, but also for several observed phenomena, making several predictions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The theory is built on the simple starting starting action in which fermion interacts with the gravitational fields 2 only A = � ddx E 1 2 ( ¯ψ γap0aψ) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' + � ddx E (α R + ˜α ˜R) , p0a = f α ap0α + 1 2E {pα, Ef α a}− , p0α = pα − 1 2Sabωabα − 1 2 ˜Sab˜ωabα , R = 1 2 {f α[af βb] (ωabα,β − ωcaα ωc bβ)} + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' , ˜R = 1 2 {f α[af βb] (˜ωabα,β − ˜ωcaα ˜ωc bβ)} + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (1) Here 1 f α[af βb] = f αaf βb − f αbf βa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' I demonstrate in this paper that in odd dimensional spaces the Clifford odd and the Clifford even objects have drastically different properties than in even dimensional spaces, offering the explanation for postulated ghost fields appearing in several theories for taking care of the singular contributions in evaluating Feynman graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2 appropriate definition of the eigenstates of the Cartan subalgebra members are presented for even dimensional spaces, and extended to odd dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='1 the internal spaces described by the Clifford odd and the Clifford even ”basis vectors” for fermion and boson fields in even dimensional spaces are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2 the internal spaces of fermion and boson fields in odd dimensional spaces are pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3, the internal spaces for fermion and boson fields in even and odd dimensional spaces for simple cases are discussed: In Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='1 for the choices d = (1 + 1), d = (3 + 1) and in Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2 for d = (2 + 1) and d = (4 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' [13, 14, 15] from 20 years ago the authors discuss the question of q time and d − q dimen- sions in odd and even dimensional spaces for any q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Using the requirements that the inner product of two fermions is unitary and invariant under Lorentz transformations the authors conclude that odd dimensional spaces are not appropriate due to the existence of fermions of both handedness and cor- respondingly not mass protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The recognition of this paper might further clarify the “effective” choice of Nature for one time and three space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 4, the main idea of this note is overviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='A, some helpful relations of the Clifford algebra can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2 Eigenstates of Cartan subalgebra members of Lorentz alge- bra for Clifford odd and Clifford even “basis vectors” In this section, the properties of the two kinds of Clifford algebra objects, γa’s and ˜γa’s, are shortly repeated following several papers [1, 2, 16, 11, 12, 7, 9, 10], in particular the reference ([8], and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 1f αa are inverted vielbeins to eaα with the properties eaαf αb = δab, eaαf βa = δβ α, E = det(eaα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Latin indices a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='., m, n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='., s, t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. denote a tangent space (a flat index), while Greek indices α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='., µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='.σ, τ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. denote an Einstein index (a curved index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Letters from the beginning of both the alphabets indicate a general index (a, b, c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. and α, β, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. ), from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m, n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. and µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='.), indexes from the bottom of the al- phabets indicate the compactified dimensions (s, t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. and σ, τ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='.).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' We assume the signature ηab = diag{1, −1, −1, · · · , −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3 The two kinds of Clifford algebra objects, γa and ˜γa, each offering 2d superposition of products of either γa or ˜γa, fulfil the relation [1, 18, 19] {γa, γb}+ = 2ηab = {˜γa, ˜γb}+ , {γa, ˜γb}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) , (γa)† = ηaa γa , (˜γa)† = ηaa ˜γa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2) Each of these two kinds of the Clifford algebra objects could be used to describe the internal spaces of fermion and boson fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' We can reduce the two possibilities to only one by deciding to describe the internal spaces of fermion and boson fields with the superposition of the Clifford odd (for fermion fields) and the Clifford even (for boson fields) products of γa’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' while using ˜γa’s to equip the irreducible representations of the Lorentz group in the internal space of fermions with the family quantum numbers by assuming {˜γaB = (−)B i Bγa} |ψoc > ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (3) with (−)B = −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' if B is (a function of) an odd product of γa’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' otherwise (−)B = 1 [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' |ψoc > is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' It is proven in [8] (App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='I, Statement 3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='b) that all the relations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2) remain valid also after the assumption of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” describing internal spaces of fermion and boson fields are chosen to be eigenstates of all the Cartan subalgebra members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' There are d 2 commuting operators of the Lorentz algebra in the even dimensional spaces, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (28), and d−1 2 in odd dimensional spaces, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' If Sab, a ̸= b, (or ˜Sab or Sab = Sab + ˜Sab) are members of the Cartan subalgebra group of the Lorentz algebra in the internal space of fermion and boson fields, then it is not difficult to find the eigenstate of each of the members just by taking into account relations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2: Sab 1 2(γa + ηaa ik γb) = k 2 1 2(γa+ ηaa ik γb) and Sab 1 2(1+ i kγaγb) = k 2 1 2(1+ i kγaγb), with k2 = ηaaηbb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The first eigenstate is nilpotent, ( 1 2(γa + ηaa ik γb))2 = 0 and the second eigenstate is projector ( 1 2(1 + i kγaγb))2 = 1 2(1 + i kγaγb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us introduce the graphic notation, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' [9, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab (k): = 1 2(γa + ηaa ik γb) , ab [k]:= 1 2(1 + i kγaγb) , ab ˜ (k): = 1 2(˜γa + ηaa ik ˜γb) , ab ˜[k]: 1 2(1 + i k ˜γa˜γb) , ( ab (k))† = ab (−k) , ( ab (k))2 = 0 , ( ab [k])† = ab [k] , ( ab [k])2 = ab [k] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (4) After taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2) the relations follow γa ab (k) = ηaa ab [−k], γb ab (k)= −ik ab [−k], γa ab [k]= ab (−k), γb ab [k]= −ikηaa ab (−k) , ˜γa ab (k) = −iηaa ab [k], ˜γb ab (k)= −k ab [k], ˜γa ab [k]= i ab (k), ˜γb ab [k]= −kηaa ab (k) , (5) More relations can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='1 Properties of Clifford odd and Clifford even “basis vectors” in even dimensional spaces In each even dimensional space there are 2 d 2 −1 members of the Clifford odd “basis vectors” appearing 2 d 2−1 families, and the same number of 2 d 2 −1 their Hermitian conjugated partners appearing in 2 d 2 −1 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 4 There are two orthogonal groups of the Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The members of each group have their Hermitian conjugated partners within the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Clifford odd “basis vectors” We find the Clifford odd “basis vectors”, describing the internal space of fermion fields, as products of odd numbers of nilpotents and the rest of projectors, if each nilpotent and each projector is the eigenstate of one of the Cartan subalgebra members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us call the Clifford odd ”basis vectors” ˆbm† f , if this is the mth member of the family f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us choose the first member ˆb1† 1 , if d = 2(2n + 1), as the product of nilpotents only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d = 2(2n + 1) , ˆb1† 1 = 03 (+i) 12 (+) 56 (+) · · · d−1 d (+) , ˆb2† 1 = 03 [−i] 12 [−] 56 (+) · · · d−1 d (+) , · · ˆb2 d 2 −1† 1 = 03 [−i] 12 [−] 56 (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d−3 d−2 [−] d−1 d [−] , · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (6) In the case that d = 4n, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='., the first member must have one projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d = 4n , ˆb1† 1 = 03 (+i) 12 (+) 56 (+) · · · d−1 d [+] , · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (7) All the rest members of the same family, 2 d 2 −1 − 1, follow by the application of all possible Sab on ˆb1† 1 , while all the rest 2 d 2 −1 − 1 families follow by the application of all possible ˜Sab on all the members of the starting family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Hermitian conjugated partners (ˆbm† f )† of the “basis vectors” ˆbm† f follow from these 2 d 2 −1 × 2 d 2 −1 “basis vectors” by replacing each nilpotent ab (k) with ab (−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Choosing the vacuum state equal to |ψoc >= 2 d 2 −1 � f=1 ˆbm f ∗Aˆbm† f | 1 > , (8) for one of the members m, anyone, of the odd irreducible representation f, with | 1 >, which is the vacuum without any structure — the identity — it follows that ˆbm f |ψoc >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Each Clifford odd “basis vector” carries the family quantum number, and so does its Hermitian conjugated partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' One correspondingly finds that the “basis vectors” and their Hermitian conjugated partners fulfil the postulates for the second quantized fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆbm f ∗A|ψoc > = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' |ψoc > , ˆbm† f ∗A|ψoc > = |ψm f > , {ˆbm f ,ˆbm′ f‘ }∗A+|ψoc > = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' |ψoc > , {ˆbm† f ,ˆbm′† f‘ }∗A+|ψoc > = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' |ψoc > , {ˆbm f ,ˆbm′† f‘ }∗A+|ψoc > = δmm′ ff‘ |ψoc > , (9) 5 where ∗A represents the algebraic multiplication of ˆbm† f and ˆbm′ f′ among themselves and with the vacuum state |ψoc > of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (9) follows by taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These “basis vectors” are not yet the representatives of the creation and annihilation operators: They must be tensor, ∗T, products of the “basis vectors” and the basis in ordinary momentum or coordinate space [8] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Clifford even “basis vectors” We can find the Clifford even “basis vectors” describing the internal space of the boson fields as products of even numbers of nilpotents and the rest of projectors if each nilpotent and each projector is the eigenstate of one of the Cartan subalgebra members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us call the Clifford even “basis vectors” iAm† f , i = I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' There are namely two groups of Clifford even basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Each group has 2 d 2 −1 × 2 d 2 −1 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us choose the starting Clifford even “basis vector”, i=IA1† 1 , to be the product of projectors ab [k], with k = i for S03, and k = 1 for the rest 2 d 2−1 − 1 members of the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' I ˆ A1† 1 = 03 [+i] 12 [+] · · · d−1 d [+] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (10) The starting Clifford even “basis vector” of the second group i=IIA1† 1 can again be the product of pro- jectors only, but in this case with 03 [−i] instead of 03 [+i] and for all the rest 2 d 2 −1−1 members of the Cartan subalgebra with k = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (This starting member can not be obtained from IA1† 1 by the application of Sab’s or ˜Sab’s, since these operators always change the eigenvalues of two Cartan subalgebra members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=') II ˆ A1† 1 = 03 [−i] 12 [+] · · · d−1 d [+] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (11) The rest of the members of each group follow from the starting member by the application of either Sab’s or ˜Sab’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Since S01 transforms 03 [+i] 12 [+] into 03 (−i) 12 (−1), while ˜S01 transforms 03 [+i] 12 [+] into 03 (+i) ab (+), we immedi- ately see that the Clifford even “basis vector” have the Hermitian conjugated partners within the same group of 2 d 2 −1 × 2 d 2 −1 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Clifford even “basis vectors” applying on Clifford odd “basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us apply IA1† 1 , which is made of projectors ab [k] only, with k = i for S03, and k = 1 for the rest members of the Cartan subalgebra, on ˆb1† 1 , which is the product of nilpotents only, with eigenvalue of S03 equal k = i and of the rest of Cartan subalgebra members equal to k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (34, 35) one sees that this application, IA1† 1 ∗A ˆb1† 1 , leaves ˆb1† 1 unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' When applying IA2† 1 , with the first two projectors transformed into two nilpotents, 03 (−i) 12 (−1), and all the rest remain the same, we see that this application transforms ˆb1† 1 into ˆb2† 1 (= 03 [−i] 12 [−1] 56 (+) 78 (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. (all the rest remains the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The application of IA2† 1 on ˆb1† 1 obviously changes the eigenvalues of S03 and of S12 of ˆb1† 1 for integer values, −i and −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2In even dimensional spaces with d = 4n, one proceeds as we did in d = 2(2n + 1) dimensional case after taking into account the requirement that the odd number of nilpotents forms the anti-commuting “basis vectors” describing the internal space of fermions: The starting “basis vector” ˆb1† 1 must have one projector, while all the rest are nilpotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Sab’s then generate all the members of one family, while ˜Sab’s generate all the families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” and their Hermitian conjugated partners fulfil on the vacuum state, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (33), the anti-commuting postulates of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 6 We conclude: The algebraic application, ∗A, of the Clifford even ”basis vectors” on the Clifford odd ”basis vectors”, describing the internal space of fermion fields, change their eigenvalues of the Cartan subalgebra members for 0 or for integer values, ±i, or ±1, leading to I ˆ Am† f‘ ∗A ˆbm′† f → � ˆbm† f , or zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (12) Clifford even “basis vectors” applying on Clifford even “basis vectors” It is not difficult to see, by taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (34, 35), that the algebraic applications of IAf† 1 ∗A IIAm′† f‘ = 0 = IIAm′† f‘ ∗A IAm† f , for all (m, m′, f, f‘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The algebraic application, ∗A, of iAm† f ∗A iAm′† f‘ within each of the two groups give in general non zero contribution, demonstrating the properties of the internal spaces of the gauge fields to the corresponding fermion fields, the internal space of which are described by the Clifford odd “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In each of the two groups, there are 2 d 2 −1 members, which are products of projectors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They are self adjoint and have the eigenvalues of all the Cartan subalgebra members equal zero: Sab = Sab + ˜Sab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' All the rest iAm† f (there are 2 d 2 −1 × (2 d 2 −1 − 1) members) appear in pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Hermitian conjugated to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Their mutual algebraic products form one of 2 d 2 −1 self-adjoint members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The algebraic multiplication of the Clifford even “basis vectors” on the Clifford even “basis vectors” lead to i ˆ Am† f ∗A i ˆ Am′† f‘ → � i ˆ Am† f‘ , or zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' i = (I, II) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (13) The reader can find in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' [7, 9] the Clifford odd and the Clifford even ”basis vectors” in the case that the dimension of the space is d = (5 + 1), describing the internal space of fermion and boson fields, respectively, illustrated by figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2 Properties of the Clifford odd and Clifford even ”basis vectors” in odd dimensional spaces In this Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2 the Clifford odd and Clifford even “basis vectors” in odd dimensional spaces [12, 9] are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' While in even dimensional spaces the Clifford odd “basis vectors” fulfil the postulates for the second quantized fermion fields, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (9), and Clifford even ”basis vectors” have all the properties of the internal spaces of their corresponding gauge fields, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (12, 13), the Clifford odd and even ”basis vectors” have in odd dimensional spaces unusual properties resembling properties of the internal spaces of the Faddeev-Popov ghosts, as we shall see in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Looking in d = (2n+1)dimensional cases, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' , for the Clifford odd and Clifford even “basis vectors” in 2n-dimensional part of space we find half of the “basis vectors” with properties presented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (6, 7, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14, 15) they are presented on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The rest of the “basis vectors” follow applying S0 2n+1 on the obtained half of the Clifford odd and the Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Since S0 2n+1 are Clifford even operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' they do not change oddness or evenness of the “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' One finds for the Clifford odd “basis vectors” correspondingly the additional 2 d−1 2 −1 members, ap- pearing in 2 d−1 2 −1 families and the same number of their Hermitian conjugated partners on the right 7 hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d = 2(2n + 1) + 1 ˆb1† 1 = 03 (+i) 12 (+) 56 (+) · · · d−2 d−1 (+) , ˆb1† 2 d−1 2 −1+1 = 03 [−i] 12 (+) 56 (+) · · · d−2 d−1 (+) γd , ˆb2† 1 = 03 [−i] 12 [−] 56 (+) · · · d−2 d−1 (+) , ˆb2† 2 d−1 2 −1+1 = 03 (+i) 12 [−] 56 (+) · · · d−2 d−1 (+) γd , · · · · ˆb2 d−1 2 −1† 1 = 03 [−i] 12 [−] 56 (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d−2 d−1 [−] , ˆb2 d−1 2 −1† 2d−12−1+1 = 03 (+i) 12 [−] 56 (+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d−2 d−1 [−] γd , · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14) The right handed half of “basis vectors” follows from the left handed “basis vectors” or from their Hermitian conjugated partners by the application of S0d on the left handed part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The application of ˜S0d on the left handed part of the “basis vectors” generates the whole set of 2d−2 members of the Clifford odd ”basis vectors” from the right hand side 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' When applying on the Clifford even “basis vectors” appearing on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15) the operators S0 2n+1 the additional two groups of 2 d−1 2 −1× 2 d−1 2 −1 “basis vectors” follow, presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15) on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The 2d−2 Clifford odd objects presented on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14), and for the special cases of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (23, 25), although they are the superposition of the Clifford odd products of γa’s, do not manifest properties of “basis vectors” and their Hermitian conjugated partners, presented on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14), and for the special cases of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( 23, 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The eigenstates appearing on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14) can be divided into two groups which are orthogonal to each other, having their Hermitian conjugated partners within the same group or are self adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Although they are Clifford odd objects they resemble the properties of the Clifford even partners of the “basis vectors”, appearing on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us see the application of the operators S0d and ˜S0d on the Clifford even “basis vectors” on the even dimensional part of the d = 2(2n + 1) + 1 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford even “basis vectors” must have an even number of nilpotents, which means that in d = 2(2n + 1), we must have at least one projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' To obtain all the Clifford even “basis vectors” we must apply on these starting Clifford even “basis vectors”, presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15) on the left hand side, the operators S0d and ˜S0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d = 2(2n + 1) + 1 IA1† 1 = 03 (+i) 12 (+) 56 (+) · · · d−2 d−1 [+] , IA1† 2d−12−1+1 = 03 [−i] 12 (+) 56 (+) · · · d−2 d−1 [+] γd , IA2† 1 = 03 [−i] 12 [−] 56 (+) · · · d−2 d−1 [+] , IA2† 2d−12−1+1 = 03 (+i) 12 [−] 56 (+) · · · d−2 d−1 [+] γd , · · · · IA2 d−1 2 −1† 1 = 03 [−i] 12 [−] 56 [−] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d−2 d−1 [+] , IA2 d−1 2 −1† 2d−12−1+1 = 03 (+i) 12 [−] 56 [−] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d−2 d−1 [+] γd , · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15) The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15), and for the special cases of the Clifford even part of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( 23, 25), are the Cliffdord even “basis vectors” as there are their left handed partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' But they resemble properties of the left handed “basis vectors”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14), and for the special cases of the Clifford odd part of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( 23, 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These Clifford even objects can be arranged into 2 d−1 2 −1 members 3The application of S0d and ˜S0d on the left hand side part of the Hermitian conjugated group to the Clifford odd ”basis vectors” generate the same 2d−2 Clifford odd “basis vectors” as the S0 d and ˜S0 d when applying on the left hand side “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Correspondingly we now have twice 2d−2 Clifford odd eigenstates of the d−1 2 Cartan subalgebra members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 8 in 2 d−1 2 −1 families of “basis vectors” and into a separate group of their Hermitian conjugated partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' However, they are the Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us point out that the Lorentz transformations in internal spaces of fermion and boson fields transform the left hand sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ((14) and of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ((15) into the corresponding right hand sides and opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' If we apply algebraically the Clifford even “basis vectors” appearing on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15) on the Clifford odd “basis vectors” appearing on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14), we end up with the Clifford odd “basis vector” appearing on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14), or on one of their Hermitian conjugated partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Or we obtain zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' If we apply algebraically the Clifford even “basis vectors” appearing on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (15) on the Clifford odd “basis vectors” appearing on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14), we end up with the Clifford odd “basis vectors” appearing on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In the next section, we discuss concrete cases to make discussions more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us conclude this section with what we have learned: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In d = 2n + 1 dimensional spaces, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' , there are two kinds of the Clifford odd “basis vectors”: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” are products of an odd number of nilpotents and the rest of the projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These “basis vectors” appear in 2 d−1 2 −1 families, each family has 2 d−1 2 −1 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They anti-commute, fulfilling together with their Hermitian conjugated partners the postulates for the second quantized fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Their Hermitian conjugated partners appear in a separate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Applying on these Clifford odd “basis vectors” the operators S0d and ˜S0d the additional two times 2 d−1 2 −1× 2 d−1 2 −1 of the Clifford odd “basis vectors” follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These Clifford odd “basis vectors” resemble the properties of the Clifford even “basis vectors” from the case b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' presented below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They form two orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The members of each group have their Hermitian conjugated partners within the same group, or they are self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In d = 2n + 1 dimensional spaces, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' , there are two kinds of the Clifford even “basis vectors”: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” are products of even number of nilpotents and the rest of the projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These “basis vectors” appear in two orthogonal groups with 2 d−1 2 −1×2 d−1 2 −1 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Each group have their Hermitian conjugated members within their own group, or they are self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They commute, fulfill- ing the postulates for the second quantized boson fields, the gauge fields of the corresponding fermion fields of the case a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Applying on these “basis vectors” the operators S0d and ˜S0d the additional two times 2 d−1 2 −1× 2 d−1 2 −1 Clifford even “basis vectors” follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These Clifford even “basis vectors” resemble the properties of the Clifford odd “basis vectors” of the case a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They form two groups with 2 d−1 2 −1 members in each of the 2 d−1 2 −1 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Their Hermitian conjugated partners appear in a separate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' But they commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' When Clifford even “basis vectors” of the kind b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' algebraically apply on the Clifford odd “basis vectors” of the kind a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' they transfer to the Clifford odd “basis vectors” the integer values of the Cartan subalgebra members (±i, ±1 or 0) or they give zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' When Clifford even basis vectors” of the kind b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' algebraically apply on the Clifford odd “basis vectors” of the kind a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' they transfer to the Clifford odd “basis vectors” the integer values of the Cartan subalgebra members, (±i, ±1 or 0) or they give zero as in the case c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' While the Clifford odd “basis vectors” in even dimensional spaces have well-defined handedness, since the operator of handedness is the Clifford even operator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (26), the eigenvectors of the operator 9 of handedness in odd dimensional spaces are the superposition of the “basis vectors” of the kind a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' and of the kind a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='. 3 “Basis vectors” in even, d = 2n for n = 1, 2, and odd, d = 2n+1 for n = 1, 2, dimensional spaces The internal spaces for fermion and boson fields in even and odd dimensional spaces for simple cases are discussed: In Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='1 for the choices d = (1+1), d = (3+1) and in Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2 for d = (0+1), d = (2 + 1) and d = (4 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' This section is meant as an illustration of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' [7, 9, 10, 8, 12, 11] the reader can find the definition of the “basis vectors” as the eigenstates of the Cartan subalgebra of the Lorentz algebra in internal spaces of fermion and boson fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' “Basis vectors” are written as superposition of the Clifford odd (for fermions) and the Clifford even (for bosons) products of γa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' “Basis vectors” for fermions have either left or right handedness, Γd (the handedness is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (26)), and appear in families (the family quantum numbers are determined by ˜γa’s, with ˜Sab = i 4{˜γa, ˜γb}−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford odd “basis vectors” have their Hermitian conjugated partners in a separate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' “Basis vectors” for bosons have no families and have their Hermitian conjugated partners within the same group, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” in odd dimensional spaces differ in properties from the “basis vectors” in even dimensional spaces, as we have concluded in the previous Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Half of the Clifford odd “basis vectors” have properties as in even dimensional spaces 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The remaining half of the Clifford odd “basis vectors” gain properties of the Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Half of the Clifford even “basis vectors” have properties as in even dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The remaining half of the Clifford even “basis vectors” gain properties of the Clifford odd “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Since the operator of handedness is is the Clifford odd object (it is the product of odd number of γa’s), only the superposition of the Clifford odd and the Clifford even “basis vectors” have a definite handedness 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='1 “Basis vectors” in even dimensional spaces: d = (1 + 1), (3 + 1) To illustrate the differences in properties of the internal spaces of fermion and boson fields in even and odd dimensional spaces, simple cases are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The definition of nilpotents and projectors and the relations among them can be found in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (4) and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d = (1 + 1) There are 4 (2d=2) “eigenvectors” of the Cartan subalgebra members, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (28), S01 and S01 of the Lorentz algebra Sab and Sab = S01 + ˜S01 (Sab = i 4{γa, γb}− ˜Sab = i 4{˜γa, ˜γb}−), representing one Clifford odd “basis vector” ˆb1† 1 = 01 (+i) (m=1), appearing in one family (f=1) and correspondingly one Hermitian conjugated partner ˆb1 1 = 01 (−i) 6 and two Clifford even “basis vector” IA1† 1 = 01 [+i] and IIA1† 1 = 01 [−i], both self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 4The same choice of the Cartan subalgebra members is made in the case d = (2n + 1) and in the case of d = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Lorentz transformations in the internal space of fermion and boson fields transform in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14, 15) the left hand sides into the right hand sides and opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 5Correspondingly the eigenvectors of the Cartan subalgebra members have both handednesses, Γ(2n+1) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 6It is our choice which one, 01 (+i) or 01 (−i), we choose as the “basis vector” ˆb1† 1 , and which one is its Hermitian conjugated partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The choice of the “basis vectors” determines the vacuum state |ψoc >, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' For ˆb1† 1 = 01 (+i), the vacuum state is |ψoc >= 01 [−i] (due to the requirement that ˆb1† 1 |ψoc > is nonzero, while ˆb1 1|ψoc > is zero), which is the Clifford even object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 10 Correspondingly we have, after using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2, 32), two Clifford odd and two Clifford even eigenvectors of the Cartan subalgebra members Clifford odd ˆb1† 1 = 01 (+i) , ˆb1 1 = 01 (−i) , Clifford even IA1† 1 = 01 [+i] , IIA1† 1 = 01 [−i] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (16) The two Clifford odd “basis vectors” are Hermitian conjugated to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The choice is made that ˆb1† 1 is the “basis vector”, the second Clifford odd object is its Hermitian conjugated partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Defining the handedness as Γ(1+1) = γ0γ1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (26), it follows, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (30), that Γ(1+1) ˆb1† 1 = ˆb1† 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1† 1 is the right handed “basis vector” 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Each of the two Clifford even “basis vectors” is self adjoint ((I,IIA1† 1 )† = I,IIA1† 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us notice, taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (30, 34), that {ˆb1 1(≡ 01 (−i)) ∗A ˆb1† 1 (≡ 01 (+i))}|ψoc > = IIA1† 1 (≡ 01 [−i])|ψoc >= |ψoc > , {ˆb1† 1 (≡ 01 (+i)) ∗A ˆb1 1(≡ 01 (−i))}|ψoc > = 0 , IA1† 1 (≡ 01 [+i]) ∗A ˆb1† 1 (≡ 01 (+i))|ψoc > = ˆb1† 1 (≡ 01 (+i))|ψoc > , IA1† 1 (≡ 01 [+i])ˆb1 1(≡ 01 (−i))|ψoc > = 0 , IA1† 1 ∗A IIA1† 1 = 0 = IIA1† 1 ∗A IA1† 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (17) The case with d = (3 + 1) is more informative: d = (3 + 1) In d = (3 + 1) there are 16 (2d=4) “eigenvectors” of the Cartan subalgebra members (S03, S12) and (S03, S12) of the Lorentz algebras Sab and Sab , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Half of them are the Clifford odd “basis vectors”, appearing in two families 2 4 2−1, f = (1, 2)), each with two (2 4 2−1, m = (1, 2)), members, ˆbm† f , and 2 4 2 −1× 2 4 2 −1 Hermitian conjugated partners ˆbm f appearing in a separate group (not reachable by Sab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' There are 2 4 2 −1 × 2 4 2−1 Clifford even ”basis vectors”, the members of the group IAm† f , which are Hermitian conjugated to each other or are self adjoint, all reachable by Sab from any starting ”basis vector” IA1† 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' And there is another group of 2 4 2 −1 × 2 4 2−1 Clifford even ”basis vectors”, they are the members of IIAm† f , again either Hermitian conjugated to each other or are self adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' All are reachable from the starting vector IIA1† 1 by the application of Sab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Choosing the right handed Clifford odd “basis vectors” as f = 1 f = 2 ˜S03 = i 2, ˜S12 = −1 2 ˜S03 = − i 2, ˜S12 = 1 2 S03 S12 ˆb1† 1 = 03 (+i) 12 [+] ˆb1† 2 = 03 [+i] 12 (+) i 2 1 2 ˆb2† 1 = 03 [−i] 12 (−) ˆb2† 2 = 03 (−i) 12 [−] − i 2 −1 2 , (18) 7We could choose left handed “basis vectors” if choosing ˆb1† 1 = 01 (−i), but the choice of handedness would remain only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 11 we find for their Hermitian conjugated partners S03 = − i 2, S12 = 1 2 S03 = i 2, S12 = −1 2 ˜S03 ˜S12 ˆb1 1 = 03 (−i) 12 [+] ˆb1 2 = 03 [+i] 12 (−) − i 2 −1 2 ˆb2 1 = 03 [−i] 12 (+) ˆb2 2 = 03 (+i) 12 [−] i 2 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (19) The vacuum state on which the Clifford odd ”basis vectors apply is equal to: |ψoc >= 1 √ 2( 03 [−i] 12 [+] + 03 [+i] 12 [+]) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us recognize that all the Clifford odd ”basis vectors” are orthogonal: ˆbm† f ∗A ˆbm′† f′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us present 2 4 2−1 × 2 4 2−1 Clifford even ”basis vectors”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' the members of the group IAm† f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' which are Hermitian conjugated to each other or are self adjoint 9 S03 S12 S03 S12 IA1† 1 = 03 [+i] 12 [+] 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA1† 2 = 03 (+i) 12 (+) i 1 IA2† 1 = 03 (−i) 12 (−) −i −1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2† 2 = 03 [−i] 12 [−] 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (20) and 2 4 2−1 × 2 4 2 −1 Clifford even ”basis vectors”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' the members of the group IIAm† f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' m = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' f = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' which are again Hermitian conjugated to each other or are self adjoint S03 S12 S03 S12 IIA1† 1 = 03 [+i] 12 [−] 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA1† 2 = 03 (+i) 12 (−) i −1 IIA2† 1 = 03 (−i) 12 (+) −i 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 2 = 03 [−i] 12 [+] 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (21) The Clifford even “basis vectors” have no families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The two groups which are not reachable by Sab are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IAm† f ∗A IIAm′† f‘ = 0, for any (m, m′, f, f‘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (22) Even dimensional spaces have the properties of the fermion and boson second quantized fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The reader can find discussions about d = (5 + 1)- dimensional case in [9, 8] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2 “Basis vectors” in odd dimensional spaces with d = (2 + 1), (4 + 1) Half of the Clifford odd and Clifford even Clifford objects in 2n + 1-dimensional cases can be found by treating the Clifford odd “basis vectors” and their Hermitian conjugated partners and the Clifford even “basis vectors” in 2(2n + 1) (or 4n) dimensional part of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The properties of these “basis vectors” are presented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (6, 7, 10, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The rest of the “basis vectors” follow by the application of S0d on the “basis vectors” determining the internal space of fermion and boson fields in 2(2n + 1) (or 4n) dimensional part of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Since S0d are the Clifford even operators, they do not change oddness or evenness of the “basis vectors” or their 8The case SO(1, 1) can be viewed as a subspace of the case SO(3, 1), recognizing the “basis vectors” 03 (+i) 12 [+] and 03 (−) 12 [−] with 03 (+i) and 03 (−i), respectively, as carrying two different handedness in d = (1 + 1), but each of them carries a different “charge” S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In the whole internal space, all the Clifford odd “basis vectors” have only one handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 9Let be repeated that Sab = Sab + ˜Sab [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 12 Hermitian conjugated partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' But they do change their properties: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In even dimensional subspace, 2(2n + 1) of d = 2(2n + 1) + 1) (or 4n of d = 4n + 1) the Clifford odd “basis vectors”, ˆbm† f , have 2 d−1 2 −1 members, m, in 2 d−1 2 −1 families, f, and their Hermitian conjugated partners appear in a separate group of 2 d−1 2 −1 members in 2 d−1 2 −1 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford even “basis vectors” appear in two mutually orthogonal groups, each with 2 d−1 2 −1× 2 d−1 2 −1 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The second part of “basis vectors” and their Hermitian conjugated partners, obtained from the first part by the application of S0d with the same number of either the Clifford odd or of the Clifford even objects as the first part, manifest: The Clifford odd “basis vectors” appear in two mutually orthogonal groups, each with 2 d−1 2 −1× 2 d−1 2 −1 members, self adjoint or with the Hermitian conjugated partners within the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford even “basis vectors” appear in 2 d−1 2 −1 members, m, in 2 d−1 2 −1 families, f, and their Hermitian conju- gated partners appear in a separate group of 2 d−1 2 −1 members in 2 d−1 2 −1 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' While ˆbm† f have in even dimensional spaces one handedness only (either right or left, depending on the definition of handedness), in odd dimensional spaces, the operator of handedness is a Clifford odd object — the product of an odd number of γa’s, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (26), (still commuting with Sab) — transforming the Clifford odd “basis vectors” into Clifford even “basis vectors” and opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Correspondingly are the eigenvectors of the operator of handedness the superposition of the Clifford odd and the Clifford even basis vectors”, offering in odd dimensional spaces the right and left handed eigenvectors of the operator of handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us illustrate the above mentioned properties of the “basis vectors” in odd dimensional spaces, starting with the simplest case: d=(2+1) In d = (2 + 1) there are 8 (2d=3) “eigenvectors” of the Cartan subalgebra members (S01) and (S01) of the Lorentz algebras Sab and Sab , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Half of the Clifford odd and Clifford even “basis vectors” and their Hermitian conjugated partners can be taken from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (16), the rest half are obtained by the application of S02 or ˜S02 on the first half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' One obtains d = 2 + 1 Clifford odd ˆb1† 1 = 01 (+i) , ˆb1† 2 = 01 [−i] γ2 , ˆb1 1 = 01 (−i) , ˆb1 2 = 01 [+i] γ2 , Clifford even IA1† 1 = 01 [+i] , IA1† 2 = 01 (−i) γ2 , IIA1† 1 = 01 [−i] , IIA1† 2 = 01 (+i) γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (23) One clearly sees that the left hand side of the Clifford odd “basiss vectors” and the right hand side of the Clifford even “basis vectors”, although the first are the Clifford odd objects and the later Clifford even objects, have similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Like: ˆb1 1 ∗A ˆb1† 1 = IA1† 2 ∗A IIA1† 2 = 01 (−i) 01 (+i)= 01 [−i]= |ψoc > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 13 And the right hand side of the Clifford odd “basis vectors” contains two self adjoint orthogonal “basis vectors” as the left hand side of the two Clifford even “basis vectors” does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us find the eigenvectors of the operator of handedness Γ(2+1) = iγ0γ1γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Since it is the Clifford odd object, its eigenvectors are the superposition of Clifford odd and Clifford even “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Γ(2+1){ 01 [−i] ±i 01 [−i] γ2} = ∓{ 01 [−i] ±i 01 [−i] γ2} , Γ(2+1){ 01 (+i) ±i 01 (+i) γ2} = ∓{ 01 (+i) ±i 01 (+i) γ2} , Γ(2+1){ 01 [+i] ±i 01 [+i] γ2} = ±{ 01 [+i] ±i 01 [+i] γ2} , Γ(2+1){ 01 (−i) γ2 ± i 01 (−i)} = ±{ 01 (−i) γ2 ± i 01 (−i)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (24) d=(4+1) In d = (4 + 1) there are 32 (2d=5) “eigenvectors” of the Cartan subalgebra members (S03, S12) and (S03, S12) of the Lorentz algebras Sab and Sab, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Half of the Clifford odd and Clifford even “basis vectors” and their Hermitian conjugated partners can be taken from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (18, 19, 20, 21), the rest half follows by the application of S05 or ˜S05 on the first half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' d = 4 + 1 Clifford odd ˆb1† 1 = 03 (+i) 12 [+] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1† 2 = 03 [+i] 12 (+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1† 3 = 03 [−i] 12 [+i] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1† 4 = 03 (−i) 12 (+) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2† 1 = 03 [−i] 12 (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2† 2 = 03 (−i) 12 [−] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2† 3 = 03 (+i) 12 (−) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2† 4 = 03 [+i] 12 [−] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1 1 = 03 (−i) 12 [+] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1 2 = 03 [+i] 12 (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1 3 = 03 [+i] 12 [+] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1 4 = 03 (−i) 12 (−) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2 1 = 03 [−i] 12 (+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2 2 = 03 (+i) 12 [−] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2 3 = 03 (+i) 12 (+) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2 4 = 03 [−i] 12 [−] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Clifford even IA1† 1 = 03 [+i] 12 [+] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA1† 2 = 03 (+i) 12 (+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA1 3 = 03 (−i) 12 [+] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA1 4 = 03 [−i] 12 (+) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2† 1 = 03 (−i) 12 (−i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2† 2 = 03 [−i] 12 [−] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2 3 = 03 [+i] 12 (−) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2 4 = 03 (+i) 12 [−] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA1† 1 = 03 [−i] 12 [+] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA1† 2 = 03 (−i) 12 (+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA1† 3 = 03 (+i) 12 [+] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA1† 4 = 03 [+i] 12 (+) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 1 = 03 (+i) 12 (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 2 = 03 [+i] 12 [−] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 3 = 03 [−i] 12 (−) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 4 = 03 (−i) 12 [−] γ5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (25) One notices that the right hand side of the Clifford odd “basis vectors” appear in two mutually orthogonal groups, each one with either self-adjoint members or with the Hermitian conjugated partners within the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The members of one group ˆb1† 3 = 03 [−i] 12 [+i] γ5 , ˆb1† 4 = 03 (−i) 12 (+) γ5 , ˆb2† 3 = 03 (+i) 12 (−) γ5 , ˆb2† 4 = 03 [+i] 12 [−] γ5 have the properties, except the commutativity (they are namely, the Clifford odd objects), as the group of Clifford even objects IIA1† 1 = 03 [−i] 12 [+] , IIA1† 2 = 03 (−i) 12 (+) , IIA2† 1 = 03 (+i) 12 (−) , IIA2† 2 = 03 [+i] 12 [−] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 14 The comparable properties also have the Clifford odd members of the group ˆb1 3 = 03 [+i] 12 [+] γ5 , ˆb1 4 = 03 (−i) 12 (−) γ5 , ˆb2 3 = 03 (+i) 12 (+) γ5 , ˆb2 4 = 03 [−i] 12 [−] γ5 , and the Clifford even members of the group IA1† 1 = 03 [+i] 12 [+] , IA1† 2 = 03 (+i) 12 (+) , IA2† 1 = 03 (−i) 12 (−i) , IA2† 2 = 03 [−i] 12 [−] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The members of both groups have Hermitian conjugated partners within the same group or are self- adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' On the other side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' the members of the Clifford even group IIA1† 3 = 03 (+i) 12 [+] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA1† 4 = 03 [+i] 12 (+) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 3 = 03 [−i] 12 (−) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IIA2† 4 = 03 (−i) 12 [−] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' have their Hermitian conjugated partners in a separate group IA1 3 = 03 (−i) 12 [+] γ5 IA1 4 = 03 [+i] 12 (−) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2 3 = 03 [−i] 12 (+) γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' IA2 4 = 03 (+i) 12 [−] γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' just like the Clifford odd objects on the left hand side ˆb1† 1 = 03 (+i) 12 [+] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1† 2 = 03 [+i] 12 (+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2† 1 = 03 [−i] 12 (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2† 2 = 03 (−i) 12 [−] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' which have their Hermitian conjugated partners in a separate group ˆb1 1 = 03 (−i) 12 [+] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb1 2 = 03 [+i] 12 (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2 1 = 03 [−i] 12 (+) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˆb2 2 = 03 (+i) 12 [−] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” of the right hand side keep oddness if they are partners of the Clifford odd “basis vectors” on left hand side, but demonstrate properties of the Clifford even objects on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The “basis vectors” of the right hand side keep evenness if they are partners of the Clifford even “basis vectors” on the left hand side, but demonstrate properties of the Clifford odd objects on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' After algebraically application of, for example, IIA1† 3 (= 03 (+i) 12 [+] γ5 on ˆb1† 4 = 03 (−i) 12 (+) γ5 we are left with ˆb1† 2 = 03 [+i] 12 (+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The eigenvectors of the operator of handedness in d = (4 + 1), Γ(4+1) = γ0γ1γ2γ3γ5, are the su- perposition of the Clifford odd and Clifford even “basis vectors”, as for example: Γ(4+1)(ˆb1† 1 [= 03 (+i) 12 [+] ] ± IIA1† 3 [= 03 (+i) 12 [+] γ5]) = ∓((ˆb1† 1 ± IIA1† 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' We can conclude that neither Clifford odd nor Clifford even “basis vectors”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' have in odd dimensional spaces the properties which they do demonstrate in even dimensional spaces: The properties which empower the Clifford odd “basis vectors” to describe the internal space of fermion fields and the Clifford even “basis vectors” to describe the internal space of the corresponding gauge fields: After enlarging the “basis vectors” in a tensor product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ∗T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' with the basis in ordinary space [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' the corresponding creation and annihilation operators manifest the properties required by the postulates for the second quantized either fermion or boson fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In odd dimensional spaces, half of the Clifford odd “basis vectors” demonstrate properties of the Clifford even “basis vectors” and half of the Clifford even “basis vectors” demonstrate properties of the Clifford odd “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Arbitrary Lorentz transformations transform the left hand sides into the right sides and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These are properties of the internal spaces of the ghost scalar fields, used in the quantum field theory to make contributions of the Feynman diagrams finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 15 4 Discussion This article discusses the properties of the internal spaces of fermion and boson fields in even and odd dimensional spaces, if the internal spaces are described by the Clifford odd and even “basis vectors”, which are the superposition of odd or even products of operators γa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' “Basis vectors” are arranged into algebraic products of nilpotents and projectors, which are eigenvectors of the Cartan subalgebra of the Lorentz algebra Sab in the internal space of fermion and bosons fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford odd “basis vectors”, which are products of an odd number of nilpotents and the rest of projectors, offer in even dimensional spaces the description of the internal space of fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Each irreducible representation of the Lorentz algebra is equipped with the family quantum number determined by the second kind of the Clifford operators ˜γa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford odd “basis vectors” anti- commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Their Hermitian conjugated partners appear in a different group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In a tensor product with the basis in ordinary space, the “basis vectors” and their Hermitian conjugated partners form the creation and annihilation operators which, applied on the vacuum state or on the Hilbert space ([8] and the references therein), fulfil the anti-commutation relations postulated for the second quantized fermion fields, offering therefore the explanation for the postulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In d = 2(2n + 1), n ≥ 7, the creation and annihilation operators, applying on the vacuum state, or the Hilbert space, offer the description of all the properties of the observed quarks and leptons and antiquarks and antileptons ([8] and the references therein) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford even “basis vectors”, which are products of an even number of nilpotents and the rest of projectors offer in even dimensional spaces the description of the internal space of boson fields, the gauge fields of the corresponding fermion fields, described by the Clifford odd “basis vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The Clifford even “basis vectors” commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' They do not appear in families and have their Hermitian conjugated partners in the same group or are self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In a tensor product with the basis in ordinary space, the Clifford even “basis vectors” form the creation and annihilation operators, which fulfil the commutation relations postulated for the second quantized boson fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In d = 2(2n + 1), n ≥ 7, these creation and annihilation operators offer the description of all the properties of the observed gauge fields as well as of Higgs’s scalar field, explaining also the Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' This way of describing the internal space of boson fields with the Clifford even “basis vectors”, although very promising, needs further studies to understand what new it can bring into understanding of the second quantization of fermion and boson fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In particular, it must be understood what new, if anything, does bring the replacement in a simple starting action in d = 2(2n + 1), n ≥ 7, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (1), of vielbeins, f aα, and the two kinds of the spin connection fields, ωabα (the gauge fields of Sab) and ˜ωabα (the gauge fields of ˜Sab) in the covariant derivative p0α p0α = pα − 1 2Sabωabα − 1 2 ˜Sab˜ωabα , with p0α = pα − � mf I ˆ Am† f ICm fα − � mf II ˆ Am† f ICm fα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The relations among I ˆ Am† f ICm fα and ωabα, and II ˆ Am† f IICm fα and ˜ωabα, not discussed directly in this article [9], need additional study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Not only that the description of the internal spaces of the fermion and boson fields with the Clifford odd and Clifford even “basis vectors” in even dimensional spaces offers an explanation for the second quantized postulates for fermion and boson fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' for all the assumptions of the standard model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' and for several so far observed phenomena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' making several predictions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' also the description of the internal spaces of the fermion and boson fields in odd dimensional spaces seems meaningful for an explanation 10Quarks and leptons and antiquarks and antileptons appear in the same irreducible representation 16 for the ghosts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' postulated by Fadeev and Popov [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' introduced into gauge quantum field theories to take care of the consistency of the path integral formulation of the quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Let us repeat what we have learned in this paper, Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2, Subsect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content='2, about properties of the Clifford even and the Clifford odd objects in odd dimensional spaces: Neither Clifford odd nor Clifford even “basis vectors” have in odd dimensional spaces the properties which they do demonstrate in even dimensional spaces, the properties which empower the Clifford odd “basis vectors” to describe the internal space of fermion fields and the Clifford even “basis vectors” to describe the internal space of the corresponding gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' In odd dimensional spaces, namely, half of the Clifford odd ”basis vectors”, although anticommuting, demonstrate properties of the Clifford even “basis vectors” in even dimensional spaces and half of the Clifford even “basis vectors”, although commuting, demonstrate properties of the Clifford odd “basis vectors” in even dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These “basis vectors” obviously resemble properties of the internal spaces of the ghost scalar fields, used in the quantum field theory to make contributions of the Feynman diagrams finite 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' These are properties of the internal spaces of the ghost scalar fields used in the quantum field theory to make contributions of the Feynman diagrams finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Also, properties of the Clifford odd and the Clifford even ”basis vectors” in odd dimensional spaces need further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' A Some useful formulas This appendix contains helpful relations needed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' For more detailed explanations, and for proofs, the reader is kindly asked to read [8] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The operator of handedness Γd is for fermions determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' Γ(d) = � a (√ηaaγa) · � (i) d 2 , for d even , (i) d−1 2 , for d odd, (26) The Clifford objects γa’s and ˜γa’s fulfil the relations {γa, γb}+ = 2ηab = {˜γa, ˜γb}+ , {γa, ˜γb}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) , (γa)† = ηaa γa , (˜γa)† = ηaa ˜γa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (27) In the paper the signature ηaa = diag(1, −1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' , −1) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' The choice of the Cartan subalgebra members is made for d even S03, S12, S56, · · · , Sd−1 d , S03, S12, S56, · · · , Sd−1 d , ˜S03, ˜S12, ˜S56, · · · , ˜Sd−1 d , Sab = Sab + ˜Sab , (28) and for d odd S03, S12, S56, · · · , Sd−2 d−1 , S03, S12, S56, · · · , Sd−2 d−1 , ˜S03, ˜S12, ˜S56, · · · , ˜Sd−2 d−1 , Sab = Sab + ˜Sab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (29) 11Arbitrary Lorentz transformations in odd dimensional spaces transform the left hand sides of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (14, 15, 23, 25) into the right sides and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' 17 Nilpotents and projectors are defined as follows [1, 18, 19] ab (k): = 1 2(γa + ηaa ik γb) , ab [k]:= 1 2(1 + i kγaγb) , (30) with k2 = ηaaηbb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' One finds, taking Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2) into account, and assuming {˜γaB = (−)B i Bγa} |ψoc > , (31) with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise (−)B = 1 [19], |ψoc > is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (33), the eigenvalues of the Cartan subalgebra operators Sab ab (k)= k 2 ab (k) , ˜Sab ab (k)= k 2 ab (k) , Sab ab [k]= k 2 ab [k] , ˜Sab ab [k]= −k 2 ab [k] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (32) The vacuum state for the Clifford odd ”basis vectors”, |ψoc >, is defined as |ψoc >= 2 d 2 −1 � f=1 ˆbm f ∗Aˆbm† f | 1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (33) Taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (2) it follows γa ab (k) = ηaa ab [−k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' γb ab (k)= −ik ab [−k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' γa ab [k]= ab (−k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' γb ab [k]= −ikηaa ab (−k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˜γa ab (k) = −iηaa ab [k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˜γb ab (k)= −k ab [k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˜γa ab [k]= i ab (k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ˜γb ab [k]= −kηaa ab (k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab (k) † = ηaa ab (−k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( ab (k))2 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab (k) ab (−k)= ηaa ab [k] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab [k] † = ab [k] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( ab [k])2 = ab [k] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab [k] ab [−k]= 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab (k) ab [k] = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab [k] ab (k)= ab (k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab (k) ab [−k]= ab (k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab [k] ab (−k)= 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ (k) † = ηaa ab ˜ (−k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( ab ˜ (k))2 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ (k) ab ˜ (−k)= ηaa ab ˜ [k] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ [k] † = ab ˜ [k] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ( ab ˜ [k])2 = ab ˜[k] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ [k] ab ˜ [−k]= 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ (k) ab ˜[k] = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ [k] ab ˜ (k)= ab ˜ (k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ (k) ab ˜ [−k]= ab ˜ (k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' ab ˜ [k] ab ˜ (−k)= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (34) One can further find Sac ab (k) cd (k) = − i 2ηaaηcc ab [−k] cd [−k] , Sac ab [k] cd [k]= i 2 ab (−k) cd (−k) , Sac ab (k) cd [k] = − i 2ηaa ab [−k] cd (−k) , Sac ab [k] cd (k)= i 2ηcc ab (−k) cd [−k] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE3T4oBgHgl3EQfWQoN/content/2301.04466v1.pdf'} +page_content=' (35) B Acknowledgment The author thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory by offering the room and computer facilities and Matjaˇz Breskvar of Beyond Semiconductor for donations, in particular for the annual workshops entitled ”What comes 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+1,971 @@ +Prompt-Based Editing for Text Style Transfer +Guoqing Luo, Yu Tong Han, Lili Mou∗, Mauajama Firdaus +Dept. Computing Science, Alberta Machine Intelligence Institute (Amii), University of Alberta +∗Canada CIFAR AI Chair, Amii +{gluo, yhan22}@ualberta.ca +{doublepower.mou, mauzama.03}@gmail.com +ABSTRACT +Prompting approaches have been recently explored in text style transfer, where a textual prompt is +used to query a pretrained language model to generate style-transferred texts word by word in an +autoregressive manner. However, such a generation process is less controllable and early prediction +errors may affect future word predictions. In this paper, we present a prompt-based editing approach +for text style transfer. Specifically, we prompt a pretrained language model for style classification +and use the classification probability to compute a style score. Then, we perform discrete search +with word-level editing to maximize a comprehensive scoring function for the style-transfer task. +In this way, we transform a prompt-based generation problem into a classification one, which is a +training-free process and more controllable than the autoregressive generation of sentences. In our +experiments, we performed both automatic and human evaluation on three style-transfer benchmark +datasets, and show that our approach largely outperforms the state-of-the-art systems that have 20 +times more parameters. Additional empirical analyses further demonstrate the effectiveness of our +approach. +1 +Introduction +Text style transfer aims to automatically rewrite a sentence by changing it from one style to another (McDonald and +Pustejovsky, 1985), such as transferring the positive-sentiment sentence “He loves eating sandwiches” into a negative +one “He hates eating sandwiches”. During the transfer, the style of the sentence must be changed, whereas the overall +content should be preserved. Text style transfer has wide real-world applications, such as personalized response +generation (Yang et al., 2017; Zheng et al., 2021), text debiasing (Nogueira dos Santos et al., 2018; Ma et al., 2020), +text simplification (Woodsend and Lapata, 2011; Kumar et al., 2020), and stylistic headline generation (Jin et al., 2020; +Zhan et al., 2022). +Early work on text style transfer mainly falls into three categories: 1) Parallel supervision with labelled source–target +sentence pairs in a sequence-to-sequence manner (Zhu et al., 2010; Rao and Tetreault, 2018; Zhang et al., 2020), 2) Non- +parallel supervision with style labels only, such as learning latent representations of style and content separately (Shen +et al., 2017; John et al., 2019; Goyal et al., 2021), and 3) Unsupervised generative methods, such as constructing +non-parallel training data for learning (Lample et al., 2018b; Luo et al., 2019; Krishna et al., 2020). +Very recently, prompting methods have been explored in text style transfer (Reif et al., 2022; Suzgun et al., 2022), as +large-scale pretrained language models (PLMs) enable us to perform various natural language generation tasks in a +zero-shot (Wei et al., 2022a; Sanh et al., 2022) or exemplar-based manner (Brown et al., 2020; Schick and Schütze, +2021a). In this paper, we also follow the prompt-based setting. This does not require any training samples or labels, but +directly performs inference with PLMs; thus, it is more challenging than the above three settings. +Previous work uses a prompt (e.g., a piece of text “Rewrite the text to be positive:”) to query a PLM, which will then +generate a style-transferred sentence in an autoregressive manner (Reif et al., 2022; Suzgun et al., 2022). However, +such autoregressive generation is less controllable, as words are generated one after another by the PLM. It has the +error accumulation problem where early prediction errors of the PLM will affect its future predictions, leading to less +satisfactory performance in general. +To this end, we propose a prompt-based editing approach to unsupervised style transfer. We prompt a PLM for style +classification and use the classification probability to compute a style score. Then, we perform steepest-ascent hill +climbing (SAHC) (Russell and Norvig, 2010) algorithm for discrete search with word-level editing (such as replacement, +insertion, and deletion) to maximize a heuristically defined scoring function for style transfer. In this way, we transform +arXiv:2301.11997v1 [cs.CL] 27 Jan 2023 + +PLM + : +} +is +The sentiment of the text + +{ +Discrete Search + Rewrite the sentence to be more positive +bland +is +taco +beef +the +Input +: +a. Prompt-Based Generation +PLM +Candidate +tasty +is +taco +beef +the +b. Prompt-Based Editing +bland +taco +beef +is +the +Input + the beef is +tasty +taco +Classification: negative/positive +Figure 1: a) Prompt-based generation: previous work (Reif et al., 2022) uses a prompt to query a PLM, which generates +a style-transferred sentence in an autoregressive manner. b) Our prompt-based editing approach involves one-word +classification (e.g., positive or negative in sentiment transfer). +a prompt-based generation problem into a classification problem, which involves only a style-word prediction and is +generally believed to be easier than multiple-word predictions for sentence generation. Our approach is a training-free +process and does not suffer from the error accumulation problem, because it performs word edits scattered throughout +the entire sentence, rather than generating a sentence word by word. Further, we are able to combine the style score +with other scoring functions such as fluency and semantic similarity, so that our generation process is more controllable. +We use Eleuther AI’s GPT-J-6B (an off-the-shelf PLM)1 and conduct both automatic and human evaluations on three +style-transfer benchmark datasets. Results show that our prompt-based editing approach largely outperforms the +state-of-the-art prompting systems that have 20 times more parameters. Further empirical analysis verifies that our +approach can achieve a balance between style transfer strength and content preservation, showing the effectiveness of +our approach. +2 +Related Work +Prompting. Prompting methods use a piece of text to query a PLM to provide desired outputs (Liu et al., 2021). The +simplest prompting method, perhaps, is zero-shot prompting (Wei et al., 2022a; Sanh et al., 2022; Suzgun et al., 2022), +which directly prompts a PLM to perform a natural language processing task (see Figure 1a), but may result in returning +less well-formatted or logical sentences (Reif et al., 2022). Another prompting method is few-shot prompting (Brown +et al., 2020; Schick and Schütze, 2021a,b; Wei et al., 2022b); it requires several task-specific exemplars for the PLMs, +but is able to achieve higher performance than zero-shot prompting, and thus is more widely applied in natural language +processing tasks (Schick and Schütze, 2021a; Brown et al., 2020; Wei et al., 2022b). +Prompting methods were initially applied to natural language classification tasks (Schick and Schütze, 2021a,b; Min +et al., 2022), where PLMs are asked to predict the masked word given a piece of text containing the token “[MASK]”, and +the predicted word is then projected to a label by a pre-defined verbalizer. With the emergence of various PLMs (Devlin +et al., 2019; Radford et al., 2019; Brown et al., 2020; Raffel et al., 2020), prompting methods have recently been widely +applied to natural language generation tasks (Liu et al., 2021), such as text style transfer (Reif et al., 2022; Suzgun et al., +2022), machine translation (Radford et al., 2019; Brown et al., 2020; Raffel et al., 2020), and generative commonsense +reasoning (Wei et al., 2022a,b). +Text style transfer. Traditional approaches to style-transfer generation can be accomplished by supervised methods +with parallel training data (Xu et al., 2012; Zhang et al., 2015; Rao and Tetreault, 2018). However, obtaining parallel +data is labor-intensive and time-consuming, which remains a significant challenge for this task. +To mitigate the need for parallel data, one line of research focuses on non-parallel supervision, where it trains the model +on a non-parallel but style-labelled corpus (Shen et al., 2017; Bao et al., 2019; Goyal et al., 2021). John et al. (2019) +train an autoencoder of disentangled representation of content and style. Goyal et al. (2021) train multiple language +models as discriminators for each of the target styles given the content representation. However, explicit separation of +content and style is not always possible, because style can only be conveyed holistically for some sentences. +1https://github.com/kingoflolz/mesh-transformer-jax +2 + +Another line of research is devoted to unsupervised generative methods, which constructs non-parallel training data for +pretraining the model (Lample et al., 2018b; Li et al., 2018; Krishna et al., 2020; Riley et al., 2021). Luo et al. (2019) +generate non-parallel training data via back-translation (Lample et al., 2018a) and apply policy gradient training to +learn one-step mappings between the corpora of source and target styles. Reid and Zhong (2021) first train an attentive +style classifier to perform synthesis of source-target style pairs, which are then used to train a Levenshtein editor and +perform multi-span edits. However, these unsupervised generative methods require a complicated training process, +which is not efficient. In addition, poor-quality data synthesis would possibly lead to low performance in general. +Recently, researchers have developed several prompt-based approaches that generate style-transferred texts in a zero- +shot (Suzgun et al., 2022) or exemplar-based manner (Reif et al., 2022). Such methods do not require a learning process +or any training labels. Reif et al. (2022) use large PLMs to understand instructions inside a prompt to generate sentences +with different styles. Suzgun et al. (2022) apply mutiple prompts to PLMs and then use a re-ranking mechanism to +choose the candidate sentence with the highest quality. +Our approach follows the prompt-based setting and directly performs style-transfer text generation without any training +procedure. However, unlike other work, we transform the generation task into a classification task and perform discrete +search, which is more controllable than autoregressive sentence generation. +3 +Approach +Given an input sentence x = (x1, · · · , xm), our goal is to generate a sentence y = (y1, · · · , yn) that transfers the style +of x. Figure 1b depicts the framework of our prompt-based editing approach, where we propose to prompt a pretrained +language model (PLM) to predict the style of a candidate sentence. Then, we perform discrete search and iteratively +edit the candidate sentence to maximize a scoring objective that involves the PLM’s classification probability. Finally, +the highest-scored candidate is taken as the style-transferred sentence. +3.1 +Prompt-Based Classifier +In previous work, researchers directly prompt a PLM to obtain style-transferred sentences (Figure 1a) (Reif et al., 2022). +However, this could be especially challenging, as the PLM has to generate the sentence in a zero-shot or exemplar-based +manner; such a process is autoregressive and less controllable. +To address this, we propose to transform prompt-based generation into prompt-based classification. We query a PLM to +obtain a style score, which involves only a one-step prediction and is much simpler than generating the whole sentence. +Given a candidate sentence [y], we intuitively design the prompt as +promptcls(y) ≡ The [t] of the text { [y] } is : +(1) +where [t] is the style-transfer task, i.e., “sentiment” or “formality” in our experiments, and “{” and “}” are text boundary +markers (Reif et al., 2022). Notice that we have not performed prompt engineering, which is beyond the scope of this +paper. Instead, our focus is to develop a prompt-based editing approach for text style transfer. +Based on the above prompt, we perform next-word prediction to obtain a style probability. +Specifically, the +PLM computes the conditional probability of the next word w in the vocabulary given the prompt, denoted by +PPLM(w | promptcls(y)). +We denote si as the representative word of the ith style. This is simply chosen to be the most intuitive style word, +namely, positive and negative for sentiment transfer and formal and informal for formality transfer. In general, the +predicted probabilities of the two styles are PPLM(s1 | promptcls(y)) and PPLM(s2 | promptcls(y)). +To compute the style score, we consider the ratio of the two styles. Suppose a sentence in style s1 is to be transferred to +s2, we design the style score as: +fsty(y) = PPLM(s2 | promptcls(y)) +PPLM(s1 | promptcls(y)) +(2) +Such a ratio measures the candidate’s relative affiliation with different styles.2 It is more robust than the predicted +target-style probability PPLM(s2| promptcls(y)), which could be affected by the data sample per se. +2While our datasets only consider the transfer between two styles, our approach can be extended to multiple styles in a one-vs-one +or one-vs-all manner. +3 + +Algorithm 1 Prompt-Based Editing +1: Input: Original sentence x, iterative steps T +2: y(0) = x +3: for t ∈ {1, . . . , T} do +4: +Enumerate all edit positions and operations +5: +Obtain the highest-scored candidate y∗ by Eqn. (3) +6: +if fsty(y∗) > 1 +▷ PLM believes style transferred +7: +or y∗ = y(t−1) +▷ Local optimum found +8: +then: return y∗ +9: +else: y(t) = y∗ +10: return y(T ) +3.2 +Search Objective +We apply an edit-based search for unsupervised style transfer. This follows the recent development of search-based text +generation (Li et al., 2020; Kumar et al., 2020; Jolly et al., 2022; Liu et al., 2022; Mou, 2022), where local edits (e.g., +word changes) are performed to maximize a heuristically defined objective function. Specifically, our objective function +involves three aspects: +f(y; x) = fsty(y) · fflu(y) · fsem(y, x) +(3) +where the style scorer fsty is designed in §3.1; fflu and fsem are fluency and semantic scorers, mostly adopted from +previous work and explained below. +Language fluency. A language fluency scorer provides an approximation of how grammatically correct a candidate +sentence y is. We follow Li et al. (2020) and use GPT2 (Radford et al., 2019) to obtain the fluency score of the candidate +y by the geometric mean of predicted probabilities: +fflu(y) = +� +� +� t� +i=1 +PGPT2(yi|y 1, +An += +(a)n−1(b)n−1(c)n−1 +(b + 1)n−1(c + 1)n−1(1)n−1 +an. +(4) +2 + +Motivated by the results in connections between various subclasses of analytic univalent +functions, by using hypergeometric functions [3, 4, 5, 6, 7, 14], and Poisson distributions +[2], we obtain the necessary and sufficient conditions on parameters for 3F2(a,b,c +b+1,c+1; z) +hypergeometric series to be in the classes M∗(λ, α) and N ∗(λ, α) and information regard- +ing the image of functions 3F2(a,b,c +b+1,c+1; z) hypergeometric series belonging to Rτ(A, B) by +applying the Hadamard product. +2. Main Results and Proofs +First, we recall the following results to prove our main theorems. +Lemma 5. [11] For some α (1 < α ≤ +4 +3) and λ (0 ≤ λ < 1), and if f ∈ V, then +f ∈ M∗(λ, α) if and only if +∞ +� +n=2 +[n − (1 + nλ − λ)α]an +≤ +α − 1. +(6) +Lemma 7. [11] For some α (1 < α ≤ +4 +3) and λ (0 ≤ λ < 1), and if f ∈ V, then +f ∈ N ∗(λ, α) if and only if +∞ +� +n=2 +n [n − (1 + nλ − λ)α]an +≤ +α − 1. +(8) +The following result is due to Miller and Paris [10] & Shpot and Srivastava [13]. +Theorem 9. For a, b, c > 0, c ̸= b and a < min(1, b + 1, c + 1), +3F2 +� +a,b,c +b+1,c+1; 1 +� += +bc +c − bΓ(1 − a) +� +Γ(b) +Γ(1 − a + b) − +Γ(c) +Γ(1 − a + c) +� +. +(10) +Now, we state the following lemma due to Chandrasekran and Prabhakaran [4] which +is useful to prove our main results. +Lemma 11. [4] Let a, b, c > 0. Then we have the following: +(1) For b, c > a − 1, we have +∞ +� +n=0 +(n + 1)(a)n (b)n (c)n +(b + 1)n (c + 1)n (1)n += +bc Γ(1 − a) +c − b +� (1 − b)Γ(b) +Γ(1 − a + b) − (1 − c)Γ(c) +Γ(1 − a + c) +� +. +(2) For b, c > a − 1, we have +∞ +� +n=0 +(n + 1)2(a)n (b)n (c)n +(b + 1)n (c + 1)n (1)n += +bc Γ(1 − a) +c − b +� (1 − b)2Γ(b) +Γ(1 − a + b) − (1 − c)2Γ(c) +Γ(1 − a + c) +� +. +(3) For b, c > a − 1, we have +∞ +� +n=0 +(n + 1)3(a)n (b)n (c)n +(b + 1)n (c + 1)n (1)n += +bc Γ(1 − a) +c − b +� (1 − b)3Γ(b) +Γ(1 − a + b) − (1 − c)3Γ(c) +Γ(1 − a + c) +� +. +3 + +(4) For a ̸= 1, b ̸= 1, and c ̸= 1 with b, c > max{0, a − 1}, we have +∞ +� +n=0 +(a)n (b)n (c)n +(b + 1)n (c + 1)n (1)(n+1) += +bc +(a − 1)(b − 1)(c − 1) +× +�Γ(2 − a) +c − b +� (c − 1)Γ(b) +Γ(1 − a + b) − (b − 1)Γ(c) +Γ(1 − a + c) +� +− 1 +� +. +Theorem 12. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b+ 1, c + 1}. A sufficient +condition for the function z 3F2 +� +a,b,c +b+1,c+1; z +� +to belong to the class M∗(λ, α), 1 < α ≤ 4 +3 +and 0 ≤ λ < 1 is that +((1 − α) − b(1 − αλ)) Γ(b) +Γ(1 − |a| + b) +≤ ((1 − α) − c(1 − αλ))) Γ(c) +Γ(1 − |a| + c) +(13) +Proof. Let f(z) = z 3F2 +� +a,b,c +b+1,c+1; z +� +, then, by Lemma 5, it is enough to show that +T1(α, λ) += +∞ +� +n=2 +[n − (1 + nλ − λ)α] |An| ≤ α − 1 +Using the fact |(a)n| ≤ (|a|)n, one can get +T1(α, λ) += +∞ +� +n=2 +[n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(1 − αλ) +∞ +� +n=2 +n +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +−α (1 − λ) +∞ +� +n=2 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(1 − αλ) +∞ +� +n=0 +�(n + 1) (|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� +− (1 − αλ) +−α (1 − λ) +∞ +� +n=0 +� +(|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� ++ α (1 − λ) +Using the result (1) of Lemma 11 and the formula (10) in above mentioned equation, we +derived that += +(1 − αλ) bc Γ(1 − |a|) +c − b +� (1 − b)Γ(b) +Γ(1 − |a| + b) − +(1 − c)Γ(c) +Γ(1 − |a| + c) +� +−α (1 − λ) bcΓ(1 − |a|) +c − b +� +Γ(b) +Γ(1 − |a| + b) − +Γ(c) +Γ(1 − |a| + c) +� ++ α − 1 += +bc Γ(1 − |a|) +c − b +�(1 − b)(1 − αλ) Γ(b) +Γ(1 − |a| + b) +− (1 − c)(1 − αλ) Γ(c) +Γ(1 − |a| + c) +−α (1 − λ) Γ(b) +Γ(1 − |a| + b) + α (1 − λ) Γ(c) +Γ(1 − |a| + c) +� ++ α − 1 +4 + += +bc Γ(1 − |a|) +c − b +�(1 − α) − b(1 − αλ)) Γ(b) +Γ(1 − |a| + b) +− ((1 − α) − c(1 − αλ)) Γ(c) +Γ(1 − |a| + c) +� ++ α − 1 +The above expression is bounded above by α − 1 if and only if the equation (13) holds, +which completes proof. +□ +Theorem 14. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b+ 1, c + 1}. A sufficient +condition for the function z 3F2 +� +a,b,c +b+1,c+1; z +� +to belong to the class N ∗(λ, α), 1 < α ≤ 4 +3 +and 0 ≤ λ < 1 is that +(b − 1)(b(1 − αλ) − (1 − α))Γ(b) +Γ(1 − |a| + b) +≤ +(c − 1) (c(1 − αλ) − (1 − α))Γ(c) +Γ(1 − |a| + c) +(15) +Proof. Let f(z) = z 3F2 +� +a,b,c +b+1,c+1; z +� +, then, by the Lemma 7, it is enough to show that +T2(α, λ) += +∞ +� +n=2 +n [n − (1 + nλ − λ)α] |An| ≤ α − 1 +Using the fact |(a)n| ≤ (|a|)n, one can get +T2(α, λ) += +∞ +� +n=2 +n [n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +∞ +� +n=2 +[n2 (1 − αλ) − α(1 − λ) n] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +Replace n = (n − 1) + 1 and n2 = (n − 1)(n − 2) + 3(n − 1) + 1 in above, we find that +T2(α, λ) += +∞ +� +n=2 +[((n − 1)(n − 2) + 3(n − 1) + 1)] +�(1 − αλ) (|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +− +∞ +� +n=2 +[α(1 − λ) ((n − 1) + 1)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(1 − αλ) +∞ +� +n=2 +�(n − 1)(n − 2) (|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� ++(3 − 2α λ − α) +∞ +� +n=2 +�(n − 1) (|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� ++(1 − α) +∞ +� +n=2 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(1 − αλ) +∞ +� +n=3 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−3 +� +5 + ++(3 − 2α λ − α) +∞ +� +n=2 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−2 +� ++(1 − α) +∞ +� +n=2 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(1 − αλ) +∞ +� +n=0 +� +(|a|)n+2 (b)n+2 (c)n+2 +(b + 1)n+2 (c + 1)n+2 (1)n +� ++(3 − 2α λ − α) +∞ +� +n=0 +� +(|a|)n+1 (b)n+1 (c)n+1 +(b + 1)n+1 (c + 1)n+1 (1)n +� ++(1 − α) +∞ +� +n=1 +� +(|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� += +(1 − αλ) +� |a|(|a| + 1)b(b + 1)c(c + 1) +(b + 1)(b + 2)(c + 1)(c + 2) +� ∞ +� +n=0 +�(|a| + 2)n (b + 2)n (c + 2)n +(b + 3)n (c + 3)n (1)n +� ++(3 − 2α λ − α) +� +abc +(b + 1)(c + 1) +� +∞ +� +n=0 +� +(|a|)n+1 (b)n+1 (c)n+1 +(b + 1)n+1 (c + 1)n+1 (1)n +� ++(1 − α) +∞ +� +n=0 +� +(|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� +− (1 − α) +Using the formula (10) in above mentioned equation, we find that += +(1 − αλ) +� |a|(|a| + 1)b(b + 1)c(c + 1) +(b + 1)(b + 2)(c + 1)(c + 2) +� �(b + 2)(c + 2)Γ(1 − (a + 2)) +(c + 2) − (b + 2) +� +× +� +Γ(b + 2) +1 − (|a| + 2) + (b + 2) − +Γ(c + 2) +1 − (|a| + 2) + (c + 2) +� ++(3 − 2α λ − α) +� +|a|bc +(b + 1)(c + 1) +� �(b + 1)(c + 1)Γ(1 − (|a| + 1)) +(c + 1) − (b + 1) +� +× +� +Γ(b + 1) +1 − (|a| + 1) + (b + 1) − +Γ(c + 1) +1 − (|a| + 1) + (c + 1) +� ++(1 − α) bcΓ(1 − |a|) +c − b +� +Γ(b) +Γ(1 − |a| + b) − +Γ(c) +Γ(1 − |a| + c) +� +− (1 − α) += +(1 − αλ) +�bc (−|a|)(−(|a| + 1)) Γ(1 − (|a| + 2)) +c − b +� �(b + 1) b Γ(b) +1 − |a| + b +− (c + 1) c Γ(c) +1 − |a| + c +� +−(3 − 2α λ − α) +�bc(−|a|)Γ(1 − (|a| + 1)) +c − b +� � +b Γ(b) +1 − |a| + b − +c Γ(c) +1 − |a| + c +� ++(1 − α) bcΓ(1 − |a|) +c − b +� +Γ(b) +Γ(1 − |a| + b) − +Γ(c) +Γ(1 − |a| + c) +� +− (1 − α) +6 + +Using Γ(1 − a) = −aΓ(−a), the aforesaid equation reduces to += +�bc Γ(1 − |a|) +c − b +� +× +�(b − 1)(b(1 − αλ) − (1 − α))Γ(b) +Γ(1 − |a| + b) +− (c − 1) (c(1 − αλ) − (1 − α))Γ(c) +Γ(1 − |a| + c) +� ++ α − 1 +The above expression is bounded above by α − 1 if and only if the equation (15) holds, +which completes proof. +□ +Lemma 16. [8] If f ∈ Rτ(A, B) is of the form (1), then +|an| +≤ +(A − B)|τ| +n , n ∈ N ∖ {1}. +(17) +The result is sharp. +Using the Lemma 16, we prove the following results: +Theorem 18. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b + 1, c + 1} and +f ∈ Rτ(A, B) ∩ V. Then Ia,b,c +b+1,c+1(f)(z) ∈ N ∗(α, λ) if +�bc Γ(1 − |a|) +c − b +�(1 − α) − b(1 − αλ)) Γ(b) +Γ(1 − |a| + b) +− ((1 − α) − c(1 − αλ)) Γ(c) +Γ(1 − |a| + c) +�� +× +� +(A − B) |τ| +(1 − (A − B) |τ|) +� +≤ +α − 1. +(19) +Proof. Let f be of the form (1) belong to the class Rτ(A, B) ∩ V. Because of Lemma 7, +it is enough to show that +∞ +� +n=2 +n [n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +|an| ≤ α − 1 +since f ∈ Rτ(A, B) ∩ V, then by Lemma 16, we have +|an| ≤ (A − B)|τ| +n , n ∈ N ∖ {1}. +Letting +T3(α, λ) += +∞ +� +n=2 +n [n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +|an| +we derived that +T3(α, λ) += +(A − B) |τ| +∞ +� +n=2 +[n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(A − B) |τ| +� +(1 − αλ) +∞ +� +n=2 +n +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +−α (1 − λ) +∞ +� +n=2 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� � +7 + += +(A − B) |τ| +� +(1 − αλ) +∞ +� +n=0 +�(n + 1) (|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� +− (1 − αλ) +−α (1 − λ) +∞ +� +n=0 +� +(|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� ++ α (1 − λ) +� +Using the result (1) of Lemma 11 and the formula (10) in above mentioned equation, we +derived that += +(A − B) |τ| +� +(1 − αλ) bc Γ(1 − |a|) +c − b +� (1 − b)Γ(b) +Γ(1 − |a| + b) − +(1 − c)Γ(c) +Γ(1 − |a| + c) +� +−α (1 − λ) bcΓ(1 − |a|) +c − b +� +Γ(b) +Γ(1 − |a| + b) − +Γ(c) +Γ(1 − |a| + c) +� ++ α − 1 +� += +(A − B) |τ| +�bc Γ(1 − |a|) +c − b +�(1 − b)(1 − αλ) Γ(b) +Γ(1 − |a| + b) +− (1 − c)(1 − αλ) Γ(c) +Γ(1 − |a| + c) +−α (1 − λ) Γ(b) +Γ(1 − |a| + b) + α (1 − λ) Γ(c) +Γ(1 − |a| + c) +� ++ α − 1 +� += +(A − B) |τ| +�bc Γ(1 − |a|) +c − b +�(1 − α) − b(1 − αλ)) Γ(b) +Γ(1 − |a| + b) +− ((1 − α) − c(1 − αλ)) Γ(c) +Γ(1 − |a| + c) +� ++α − 1 +� +The above expression is bounded above by α − 1 if and only if the equation (19) holds, +which completes proof. +□ +Theorem 20. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b + 1, c + 1} and +f ∈ Rτ(A, B) ∩ V. Then Ia,b,c +b+1,c+1(f)(z) ∈ M∗(α, λ) if +�(1 − αλ) bc Γ(1 − |a|) +c − b +� +Γ(b) +Γ(1 − |a| + b) − +Γ(c) +Γ(1 − |a| + c) +� +− +� +α (1 − λ) bc +(|a| − 1)(b − 1)(c − 1) +� �Γ(2 − |a|) +c − b +� (c − 1)Γ(b) +Γ(1 − |a| + b) − +(b − 1)Γ(c) +Γ(1 − |a| + c) +� +− 1 +� � +× +� +(A − B) |τ| +(1 − (A − B) |τ|) +� +≤ α − 1. +(21) +Proof. Let f be of the form (1) belong to the class Rτ(A, B) ∩ V. Because of Lemma 5, +it is enough to show that +∞ +� +n=2 +[n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +|an| ≤ α − 1 +since f ∈ Rτ(A, B) ∩ V, then by Lemma 16 the inequality (17) holds. Letting +T4(α, λ) += +∞ +� +n=2 +[n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +|an| +8 + +We get +T4(α, λ) += +(A − B) |τ| +∞ +� +n=2 +1 +n [n(1 − αλ) − α(1 − λ)] +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� += +(A − B) |τ| +� +(1 − αλ) +∞ +� +n=2 +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� +−α (1 − λ) +∞ +� +n=2 +1 +n +� +(|a|)n−1 (b)n−1 (c)n−1 +(b + 1)n−1 (c + 1)n−1 (1)n−1 +� � += +(A − B) |τ| +� +(1 − αλ) +∞ +� +n=0 +� +(|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n +� +− (1 − αλ) +−α (1 − λ) +∞ +� +n=0 +� +(|a|)n (b)n (c)n +(b + 1)n (c + 1)n (1)n+1 +� ++ α (1 − λ) +� +Using the formula (10) and the result (4) of Lemma 11 in above mentioned equation, we +have += +(A − B) |τ| +� +(1 − αλ) bc Γ(1 − |a|) +c − b +� +Γ(b) +Γ(1 − a + b) − +Γ(c) +Γ(1 − |a| + c) +� +− +� +α (1 − λ) bc +(|a| − 1)(b − 1)(c − 1) +� �Γ(2 − |a|) +c − b +� (c − 1)Γ(b) +Γ(1 − |a| + b) − +(b − 1)Γ(c) +Γ(1 − |a| + c) +� +− 1 +� ++α − 1 +� +The above expression is bounded above by α − 1 if and only if the equation (21) holds, +which completes proof. +□ +References +[1] G.E.Andrews, R.Askey and R.Roy 1999, Special functions, Encyclopedia of Mathematics and its +Applications, 71, Cambridge University Press, Cambridge. +[2] T.Bulboaca and G.Murugusundaramoorthy, (2020), Univalent functions with positive coefficients +involving Pascal distribution series, Commun. Korean Math. Soc. 35, no. 3, pp. 867–877. +[3] K.Chandrasekran and D.J.Prabhakaran, Geometric Properties of Generalized Hypergeometric +Functions and Stable Functions, Ph.D. Thesis, May 2022. +[4] K.Chandrasekran and D.J.Prabhakaran, Geometric Properties of Clausen’s Hypergeometric Func- +tions, Preprint. +[5] K.Chandrasekran and D.J.Prabhakaran, Hohlov Type Integral Operator involving Clausen’s Hy- +pergeometric Functions, Preprint. +[6] K.Chandrasekran and D.J.Prabhakaran, Univalence, Starlikeness and Convexity properties of +4F3(a1, a2, a3, a4 +b1, b2, b3 +; z) Hypergeometric Functions using convolution technique, Preprint. +[7] K.Chandrasekran and D.J.Prabhakaran, Convolutions with Generalized Hypergeometric Functions, +Preprint. +[8] K. K. Dixit and S. K. Pal, On a class of univalent functions related to complex order, Indian J. +Pure Appl. Math. 26 (1995), no. 9, 889–896. +[9] A. W. Goodman, Univalent functions, Vol.I and Vol.II, Tampa Florida Mariner Publishing Com- +pany, (1983). +9 + +[10] A. R. Miller and R. B. Paris, Clausen’s series 3F2(1) with integral parameter differences and trans- +formations of the hypergeometric function 2F2(x), Integral Transforms Spec. Funct. 23 (2012), +no. 1, 21–33. +[11] G. Murugusundaramoorthy, Univalent functions with positive coefficients involving Poisson distri- +bution series, Honam Math. J. 40 (2018), no. 3, 529–538. +[12] K. S. Padmanabhan, On a certain class of functions whose derivatives have a positive real part in +the unit disc, Ann. Polon. Math. 23 (1970/71), 73–81. +[13] M. A. Shpot and H. M. Srivastava, The Clausenian hypergeometric function 3F2 with unit argument +and negative integral parameter differences, Appl. Math. Comput. 259 (2015), 819–827. +[14] B. A. Uralegaddi, M. D. Ganigi and S. M. Sarangi, (1994), Univalent functions with positive +coefficients, Tamkang J. Math. 25, no. 3, pp. 225–230. +K. Chandrasekran, Research Scholar, Department of Mathematics, MIT Campus, Anna +University, Chennai 600 044, India +Email address: kchandru2014@gmail.com +G. Murugusundaramoorthy, School of Advanced Sciences, Vellore Institute of Tech- +nology, Vellore-632014, India +Email address: gmsmoorthy@yahoo.com +D. J. Prabhakaran, Department of Mathematics, MIT Campus, Anna University, Chen- +nai 600 044, India +Email address: asirprabha@gmail.com +10 + diff --git a/JtFOT4oBgHgl3EQfyTQV/content/tmp_files/load_file.txt b/JtFOT4oBgHgl3EQfyTQV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b13fa700de05e48eb11aae345922a082b0202e3a --- /dev/null +++ b/JtFOT4oBgHgl3EQfyTQV/content/tmp_files/load_file.txt @@ -0,0 +1,659 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf,len=658 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='12927v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='CV] 30 Jan 2023 UNIVALENT FUNCTIONS WITH NON-NEGATIVE COEFFICIENTS INVOLVING CLAUSEN’S HYPERGEOMETRIC FUNCTION K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' CHANDRASEKRAN, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' MURUGUSUNDARAMOORTHY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' PRABHAKARAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' In this work, we derived the necessary and sufficient conditions on param- eters for 3F2(a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) Hypergeometric Function to be in the classes M∗(λ, α) and N ∗(λ, α) and information regarding the image of function 3F2(a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) belonging to Rτ(A, B) by applying the convolution operator in open unit disc D = {z : |z| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Introduction Let D = {z ∈ C : |z| < 1} be the open unit disc in the complex plane C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let H denote the class of all analytic functions in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let A denote the family of analytic functions f of the form f(z) = z + ∞ � n=2 an zn, z ∈ D (1) with f(0) = 0 and f ′(0) = 1 in the open unit disc D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Which is the subclass of H and Let, S ⊂ A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' S denotes the class of all normalised functions that are analytic and univalent in open unit disc D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' For the function f is given by (1) in A and g ∈ A with g(z) = z + ∞ � n=2 bn zn, the convolution product of f and g is defined by (f ∗ g)(z) = z + ∞ � n=2 an bn zn, z ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Note that the convolution product is called Hadamard Product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' For more details refer [9] Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' The subclass V of A consisting of functions of the form f(z) = z + ∞ � n=2 an zn, z ∈ D, with an ≥ 0, n ∈ N, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' In [14], Uralegaddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' introduced the following two classes which are stated as: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [14] The class M(α) of starlike functions of order α, with 1 < α ≤ 4 3, defined by M(α) = � f ∈ A : ℜ �zf ′(z) f(z) � < α, z ∈ D � 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' 30C45, 33C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Generalized Hypergeometric Series, Univalent Functions, Starlike Functions, Convex Functions and Alexander Integral Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Final Version as on 30-01-2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' 1 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [14] The class N (α) of convex functions of order α, with 1 < α ≤ 4 3, defined by N (α) = � f ∈ A : ℜ � 1 + zf ′′(z) f ′(z) � < α, z ∈ D � = {f ∈ A : zf ′(z) ∈ M(α)} In this paper, we considere the two subclasses M(λ, α) and N (λ, α) of to discuss some inclusion properties based on Clausen’s Hypergeometric Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' These two subclasses was introduced by Bulboaca and Murugusundaramoorthy [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' which are stated as follows: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [2] For some α � 1 < α ≤ 4 3 � and λ (0 ≤ λ < 1), the functions of the form (1) be in the subclass M(λ, α) of S is M(λ, α) = � f ∈ A : ℜ � zf ′(z) (1 − λ)f(z) + λz f ′(z) � < α, z ∈ D � Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [2] For some α � 1 < α ≤ 4 3 � and λ (0 ≤ λ < 1), the functions of the form (1) be in the subclass N (λ, α) of S is N (λ, α) = � f ∈ A : ℜ � f ′(z) + zf ′′(z) f ′(z) + λz f ′′(z) � < α, z ∈ D � Also, let M∗(λ, α) ≡ M(λ, α) ∩ V and N ∗(λ, α) ≡ N (λ, α) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [8] A function f ∈ A is said to be in the class Rτ(A, B), with τ ∈ C\\{0} and −1 ≤ B ≤ A ≤ 1, if it satisfies the inequality ���� f ′(z) − 1 (A − B)τ − B[f ′(z) − 1] ���� < 1, z ∈ D Dixit and Pal [8] introduced the Class Rτ(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Which is stated as in the definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' If we substitute τ = 1, A = β and B = −β, (0 < β ≤ 1) in the definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='6, then we obtain the class of functions f ∈ A satisfying the inequality ���� f ′(z) − 1 f ′(z) + 1 ���� < β, z ∈ D which was studied by Padmanabhan [12] and others subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [1] The 3F2(a, b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' d, e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) hypergeometric series is defined as 3F2(a, b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' d, e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) = ∞ � n=0 (a)n(b)n(c)n (d)n(e)n(1)n zn, a, b, c, d, e ∈ C, (2) provided d, e ̸= 0, −1, −2, −3 · · · , which is an analytic function in open unit disc D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' We consider the linear operator Ia,b,c b+1,c+1(f) : A → A defined by convolution product Ia,b,c b+1,c+1(f)(z) = z 3F2(a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) ∗ f(z) = z + ∞ � n=2 An zn (3) where A1 = 1 and for n > 1, An = (a)n−1(b)n−1(c)n−1 (b + 1)n−1(c + 1)n−1(1)n−1 an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (4) 2 Motivated by the results in connections between various subclasses of analytic univalent functions, by using hypergeometric functions [3, 4, 5, 6, 7, 14], and Poisson distributions [2], we obtain the necessary and sufficient conditions on parameters for 3F2(a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) hypergeometric series to be in the classes M∗(λ, α) and N ∗(λ, α) and information regard- ing the image of functions 3F2(a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z) hypergeometric series belonging to Rτ(A, B) by applying the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Main Results and Proofs First, we recall the following results to prove our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [11] For some α (1 < α ≤ 4 3) and λ (0 ≤ λ < 1), and if f ∈ V, then f ∈ M∗(λ, α) if and only if ∞ � n=2 [n − (1 + nλ − λ)α]an ≤ α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (6) Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [11] For some α (1 < α ≤ 4 3) and λ (0 ≤ λ < 1), and if f ∈ V, then f ∈ N ∗(λ, α) if and only if ∞ � n=2 n [n − (1 + nλ − λ)α]an ≤ α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (8) The following result is due to Miller and Paris [10] & Shpot and Srivastava [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' For a, b, c > 0, c ̸= b and a < min(1, b + 1, c + 1), 3F2 � a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' 1 � = bc c − bΓ(1 − a) � Γ(b) Γ(1 − a + b) − Γ(c) Γ(1 − a + c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (10) Now, we state the following lemma due to Chandrasekran and Prabhakaran [4] which is useful to prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [4] Let a, b, c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Then we have the following: (1) For b, c > a − 1, we have ∞ � n=0 (n + 1)(a)n (b)n (c)n (b + 1)n (c + 1)n (1)n = bc Γ(1 − a) c − b � (1 − b)Γ(b) Γ(1 − a + b) − (1 − c)Γ(c) Γ(1 − a + c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (2) For b, c > a − 1, we have ∞ � n=0 (n + 1)2(a)n (b)n (c)n (b + 1)n (c + 1)n (1)n = bc Γ(1 − a) c − b � (1 − b)2Γ(b) Γ(1 − a + b) − (1 − c)2Γ(c) Γ(1 − a + c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (3) For b, c > a − 1, we have ∞ � n=0 (n + 1)3(a)n (b)n (c)n (b + 1)n (c + 1)n (1)n = bc Γ(1 − a) c − b � (1 − b)3Γ(b) Γ(1 − a + b) − (1 − c)3Γ(c) Γ(1 − a + c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' 3 (4) For a ̸= 1, b ̸= 1, and c ̸= 1 with b, c > max{0, a − 1}, we have ∞ � n=0 (a)n (b)n (c)n (b + 1)n (c + 1)n (1)(n+1) = bc (a − 1)(b − 1)(c − 1) × �Γ(2 − a) c − b � (c − 1)Γ(b) Γ(1 − a + b) − (b − 1)Γ(c) Γ(1 − a + c) � − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let a ∈ C\\{0}, b, c > 0, c ̸= b and |a| < min{1, b+ 1, c + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' A sufficient condition for the function z 3F2 � a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z � to belong to the class M∗(λ, α), 1 < α ≤ 4 3 and 0 ≤ λ < 1 is that ((1 − α) − b(1 − αλ)) Γ(b) Γ(1 − |a| + b) ≤ ((1 − α) − c(1 − αλ))) Γ(c) Γ(1 − |a| + c) (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let f(z) = z 3F2 � a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' by Lemma 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' it is enough to show that T1(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) = ∞ � n=2 [n − (1 + nλ − λ)α] |An| ≤ α − 1 Using the fact |(a)n| ≤ (|a|)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' one can get T1(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='[n(1 − αλ) − α(1 − λ)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(n + 1) (|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Using the result (1) of Lemma 11 and the formula (10) in above mentioned equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='derived that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) bc Γ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� (1 − b)Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − c)Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) bcΓ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='bc Γ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(1 − b)(1 − αλ) Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − c)(1 − αλ) Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) + α (1 − λ) Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='bc Γ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(1 − α) − b(1 − αλ)) Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− ((1 − α) − c(1 − αλ)) Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='The above expression is bounded above by α − 1 if and only if the equation (13) holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' which completes proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' □ Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let a ∈ C\\{0}, b, c > 0, c ̸= b and |a| < min{1, b+ 1, c + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' A sufficient condition for the function z 3F2 � a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z � to belong to the class N ∗(λ, α), 1 < α ≤ 4 3 and 0 ≤ λ < 1 is that (b − 1)(b(1 − αλ) − (1 − α))Γ(b) Γ(1 − |a| + b) ≤ (c − 1) (c(1 − αλ) − (1 − α))Γ(c) Γ(1 − |a| + c) (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let f(z) = z 3F2 � a,b,c b+1,c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' z � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' by the Lemma 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' it is enough to show that T2(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) = ∞ � n=2 n [n − (1 + nλ − λ)α] |An| ≤ α − 1 Using the fact |(a)n| ≤ (|a|)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' one can get T2(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) = ∞ � n=2 n [n(1 − αλ) − α(1 − λ)] � (|a|)n−1 (b)n−1 (c)n−1 (b + 1)n−1 (c + 1)n−1 (1)n−1 � = ∞ � n=2 [n2 (1 − αλ) − α(1 − λ) n] � (|a|)n−1 (b)n−1 (c)n−1 (b + 1)n−1 (c + 1)n−1 (1)n−1 � Replace n = (n − 1) + 1 and n2 = (n − 1)(n − 2) + 3(n − 1) + 1 in above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' we find that T2(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='[((n − 1)(n − 2) + 3(n − 1) + 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(1 − αλ) (|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='[α(1 − λ) ((n − 1) + 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(n − 1)(n − 2) (|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(3 − 2α λ − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(n − 1) (|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(3 − 2α λ − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n+2 (b)n+2 (c)n+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n+2 (c + 1)n+2 (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(3 − 2α λ − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n+1 (b)n+1 (c)n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n+1 (c + 1)n+1 (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� |a|(|a| + 1)b(b + 1)c(c + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)(b + 2)(c + 1)(c + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(|a| + 2)n (b + 2)n (c + 2)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 3)n (c + 3)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(3 − 2α λ − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='abc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)(c + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n+1 (b)n+1 (c)n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n+1 (c + 1)n+1 (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Using the formula (10) in above mentioned equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' we find that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� |a|(|a| + 1)b(b + 1)c(c + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)(b + 2)(c + 1)(c + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� �(b + 2)(c + 2)Γ(1 − (a + 2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(c + 2) − (b + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(b + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − (|a| + 2) + (b + 2) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(c + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − (|a| + 2) + (c + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(3 − 2α λ − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='|a|bc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)(c + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� �(b + 1)(c + 1)Γ(1 − (|a| + 1)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(c + 1) − (b + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(b + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − (|a| + 1) + (b + 1) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(c + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − (|a| + 1) + (c + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(1 − α) bcΓ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�bc (−|a|)(−(|a| + 1)) Γ(1 − (|a| + 2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� �(b + 1) b Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − |a| + b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (c + 1) c Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − |a| + c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−(3 − 2α λ − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�bc(−|a|)Γ(1 − (|a| + 1)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='b Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − |a| + b − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 − |a| + c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+(1 − α) bcΓ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Using Γ(1 − a) = −aΓ(−a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' the aforesaid equation reduces to = �bc Γ(1 − |a|) c − b � × �(b − 1)(b(1 − αλ) − (1 − α))Γ(b) Γ(1 − |a| + b) − (c − 1) (c(1 − αλ) − (1 − α))Γ(c) Γ(1 − |a| + c) � + α − 1 The above expression is bounded above by α − 1 if and only if the equation (15) holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' which completes proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' □ Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' [8] If f ∈ Rτ(A, B) is of the form (1), then |an| ≤ (A − B)|τ| n , n ∈ N ∖ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (17) The result is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Using the Lemma 16, we prove the following results: Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let a ∈ C\\{0}, b, c > 0, c ̸= b and |a| < min{1, b + 1, c + 1} and f ∈ Rτ(A, B) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Then Ia,b,c b+1,c+1(f)(z) ∈ N ∗(α, λ) if �bc Γ(1 − |a|) c − b �(1 − α) − b(1 − αλ)) Γ(b) Γ(1 − |a| + b) − ((1 − α) − c(1 − αλ)) Γ(c) Γ(1 − |a| + c) �� × � (A − B) |τ| (1 − (A − B) |τ|) � ≤ α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let f be of the form (1) belong to the class Rτ(A, B) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Because of Lemma 7, it is enough to show that ∞ � n=2 n [n(1 − αλ) − α(1 − λ)] � (|a|)n−1 (b)n−1 (c)n−1 (b + 1)n−1 (c + 1)n−1 (1)n−1 � |an| ≤ α − 1 since f ∈ Rτ(A, B) ∩ V, then by Lemma 16, we have |an| ≤ (A − B)|τ| n , n ∈ N ∖ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Letting T3(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) = ∞ � n=2 n [n(1 − αλ) − α(1 − λ)] � (|a|)n−1 (b)n−1 (c)n−1 (b + 1)n−1 (c + 1)n−1 (1)n−1 � |an| we derived that T3(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='[n(1 − αλ) − α(1 − λ)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(n + 1) (|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Using the result (1) of Lemma 11 and the formula (10) in above mentioned equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='derived that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) bc Γ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� (1 − b)Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − c)Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) bcΓ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�bc Γ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(1 − b)(1 − αλ) Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − c)(1 − αλ) Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) + α (1 − λ) Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�bc Γ(1 − |a|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='c − b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='�(1 − α) − b(1 − αλ)) Γ(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− ((1 − α) − c(1 − αλ)) Γ(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Γ(1 − |a| + c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='The above expression is bounded above by α − 1 if and only if the equation (19) holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' which completes proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' □ Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let a ∈ C\\{0}, b, c > 0, c ̸= b and |a| < min{1, b + 1, c + 1} and f ∈ Rτ(A, B) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Then Ia,b,c b+1,c+1(f)(z) ∈ M∗(α, λ) if �(1 − αλ) bc Γ(1 − |a|) c − b � Γ(b) Γ(1 − |a| + b) − Γ(c) Γ(1 − |a| + c) � − � α (1 − λ) bc (|a| − 1)(b − 1)(c − 1) � �Γ(2 − |a|) c − b � (c − 1)Γ(b) Γ(1 − |a| + b) − (b − 1)Γ(c) Γ(1 − |a| + c) � − 1 � � × � (A − B) |τ| (1 − (A − B) |τ|) � ≤ α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' (21) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Let f be of the form (1) belong to the class Rτ(A, B) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Because of Lemma 5, it is enough to show that ∞ � n=2 [n(1 − αλ) − α(1 − λ)] � (|a|)n−1 (b)n−1 (c)n−1 (b + 1)n−1 (c + 1)n−1 (1)n−1 � |an| ≤ α − 1 since f ∈ Rτ(A, B) ∩ V, then by Lemma 16 the inequality (17) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Letting T4(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) = ∞ � n=2 [n(1 − αλ) − α(1 − λ)] � (|a|)n−1 (b)n−1 (c)n−1 (b + 1)n−1 (c + 1)n−1 (1)n−1 � |an| 8 We get T4(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n [n(1 − αλ) − α(1 − λ)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n−1 (b)n−1 (c)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n−1 (c + 1)n−1 (1)n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(A − B) |τ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='− (1 − αλ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='−α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='n=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(|a|)n (b)n (c)n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='(b + 1)n (c + 1)n (1)n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='+ α (1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Using the formula (10) and the result (4) of Lemma 11 in above mentioned equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' we have = (A − B) |τ| � (1 − αλ) bc Γ(1 − |a|) c − b � Γ(b) Γ(1 − a + b) − Γ(c) Γ(1 − |a| + c) � − � α (1 − λ) bc (|a| − 1)(b − 1)(c − 1) � �Γ(2 − |a|) c − b � (c − 1)Γ(b) Γ(1 − |a| + b) − (b − 1)Γ(c) Γ(1 − |a| + c) � − 1 � +α − 1 � The above expression is bounded above by α − 1 if and only if the equation (21) holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' which completes proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' □ References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Andrews, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Askey and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='Roy 1999, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='com G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Murugusundaramoorthy, School of Advanced Sciences, Vellore Institute of Tech- nology, Vellore-632014, India Email address: gmsmoorthy@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='com D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content=' Prabhakaran, Department of Mathematics, MIT Campus, Anna University, Chen- nai 600 044, India Email address: asirprabha@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} +page_content='com 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFOT4oBgHgl3EQfyTQV/content/2301.12927v1.pdf'} diff --git a/KtAyT4oBgHgl3EQfTvdX/content/2301.00111v1.pdf b/KtAyT4oBgHgl3EQfTvdX/content/2301.00111v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..28bac3ea1c9e73967129afe327bdf328f130ae26 --- /dev/null +++ b/KtAyT4oBgHgl3EQfTvdX/content/2301.00111v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6f9e5fe11da95da01ea9dafa367dfd1e4e902890f17d0de619ff5fc858249825 +size 6671592 diff --git a/M9AyT4oBgHgl3EQf6_oJ/vector_store/index.pkl b/M9AyT4oBgHgl3EQf6_oJ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b8fe783ec4c9e483d5d83d42b97acdc357c01c3a --- /dev/null +++ b/M9AyT4oBgHgl3EQf6_oJ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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Yao1,2 , Xiaoyang Tan1,2 +1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics +2MIIT Key Laboratory of Pattern Analysis and Machine Intelligence +{ke jiang, jiayu yao, x.tan}@nuaa.edu.cn +Abstract +Offline reinforcement learning learns an effective +policy on offline datasets without online interac- +tion, and it attracts persistent research attention due +to its potential of practical application. However, +extrapolation error generated by distribution shift +will still lead to the overestimation for those ac- +tions that transit to out-of-distribution(OOD) states, +which degrades the reliability and robustness of +the offline policy. +In this paper, we propose +Contextual Conservative Q-Learning(C-CQL) to +learn a robustly reliable policy through the contex- +tual information captured via an inverse dynamics +model. With the supervision of the inverse dynam- +ics model, it tends to learn a policy that generates +stable transition at perturbed states, for the fact that +pertuebed states are a common kind of OOD states. +In this manner, we enable the learnt policy more +likely to generate transition that destines to the em- +pirical next state distributions of the offline dataset, +i.e., robustly reliable transition. Besides, we theo- +retically reveal that C-CQL is the generalization of +the Conservative Q-Learning(CQL) and aggressive +State Deviation Correction(SDC). Finally, exper- +imental results demonstrate the proposed C-CQL +achieves the state-of-the-art performance in most +environments of offline Mujoco suite and a noisy +Mujoco setting. +1 +Introduction +Recently, many significant advances in offline reinforce- +ment learning (RL) range from robotics to autonomous driv- +ing and recommendation systems[Lobbezoo et al., 2021; +Kiran et al., 2022; Afsar et al., 2021]. The purpose of offline +reinforcement learning is to learn an effective policy from the +offline dataset without online interaction. Directly imitating +from the datasets without proper constraint suffers from per- +formance degrading caused by distribution shift between the +behavior of the learnt policy and that of the behavior policy. +Thus methods like Conservative Q-Learning(CQL) [Kumar +et al., 2020] are proposed to narrow the gap between the be- +havior policy and the learnt policy, dealing with distribution +shift. However, traditional algorithms try to avoid entering +the OOD states, while extrapolation error generated by distri- +bution shift would still overestimate those actions that transit +to out-of-distribution(OOD) states[?], weakening the reliabil- +ity and robustness of the offline policy. Therefore, it may not +be a good idea to singly prevent entering OOD states while +constrain nothing of the learnt policy at those OOD states. +A reliable policy is supposed to generate predictable +consequences[?]. To be specific, once the agent entering an +OOD state, it may predict its potential destinations and try +to reach the ’safest’ one. But CQL, the representative work +of the conventional offline RL, ignores the predictability for +the transitions, overestimating the Q-values at the perturbed +states which leads to state deviation[Zhang et al., 2022] and +unreliable OOD trajectories. This phenomenon could also be +considered as the poor reliability caused by the lack of ro- +bustness of the learnt policy. Specifically, when the learnt +policy is constrained to generate stable transitions from per- +turbed states to the empirical next state distributions of the +offline dataset, then the policy is robustly reliable. To allevi- +ate this issue, State Deviation Correction(SDC)[Zhang et al., +2022] trains the learnt policy by adding a regularization to +minimize the MMD distance between the distributions of the +next state generated by learnt policy and the offline dataset, so +the agent could return to in-distribution states from perturbed +states, enhancing the reliability and robustness of the learnt +policy. However, SDC is too conservative because mechan- +ically returning to in-distribution states may not be optimal +behavior; on the other, SDC ignores its generality with CQL, +formed as an extra regularization added onto CQL, increasing +the difficulty of tuning such a complex system. +In this paper, we propose a novel and effective method, +Contextual Conservative Q-Learning(C-CQL), to better uti- +lize the offline contextual information for training a more ro- +bust and reliable policy. The main idea is to estimate the +action distribution with the specific transition from the per- +turbed state to the similar succeed state distribution with the +corresponding unperturbed previous state. We introduce an +inverse dynamics model [Allen et al., 2021] to estimate the +action distribution most likely to explain the transition of two +consecutive states. In detail, we first sample a (previous) state +and its succeed state from the offline dataset and perturb the +previous state. Then, via the inverse dynamics model, we +estimate the action distribution that is most suitable for the +transition from the perturbed state to the unperturbed suc- +arXiv:2301.01298v1 [cs.LG] 3 Jan 2023 + +Figure 1: The learnt policy generates C-CQL transition at perturbed +state ˆs, reaching a trade-off between destining to in-distribution next +state s′ and maximizing the one-step reward. +ceed state. To simulate the behavior of the inverse dynamics +model, we overestimate the Q-value of the perturbed state and +the action estimated by the inverse dynamics model to opti- +mize the learnt policy indirectly. In this manner, we declare +the behavior of the learnt policy is robustly reliable. Besides, +the overestimation for the Q-value of the perturbed state and +the action generated by the inverse dynamics model implies +to maximize the one-step reward additionally, which is more +aggressive than traditional SDC. As is shown in Figure 1, at +perturbed states, the policy learnt by C-CQL generates transi- +tion to reach a trade-off between SDC transition and an one- +step greedy transition. +Theoretical results show that C-CQL could be considered +as a generalization of CQL and SDC. If we only use the un- +perturbed dataset for training, C-CQL would degenerate into +traditional CQL; otherwise, C-CQL is approximately equiva- +lent to reaching a trade-off between the EM distance version +of SDC and one-step rewards, i.e. an aggressive SDC. We +implement our C-CQL with actor-critic framework. Exper- +imental results show that C-CQL performs better than most +of the traditional offline RL algorithms and SDC in a offline +Mujoco control suite and a multi-level noisy Mujoco setting. +To sum up, our contributions can be summarized as fol- +lows: +• We propose a novel and effective offline RL algorithm, +Contextual Conservative Q-Learning(C-CQL), to learn a +more robust and reliable policy via an inverse dynamics +model. +• Theoretical results show that the proposed algorithm C- +CQL could be considered as a generalization of Conser- +vative Q-Learning and aggressive State deviation correc- +tion. +• Sufficient experiments conducted show that C-CQL per- +forms better than other SOTA methods and makes the +agent generate more reliable trajectories in obervation- +perturbed environments. +2 +Related work +Conservatism in offline reinforcement learning. +Offline +reinforcement learning [Riedmiller, 2005; Lange et al., 2012] +aims to learn a powerful decision making engine from an pre- +viously collected dataset [Levine et al., 2020]. To deal with +the distributional shift between testing data and the offline +datasets, previous methods introduce the conservatism by un- +derestimating the values of OOD states to alleviate the over- +estimation bias [Kumar et al., 2020; Kuznetsov et al., 2020; +Kumar et al., 2019a; Siegel et al., 2020]. However, these +methods lack of the utilization of contextual information and +the constraint of the behaviors at those perturbed or OOD +states, lacking reliability and robustness. State deviation cor- +rection [Zhang et al., 2022] builds a dynamics model and a +transition model to enable the agent generate specific transi- +tions at perturbed states, enhancing the reliability and robust- +ness. +Inverse dynamics model. +An inverse dynamics model +I(a|s′, s) predicts the action distribution that could explain +the transition between a given pair of states. Inverse dynam- +ics models haven been applied to improving generalization to +real-world problems [Christiano et al., 2016], Markov repre- +sentation learning [Allen et al., 2021], defining intrinsic re- +wards for exploration [Choi et al., 2019]. In our work, the +inverse dynamics model is considered as a contextual pol- +icy that generates action distributions with predictable conse- +quences for the supervision to train the learnt policy. +3 +Background +Given a reinforcement learning problem, we usually model +it as a Markov Decision Process (MDP) and it can be repre- +sented by a tuple of the form (S, A, P, R, γ), where S is the +state space, A is the action space, P is the transition probabil- +ity matrix, R and γ are the reward function and the discount +factor. A policy is defined as π : S → A and trained to +maximize the expected cumulative discounted reward in the +MDP: +max +π +E +� ∞ +� +t=0 +γR(st, π(at|st)) +� +(1) +In general, we define a Q-value function Qπ(s, a) += +E[�∞ +t=0 γR(st, π(at|st))|s, a] to represent the expected cu- +mulative rewards. Q-Learning is a classic method that trains +the Q-value function by minimizing the Bellman error over +Q[Watkins and Dayan, 1992]: +Q ← arg min +Q E +� +R(s, a) + γ max +a′ Q(s′, a′) − Q(s, a) +� +(2) +3.1 +Conservative Q-Learning +In offline setting, Q-Learning algorithms need to learn a Q- +value function Qπ(s, a) and a policy π from a from dataset +D, which is collected by a behavior policy πβ. Since there is +always a state distribution shift between the stationary state +distributions of the learnt policy π and the behavior policy +πβ, this basic recipe falls to estimate the Q-values for OOD +state-action pairs. CQL try to underestimate the Q-values for +OOD state-action pairs to prevent the agent from enter the +OOD states[Kumar et al., 2020]: +Q ← arg min +Q α · +� +Es∼D,a∼π(a|s)Q(s, a) − Es,a∼DQ(s, a) +� ++ 1 +2 +� +R(s, a) + γ max +a′ Q(s′, a′) − Q(s, a) +�2 +(3) + +perturbed state +out-of-distribution +ΛS +greedy transition +one-step +reward +C-CQL transition +perturb +(a trade-off) +SDC transition +S +S' +in-distribution transition +previous state +in-distribution/dataset +succeed statewhere α is a weight for CQL regularization. +3.2 +State Deviation Correction +There is always a distributional shift between the empiri- +cal distribution of the dynamics of the offline dataset ˆP and +the true dynamics ˆP. For some perturbed state ˆs and a, if +ˆP(s′|ˆs, a) = 0 while P(s′|ˆs, a) > 0, the agent may enter +its unfamiliar state s′ when it executes action a at the state s. +This phenomenon is defined as dynamics bias, which might +lead to state deviation at each time step. Besides, the approxi- +mation error of the deep models would generate the deviation +as well. +To deal with this problem, State Deviation Correc- +tion(SDC) proposed in [Zhang et al., 2022] expects to train +the policy leading the agent to a predictable transition, which +make the agent closer to the dataset. SDC trains a dynamics +model M and a transition model U and optimizes the policy +as: +π(·|s) = max +π +Ea∼π(·|s)Q(s, a) ++ λ +� +Es∼DMMD +� +M(·|ˆs, π(·|ˆs)), U(·|s) +� +− η +� +(4) +where ˆs = s + β · ϵ, β is the magnitude and ϵ is a noise sam- +pled from a Gaussian distribution N(0, 1). However, SDC +ignores its relationship with CQL but is proposed as a regu- +larization onto CQL, introducing an extra hyperparameter λ, +which makes it difficult to tune such a complex system. In +addition, SDC may be too conservative to mechanically force +the agent returning to the dataset while ignoring the reward +may be achieved at perturbed states. +4 +Contextual Conservative Q-Learning +Consider a noisy state setting, that is, the agent should make +decisions on noisy states during test period. Traditional CQL +fails to deal with this setting because it lacks consideration +of context, leading to the uncertainty of the consequences of +the decisions at perturbed states. To deal with this, we in- +troduce the Contextual Conservative Q-Learning(C-CQL) in +detail in this section. First, we introduce how to generate the +mixed dataset. Then we introduce how to use the contextual +module to supervise our policy to make propoer decisions at +perturbed states. Finally, the loss functions for implementa- +tion would be introduced. The figure of framework is shown +in Appendix A.2.2. +4.1 +Mixed Dataset Generation +Given an offline dataset D, which consists of quadruples +(s, a, r, s′), we perform data augmentation on it in this sec- +tion. Given a quadruple (s, a, r, s′), we first perturb the state +s by a noise ϵ with magnitude β and get an noisy state: +ˆs = s + β · ϵ +(5) +where ϵ is sampled from a Gaussian distribution N(0, 1) and +β is usually a small constant. Therefore, we could get the +perturbed quadruples (ˆs, a, r, s′), and make them into a new +perturbed dataset ˜D. In particular, we do not perturb the next +state s′, to preserve the destination we wish the agent to jump +from ˆs. Then we combine the two datasets D and ˜D to form +a new dataset Dtot, which consists of both perturbed and un- +perturbed samples. +Our work is based on the mixed dataset Dtot. To be spe- +cific, we have: +Dtot = D + ˜D +(6) +and the quadruples in Dtot is noted by (˜s, a, r, s′), where {˜s} +could be divided into unperturbed {s} and perturbed {ˆs}. +4.2 +Contextual Supervision +The inverse dynamics model Iπβ(a|s, s′) is defined interms +of the dynamics function P(s′|s, a) and the transition func- +tion P(s′|s, πβ) via Bayes’ theorem: +Iπβ(a|s, s′) = πβ(a|s)P(s′|s, a) +P(s′|s, πβ) +(7) +where P(s′|s, πβ) = � +a∈A +P(s′|s, a)πβ(a|s). +For each quadruple (˜s, a, r, s′) in Dtot, the task for the +learnt policy π is to generate an action distribution that is +suitable for the transition between ˜s and s′. This action dis- +tribution could be estimated by the inverse dynamics model +Iπβ(a|˜s, s′). Therefore, we overestimate the Q-value of the +action sampled from Iπβ(a|˜s, s′) to indirectly guide the learnt +policy to produce action distribution that could reach s′ with +the best probability, while underestimate the Q-values of the +other actions. In this manner, we have: +max +Q E˜s,s′∼Dtot +� +Ea∼Iπβ (·|˜s,s′)Q(˜s, a) +−Ea∼π(·|˜s)Q(˜s, a) +� +(8) +By the way, the predecessor states in the mixed dataset D +defined in (6) could be divided into two sets: perturbed states +{ˆs} and unperturbed states {s}. Then ˜s, s′ ∼ Dtot in (8) +could also be considered as ˜s ∼ Dtot, s′ ∼ P(s′|s, πβ), +where s is the original state of ˜s if ˜s ∈ {ˆs}, of which the +items are perturbed, or s = ˜s if ˜s ∈ {s}, of which the items +are not perturbed. +4.3 +Implementation +We implement our method introduced in previous setions on +an actor-critic framework, and name it as Contextual Conser- +vative Q-Learning(C-CQL). +The actor loss function is translated into: +Lπ = − E˜s∼Dtot,ˆa∼π(·|˜s)[Q(˜s, ˆa)] +(9) +where Dtot is the mixed dataset defined in (6) and π is the +learnt policy. It is the same form as original actor loss func- +tion, except for the replacement of dataset. +And the critic loss function is translated into: +LQ = α · +� +E(˜s,s′)∼Dtot +� +Eˆatar∼Iπβ (·|˜s,s′)Q(˜s, ˆatar) +−Eˆa∼π(·|˜s)Q(˜s, ˆa) +�� ++1 +2Es,a,r,s′∼D +� +r + γ max +a′ Q(s′, a′) − Q(s, a) +�2 +(10) + +where ˜s and s′ are included in the same quadruple sampled +from Dtot; Iπβ(a|s, s′) is the inverse dynamics model trained +on the dataset generated by the behavior policy πβ. +To be specific, we use a conditional variational auto- +ecoder(CVAE) in [Kingma and Welling, 2014] to model the +inverse dynamics network. The inverse dynamics network +could sample the actions given current states and destination +states as a supervision for the learnt policy to generate pre- +dictable transitions. The second term is an one-step Bellman +error, as shown in (2), and α is a weight for C-CQL regular- +ization. +To sum up, C-CQL optimizes both actor loss Lπ to up- +date the policy network π and critic loss LQ to update our +Q-networks, and outputs the learnt policy network π finally. +The whole process is summarized in Algorithm 1. +Algorithm 1 Contextual Conservative Q-Learning +Input: offline dataset Dtot, a pretrained inverse dynamics +model I, maximal update iterations T, +Parameter: policy network π, Q-networks Q1, Q2, noise- +level β, +Output: learnt policy network π +1: Initialize the policy network, Q-networks and the inverse +dynamics model. +2: Let t = 0. +3: while t < T do +4: +Sample mini-batch of N samples (˜s, a, r, s′) from +Dtot. +5: +Feed ˜s to the policy network π and get ˆa. +6: +Feed ˜s and s′ to the inverse dynamics model I and get +ˆatar. +7: +Update the policy network π according to (9). +8: +Update the Q-networks according to (10). +9: end while +10: return learnt policy network π. +5 +Theoretical Justification +In this section, we perform a theoretical analysis on our pro- +posed C-CQL around Claim 1. +Claim 1. C-CQL is the generalization of CQL and the EM +distance version of aggressive SDC. +First of all, a definition of aggressive SDC is given. Then, +Assumption 1 is given on the properties of transition func- +tion P(s′|s, πβ). From Theorem 1 and Proposition 3, we +conclude that C-CQL approximately equals to the aggressive +SDC based on Assumption 1 at perturbed states, and equals +to CQL at unperturbed states. Finally, we conclude Claim 1 +from Proposition 2,3 and Theorem 1. +5.1 +Aggressive State Deviation Correction +The goal of State deviation correction(SDC) is to learn a pol- +icy that generates stable transitions at perturbed states[Zhang +et al., 2022]. The regularization of SDC is: +min +π λ +� +Es∼DD +� +M(·|ˆs, π(·|ˆs)), U(·|s) +�� +where M is the dynamics model and U is the transition +model. D is some kind of distance. However, no evidence has +shown that returning in-distribution state s′ from perturbed +state ˆs is the optimal behavior to clone, because this may be +too conservative. Therefore, we consider relaxing the conser- +vatism and a most intuitive way is to maximize the one-step +reward at the same time: +min +π λ +� +Es∼DD +� +M(·|ˆs, π(·|ˆs)), U(·|s) +� +− Rπ(ˆs) +� +where Rπ(ˆs) = � +a π(a|ˆs)R(ˆs, a) and R(s, a) is the reward +function according to s, a. This could be condered as a trade- +off between SDC and one-step reward, i.e., aggressive SDC. +5.2 +Connection between CQL, aggressive SDC and +C-CQL +Definition 1. (EM distance) The Earth-Mover(EM) distance +of two distribution p and q can be defined as: +EM +� +p(x) +����p(y) +� += sup +∥f∥≤1 +Ex∼p(x)f(x) − Ey∼q(y)f(y) +(11) +EM distance is also known as Wasserstein distance[Ar- +jovsky et al., 2017]. +Assumption 1. The transition function P(s′|s, πβ) could be +considered as a function according to variable s, that is f(s). +Then we assume this function is continuous: +∀ϵ, ∃δ, s.t.∥s − ˆs∥ ≤ ϵ ⇒ +∥P(s′|s, πβ) − P(s′|ˆs, πβ)∥ ≤ δ +and ∀s, s′, we have P(s′|s, πβ) > 0. +Proposition 1. By Assumption 1, we could easily have: +∀ϵ, ∃δ, s.t.∥s − ˆs∥ ≤ ϵ ⇒ +1 − δ ≤ P(s′|s, πβ) +P(s′|ˆs, πβ) ≤ 1 + δ +Proposition 2. (8) is equivalent to the expected EM distance +between Iπβ(·|˜s, s′) and π(·|˜s): +E˜s∼DtotEM +� +Es′∼P (s′|s,πβ)Iπβ(a|˜s, s′) +����π(a|˜s) +� +(12) +Proof. The proof is easily to be conducted by the definition +in Definition 1: +max +Q E˜s,s′∼Dtot +� +Ea∼Iπβ (·|˜s,s′)Q(˜s, a) +− Ea∼π(·|˜s)Q(˜s, a) +� +⇔ max +Q E˜s∼DtotEs′∼P (s′|s,πβ) +� +Ea∼Iπβ (·|˜s,s′)Q(˜s, a) +− Ea∼π(·|˜s)Q(˜s, a) +� +⇔ max +Q E˜s∼Dtot +� +Es′∼P (s′|s,πβ)Ea∼Iπβ (·|˜s,s′)Q(˜s, a) +− Ea∼π(·|˜s)Q(˜s, a) +� +⇔E˜s∼DtotEM +� +Es′∼P (s′|s,πβ)Iπβ(a|˜s, s′) +����π(a|˜s) +� +Complete the proof. + +Task name +SAC +BC +BEAR +BRAC-p +BRAC-v +CQL +MOPO +SDC +C-CQL(Ours) +Halfcheetah-random +30.5 +2.1 +25.5 +23.5 +28.1 +35.4 +31.9 +36.2 +35.9± 1.0 +Walker2d-random +4.1 +1.6 +6.7 +0.8 +0.5 +7.0 +13.3 +14.3 +16.1± 3.9 +Hopper-random +11.3 +9.8 +9.5 +11.1 +12.0 +10.8 +13.0 +10.6 +23.5± 3.4 +Halfcheetah-medium +-4.3 +36.1 +38.6 +44.0 +45.4 +44.4 +40.2 +47.1 +49.1± 0.4 +Walker2d-medium +0.9 +6.6 +33.2 +72.7 +81.3 +79.2 +26.5 +81.1 +85.1± 0.7 +Hopper-medium +0.8 +29.0 +47.6 +31.2 +32.3 +58.0 +14.0 +91.3 +95.2± 1.1 +Halfcheetah-medium-replay +-2.4 +38.4 +36.2 +45.6 +46.9 +46.2 +54.0 +47.3 +48.5± 0.6 +Walker2d-medium-replay +1.9 +11.3 +10.8 +-0.3 +0.9 +26.7 +92.5 +30.3 +87.8± 1.0 +Hopper-medium-replay +3.5 +11.8 +25.3 +0.7 +0.8 +48.6 +42.7 +48.2 +101.2± 0.8 +Halfcheetah-medium-expert +1.8 +35.8 +51.7 +43.8 +45.3 +62.4 +57.9 +101.3 +99.1± 0.4 +Walker2d-medium-expert +1.9 +11.3 +10.8 +-0.3 +0.9 +98.7 +51.7 +105.3 +112.9± 0.8 +Hopper-medium-expert +1.6 +111.9 +4.0 +1.1 +0.8 +111.0 +55.0 +112.9 +113.2± 0.6 +Halfcheetah-expert +-1.9 +107.0 +108.2 +3.8 +-1.1 +104.8 +- +106.6 +102.1± 0.8 +Walker2d-expert +-0.3 +125.7 +106.1 +-0.2 +0.0 +153.9 +- +108.3 +111.4± 0.8 +Hopper-expert +0.7 +109.0 +110.3 +6.6 +3.7 +109.9 +- +112.6 +112.9± 0.4 +Average-score +3.3 +43.2 +41.6 +18.9 +19.9 +66.5 +41.1 +70.2 +79.6 +Average-ranking +7.5 +6.0 +5.5 +6.9 +6.1 +3.6 +4.1 +2.7 +1.7 +Table 1: Results of C-CQL, SDC, SAC, BC, BEAR, BRAC, and CQL on offline Mujoco control suite tasks, on the normalized return metric, +averaged over four seeds. Note that C-CQL performs better or similar to other methods on most of the tasks. +Proposition 3. When training on an unperturbed dataset D, +C-CQL is equivalent to CQL. +Proof. The proof is performed on the equivalence of the C- +CQL and CQL regularizations given an unperturbed state s: +max +Q Es′∼P (s′|s,πβ)Ea∼Iπβ (a|s,s′)Q(s, a) +− Ea∼π(a|s)Q(s, a) +⇔ max +Q +� +s′ +P(s′|s, πβ) +� +a +πβ(a|s)P(s′|s, a) +P(s′|s, πβ) +Q(s, a) +− Ea∼π(a|s)Q(s, a) +⇔ max +Q +� +a +πβ(a|s)Q(s, a) − Ea∼π(a|s)Q(s, a) +⇔ max +Q Ea∼π(a|s)Q(s, a) − Ea∼π(a|s)Q(s, a) +⇔ +CQLregularization +Complete the proof. +Proposition 2 shows that C-CQL regularization proposed +in (8) has another form of the EM distance between +Iπβ(·|˜s, s′) and π(·|˜s), which is used in subsequent content. +Proposition 3 shows that C-CQL degrades into CQL at the +unperturbed states. And then, on the other, we need to show +that C-CQL is equivalent to the aggressive SDC at the per- +turbed states, or maybe approximately, to conclude Claim 1. +Theorem 1. Given a state s, its perturbed state ˆs and the +behavior policy πβ. Define Q(s, a) as a approximal Q-value +function and V (s) as a approximal value function. Given an +inverse dynamics model Iπβ(a|s, s′). By Assumption 1 and +Proposition 1, the following two optimization formulas are +approximately equivalent: +min +π EM +� +Es′∼P (s′|s,πβ)Iπβ(a|ˆs, s′) +����π(a|ˆs) +� +(13) +∼ +⇐⇒ min +π EM +� +P(s′|s, πβ) +����P(s′|ˆs, π) +� +− Ea∼π(a|ˆs)R(ˆs, a) +(14) +, that is, they imply the same policy approximately. +The proof of Theorem 1 is shown in Appendix A.1. The- +orem 1 shows that the policy optimized by our method is ap- +proximately equivalent to minimizing the objective of an EM +distance version of SDC while maximizing the expected im- +mediate reward, i.e., aggressive SDC. +Proposition 4. The gap between the upper and lower bounds +mentioned in Theorem 1 is less than 2δRπβ(ˆs). If we have +∥R∥∞ ≤ 1, then the gap is less than 2δ. +Proof. This proposition can be easily obtained by subtracting +the lower bound from the upper bound. +Although there is actually a little gap between (13) and (14) +which is caused by the existence of the continuous gap δ, as is +shown in Proposition 4. When δ is large, (13) and (14) are not +strictly equivalent. However, by the conservative selection of +the magnitude of the noise ϵ, the δ would be a quite tiny value +in practice. Therefore, we consider that the result of Theorem +1 is reasonable. +Combing Proposition 2, Proposition 3 and Theorem 1, we +conclude Claim 1. +6 +Experiments +In this section, we will describe the dataset D4RL[Fu et al., +2020] and information about experimental setup. We con- +duct a comparative experiment on the Mujoco datasets in the +D4RL benchmarks. Then we design a noisy Mujoco setting +to test the robustness of CQL, SDC and C-CQL we proposed. +Then, to understand the utility of our method, we visualize the +state distributions of CQL, SDC and our methods in the two + +Figure 2: Performance decrease of CQL, SDC and C-CQL’s performance on the noisy Mujoco setting compared with noisless Mujoco tasks. +The best method of each benchmark is marked by red star. +noisy environments with obvious performance gaps. Finally, +we perform a sensitive analysis to learn how the hyperparam- +eters β in (5) and α in (8) affect the performance of C-CQL. +To sum up, the experiments are designed to answer the fol- +lowing: +• Does C-CQL outperform the state-of-the-art methods +across a variety of tasks? +• Is C-CQL robust for different levels of noise? +• Is C-CQL able to generate reliable trajectories, i.e., in- +distribution trajectories? +D4RL. +There are three types of control environments +we use in this paper: +Hopper, Halfcheeta and Walker2d +benchmarks in D4RL and five kinds of datasets for each +type of benchmark: ‘random’, ‘medium’, ‘medium-replay’, +‘medium-expert’ and ‘expert’. The ‘random’ dataset is gen- +erated by a randomly initialized policy and the ‘medium’ +dataset consists of the data collected by a soft actor- +critic(SAC)[Haarnoja et al., +2018] policy insufficiently +trained. The ‘medium-replay’ dataset collects the data in the +replay buffer when the behavior policy reaches 50% perfor- +mance of the expert policy. The ‘expert’ dataset is produced +by a completely trained SAC. The ‘medium-expert’ is a mix- +ture of ‘medium’ dataset and ‘expert’ dataset in half. All the +levels of datasets mentioned above contain 1,000,000 sam- +ples. +CQL +C-CQL +SDC +Task name +score +score +score +Halfcheetah-noisy-s +56.2 +100.9 +99.1 +Halfcheetah-noisy-m +52.9 +99.8 +98.4 +Halfcheetah-noisy-l +42.1 +83.3 +79.8 +Walker2d-noisy-s +81.1 +102.7 +110.3 +Walker2d-noisy-m +77.4 +102.2 +110.9 +Walker2d-noisy-l +52.0 +95.6 +106.7 +Hopper-noisy-s +110.4 +112.3 +112.5 +Hopper-noisy-m +97.3 +110.8 +112.2 +Hopper-noisy-l +74.5 +74.3 +71.0 +Table 2: Results of CQL, SDC and C-CQL on the noisy Mujoco +setting, on the normalized return metric, averaged over four seeds. +6.1 +Offline Mujoco Control +In this section, we compare our C-CQL with several signif- +icant methods, including BC, SAC[Haarnoja et al., 2018], +BEAR[Kumar et al., 2019b], BRAC[Wu et al., 2019], +MOPO[Yu et al., 2020] and SDC[Zhang et al., 2022], on +the D4RL dataset in the Mujoco benchmarks. The results +for SDC, SAC, BC, BEAR, BRAC, and CQL are obtained +by [Zhang et al., 2022]. Results are shown in Table 1. It +is shown that C-CQL we proposed achieve the similar or +better performance than other methods on most tasks, and +acheve the highest average score and the best average ranking +among these methods. In addition, our methods outperforms +the other methods by a large margin in ‘Walker2d-medium’, +’Hopper-medium’, ‘Hopper-medium-replay’ and ‘Walker2d- +medium-expert’. The details of training process is shown in +Appendix A.2.2. +6.2 +Noisy Mujoco setting +In this setting, we add different magnitude of noise on the +three standard Mujoco benchmarks: Halfcheetah, Walker2d +and Hopper. The magnitude of noise could be devide into 3 +levels: slight(s), moderate(m) and large(l). The magnitude of +slight(s) noise is about 1e − 3 in our experiment. The mod- +erate(m) magnitude is range from 1e − 2 to 2e − 2 and the +large(l) is range from 5e − 2 to 1e − 1. +We use the policies trained by CQL, SDC and CQL on +‘medium-expert’ datasets, and then run in the noisy bench- +marks. The score and performance decrease of these poli- +cies in the 9 noisy benchmarks are respectively recorded as +is shown in Table 2 and Figure 2. +The performance de- +crease refers to the percentage reduction from the scores of +CQL, SDC and C-CQL in observation-noisy environment, as +is shown in Table 2 to those shown in Table 1. We could +conclude that our method, C-CQL, exceeds CQL and SDC, +achieving much less performance degradation in most noisy +benchmarks, i.e., is more robust. +Besides, C-CQL is not +strictly anti-explored as SDC, so C-CQL is able to infer the +reward at those OOD state, which may be the reason of the +less performance degradation than SDC. +To help understand the reason why our approach performs +better, we visualize the state distributions of some bench- +marks, as is shown in Figure 3, with t-Distributed Stochas- +tic Neighbour Embedding(t-SNE)[Hinton and Roweis, 2002]. +The green points note the offline dataset, while the red points + +CQL/decrease(%) SDC/decrease(%) Ours/decrease(%) +50.0 +47.3 +大 +40.0 +37.3 +32.5 +32.8 +34.2 +30.0 +大 +21.6 +19.5 +20.0 +17.8 +17.8 +15.3 +大 +12.3 +10.0 +9.2 +10.0 +大 +大 +大 +5.5 +大 +2.5 +2.9 +0.4 +1.5 +0.7 +2.3 +1.8 +0.6 +0.5 +1.9 +0.0 +0.6 +0.9 +0.0 +halfcheetah-noisy-s +halfcheetah-noisy-m +halfcheetah-noisy-l +Walker2d-noisy-s +Walker2d-noisy-m +Walker2d-noisy-l +hopper-noisy-s +hopper-noisy-m +hopper-noisy-l +-10.0Figure 3: The visualizations of the state distributions of exper- +iments on the ‘Walker2d-noisy-l’ and ‘Walker2d-noisy-m’. +The +(a), (d) are CQL(red) vs dataset(green); the (b), (e) are SDC(red) +vs dataset(green); and the (c), (f) are our method C-CQL(red) vs +dataset(green). +in (a), (d) note the data generated by the CQL policy, in (b), +(e) note the data generated by the SDC policy and in (b), (d) +note the data generated by the C-CQL(Ours) policy. The re- +sults show that CQL tends to deviate from the training data +when it makes decisions at perturbed states, because it gener- +ates a lot of OOD samples. However, SDC and our method +C-CQL produces much less OOD samples and always fol- +lows the dataset. +Therefore, we conclude that the method we propose gener- +ate more robust and reliable trajectories. +6.3 +Sensitive Analysis +In this section, we perform sensitive analysis on two key hy- +perparameters: β in (5) and α in (8) to evaluate how these +hyperparameters influence the performance of C-CQL. +Sensitive analysis on β. +The β in (5) is the hyperparameter +that control the magnitude of the noise that add to our mixed +dataset Dtot for training. Its influence to C-CQL is as shown +in Figure 4. The performance of the learnt policy is best when +the β is 1e-3; takes second place when the β equals to 5e-3; is +worst when the β is 1e-2. From the results, we conclude that +when β is too large, the agent would fail to be well trained +over some or all seeds. Therefore, we ought to conservatively +choose the value of β and experience suggests the β should +be no bigger than 2e-3 for ’walker2d’ benchmark, 5e-3 for +’hopper’ and ’3e-3’ for ’halfcheetah’. +Figure 4: The sensitive results of β. (a) is on ’walker2d-expert’ and +(b) is on ’hopper-expert’. +Figure 5: The sensitive results of α. (a) is on ’walker2d-medium- +expert’ and (b) is on ’hopper-medium-expert’. +Sensitive analysis on α. +The α in (8) is the weight for the +C-CQL regularization that affects the magnitude of conser- +vatism of C-CQL. The sensitivity of this hyperparameter to +the performance of C-CQL is as shown in Figure 5. The learnt +policy performs best when α equals to 10.0; slight poor when +α is 5.0; worst when α is 1.0. Therefore, we can conclude +that if α is too small, C-CQL may fail to train the policy, and +experience suggests that the α should range from 5.0 to 10.0. +7 +Conclusion +In this paper, we proposed a simple and novel yet effective +method, Contextual Conservative Q-Learning, to learn a ro- +bustly reliable policy. With the supervision of a single in- +verse dynamics model, the learnt policy is able to generate +consequence-predictable and consequence-stable action dis- +tributions. +Besides, we have theoretically shown that our +method is a generalization of CQL and the aggressive SDC +for the first time. Finally, the C-CQL achieves the SOTA +performance across multiple offline Mujoco settings, and we +suppose the robustness & reliability of the offline reinforce- +ment learning algorithms would be critical for future work. + +SDC +-CQL +Training data +Training data +Training data +SDC +C-CQL +Training data +Training data +Training dataβ=le-2 +β=1e-2 +1.0 +β=5e-3 +β=5e-3 +1.0- +β=1e-3 +β=1e-3 +0.8 +SCor + scor +0.8 +lized +0.6 +d +e +normalize +0.6 +normali +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +0 +20 +40 +60 +80 +100 +120 +140 +160 +10-steps +10-stepsα=1.0 +α=1.0 +α=5.0 +1.0 - +1.0 +α=5.0 +α=10.0 +α=10.0 + score +normalized score +0.8 +0.8 +normalized +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +10-steps +10-stepsReferences +[Afsar et al., 2021] Mohammad +Mehdi +Afsar, +Trafford +Crump, and Behrouz H. 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AAAI +Press, 2022. + diff --git a/NdAzT4oBgHgl3EQfWPyB/content/tmp_files/load_file.txt b/NdAzT4oBgHgl3EQfWPyB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e255ea5e28c397ea242fd52693d209357527d9cb --- /dev/null +++ b/NdAzT4oBgHgl3EQfWPyB/content/tmp_files/load_file.txt @@ -0,0 +1,680 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf,len=679 +page_content='Contextual Conservative Q-Learning for Offline Reinforcement Learning Ke Jiang1,2 , Jiayu Yao1,2 , Xiaoyang Tan1,2 1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics 2MIIT Key Laboratory of Pattern Analysis and Machine Intelligence {ke jiang, jiayu yao, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='tan}@nuaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='cn Abstract Offline reinforcement learning learns an effective policy on offline datasets without online interac- tion, and it attracts persistent research attention due to its potential of practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, extrapolation error generated by distribution shift will still lead to the overestimation for those ac- tions that transit to out-of-distribution(OOD) states, which degrades the reliability and robustness of the offline policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In this paper, we propose Contextual Conservative Q-Learning(C-CQL) to learn a robustly reliable policy through the contex- tual information captured via an inverse dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' With the supervision of the inverse dynam- ics model, it tends to learn a policy that generates stable transition at perturbed states, for the fact that pertuebed states are a common kind of OOD states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In this manner, we enable the learnt policy more likely to generate transition that destines to the em- pirical next state distributions of the offline dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', robustly reliable transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Besides, we theo- retically reveal that C-CQL is the generalization of the Conservative Q-Learning(CQL) and aggressive State Deviation Correction(SDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Finally, exper- imental results demonstrate the proposed C-CQL achieves the state-of-the-art performance in most environments of offline Mujoco suite and a noisy Mujoco setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 1 Introduction Recently, many significant advances in offline reinforce- ment learning (RL) range from robotics to autonomous driv- ing and recommendation systems[Lobbezoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Kiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Afsar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The purpose of offline reinforcement learning is to learn an effective policy from the offline dataset without online interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Directly imitating from the datasets without proper constraint suffers from per- formance degrading caused by distribution shift between the behavior of the learnt policy and that of the behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Thus methods like Conservative Q-Learning(CQL) [Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020] are proposed to narrow the gap between the be- havior policy and the learnt policy, dealing with distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, traditional algorithms try to avoid entering the OOD states, while extrapolation error generated by distri- bution shift would still overestimate those actions that transit to out-of-distribution(OOD) states[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' ], weakening the reliabil- ity and robustness of the offline policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, it may not be a good idea to singly prevent entering OOD states while constrain nothing of the learnt policy at those OOD states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' A reliable policy is supposed to generate predictable consequences[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To be specific, once the agent entering an OOD state, it may predict its potential destinations and try to reach the ’safest’ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' But CQL, the representative work of the conventional offline RL, ignores the predictability for the transitions, overestimating the Q-values at the perturbed states which leads to state deviation[Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022] and unreliable OOD trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' This phenomenon could also be considered as the poor reliability caused by the lack of ro- bustness of the learnt policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Specifically, when the learnt policy is constrained to generate stable transitions from per- turbed states to the empirical next state distributions of the offline dataset, then the policy is robustly reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To allevi- ate this issue, State Deviation Correction(SDC)[Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022] trains the learnt policy by adding a regularization to minimize the MMD distance between the distributions of the next state generated by learnt policy and the offline dataset, so the agent could return to in-distribution states from perturbed states, enhancing the reliability and robustness of the learnt policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, SDC is too conservative because mechan- ically returning to in-distribution states may not be optimal behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' on the other, SDC ignores its generality with CQL, formed as an extra regularization added onto CQL, increasing the difficulty of tuning such a complex system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In this paper, we propose a novel and effective method, Contextual Conservative Q-Learning(C-CQL), to better uti- lize the offline contextual information for training a more ro- bust and reliable policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The main idea is to estimate the action distribution with the specific transition from the per- turbed state to the similar succeed state distribution with the corresponding unperturbed previous state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' We introduce an inverse dynamics model [Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2021] to estimate the action distribution most likely to explain the transition of two consecutive states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In detail, we first sample a (previous) state and its succeed state from the offline dataset and perturb the previous state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then, via the inverse dynamics model, we estimate the action distribution that is most suitable for the transition from the perturbed state to the unperturbed suc- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='01298v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='LG] 3 Jan 2023 Figure 1: The learnt policy generates C-CQL transition at perturbed state ˆs, reaching a trade-off between destining to in-distribution next state s′ and maximizing the one-step reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' ceed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To simulate the behavior of the inverse dynamics model, we overestimate the Q-value of the perturbed state and the action estimated by the inverse dynamics model to opti- mize the learnt policy indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In this manner, we declare the behavior of the learnt policy is robustly reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Besides, the overestimation for the Q-value of the perturbed state and the action generated by the inverse dynamics model implies to maximize the one-step reward additionally, which is more aggressive than traditional SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' As is shown in Figure 1, at perturbed states, the policy learnt by C-CQL generates transi- tion to reach a trade-off between SDC transition and an one- step greedy transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Theoretical results show that C-CQL could be considered as a generalization of CQL and SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' If we only use the un- perturbed dataset for training, C-CQL would degenerate into traditional CQL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' otherwise, C-CQL is approximately equiva- lent to reaching a trade-off between the EM distance version of SDC and one-step rewards, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' an aggressive SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' We implement our C-CQL with actor-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Exper- imental results show that C-CQL performs better than most of the traditional offline RL algorithms and SDC in a offline Mujoco control suite and a multi-level noisy Mujoco setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To sum up, our contributions can be summarized as fol- lows: We propose a novel and effective offline RL algorithm, Contextual Conservative Q-Learning(C-CQL), to learn a more robust and reliable policy via an inverse dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Theoretical results show that the proposed algorithm C- CQL could be considered as a generalization of Conser- vative Q-Learning and aggressive State deviation correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Sufficient experiments conducted show that C-CQL per- forms better than other SOTA methods and makes the agent generate more reliable trajectories in obervation- perturbed environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 2 Related work Conservatism in offline reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Offline reinforcement learning [Riedmiller, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2012] aims to learn a powerful decision making engine from an pre- viously collected dataset [Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To deal with the distributional shift between testing data and the offline datasets, previous methods introduce the conservatism by un- derestimating the values of OOD states to alleviate the over- estimation bias [Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Kuznetsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Siegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, these methods lack of the utilization of contextual information and the constraint of the behaviors at those perturbed or OOD states, lacking reliability and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' State deviation cor- rection [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022] builds a dynamics model and a transition model to enable the agent generate specific transi- tions at perturbed states, enhancing the reliability and robust- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Inverse dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' An inverse dynamics model I(a|s′, s) predicts the action distribution that could explain the transition between a given pair of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Inverse dynam- ics models haven been applied to improving generalization to real-world problems [Christiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2016], Markov repre- sentation learning [Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2021], defining intrinsic re- wards for exploration [Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In our work, the inverse dynamics model is considered as a contextual pol- icy that generates action distributions with predictable conse- quences for the supervision to train the learnt policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 3 Background Given a reinforcement learning problem, we usually model it as a Markov Decision Process (MDP) and it can be repre- sented by a tuple of the form (S, A, P, R, γ), where S is the state space, A is the action space, P is the transition probabil- ity matrix, R and γ are the reward function and the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' A policy is defined as π : S → A and trained to maximize the expected cumulative discounted reward in the MDP: max π E � ∞ � t=0 γR(st, π(at|st)) � (1) In general, we define a Q-value function Qπ(s, a) = E[�∞ t=0 γR(st, π(at|st))|s, a] to represent the expected cu- mulative rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Q-Learning is a classic method that trains the Q-value function by minimizing the Bellman error over Q[Watkins and Dayan, 1992]: Q ← arg min Q E � R(s, a) + γ max a′ Q(s′, a′) − Q(s, a) � (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 Conservative Q-Learning In offline setting, Q-Learning algorithms need to learn a Q- value function Qπ(s, a) and a policy π from a from dataset D, which is collected by a behavior policy πβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Since there is always a state distribution shift between the stationary state distributions of the learnt policy π and the behavior policy πβ, this basic recipe falls to estimate the Q-values for OOD state-action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' CQL try to underestimate the Q-values for OOD state-action pairs to prevent the agent from enter the OOD states[Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=", 2020]: Q ← arg min Q α · � Es∼D,a∼π(a|s)Q(s, a) − Es,a∼DQ(s, a) � + 1 2 � R(s, a) + γ max a′ Q(s′, a′) − Q(s, a) �2 (3) perturbed state out-of-distribution ΛS greedy transition one-step reward C-CQL transition perturb (a trade-off) SDC transition S S' in-distribution transition previous state in-distribution/dataset succeed statewhere α is a weight for CQL regularization." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 State Deviation Correction There is always a distributional shift between the empiri- cal distribution of the dynamics of the offline dataset ˆP and the true dynamics ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' For some perturbed state ˆs and a, if ˆP(s′|ˆs, a) = 0 while P(s′|ˆs, a) > 0, the agent may enter its unfamiliar state s′ when it executes action a at the state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' This phenomenon is defined as dynamics bias, which might lead to state deviation at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Besides, the approxi- mation error of the deep models would generate the deviation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To deal with this problem, State Deviation Correc- tion(SDC) proposed in [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022] expects to train the policy leading the agent to a predictable transition, which make the agent closer to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' SDC trains a dynamics model M and a transition model U and optimizes the policy as: π(·|s) = max π Ea∼π(·|s)Q(s, a) + λ � Es∼DMMD � M(·|ˆs, π(·|ˆs)), U(·|s) � − η � (4) where ˆs = s + β · ϵ, β is the magnitude and ϵ is a noise sam- pled from a Gaussian distribution N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, SDC ignores its relationship with CQL but is proposed as a regu- larization onto CQL, introducing an extra hyperparameter λ, which makes it difficult to tune such a complex system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In addition, SDC may be too conservative to mechanically force the agent returning to the dataset while ignoring the reward may be achieved at perturbed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 4 Contextual Conservative Q-Learning Consider a noisy state setting, that is, the agent should make decisions on noisy states during test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Traditional CQL fails to deal with this setting because it lacks consideration of context, leading to the uncertainty of the consequences of the decisions at perturbed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To deal with this, we in- troduce the Contextual Conservative Q-Learning(C-CQL) in detail in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' First, we introduce how to generate the mixed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then we introduce how to use the contextual module to supervise our policy to make propoer decisions at perturbed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Finally, the loss functions for implementa- tion would be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The figure of framework is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 Mixed Dataset Generation Given an offline dataset D, which consists of quadruples (s, a, r, s′), we perform data augmentation on it in this sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Given a quadruple (s, a, r, s′), we first perturb the state s by a noise ϵ with magnitude β and get an noisy state: ˆs = s + β · ϵ (5) where ϵ is sampled from a Gaussian distribution N(0, 1) and β is usually a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we could get the perturbed quadruples (ˆs, a, r, s′), and make them into a new perturbed dataset ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In particular, we do not perturb the next state s′, to preserve the destination we wish the agent to jump from ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then we combine the two datasets D and ˜D to form a new dataset Dtot, which consists of both perturbed and un- perturbed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Our work is based on the mixed dataset Dtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To be spe- cific, we have: Dtot = D + ˜D (6) and the quadruples in Dtot is noted by (˜s, a, r, s′), where {˜s} could be divided into unperturbed {s} and perturbed {ˆs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 Contextual Supervision The inverse dynamics model Iπβ(a|s, s′) is defined interms of the dynamics function P(s′|s, a) and the transition func- tion P(s′|s, πβ) via Bayes’ theorem: Iπβ(a|s, s′) = πβ(a|s)P(s′|s, a) P(s′|s, πβ) (7) where P(s′|s, πβ) = � a∈A P(s′|s, a)πβ(a|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' For each quadruple (˜s, a, r, s′) in Dtot, the task for the learnt policy π is to generate an action distribution that is suitable for the transition between ˜s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' This action dis- tribution could be estimated by the inverse dynamics model Iπβ(a|˜s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we overestimate the Q-value of the action sampled from Iπβ(a|˜s, s′) to indirectly guide the learnt policy to produce action distribution that could reach s′ with the best probability, while underestimate the Q-values of the other actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In this manner, we have: max Q E˜s,s′∼Dtot � Ea∼Iπβ (·|˜s,s′)Q(˜s, a) −Ea∼π(·|˜s)Q(˜s, a) � (8) By the way, the predecessor states in the mixed dataset D defined in (6) could be divided into two sets: perturbed states {ˆs} and unperturbed states {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then ˜s, s′ ∼ Dtot in (8) could also be considered as ˜s ∼ Dtot, s′ ∼ P(s′|s, πβ), where s is the original state of ˜s if ˜s ∈ {ˆs}, of which the items are perturbed, or s = ˜s if ˜s ∈ {s}, of which the items are not perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 Implementation We implement our method introduced in previous setions on an actor-critic framework, and name it as Contextual Conser- vative Q-Learning(C-CQL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The actor loss function is translated into: Lπ = − E˜s∼Dtot,ˆa∼π(·|˜s)[Q(˜s, ˆa)] (9) where Dtot is the mixed dataset defined in (6) and π is the learnt policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' It is the same form as original actor loss func- tion, except for the replacement of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' And the critic loss function is translated into: LQ = α · � E(˜s,s′)∼Dtot � Eˆatar∼Iπβ (·|˜s,s′)Q(˜s, ˆatar) −Eˆa∼π(·|˜s)Q(˜s, ˆa) �� +1 2Es,a,r,s′∼D � r + γ max a′ Q(s′, a′) − Q(s, a) �2 (10) where ˜s and s′ are included in the same quadruple sampled from Dtot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Iπβ(a|s, s′) is the inverse dynamics model trained on the dataset generated by the behavior policy πβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To be specific, we use a conditional variational auto- ecoder(CVAE) in [Kingma and Welling, 2014] to model the inverse dynamics network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The inverse dynamics network could sample the actions given current states and destination states as a supervision for the learnt policy to generate pre- dictable transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The second term is an one-step Bellman error, as shown in (2), and α is a weight for C-CQL regular- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To sum up, C-CQL optimizes both actor loss Lπ to up- date the policy network π and critic loss LQ to update our Q-networks, and outputs the learnt policy network π finally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The whole process is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Algorithm 1 Contextual Conservative Q-Learning Input: offline dataset Dtot, a pretrained inverse dynamics model I, maximal update iterations T, Parameter: policy network π, Q-networks Q1, Q2, noise- level β, Output: learnt policy network π 1: Initialize the policy network, Q-networks and the inverse dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 2: Let t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 3: while t < T do 4: Sample mini-batch of N samples (˜s, a, r, s′) from Dtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 5: Feed ˜s to the policy network π and get ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 6: Feed ˜s and s′ to the inverse dynamics model I and get ˆatar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 7: Update the policy network π according to (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 8: Update the Q-networks according to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 9: end while 10: return learnt policy network π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 5 Theoretical Justification In this section, we perform a theoretical analysis on our pro- posed C-CQL around Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' C-CQL is the generalization of CQL and the EM distance version of aggressive SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' First of all, a definition of aggressive SDC is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then, Assumption 1 is given on the properties of transition func- tion P(s′|s, πβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' From Theorem 1 and Proposition 3, we conclude that C-CQL approximately equals to the aggressive SDC based on Assumption 1 at perturbed states, and equals to CQL at unperturbed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Finally, we conclude Claim 1 from Proposition 2,3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 Aggressive State Deviation Correction The goal of State deviation correction(SDC) is to learn a pol- icy that generates stable transitions at perturbed states[Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The regularization of SDC is: min π λ � Es∼DD � M(·|ˆs, π(·|ˆs)), U(·|s) �� where M is the dynamics model and U is the transition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' D is some kind of distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, no evidence has shown that returning in-distribution state s′ from perturbed state ˆs is the optimal behavior to clone, because this may be too conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we consider relaxing the conser- vatism and a most intuitive way is to maximize the one-step reward at the same time: min π λ � Es∼DD � M(·|ˆs, π(·|ˆs)), U(·|s) � − Rπ(ˆs) � where Rπ(ˆs) = � a π(a|ˆs)R(ˆs, a) and R(s, a) is the reward function according to s, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' This could be condered as a trade- off between SDC and one-step reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', aggressive SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 Connection between CQL, aggressive SDC and C-CQL Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' (EM distance) The Earth-Mover(EM) distance of two distribution p and q can be defined as: EM � p(x) ����p(y) � = sup ∥f∥≤1 Ex∼p(x)f(x) − Ey∼q(y)f(y) (11) EM distance is also known as Wasserstein distance[Ar- jovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The transition function P(s′|s, πβ) could be considered as a function according to variable s, that is f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then we assume this function is continuous: ∀ϵ, ∃δ, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='∥s − ˆs∥ ≤ ϵ ⇒ ∥P(s′|s, πβ) − P(s′|ˆs, πβ)∥ ≤ δ and ∀s, s′, we have P(s′|s, πβ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' By Assumption 1, we could easily have: ∀ϵ, ∃δ, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='∥s − ˆs∥ ≤ ϵ ⇒ 1 − δ ≤ P(s′|s, πβ) P(s′|ˆs, πβ) ≤ 1 + δ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' (8) is equivalent to the expected EM distance between Iπβ(·|˜s, s′) and π(·|˜s): E˜s∼DtotEM � Es′∼P (s′|s,πβ)Iπβ(a|˜s, s′) ����π(a|˜s) � (12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The proof is easily to be conducted by the definition in Definition 1: max Q E˜s,s′∼Dtot � Ea∼Iπβ (·|˜s,s′)Q(˜s, a) − Ea∼π(·|˜s)Q(˜s, a) � ⇔ max Q E˜s∼DtotEs′∼P (s′|s,πβ) � Ea∼Iπβ (·|˜s,s′)Q(˜s, a) − Ea∼π(·|˜s)Q(˜s, a) � ⇔ max Q E˜s∼Dtot � Es′∼P (s′|s,πβ)Ea∼Iπβ (·|˜s,s′)Q(˜s, a) − Ea∼π(·|˜s)Q(˜s, a) � ⇔E˜s∼DtotEM � Es′∼P (s′|s,πβ)Iπβ(a|˜s, s′) ����π(a|˜s) � Complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Task name SAC BC BEAR BRAC-p BRAC-v CQL MOPO SDC C-CQL(Ours) Halfcheetah-random 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 28.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='4 Average-score 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 Average-ranking 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='7 Table 1: Results of C-CQL, SDC, SAC, BC, BEAR, BRAC, and CQL on offline Mujoco control suite tasks, on the normalized return metric, averaged over four seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Note that C-CQL performs better or similar to other methods on most of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' When training on an unperturbed dataset D, C-CQL is equivalent to CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The proof is performed on the equivalence of the C- CQL and CQL regularizations given an unperturbed state s: max Q Es′∼P (s′|s,πβ)Ea∼Iπβ (a|s,s′)Q(s, a) − Ea∼π(a|s)Q(s, a) ⇔ max Q � s′ P(s′|s, πβ) � a πβ(a|s)P(s′|s, a) P(s′|s, πβ) Q(s, a) − Ea∼π(a|s)Q(s, a) ⇔ max Q � a πβ(a|s)Q(s, a) − Ea∼π(a|s)Q(s, a) ⇔ max Q Ea∼π(a|s)Q(s, a) − Ea∼π(a|s)Q(s, a) ⇔ CQLregularization Complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proposition 2 shows that C-CQL regularization proposed in (8) has another form of the EM distance between Iπβ(·|˜s, s′) and π(·|˜s), which is used in subsequent content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proposition 3 shows that C-CQL degrades into CQL at the unperturbed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' And then, on the other, we need to show that C-CQL is equivalent to the aggressive SDC at the per- turbed states, or maybe approximately, to conclude Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Given a state s, its perturbed state ˆs and the behavior policy πβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Define Q(s, a) as a approximal Q-value function and V (s) as a approximal value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Given an inverse dynamics model Iπβ(a|s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' By Assumption 1 and Proposition 1, the following two optimization formulas are approximately equivalent: min π EM � Es′∼P (s′|s,πβ)Iπβ(a|ˆs, s′) ����π(a|ˆs) � (13) ∼ ⇐⇒ min π EM � P(s′|s, πβ) ����P(s′|ˆs, π) � − Ea∼π(a|ˆs)R(ˆs, a) (14) , that is, they imply the same policy approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The proof of Theorem 1 is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The- orem 1 shows that the policy optimized by our method is ap- proximately equivalent to minimizing the objective of an EM distance version of SDC while maximizing the expected im- mediate reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', aggressive SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The gap between the upper and lower bounds mentioned in Theorem 1 is less than 2δRπβ(ˆs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' If we have ∥R∥∞ ≤ 1, then the gap is less than 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' This proposition can be easily obtained by subtracting the lower bound from the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Although there is actually a little gap between (13) and (14) which is caused by the existence of the continuous gap δ, as is shown in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' When δ is large, (13) and (14) are not strictly equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, by the conservative selection of the magnitude of the noise ϵ, the δ would be a quite tiny value in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we consider that the result of Theorem 1 is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Combing Proposition 2, Proposition 3 and Theorem 1, we conclude Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 6 Experiments In this section, we will describe the dataset D4RL[Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020] and information about experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' We con- duct a comparative experiment on the Mujoco datasets in the D4RL benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then we design a noisy Mujoco setting to test the robustness of CQL, SDC and C-CQL we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Then, to understand the utility of our method, we visualize the state distributions of CQL, SDC and our methods in the two Figure 2: Performance decrease of CQL, SDC and C-CQL’s performance on the noisy Mujoco setting compared with noisless Mujoco tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The best method of each benchmark is marked by red star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' noisy environments with obvious performance gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Finally, we perform a sensitive analysis to learn how the hyperparam- eters β in (5) and α in (8) affect the performance of C-CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To sum up, the experiments are designed to answer the fol- lowing: Does C-CQL outperform the state-of-the-art methods across a variety of tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Is C-CQL robust for different levels of noise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Is C-CQL able to generate reliable trajectories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', in- distribution trajectories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' D4RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' There are three types of control environments we use in this paper: Hopper, Halfcheeta and Walker2d benchmarks in D4RL and five kinds of datasets for each type of benchmark: ‘random’, ‘medium’, ‘medium-replay’, ‘medium-expert’ and ‘expert’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The ‘random’ dataset is gen- erated by a randomly initialized policy and the ‘medium’ dataset consists of the data collected by a soft actor- critic(SAC)[Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2018] policy insufficiently trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The ‘medium-replay’ dataset collects the data in the replay buffer when the behavior policy reaches 50% perfor- mance of the expert policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The ‘expert’ dataset is produced by a completely trained SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The ‘medium-expert’ is a mix- ture of ‘medium’ dataset and ‘expert’ dataset in half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' All the levels of datasets mentioned above contain 1,000,000 sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' CQL C-CQL SDC Task name score score score Halfcheetah-noisy-s 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 Halfcheetah-noisy-m 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='4 Halfcheetah-noisy-l 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 Walker2d-noisy-s 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='7 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 Walker2d-noisy-m 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='4 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 Walker2d-noisy-l 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='7 Hopper-noisy-s 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='4 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 Hopper-noisy-m 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 Hopper-noisy-l 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 Table 2: Results of CQL, SDC and C-CQL on the noisy Mujoco setting, on the normalized return metric, averaged over four seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='1 Offline Mujoco Control In this section, we compare our C-CQL with several signif- icant methods, including BC, SAC[Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2018], BEAR[Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2019b], BRAC[Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2019], MOPO[Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2020] and SDC[Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022], on the D4RL dataset in the Mujoco benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The results for SDC, SAC, BC, BEAR, BRAC, and CQL are obtained by [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' It is shown that C-CQL we proposed achieve the similar or better performance than other methods on most tasks, and acheve the highest average score and the best average ranking among these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In addition, our methods outperforms the other methods by a large margin in ‘Walker2d-medium’, ’Hopper-medium’, ‘Hopper-medium-replay’ and ‘Walker2d- medium-expert’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The details of training process is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 Noisy Mujoco setting In this setting, we add different magnitude of noise on the three standard Mujoco benchmarks: Halfcheetah, Walker2d and Hopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The magnitude of noise could be devide into 3 levels: slight(s), moderate(m) and large(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The magnitude of slight(s) noise is about 1e − 3 in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The mod- erate(m) magnitude is range from 1e − 2 to 2e − 2 and the large(l) is range from 5e − 2 to 1e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' We use the policies trained by CQL, SDC and CQL on ‘medium-expert’ datasets, and then run in the noisy bench- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The score and performance decrease of these poli- cies in the 9 noisy benchmarks are respectively recorded as is shown in Table 2 and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The performance de- crease refers to the percentage reduction from the scores of CQL, SDC and C-CQL in observation-noisy environment, as is shown in Table 2 to those shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' We could conclude that our method, C-CQL, exceeds CQL and SDC, achieving much less performance degradation in most noisy benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Besides, C-CQL is not strictly anti-explored as SDC, so C-CQL is able to infer the reward at those OOD state, which may be the reason of the less performance degradation than SDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' To help understand the reason why our approach performs better, we visualize the state distributions of some bench- marks, as is shown in Figure 3, with t-Distributed Stochas- tic Neighbour Embedding(t-SNE)[Hinton and Roweis, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The green points note the offline dataset, while the red points CQL/decrease(%) SDC/decrease(%) Ours/decrease(%) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 大 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 大 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 大 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 大 大 大 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 大 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 halfcheetah-noisy-s halfcheetah-noisy-m halfcheetah-noisy-l Walker2d-noisy-s Walker2d-noisy-m Walker2d-noisy-l hopper-noisy-s hopper-noisy-m hopper-noisy-l 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0Figure 3: The visualizations of the state distributions of exper- iments on the ‘Walker2d-noisy-l’ and ‘Walker2d-noisy-m’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The (a), (d) are CQL(red) vs dataset(green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' the (b), (e) are SDC(red) vs dataset(green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' and the (c), (f) are our method C-CQL(red) vs dataset(green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' in (a), (d) note the data generated by the CQL policy, in (b), (e) note the data generated by the SDC policy and in (b), (d) note the data generated by the C-CQL(Ours) policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The re- sults show that CQL tends to deviate from the training data when it makes decisions at perturbed states, because it gener- ates a lot of OOD samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' However, SDC and our method C-CQL produces much less OOD samples and always fol- lows the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we conclude that the method we propose gener- ate more robust and reliable trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='3 Sensitive Analysis In this section, we perform sensitive analysis on two key hy- perparameters: β in (5) and α in (8) to evaluate how these hyperparameters influence the performance of C-CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Sensitive analysis on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The β in (5) is the hyperparameter that control the magnitude of the noise that add to our mixed dataset Dtot for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Its influence to C-CQL is as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The performance of the learnt policy is best when the β is 1e-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' takes second place when the β equals to 5e-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' is worst when the β is 1e-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' From the results, we conclude that when β is too large, the agent would fail to be well trained over some or all seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we ought to conservatively choose the value of β and experience suggests the β should be no bigger than 2e-3 for ’walker2d’ benchmark, 5e-3 for ’hopper’ and ’3e-3’ for ’halfcheetah’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Figure 4: The sensitive results of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' (a) is on ’walker2d-expert’ and (b) is on ’hopper-expert’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Figure 5: The sensitive results of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' (a) is on ’walker2d-medium- expert’ and (b) is on ’hopper-medium-expert’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Sensitive analysis on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The α in (8) is the weight for the C-CQL regularization that affects the magnitude of conser- vatism of C-CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The sensitivity of this hyperparameter to the performance of C-CQL is as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' The learnt policy performs best when α equals to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' slight poor when α is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' worst when α is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Therefore, we can conclude that if α is too small, C-CQL may fail to train the policy, and experience suggests that the α should range from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' 7 Conclusion In this paper, we proposed a simple and novel yet effective method, Contextual Conservative Q-Learning, to learn a ro- bustly reliable policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' With the supervision of a single in- verse dynamics model, the learnt policy is able to generate consequence-predictable and consequence-stable action dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Besides, we have theoretically shown that our method is a generalization of CQL and the aggressive SDC for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' Finally, the C-CQL achieves the SOTA performance across multiple offline Mujoco settings, and we suppose the robustness & reliability of the offline reinforce- ment learning algorithms would be critical for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' SDC CQL Training data Training data Training data SDC C-CQL Training data Training data Training dataβ=le-2 β=1e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0 β=5e-3 β=5e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='0- β=1e-3 β=1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content='8 SCor scor 0.' metadata={'source': 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timization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Sys- tems 33: Annual Conference on Neural Information Pro- cessing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=', 2022] Hongchang Zhang, Jianzhun Shao, Yuhang Jiang, Shuncheng He, Guanwen Zhang, and Xi- angyang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' State deviation correction for offline reinforce- ment learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' In Thirty-Sixth AAAI Conference on Arti- ficial Intelligence, AAAI 2022, pages 9022–9030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} +page_content=' AAAI Press, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAzT4oBgHgl3EQfWPyB/content/2301.01298v1.pdf'} diff --git a/R9E2T4oBgHgl3EQfBwbf/vector_store/index.pkl b/R9E2T4oBgHgl3EQfBwbf/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e9b894d411ab7a15d22dfe70e5ed9f138342e1a1 --- /dev/null +++ b/R9E2T4oBgHgl3EQfBwbf/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:da2bbbed9a1f8350a4857bab0a527d4fb9c0f7b84d5e27d8007741a65ba15a0d +size 253722 diff --git a/TNFKT4oBgHgl3EQfkS62/content/tmp_files/2301.11849v1.pdf.txt b/TNFKT4oBgHgl3EQfkS62/content/tmp_files/2301.11849v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7020d12e5d058de6e61aade6a2834220e9ec58d --- /dev/null +++ b/TNFKT4oBgHgl3EQfkS62/content/tmp_files/2301.11849v1.pdf.txt @@ -0,0 +1,1260 @@ +arXiv:2301.11849v1 [cs.GT] 27 Jan 2023 +Complexity of equilibria in binary public goods games on +undirected graphs +Max Klimm1 +Maximilian J. Stahlberg1 +1Technische Universit¨at Berlin, Germany +{klimm, stahlberg}@math.tu-berlin.de +Abstract +We study the complexity of computing equilibria in binary public goods games on undi- +rected graphs. In such a game, players correspond to vertices in a graph and face a binary +choice of performing an action, or not. Each player’s decision depends only on the number +of neighbors in the graph who perform the action and is encoded by a per-player binary +pattern. We show that games with decreasing patterns (where players only want to act up +to a threshold number of adjacent players doing so) always have a pure Nash equilibrium and +that one is reached from any starting profile by following a polynomially bounded sequence +of best responses. For non-monotonic patterns of the form 10k10∗ (where players want to +act alone or alongside k +1 neighbors), we show that it is NP-hard to decide whether a pure +Nash equilibrium exists. We further investigate a generalization of the model that permits +ties of varying strength: an edge with integral weight w behaves as w parallel edges. While, +in this model, a pure Nash equilibrium still exists for decreasing patters, we show that the +task of computing one is PLS-complete. +1 +Introduction +Public goods are resources that can be freely accessed and simultaneously used by many in- +dividuals. In the physical world, they comprise abundant natural resources like sunlight and +breathable air alongside artificial goods such as cultural heritage, public art, or early warning +systems. Public goods have become ubiquitous in the information age, when data can be re- +produced at a negligible cost, yet remains valuable to its users. Examples from this domain +are open source software, public databases, radio transmissions, and the diffusion of scientific +knowledge through open channels. However, not all public goods are universal: a population +warning system profits a region, herd immunity to a virus is enjoyed on the basis of human con- +tact, and a scientific manuscript may be legible only to a group of peers. For these scenarios, +the possibility of access may be represented by a graph, where a vertex can only enjoy goods +that are provided by itself or by one of its neighbors. In the classical setting of Bramoull´e and +Kranton [2], each vertex may produce any amount of a fixed good and experiences utility based +on the total amount of the good that is available in the closed neighborhood, minus a cost +associated with local production. In the public goods game, vertices aim to configure their own +production to maximize the resulting utility. Whether such strategic decisions allow for a stable +arrangement and, if so, how one can be found and proposed to the actors are natural questions +for policy makers. Bramoull´e and Kranton show for strictly concave benefits and linear costs +the existence of specialized equilibria, in which the production of each vertex is either zero or +a fixed, positive amount. This motivates the study of a binary variant of the game, wherein +vertices have only two options: to be active and produce (one unit of) the good, or to remain +inactive [6, 7, 9, 13, 14]. In general, costs and utility functions can be defined in a way that +allows vertices to be indifferent to this choice. Yang and Wang [13] use this to show that it +1 + +Complexity +Pattern class +Reference +O(1) +1+0+ +Theorem 3.3† +O(poly(n)) +(10)∗ +[11, Lemma 3.3] +NP-complete +10+10∗ +Theorem 5.11 +(10)+10∗ +Corollary 5.12‡ +11.∗0.∗10∗ +[6, Theorem 5] +10+11.∗0+ +[6, Theorem 6] +Table 1: Complexity of deciding the existence of equilibria in the homogeneous binary public +goods games on undirected graphs, for varying classes of non-trivial (not starting with zero) +best response patterns. The pattern class notation is defined in Section 2.1. †The special case +of 10∗ has been shown in [2], that of 110∗ in [6]. ‡Uses [6, Theorem 7]. +is NP-hard to decide whether an equilibrium exists, already for a monotone benefit function +common to all agents. In response, Maiti and Dey [9] introduce—and Gilboa and Nisan [6] +shift the focus towards—restricted games in which every vertex i always has a strict preference +with respect to its options. As this preference depends only on the number of active neighbors, +we may encode it by an infinite binary sequence T i, called a best response pattern [6], where +T i +ℓ is the activity response of i to exactly ℓ active neighbors. When there is a single pattern +common to each vertex, this is called the fully homogeneous case [14], otherwise we call the +game inhomogeneous. +1.1 +Our results +We investigate the computational hardness of deciding whether a game admits a pure Nash +equilibrium, for two natural classes of patterns for which this question was unresolved [6]. +Decreasing patterns are those that start with a positive number of ones and are zero there- +after. We show that for this class, even the inhomogeneous binary public goods game is equiva- +lent to a congestion game, which implies that best-response dynamics converge to a pure Nash +equilibrium in polynomial time. For an extension of the game in which edges have an integral +weight, however, we show that finding an equilibrium is PLS-complete. +We call a picky pattern one where Tℓ = 1 if and only if ℓ ∈ {1, k + 1}, for some k ≥ 1. Here, +vertices wish to act only alone or alongside k + 1 neighbors. For example, when k = 1 for every +vertex, then the Graph induced by the active vertices of an equilibrium consists of singletons and +cycles. We show that deciding the existence of such an equilibrium is an NP-complete problem, +even in the fully homogeneous case. A notable corollary is the following: While deciding the +infinite alternating sequence T = (1, 0, 1, 0, . . .) is in P [11], any truncation of this sequence in +which it is zero after alternating at least two times corresponds to a hard problem. +Table 1 shows a summary of our results for homogeneous patterns alongside previous results. +1.2 +Related work +Let G = (V, E) be an undirected graph and denote the open neighborhood of vertex i ∈ V by +N(i) := {j ∈ V : {i, j} ∈ E}. Bramoull´e and Kranton [2] study a model where each vertex is +a player whose strategy is to pick a production effort si ∈ Si := R≥0. For a strategy vector +s = (s1, . . . , sn), the utility of each player is given by ui(s) := b(si + � +j∈N(i) sj) − csi, where +b: R≥0 → R≥0 is a benefit function with b(0) = 0, b′ > 0 and b′′ < 0, and where c > 0 denotes +the cost of production. The authors show the existence of a particular kind of Nash equilibrium, +s with si ∈ {0, (b′)−1(c)}, that they call a specialized equilibrium. They translate this result also +to the more general case where bi and ci depend on the player i. +2 + +Yu et al. [14] study a binary variant of the game with Si := {0, 1} and where they allow for +arbitrary non-decreasing and player-specific bi, and for player-specific ci. They aim to show that +in this setting, deciding the existence of a pure Nash equilibrium is an NP-complete problem. +Yang and Wang [13] point out a flaw in the proof and provide a corrected version. They further +extend this result to homogenous games where all functions bi and all parameters ci are equal. +As noted by Gilboa and Nisan [6], the hardness result of Yang and Wang hinges on the fact +that some players are indifferent to their binary decisions, i.e., b(k + 1) − b(k) = c for some +value of k. Gilboa and Nisan exclude this possibility by studying best-response patterns. These +are binary sequences that explicitly give the best response of a player i based on the number +of neighbors j ∈ N(i) with sj = 1. The authors settle the complexity for a number of patterns, +including some that start with 0, for which they consider non-trivial equilibria. They also show +that hard patterns that start with 1 remain hard when they are prefixed with (1, 0). See Table 1 +for results on patterns starting with 1. Maiti and Dey [9] provide a parametrized complexity +perspective wherein they study natural graph parameters such as the maximum degree and the +treewidth. +L´opez-Pintado [8] initiates the study of public goods games on directed graphs. The author +analyzes a restricted binary model where each agent i prefers to play si = 1 if and only if none +of their in-neighbors j plays sj = 1. Papadimitriou and Peng [11] perform a complexity analysis +of the binary variant on directed graphs for more general utility functions. They investigate in +particular the setting where all players share a utility function that is monotone in the number +of in-neighbors j who play sj = 1. This allows for a pattern-wise analysis akin to that of Gilboa +and Nisan [6]. +Here the authors provide a complete characterization by showing that only +the trivial all-zero and all-ones patterns as well as the infinite alternating sequence starting +with 1 are polynomial-time decidable. They further provide a PPAD-hardness result for the +computation of approximate mixed Nash equilibria. +Kempe et al. [7] are interested in modifying the graph in such a way that one element of +a given set of strategy vectors is a pure Nash equilibrium of the binary game. The motivation +behind this is to enforce a socially preferable equilibrium through the intervention of a policy +maker. Finally, Galeotti et al. [4] study a variant of the game where players have only partial +knowledge of the network structure. +2 +Preliminaries +For n ∈ Z≥1 we write [n] short for {1, . . . , n}. For a set A, we write 1A for the indicator function +of A, that is 1A(x) := 1, if x ∈ A, and 1A(x) := 0, if x ̸∈ A. For an undirected and simple graph +G = (V, E), we write ∆(G) for its maximum vertex degree, deg(v) for the degree of v ∈ V , +G[X] for the subgraph induced by X ⊆ V , NG(v) for the open neighborhood of v ∈ V , and +NG(X) := {v ∈ V \ X | ∃x ∈ X : {v, x} ∈ E} for the open neighborhood of X ⊆ V . We drop +suffixes when the graph in question is clear. +When s ∈ {0, 1}n with n = |V | denotes a strategy profile of the binary public goods game +played on G = (V, E) with best-response pattern T, then we say that a vertex v ∈ V is active +(in s), if sv = 1, and inactive, if sv = 0. We further write degs(v) := |{u ∈ NG(v) | su = 1}| +and call this the active degree of v. Finally, we say that a vertex v ∈ V has a stable strategy +assignment (under s) if it plays its best response, that is sv = Tdegs(v). +2.1 +Response pattern notation +We use regular expressions (REs) over the alphabet A := {0, 1} to describe classes of best +response patterns. We first define their syntax: Every c ∈ A and the dot . are RE. If e and f +are REs, then so are the concatenation ef, the k-fold concatenation of e with itself, (e)k, the +zero-or-more operation (e)∗, and the one-or-more operation (e)+. Parentheses may be omitted: +3 + +superscripts then take precedence over concatenation, for example 10∗ = 1(0)∗ ̸= (10)∗. +The language L(e) generated by an RE e is the following: For the dot we have L(.) := +{(a) | a ∈ A}. For c ∈ A, it is L(c) := {(c)}. For REs e and f, it is L(ef) := L(e) × L(f), +L(ek) := L(e)k, L(e+) := �∞ +i=1 L(e) and L(e∗) := L(e+) ∪ {()}, where () is the empty sequence +over A and where (c) = c is a sequence of length one. +Extensible REs end in (e)∗ or (e)+ for some sub-expression e. For the set of best response +patterns P(e) generated by an extensible RE e, we have (x)∞ +i=1 ∈ P(e) if and only if there +is a threshold N such that (x)n +i=1 ∈ L(e) for all n ≥ N. For example, P(1∗0+) = P(1∗00∗) +contains all infinite sequences that start with a nonnegative, finite number of ones and are zero +thereafter, while P(1∗0∗) contains in addition the infinite sequence of ones. In the following, we +write T ∈ e short for T ∈ P(e). We also write T = e instead of T ∈ e to signify that |P(e)| = 1. +2.2 +Best-response games, strategic games, and consistent representations +In the following we make explicit the relation between the binary public goods game that +is specified by the players’ best-response behaviors and the classical definition where players +associate a utility value with their two choices. This work adopts the former representation as +a best-response game that captures precisely the strict [9] public goods games on graphs. To +still leverage established notions of game equivalence that are based on utility functions, we +introduce the notion of a consistent representation of a best-response game as a strategic game +in which players experience utility. We start the introduction of these concepts with the class +of best-response games: +Definition 2.1 (Best-response game). A best-response game (BRG) is described by a triple +(n, (Si)n +i=1, (βi)n +i=1) where n is the number of players, Si is the strategy set of player i ∈ [n], +and βi : �i−1 +j=1 Sj × �n +j=i+1 Sj → Si is a function such that βi(s−i) denotes the best response of +player i given that the other players play s−i. +For a strategy profile s ∈ �n +j=1 Sj and i ∈ [n], we write s−i short for �i−1 +j=1{sj}×�n +j=i+1{sj} +and we call s a pure Nash equilibrium (PNE) if βi(s−i) = si for all i ∈ [n]. +It is straightforward to formalize the binary public goods game as a BRG. +Definition 2.2 (Binary public goods game). For an undirected simple graph G = (V, E) with +n vertices V = [n] and for best-response patterns T i = (τ i +ℓ)∞ +ℓ=1 for all i ∈ V , we call the +best-response game (n, ({0, 1})n +i=1, (βi)n +i=1) with βi(s−i) := T i +xi(s−i) and xi(s−i) := � +j∈NG(i) sj +a (binary) public goods game (PGG). We also refer to just G and the T i as a PGG. +If T i = T for all i ∈ V and for some pattern T, then we refer to the associated PGG as a +homogeneous PGG with pattern T. +While in a best-response game a player’s response is defined uniquely, in strategic games +players choose a strategy among all that maximize a utility function: +Definition 2.3 (Strategic game). A strategic game is a triple (n, (Si)n +i=1, (ui)n +i=1) where n is +the number of players, Si is the strategy set of player i ∈ [n], and ui : �n +j=1 Sj → Q is a +function such that ui(s) denotes the utility experienced by player i ∈ [n] when the strategy +profile s ∈ �n +j=1 Sj is played. +For a strategy profile s ∈ �n +j=1 Sj, i ∈ [n], and a ∈ Si, we write (s−i, a) short for �i−1 +j=1{sj}× +{a} × �n +j=i+1{sj}. We call a strategy profile s ∈ �n +j=1 Sj with ui(s) ≥ ui(s−i, a) for all i ∈ [n] +and all a ∈ Si a pure Nash equilibrium (PNE). +With a suited utility function, we can represent any BRG as a strategic game with equivalent +best-response dynamics (cf. [9]): +4 + +Definition 2.4 (Consistent representation). Let A = (n, (Si)n +i=1, (βi)n +i=1) be a BRG. We call a +strategic game B = (n, (Si)n +i=1, (ui)n +i=1) a consistent representation of A if for all s ∈ �n +j=1 Sj +and all i ∈ [n], it is ui(s−i, a) > ui(s−i, a′) for all a′ ∈ Si \ {a} where a := βi(s−i) is the best +response of i in A. +Consistent representations are closely related to the notion of a weak isomorphism between +strategic games [3]. In particular, they also preserve the structure of pure Nash equilibria: +Lemma 2.5. Let A = (n, (Si)n +i=1, (βi)n +i=1) be a BRG and B = (n, (Si)n +i=1, (ui)n +i=1) a consistent +representation of A. Then, s ∈ �n +j=1 Sj is a PNE of A if and only if s is a PNE of B. +Proof. Let first s be a PNE of A and consider a player i ∈ [n] of A and their best response +a := βi(s−i). Since A is a PNE, it is a = si. As further B is a consistent representation, +ui(s) = ui(s−i, si) = ui(s−i, a) > ui(s−i, a′) +for all a′ ∈ Si \ {a}. Thus, ui(s) ≥ ui(s−i, a′′) for all a′′ ∈ Si, so s is a PNE of B. +Let next s be a PNE of B and consider a player i ∈ [n] of B playing a := si. Assume towards +a contradiction that si ̸= βi(s−i) =: b. Since B is a PNE, ui(s−i, a) = ui(s) ≥ ui(s−i, a′) for all +a′ ∈ Si. From b ∈ Si, it follows in particular that ui(s−i, a) ≥ ui(s−i, b). As B is a consistent +representation and a ̸= b, ui(s−i, b) > ui(s−i, a), a contradiction. Hence, si = βi(s−i), so s is a +PNE of A. +We are further interested in the dynamics that may lead to a PNE: +Definition 2.6 (Better-response sequence). For a best-response game (n, (Si)n +i=1, (βi)n +i=1), a +better-response sequence is a sequence of strategy profiles (si)N +i=1 such that for all i ∈ [N − 1], +it is si+1 = (si +−j, a) for some j ∈ [n] and a ∈ Sj with βj(si) = a and a ̸= si +j. +For a strategic game (n, (Si)n +i=1, (ui)n +i=1), a better-response sequence is a sequence of strategy +profiles (si)N +i=1 such that for all i ∈ [N − 1], it is si+1 = (si +−j, a) for some j ∈ [n] and a ∈ Sj +such that uj(si +−j, a) > uj(si). +It is immediate from Definition 2.4 that consistent representations preserve better-response +sequences: +Lemma 2.7. If A is a BRG and B a consistent representation of A, then any better-response +sequence in A is also a better-response sequence in B. +2.3 +Game isomorphisms +We will make use of the following notion of equivalence of strategic games: +Definition 2.8 (Strong isomorphism [3]). Two strategic games A = (n, (Si)n +i=1, (ui)n +i=1) and +B = +� +n, (S′ +i)n +i=1, (u′ +i)n +i=1 +� +are called (strongly) isomorphic when there are bijections π: [n] → [n] +and φi : Si → S′ +π(i) for all i ∈ [n] such that for all players i ∈ [n] (of A) and strategy profiles +s ∈ �n +j=1 Sj, it is ui(s) = u′ +π(i) (φ(s)) where φ(s) := +� +φπ−1(j)(sπ−1(j)) +�n +j=1. +Note that for π = id[n], it is φ(s) = (φj(sj))n +j=1. Informally, an isomorphism describes a +one-to-one mapping between the players and strategy profiles of two games that preserves the +players’ utility and, by extension, the players’ strategy preferences and the PNE structure of +both games. In particular: +Lemma 2.9. If two games A and B are isomorphic and A admits a PNE, then so does B. +Proof. If A = (n, (Si)n +i=1, (ui)n +i=1) has a PNE s ∈ �n +i=1 Si, then ui(t) ≤ ui(s) for all i ∈ [n] and +t ∈ �n +j=1 Sj. As A and B are isomorphic, there exist bijections π and φ such that +u′ +π(i) (φ(t)) = ui(t) ≤ ui(s) = u′ +π(i) (φ(s)) +for all i and t as above. As both π and φ are surjective, s′ is a PNE of B. +5 + +2.4 +Encoding +For the homogeneous PGG, we assume in line with previous work that the common best response +pattern is part of the problem definition and not of the problem input. For the inhomogeneous +variant, this approach is not adequate as each player might have a distinct pattern, so here we +assume that the patterns are part of the input and give the encoding explicitly. +3 +Existence of equilibria for decreasing patterns +Decreasing best-response patterns are those that begin with a positive number of ones and are +zero thereafter. We show that for this family of patterns, formally P(1+0+), the public goods +game is equivalent to a congestion game: +Definition 3.1 (Congestion game). Let E be a set of goods equipped with a delay function +de : Z≥1 → Q for every e ∈ E. Then, we call a strategic game (n, (Si)n +i=1, (ui)n +i=1) with Si ⊆ E +and ui(s) = − � +e∈si de(xs(e)) with xs(e) := �n +j=1 1sj(e) for all i ∈ [n] a congestion game. +More precisely, we show equivalence in the following sense: +Proposition 3.2. The binary public goods game on undirected graphs with best-response pat- +terns T i ∈ 1+0+, i ∈ V , has a consistent representation that is isomorphic to a congestion +game. +Proof. Let G = (V, E) be an instance of the PGG with patterns T i = 1ki0∗, ki ≥ 1, for all +i ∈ V = [n]. Define the strategic game Γ := (n, (Si)n +i=1, (ui)n +i=1) with +Si := {0, 1} and +ui(s) := +� +−(ki − 1 +2), +if si = 0, +− degs(i), +if si = 1, +(1) +for all i ∈ V . +We first show that Γ is a consistent representation of G. Let s ∈ {0, 1}n be a strategy profile +and i ∈ V a player of G. Let further a := βi(s−i) be the best response of i and a′ := 1 − a its +unique alternative. We show that then ui(s−i, a) > ui(s−i, a′). If a = 0, then degs(i) ≥ ki, thus +ui(s−i, a) = − +� +ki − 1 +2 +� +> −ki ≥ − degs(i) = ui(s−i, a′). +On the other hand, if a = 1, then degs(i) ≤ ki − 1 and +ui(s−i, a) = − degs(i) ≥ −(ki − 1) > − +� +ki − 1 +2 +� += ui(s−i, a′). +Consider next the congestion game C = +� +n, (S′ +i)n +i=1, (u′ +i)n +i=1 +� +defined by the set of goods +V ∪ E with delays dv(x) := kv − 1 +2, for v ∈ V , and de(x) := x − 1, for e ∈ E, and by the +strategies S′ +v := {{v}, {e ∈ E | v ∈ e}} for all v ∈ V . The strategy {v} corresponds to v +remaining inactive, which has no effect on adjacent vertices, while {e ∈ E | v ∈ e} corresponds +to v being active, which makes being active more expensive for v’s neighbors by “congesting” +incident edges. +We show that C is isomorphic to Γ as witnessed by the bijections π := id[n] and φv : Sv → S′ +v +with φv(0) = {v} and φv(1) = {e ∈ E | v ∈ e} for all v ∈ V . To this end, let v ∈ V be a player +and s ∈ �n +i=1 Si a strategy profile of Γ. If sv = 0, then φv(sv) = {v}, and thus +uv(s) = − +� +kv − 1 +2 +� += −dv(xs({v})) = u′ +v(φ(s)). +6 + +If sv = 1, then φv(sv) = {e ∈ E | v ∈ e}, so that +uv(s) = +− degs(v) += +− +� +{v,i}∈E +si += +− +� +{v,i}∈E +��� +� +j ∈ [n] \ {v} | sj = 1 ∧ {v, i} ∈ {e ∈ E | j ∈ e} +� +�� +� +j=i as j̸=v +���� += +− +� +{v,i}∈E +|{j ∈ [n] | sj = 1 ∧ {v, i} ∈ {e ∈ E | j ∈ e}}| ++ +� +{v,i}∈E +��� +� +j ∈ {v} | sj = 1 ∧ {v, i} ∈ {e ∈ E | j ∈ e} +� +�� +� +true as j=v +���� +by the definitions of uv for sv = 1, and degs. It follows that +uv(s) = +− + + � +{v,i}∈E +|{j ∈ [n] | {v, i} ∈ φj(sj)}| − degG(v) + + += +− +� +{v,i}∈E + + +n +� +j=1 +1φj(sj)({v, i}) − 1 + + +by the definition of φj for sj = 1. Further, +uv(s) = +− +� +{v,i}∈E +d{v,i} + + +n +� +j=1 +1φj(sj)({v, i}) + + += +− +� +e∈{e∈E|v∈e} +de +� n +� +i=1 +1φ(s)i(e) +� +follows from the definition of de for e ∈ E. Finally, +uv(s) = +− +� +e∈φv(sv) +de +� +xφ(s)(e) +� += u′ +v(φ(s)), +which concludes the proof. +This observation allows us to state the main theorem of this section, which answers the first +out of two open questions in [6]: +Theorem 3.3. The binary public goods game on undirected graphs with best-response patterns +T i ∈ 1+0+, i ∈ V , always admits a PNE; the associated decision problem is thus in P. +Proof. As every congestion game has a PNE [12], the claim follows from Lemma 2.9 and Propo- +sition 3.2. +Additionally, the representation as a congestion game brings with it convergence guarantees: +Corollary 3.4. In the game of Theorem 3.3, best-response dynamics converge to a PNE after +at most O(n2) improving steps. +7 + +Proof. Let kmax := maxi∈V ki. Any vertex can have at most ∆(G) active neighbors, so the game +for kmax > ∆(G)+1 is equivalent to the one for kmax = ∆(G)+1. Let thus kmax ≤ ∆(G)+1 ≤ n, +without loss of generality. +The congestion game C in the proof of Proposition 3.2 is an exact potential game [10] with +potential function +Φ(s) = +� +g∈V ∪E +xs(g) +� +ℓ=1 +de(ℓ) = +� +{u,v}∈E +susv + +� +v∈V +� +kv − 1 +2 +� +sv ≤ |E| + kmax|V |. +From de(ℓ) ≥ 0 for all ℓ ≥ 1 and e ∈ V ∪E, it follows that Φ(s) ≥ 0 for all s ∈ � +i∈[n] S′ +i. On the +other hand, it is Φ(s) ≤ |E| + kmax|V | ≤ 2n2. As 2Φ is integral by construction, these bounds +imply that any decreasing sequence Φ(s1) > . . . > Φ(sN) has length at most O(n2). The claim +follows as any better-response sequence in G corresponds to a better-response sequence in C, +which has decreasing potential. +4 +Hardness of decreasing patterns with weighted edges +We next investigate a natural extension of the public goods game, in which ties can have varying +importance: +Definition 4.1 (Weighted binary public goods game). For an undirected simple graph G = +(V, E) with n vertices V = [n], edge weights we ∈ Z≥1 for all e ∈ E, and best-response +patterns T i = (τ i +ℓ)∞ +ℓ=1 for all i ∈ V , we call the best-response game (n, ({0, 1})n +i=1, (βi)n +i=1) with +βi(s−i) := T i +xi(s−i) and xi(s−i) := � +j∈NG(i) w{i,j}sj an (edge-)weighted binary public goods +game. +We first notice that a straightforward adaption of the proof of Proposition 3.2 yields that +also games with weighted edges are isomorphic to a congestion game: +Proposition 4.2. The weighted binary public goods game on undirected graphs with best- +response patterns T i ∈ 1+0+, i ∈ V , has a consistent representation that is isomorphic to +a congestion game. +Proof (Sketch). We use the same construction as in the proof of Proposition 3.2 except that the +utilities in the definition of Γ (Eq. (1)) are given by ui(s) := −xi(s−1) for si = 1 and the delay +functions are defined as de(x) := we(x − 1) for all e ∈ E. +In Corollary 3.4, we have shown that best-response dynamics converge in polynomial time +to a PNE for unweighted binary public goods games. In contrast, we show that for weighted +games, the computation of a PNE is PLS-complete. For the proof, we use a straightforward +reduction to the problem of computing a pure Nash equilibrium in a threshold game, a particular +kind of congestion game where each player has two strategies only. The only non-trivial part +in the reduction is the fact that in a threshold game, a player may be indifferent between their +two strategies, while in a weighted binary public goods game this cannot occur. Intuitively, the +absence of indifference makes the computation of equilibria for public goods games only harder. +For the proof, we formalize this intuition. +Theorem 4.3. Computing a pure Nash equilibrium of a weighted binary public goods game on +an undirected graph with patterns T i ∈ 1+0+ for all i ∈ [n] is PLS-complete when patterns are +encoded through the number of leading ones as a binary number. +Proof. Ackermann et al. [1, Theorem 4.1] have shown that the computation of a pure Nash +equilibrium for a threshold game is PLS-complete. A threshold game is a congestion game where +8 + +the set of goods E is partitioned into two disjoint sets Ein and Eout. The set Eout := {ei | i ∈ [n]} +contains a good ei for every player i. The set Eout contains a good ei,j for every unordered pair +of players {i, j} ⊆ [n] with i ̸= j. The delay function of the goods ei ∈ Eout is constant, i.e., for +every player i there is a constant θi ∈ R>0 such that dei(x) = θi for all x ∈ Z≥1. As explained +in [1, Remark 4.2], the proof of [1, Theorem 4.1] only uses delay functions of the form +dei,j(x) = ai,j(x − 1) +for all {i, j} ⊆ [n] with i ̸= j, where ai,j > 0. +(2) +A closer examination of the proof further reveals that the values ai,j can in fact be chosen to +be integer. Thus, the PLS-completeness also holds for this special case. The set of strategies of +each player i is given by Si = {sout +i +, sin +i } with sout +i += {ei} and sin +i = {ei,j | j ∈ [n], j ̸= i}. +For the reduction, let an instance of a threshold game with the delay functions as in (2) +be given. We construct a corresponding instance of a weighted binary public goods game as +follows. The game is played on a complete graph G = (V, E) with edge weights we = ai,j ∈ Z≥1 +for all e = {i, j} ∈ E. We further set T i := 1⌊θi⌋0∗ for all i ∈ [n]. Let s be a pure Nash +equilibrium of the thus defined weighted binary public goods game. We claim that ¯s defined +for all i ∈ [n] as ¯si = sout +i +, if si = 0, and ¯si = sin +i , if si = 1, is a pure Nash equilibrium of the +threshold game. By the definition of ¯s and the fact that s is a pure Nash equilibrium of the +weighted binary public goods game, we have ¯si = sout +i +whenever xi(s−i) ≥ ⌊θi⌋ + 1 and ¯si = sin +i +whenever xi(s−i) ≤ ⌊θi⌋. A player i with ¯si = sout +i +has utility −θi. After a deviation to sin +i , the +utility would be −xi(s−i) ≤ −(⌊θi⌋ + 1) ≤ −θi, so that this deviation is not profitable. On the +other hand, a player i with ¯si = sin +i has utility −xi(s−i). After a deviation to sout +i +, the utility +would be −θi ≤ −⌊θi⌋ ≤ −xi(s−i), so that also this deviation is not profitable. This concludes +the PLS-reduction. +5 +Hardness of the picky pattern +In a picky pattern, players want to perform the action if either none or a particular number of +neighbors also does so, formally T = 10k10∗ with k ≥ 1. We define a number of graph gadgets +that are used in a polynomial-time reduction from the NP-hard [5] POSITIVE-1IN3-SAT problem: +Definition 5.1. An instance of the POSITIVE-1IN3-SAT problem is a collection of ℓ clauses +I := +� +{li +1, li +2, li +3} | i ∈ [ℓ] +� +with li +j ∈ X ∪ {⊥} for all (i, j) ∈ [ℓ] × [3], where X = {ξ1, . . . , ξm} is +a set of boolean variables and where ⊥ denotes unconditional falsity. The problem is to decide +whether there is a truth assignment to the variables in X satisfying +Φ(X) := +� +i∈[ℓ] +�� +li +1 ∨ li +2 ∨ li +3 +� +∧ ¬ +� +li +1 ∧ li +2 +� +∧ ¬ +� +li +2 ∧ li +3 +� +∧ ¬ +� +li +3 ∧ li +1 +�� +. +In the reduction, the literals of a POSITIVE-1IN3-SAT instance (either non-negated variables +or falsity) are represented by literal vertices and truth assignments to the underlying boolean +variables are encoded by these vertices’ strategies. The gadgets represent logical operators and +connect the literal vertices in such a way that the resulting graph admits a PNE if and only if +the POSITIVE-1IN3-SAT formula is satisfiable. To this end, every gadget has a set of operand +vertices that are identified with a subset of the literal vertices; no other gadget vertex has an +edge leaving the gadget. We refer to gadget vertices that are adjacent to the operator vertices +as membrane vertices. Three out of four gadgets are constructed in such a way that membrane +vertices are inactive in any PNE on a graph containing the gadget. We call this property safety +as it rules out potential side effects when the gadget is added to a graph. In particular, this +ensures that adding a gadget to a graph that admits no PNE will not allow a PNE to exist in +the resulting graph: +9 + +w +y +y′ +z +z′ +q +x1 +xℓ +X +(a) NEAR-OR gadget for k = 1. +w +y +z +y1 +yk +zk +z1 +x1 +xℓ +X +Y +Z +(b) NEAR-OR gadget for k ≥ 2. +Figure 1: NEAR-OR gadget: In any PNE s on a graph that contains this gadget as a subgraph in +such a way that non-black vertices are connected only to other gadget vertices, it is �ℓ +i=1 sxi ̸∈ +{0, k + 1}. We refer to black vertices as operand vertices, to gray vertices as membrane vertices, +and to white vertices as internal vertices. +(a) A graph gadget comprising the 3-sun graph S3 and ℓ additional vertices X attached to its +top corner, which is labeled w. The middle level of the S3 is labeled y and z, the lower level y′, +q, and z′. (b) A triangle with additional vertex groups X, Y , and Z attached to its corners. +Lemma 5.2. Let G◦ = (V◦, E◦) be a graph gadget with operand vertices X ⊆ V◦ and membrane +vertices M := NG◦(X) such that for all graphs G′ = (V ′, E′) with V ′ ∩V◦ = X, for all PNE s on +G′, and for all m ∈ M, it is sm = 0. Let further G = (V, E) be a fixed graph with V ∩ V◦ = X +that admits no PNE. Then, also H = (V ∪ V◦, E ∪ E◦) admits no PNE. +Proof. Let G◦, X, M, G, and H as in the lemma and assume towards a contradiction that +H admits a PNE s. Consider the strategy profile t obtained by limiting s to vertices in G. If +degt(v) = degs(v) for all v ∈ V , then s is a PNE on G, so there is a v ∈ V with degt(v) ̸= degs(v). +If NH(v) ⊆ V , then degt(v) = degs(v) by construction of t, so there is further a u ∈ NH(v) \ V +with tu = 1. Since u ̸∈ V , {u, v} ̸∈ E, so {u, v} ∈ E◦, implying u, v ∈ V◦. From v ∈ V it +follows that v ∈ X and from u ∈ NH(v) it follows that u ∈ NG◦(v) and thus u ∈ NG◦(X) = M, +contradicting tu = 1. +5.1 +The NEAR-OR gadget +The NEAR-OR gadget (Fig. 1) is used to ensure (under the assumption that a PNE exists) that at +least one literal in each clause of a POSITIVE-1IN3-SAT instance evaluates to true. The “near” +in its name stems from the fact that, when used to represent an ℓ-ary logical operator with +ℓ > k, the gadget cannot distinguish between a total of 0 or k + 1 operands evaluating to true; +in both cases the gadget will prevent a PNE. The gadget further appears as a building block in +other gadgets, where we make use of this property. We call the NEAR-OR gadget for ℓ = 1 the +TRUE gadget as it forces its single operand vertex to be active in any PNE. +The following lemma implies that the NEAR-OR gadget forbids a PNE in which none of its +operand vertices are active. +Lemma 5.3. The NEAR-OR gadget with operand vertices removed admits no PNE. +Proof for k = 1. Let G be the graph of Fig. 1a without the vertices in X and assume towards a +contradiction that G admits a PNE s. Assume further that degs(v) ≥ 4 for some v ∈ V . Then, +v is a vertex with deg(v) = 4 and v is inactive in s. Consider {u, v} ∈ E with deg(u) = 2. Then, +u is active with degs(u) = 1 as N(u) = {v, v′} ∈ E, contradicting that s is a PNE. We have +thus degs(v) ≤ 3 for all v ∈ V . In particular, v ∈ V is active if and only if degs(v) is even. Let +next A ⊆ V active and I = V \ A inactive. Since � +a∈A degs(a) + � +i∈I degs(i) = � +a∈A deg(a) +is even, also |I| and by extension |A| = 6 − |I| are even. The cases of |A| ∈ {0, 6} are easily +ruled out, so either |A| = 2 or |I| = 2. If A = {a1, a2} ∈ E, degs(a1) = 1 contradicts a1 +10 + +active. If A = {a1, a2} ̸∈ E, diam(G) = 2 implies an i ∈ I with A ⊆ N(i) so that degs(i) = 2 +contradicts i inactive. If I = {i1, i2} ∈ E, then G[A] is either the paw graph or the union of a +P3 and a singleton. Both have a vertex of degree one, implying a ∈ A with degs(a) = 1, which +contradicts a being active. Finally, if I = {i1, i2} ̸∈ E, then degs(i1) = deg(i1) ∈ {2, 4}. As we +ruled out degs(i1) = 4 earlier, this contradicts i1 inactive. +Proof for k ≥ 2. Let G be the graph of Fig. 1b without the vertices in X and assume towards +a contradiction that G admits a PNE s. If both y and z are inactive, then w and all vertices +in Y and in Z have no active neighbors and are active. +This contradicts y being inactive, +as degs(y) = |Y | + 1 = k + 1. If both y and z are active, then degs(v) ∈ {1, 2} for all v in +{w}∪Y ∪Z, so all vertices other than y and z are inactive. Thus, degs(y) = 1, which contradicts +y being active. If exactly one of y and z is active, say y, then all vertices in Y are inactive and +all in Z are active. If further w is inactive, then degs(z) = k + 1 contradicts z being inactive. +If however w is active, then degs(y) = 1 contradicts y being active. +The next lemma states that the NEAR-OR gadget also forbids a PNE in which exactly k + 1 +of its operand vertices are active. +Lemma 5.4. Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph +with V ∩ V∨ = {xi | i ∈ [ℓ]}, and H := (V ∪ V∨, E ∪ E∨). Then, H admits no PNE s with +�ℓ +i=1 sxi = k + 1. +Proof for k = 1. Assume towards a contradiction that s is a PNE of H with �ℓ +i=1 sxi = k + 1. +If w is active, then both y and z are inactive as otherwise degs(w) > k + 1. If further q is +inactive, then both y′ and z′ are active as degs(y′) = degs(z′) = 0. This contradicts q inactive, +as degs(q) = 2, so q is active. If y′ is inactive, then degs(y) = 2 contradicts y inactive. So y′ +must be active, contradicting degs(y′) = 1. If w is inactive, then degs(w) > k + 1, so y or z +or both are active. If both y and z are active, then exactly one of y′ and q must be active, +otherwise degs(y) ∈ {1, 3}. If q is active, degs(y′) = 2 contradicts y′ inactive. If y′ is active, this +contradicts degs(y′) = 1. Thus, exactly one of y and z is active, say y. Since degs(y) ∈ {0, 2} +with z and w both inactive, it follows that y′ and q are either both active or both inactive. +If both are active, this contradicts z inactive as then degs(z) = 2. If both are inactive, it is +degs(z′) = 0, so z′ is active. This again implies degs(z) = 2, contradicting z inactive. +Proof for k ≥ 2. Assume towards a contradiction that s is a PNE of H with �ℓ +i=1 sxi = k + 1. +Analogous to the proof for k = 1, we have that either w is active and both y and z are inactive, +or that w is inactive and at least one of y and z is active. In the former case, all vertices in Y +have no active neighbors and are active, contradicting y inactive as degs(y) = |Y | + 1 = k + 1. +In the latter case, if both y and z are active, then all vertices in Y are inactive, contradicting +y active as degs(y) = 1. If only z is active, then all vertices in Y are active, contradicting y +inactive as again degs(y) = |Y | + 1. A symmetric argument rules out that only y is active. +Next, we show that the NEAR-OR gadget permits a PNE in every other case. +Lemma 5.5. Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with +V ∩V∨ = X, and H := (V ∪V∨, E ∪E∨). Then, if G admits a PNE s with �ℓ +i=1 sxi ̸∈ {0, k+1}, +then also H admits a PNE t with tv = sv for all v ∈ V . +Proof for k = 1. We claim that t with +tv := + + + + + +sv, +if v ∈ V, +1, +if v ∈ {q}, +0, +if v ∈ V∨ \ (X ∪ {q}) . +11 + +is a PNE on H. Since tv = sv and, due to tw = 0, also degt(v) = degs(v) holds for all v ∈ V by +construction, it remains to show that vertices in (V ∪ V∨) \ V = V∨ \ X have a stable strategy +assignment. This is the case as degt(q) = 0 and q is active, degt(w) = m ̸∈ {0, k + 1} and w is +inactive, and for all v ∈ {y, z, y′, z′}, degt(v) = 1 and v is inactive. +Proof for k ≥ 2. Let Q := Y ∪ Z. We claim that t with +tv := + + + + + +sv, +if v ∈ V, +1, +if v ∈ Q, +0, +if v ∈ V∨ \ (X ∪ Q) . +is a PNE on H. Again, we only need to show that vertices in V∨ \ X have a stable strategy +assignment. This is the case as for all q ∈ Q, degt(q) = 0 and q is active, degt(w) = m ̸∈ {0, k+1} +and w is inactive, and degt(y) = degt(z) = k and y and z are both inactive. +Additionally, we argue that the NEAR-OR gadget is safe. +Lemma 5.6. Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with +V ∩ V∨ = X, and H := (V ∪ V∨, E ∪ E∨). Then, H admits no PNE in which w, the unique +vertex in NG∨(X), is active. +Proof for k = 1. Let t be a PNE of H and assume towards a contradiction that tw = 1. By +Lemmas 5.3 and 5.4, �ℓ +i=1 txi =: m ̸∈ {0, k + 1}. Since w is active and degt(w) ≥ m > 0, it is +degt(w) = k+1 = 2. Therefor, at least one of y and z must be active, as otherwise degt(w) = m. +If both y and z are active, then neither y′ nor q can be active as otherwise degs(y) > 2. An +analogous argument rules out that z′ is active. This implies deg(q) = 2, contradicting q inactive. +Thus, exactly one of y and z is active, say y. If further q is active, also y′ is active as degs(y′) = 2 +but this contradicts y active as then degs(y) = 3. So q is inactive, implying that z′ is active +due to degs(z′) = 0. If further y′ is inactive, then degs(q) = 2 contradicts q inactive, so also y′ +is active, contradicting degs(y′) = 1. +Proof for k ≥ 2. Let t be a PNE of H and assume towards a contradiction that tw = 1. By +analogy with the proof for k = 1, we have degt(w) = k + 1 > m so that at least one of y and z +must be active. If y is active, then degt(y′) = 1 for all y′ ∈ Y , so all vertices in Y are inactive +and degt(y) ∈ {1, 2}. Since 2 < k + 1, this contradicts y active. An analogous argument rules +out that z is active. +Finally, we summarize the behavior of the NEAR-OR gadget. +Corollary 5.7. Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with +V ∩ V∨ = X, and H := (V ∪ V∨, E ∪ E∨). Then: +1. NEAR-OR is permissive: If G admits a PNE s with � +x∈X sx ̸∈ {0, k + 1}, then also H +admits a PNE t with tv = sv for all v ∈ V . +2. NEAR-OR is restrictive: If H admits a PNE t, then � +x∈X tx ̸∈ {0, k + 1}. +3. NEAR-OR is safe: In any PNE t on H and for all m ∈ NG∨(x), tm = 0. +Proof. Permissiveness was shown in Lemma 5.5, restrictiveness is the sum of Lemmas 5.3 +and 5.4, and safety follows from Lemma 5.6. +12 + +x y1 +yk +TRUE +TRUE +NEAR-OR +(a) FALSE gadget: In any PNE s on a +graph defined as for Fig. 1, sx = 0. +y +x1 +x2 +z1 +zk +FALSE +TRUE +TRUE +(b) EQUIV gadget: In any PNE s on a +graph defined as for Fig. 1, sx1 = sx2. +Figure 2: FALSE and EQUIV gadgets. Only the name and operand vertices of auxiliary gadgets +are shown. +(a) A graph gadget composed of NEAR-OR gadgets: An outer (k + 1)-ary one and k inner TRUE +gadgets with common operand vertices. The remaining outer operand vertex is labeled x. (b) A +FALSE gadget with four vertices attached: x1, x2, and two operand vertices of additional TRUE +gadgets. +5.2 +The FALSE gadget +In a POSITIVE-1IN3-SAT instance, literals are either non-negated boolean variables or falsity +(⊥). To represent the latter, we introduce a gadget that forces a vertex to be inactive in any +PNE (Fig. 2a). +Lemma 5.8. Let G⊥ = (V⊥, E⊥) an instance of the FALSE gadget, G = (V, E) a graph with +V ∩ V⊥ = {x}, and H := (V ∪ V⊥, E ∪ E⊥). Then: +1. FALSE is permissive: If G admits a PNE s with sx = 0, then also H admits a PNE t with +tv = sv for all v ∈ V . +2. FALSE is restrictive: If H admits a PNE t, then tx = 0. +3. FALSE is safe: In any PNE t on H and for all m ∈ NG⊥(x), tm = 0. +Proof. Permissiveness. Let G admit a PNE s with sx = 0. Then, the partial strategy profile p +with pv = sv, for all v ∈ V , and pyi = 1, for all i ∈ [k], can be extended to a PNE for H by the +permissiveness of the NEAR-OR gadget. +Restrictiveness. +Assume towards a contradiction that H admits a PNE t with tx = 1. +Since TRUE is restrictive, tyi = 1 for all i ∈ [k]. This contradicts NEAR-OR being restrictive, as +tx + �k +i=1 tyi = k + 1. +Safety. Follows from the safety of the NEAR-OR gadget as x is identified with an operand +vertex of the (k + 1)-ary NEAR-OR gadget in G⊥. +5.3 +The EQUIV gadget +Next, we introduce a gadget to identify equal variables in distinct clauses (Fig. 2b). +Lemma 5.9. Let G↔ = (V↔, E↔) an instance of the EQUIV gadget, G = (V, E) a graph with +V ∩ V↔ = {x1, x2}, and H := (V ∪ V↔, E ∪ E↔). Then: +1. EQUIV is permissive: If G admits a PNE s with sx1 = sx2, then also H admits a PNE t +with tv = sv for all v ∈ V . +2. EQUIV is restrictive: If H admits a PNE t, then tx1 = tx2. +3. EQUIV is safe: In any PNE t on H and for all m ∈ NG⊥({x1, x2}), tm = 0. +13 + +t1 +t2 +t3 +x1 +y1 +z1 +x3 +y3 +z3 +NEAR-OR +(a) CLAUSE gadget for k = 1. +t1 +t2 +t3 +(b) CLAUSE gadget for k ≥ 2. +Figure 3: CLAUSE gadget: Admits three symmetric PNE with st1 + st2 + st3 = 1. +(a) A NEAR-OR gadget whose operand vertices t1, t2, and t3 are pairwise connected by three +parallel paths with one inner vertex each. (b) A triangle with the same operand vertex labels. +Proof. Permissiveness. Let G admit a PNE s with sx1 = sx2. We first show that the strategy +profile p with +pv := + + + + + +sv, +if v ∈ V, +0, +if v = y, +1, +if v ∈ {zi | i ∈ [k]}, +is a PNE in the graph G′ that is obtained by removing all vertices in V↔\({x1, x2, y} ∪ {zi | i ∈ [k]}) +from H. Since py = 0, p is a stable assignment for all v ∈ V , so it remains to show that p is +stable also for y and for zi with i ∈ [k]. This is the case as for all i ∈ [k], degp(zi) = 0 and +zi is active, while degp(y) ∈ {k, k + 2} with k ≥ 1 and y is inactive. Since p is a PNE on G′, +it follows from the permissiveness of the TRUE and FALSE gadgets that also H admits a PNE t +with tv = pv = sv for all v ∈ V . +Restrictiveness. Assume towards a contradiction that H admits a PNE t with tx1 ̸= tx2. +Without loss of generality let tx1 = 1 and tx2 = 0. By the restrictiveness of the TRUE and FALSE +gadgets, ty = 0 and tzi = 1 for all i ∈ [k]. Thus, degt(y) = k + 1, contradicting y inactive. +Safety. It is NG⊥({x1, x2}) = {y} and, in any PNE t on H, ty = 0 as FALSE is restrictive. +5.4 +The CLAUSE gadget +Finally, we represent each clause of a POSITIVE-1IN3-SAT instance by a CLAUSE gadget (Fig. 3). +Here, we do not require the familiar trio of properties. Instead, the gadget is designed to admit +three symmetric PNE, each of which has one distinct vertex from {t1, t2, t3} in active state. We +prove a slightly weaker claim, which is sufficient for the reduction. +Lemma 5.10. Let G be an instance of the CLAUSE gadget. Then, +• for every t ∈ {t1, t2, t3}, G admits a PNE s with st = 1, and +• in any PNE s on G, st1 + st2 + st3 = 1. +Proof. The case of k ≥ 2 is trivial: Exactly the profiles in which exactly one of the three vertices +is active are PNE. Let thus k = 1 in the following. +We show the first claim only for t = t2 as the other cases are analogous. We claim that the +partial strategy profile s with +sv := +� +1, +if v ∈ {t2, z1, z2, z3}, +0, +if v ∈ {t1, t3, x1, x2, x3, y1, y2, y3}, +can be extended to a PNE on G. +We first argue that s is a PNE on the graph obtained +by removing all non-operand vertices of the NEAR-OR gadget from G. +On this graph it is +14 + +degs(t2) = 0 and t2 is active while degs(t1) = degs(t3) = 3 and both t1 and t3 are inactive. +Further, degs(z) = 0 and z is active for every z ∈ {z1, z2, z3} while degs(v) = 1 and v is inactive +for all v ∈ {x1, x2, x3, y1, y2, y3}. Since t1 + t2 + t3 = 1 ̸∈ {0, k + 1}, it follows from Lemma 5.5 +that s can be extended to a PNE on G. +Next, let s be any PNE on G and assume towards a contradiction that ℓ := st1 +st2 +st3 ̸= 1. +The cases of ℓ = 0 and ℓ = 3 are ruled out by the restrictiveness of the NEAR-OR gadget, so it +remains to rule out ℓ = 2. Without loss of generality let st1 = st2 = 1 and st3 = 0; the other +cases are analogous. Then, degs(xi) = 2 and xi is active for all i ∈ [3]. Thus, degs(t1) ≥ 3, +contradicting t1 active. +5.5 +Reduction +With our assortment of gadgets, we can prove the main result of this section, which answers +the second open question posed in [6]: +Theorem 5.11. The homogeneous binary public goods game equilibrium decision problem on +undirected graphs with best-response pattern T ∈ 10+10∗ is NP-complete. +Proof. Containment in NP is obvious. +For NP-hardness, we describe a polynomial-time re- +duction from POSITIVE-1IN3-SAT. In the following, we consider the best-response pattern +T = 10k10∗ for a fixed k ≥ 1. To ease notation, we write s(v) instead of sv to denote the +strategy of a vertex v in a strategy profile s. +Let I := +� +{li +1, li +2, li +3} | i ∈ [ℓ] +� +with li +j ∈ X ∪ {⊥} for all (i, j) ∈ [ℓ] × [3], ℓ ∈ Z≥1, and +X = {ξ1, . . . , ξm} a set of boolean variables, be an instance of POSITIVE-1IN3-SAT. Recall that +I is a yes-instance if and only if the formula +Φ(X) := +� +i∈[ℓ] +�� +li +1 ∨ li +2 ∨ li +3 +� +∧ ¬ +� +li +1 ∧ li +2 +� +∧ ¬ +� +li +2 ∧ li +3 +� +∧ ¬ +� +li +3 ∧ li +1 +�� +is satisfiable. +We construct an instance G = (V, E) of the binary public goods game that +has a PNE if and only if this is the case. +Starting from the empty graph, we introduce a +disjoint CLAUSE gadget Ci = (V i, Ei) for every {li +1, li +2, li +3} ∈ I, whose vertices we relabel with a +superscript i. We call the resulting graph G′. Note that for some i ̸= i′ ∈ [ℓ] and j, j′ ∈ [3], it +may be the case that li +j = li′ +j′ ∈ X while ti +j ̸= ti′ +j′ are disjoint vertices. For every such quadruple +(i, j, i′, j′), we add an EQUIV gadget on fresh non-operand vertices whose operand vertices are +identified with ti +j and ti′ +j′. We call the graph at this point G′′. Next, for every (i, j) ∈ [ℓ] × [3] +with li +j = ⊥, we add a FALSE gadget on fresh non-operand vertices whose operand vertex we +identify with ti +j. This yields the graph G. Clearly, the number of vertices added is polynomial +in the number of clauses ℓ, so that this construction can be accomplished in time polynomial in +the size of I. In the following we show decision equivalence. +Let first I be a yes-instance. Then, there is a truth assignment σ: X → {0, 1} satisfying Φ. +We claim that the partial strategy assignment s0 with +s0(ti +j) := +� +σ(ξ), +if li +j = ξ ∈ X, +0, +if li +j = ⊥, +for all (i, j) ∈ [ℓ] × [3] can be extended to a PNE on G. To this end consider first the CLAUSE +gadgets and the associated subgraph G′ of G. Since Φ is satisfied, we have s0(ti +1) + s0(ti +2) + +s0(ti +3) = 1 for every i ∈ [ℓ]. By Lemma 5.10, it follows that G′ admits a PNE s′ with s′(ti +j) = +s0(ti +j) for all (i, j) ∈ [ℓ] × [3]. Consider next the EQUIV gadgets and the associated subgraph +G′′ of G. Let ti +j and ti′ +j′ be the operand vertices of an EQUIV gadget. Then, li +j = li′ +j′ ∈ X by +construction so that s′(ti +j) = s0(ti +j) = s0(ti′ +j′) = s′(ti′ +j′) by definition of s′ and s0. From the +permissiveness of the EQUIV gadgets (applied iteratively), it follows that G′′ admits a PNE +15 + +s′′ with s′′(ti +j) = s′(ti +j) = s0(ti +j) for all (i, j) ∈ [ℓ] × [3]. Consider next the FALSE gadgets in +G. Let ti +j be the operand vertex of such a gadget. Then, by construction, li +j = ⊥ and thus +s′′(ti +j) = s0(ti +j) = 0. By the permissiveness of the FALSE gadgets (applied iteratively), we have +that G admits a PNE as required. +Let next G admit a PNE s. We show that the truth assignment σ: X → {0, 1} given by +σ(li +j) := s(ti +j) for all (i, j) ∈ [ℓ]×[3] with li +j ∈ X satisfies Φ. First, we show that σ is well-defined. +To this end assume towards a contradiction that there are (i, j) ̸= (i′, j′) ∈ [ℓ] × [3] such that +li +j = li′ +j′ but s(ti +j) ̸= s(ti′ +j′). By construction, there is an EQUIV gadget in G with operand vertices +ti +j and ti′ +j′, whose restrictiveness contradicts s being a PNE. Next, we show that Φ is satisfied. +Let i ∈ [ℓ] index a clause {li +1, li +2, li +3} ∈ I and consider the associated CLAUSE gadget Ci with +operand vertices V i = {ti +1, ti +2, ti +3}. Since EQUIV and FALSE are safe gadgets, implying s(v) = 0 +for all v ∈ NG(V i), it follows that s limited to V i is a PNE on G. By Lemma 5.10, this implies +s(ti +1)+s(ti +2)+s(ti +3) = 1. Without loss of generality, assume that s(ti +1) = 1 and s(ti +2) = s(ti +3) = 0. +We establish that li +1 ∈ X. To this end assume towards a contradiction that li +1 = ⊥. Then, by +construction, there is a FALSE gadget in G whose operand vertex is ti +1. Since FALSE is restrictive, +s(ti +1) = 1 contradicts s being a PNE. Since li +1 ∈ X, we have σ(li +1) = s(ti +1) = 1, so +� +li +1 ∨ li +2 ∨ li +3 +� +is satisfied by σ. It remains to show that both li +2 and li +3 are false under σ. We do so for li +2; the +argument for li +3 is analogous. The case of li +2 = ⊥ is clear, so let li +2 ∈ X. Then, σ(li +2) = s(ti +2) = 0. +It follows that also ¬ +� +li +1 ∧ li +2 +� +∧¬ +� +li +2 ∧ li +3 +� +∧¬ +� +li +3 ∧ li +1 +� +and, by extension, Φ is satisfied by σ. +Recall that we assume that the pattern of a homogeneous game is part of the problem +definition. If we require instead that the number of intermediate zeros, k, is part of the problem +input, then we only obtain a weak NP-hardness result as the reduction above produces a graph +with maximum degree in Ω(k). +This is consistent with our argumentation in the proof of +Corollary 3.4: for k > ∆(G), the precise value of k becomes irrelevant as no vertex can have +k + 1 or more active neighbors. +In [6] it was shown that any NP-hard pattern starting with a 1 remains NP-hard when it is +prefixed with the sequence 10. This lets us identify another natural family of patterns that is +NP-hard to decide, that of all truncated alternating sequences: +Corollary 5.12. The homogeneous binary public goods game equilibrium decision problem on +undirected graphs with best-response pattern T ∈ (10)+10∗ is NP-complete. +What makes this family interesting is that its “limit case”, the alternating sequence T = +(10)∗, is polynomial-time decidable already for the more general case of directed graphs [11]. +6 +Conclusions +We studied equilibria of the binary public goods game from the perspective of computational +complexity. We have resolved two open questions posed by Gilboa and Nisan [6] that concern +the best-response patterns 1k0∗ for k ≥ 3 and 10k10∗ for k ≥ 0. For the former family, we +discovered a connection to congestion games, which guarantees the existence of equilibria and +yields a straightforward polynomial time algorithm to compute one: any sequence of better +responses will converge to an equilibrium after at most O(n2) steps. While this holds already +for the inhomogeneous case where each player may follow a different pattern from this family, +the problem becomes PLS-complete when we consider in addition links of varying strength. For +the latter family of patterns, we proved that it is NP-hard to decide whether an equilibrium +exists. The special case of 1010∗ together with a result in [6] shows that the family of truncated +alternating patterns, (10)+10∗, also induces a hard problem. This complements nicely a positive +result for (10)∗ given in [11]. +16 + +References +[1] H. Ackermann, H. R¨oglin, and B. V¨ocking. On the impact of combinatorial structure on +congestion games. Journal of the ACM, 55(6):1–22, 2008. doi: 10.1145/1455248.1455249. +[2] Y. Bramoull´e and R. Kranton. Public goods in networks. Journal of Economic Theory, +135(1):478–494, 2007. doi: 10.1016/j.jet.2006.06.006. +[3] J. Gabarr´o, A. Garc´ıa, and M. Serna. The complexity of game isomorphism. Theoretical +Computer Science, 412(48):6675–6695, Nov. 2011. +ISSN 1879-2294. +doi: 10.1016/j.tcs. +2011.07.022. +[4] A. Galeotti, S. Goyal, M. O. Jackson, F. Vega-Redondo, and L. Yariv. Network games. +Review of Economic Studies, 77:218–244, 2010. doi: 10.1111/j.1467-937X.2009.00570.x. +[5] M. R. Garey and D. S. Johnson. Computers and intractability. W. H. Freeman, New York, +1979. ISBN 0716710455. +[6] M. Gilboa and N. Nisan. +Complexity of public goods games on graphs. +In P. Kanel- +lopoulos, M. Kyropoulou, and A. Voudouris, editors, Algorithmic Game Theory, volume +13584 of Lecture Notes in Computer Science, pages 151–168, Cham, Sept. 2022. Springer +International Publishing. ISBN 978-3031157141. doi: 10.1007/978-3-031-15714-1 9. +[7] D. Kempe, S. Yu, and Y. Vorobeychik. +Inducing equilibria in networked public goods +games through network structure modification. In A. E. F. Seghrouchni, G. Sukthankar, +B. An, and N. Yorke-Smith, editors, Proceedings of the 19th International Conference on +Autonomous Agents and Multiagent Systems (AAMAS), pages 611–619, Richland, SC, May +2020. International Foundation for Autonomous Agents and Multiagent Systems. ISBN +978-1450375184. +[8] D. L´opez-Pintado. +Public goods in directed networks. +Economic Letters, 121:160–162, +2013. doi: 10.1016/j.econlet.2013.08.003. +[9] A. Maiti and P. Dey. On parameterized complexity of binary networked public goods game. +In P. Faliszewski, V. Mascardi, C. Pelachaud, and M. E. Taylor, editors, Proceedings of the +21st International Conference on Autonomous Agents and Multiagent Systems (AAMAS), +pages 871–879, Richland, SC, May 2022. International Foundation for Autonomous Agents +and Multiagent Systems. ISBN 978-1450392136. +[10] D. Monderer and L. S. Shapley. Potential games. Games and Economic Behavior, 14(1): +124–143, May 1996. ISSN 1090-2473. doi: 10.1006/game.1996.0044. +[11] C. Papadimitriou and B. Peng. Public goods games in directed networks. In Proceedings +of the 22nd ACM Conference on Economics and Computation (EC), pages 745–762, New +York, July 2021. Association for Computing Machinery. ISBN 978-1450385541. doi: 10. +1145/3465456.3467616. +[12] R. W. Rosenthal. A class of games possessing pure-strategy Nash equilibria. International +Journal of Game Theory, 2(1):65–67, 1973. doi: 10.1007/BF01737559. +[13] Y. Yang and J. Wang. A refined study of the complexity of binary networked public goods +games. CoRR, 2020. doi: 10.48550/arXiv.2012.02916. +[14] S. Yu, K. Zhou, P. J. Brantingham, and Y. Vorobeychik. Computing equilibria in binary +networked public goods games. In Proceedings of the 34th AAAI Conference on Artificial +Intelligence (AAAI), pages 2310–2317, Palo Alto, CA, 2020. AAAI Press. doi: 10.1609/ +aaai.v34i02.5609. +17 + diff --git a/TNFKT4oBgHgl3EQfkS62/content/tmp_files/load_file.txt b/TNFKT4oBgHgl3EQfkS62/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5c50d23c8baf70b9cd5869d0e762d5e3e4024a2 --- /dev/null +++ b/TNFKT4oBgHgl3EQfkS62/content/tmp_files/load_file.txt @@ -0,0 +1,762 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf,len=761 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='11849v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='GT] 27 Jan 2023 Complexity of equilibria in binary public goods games on undirected graphs Max Klimm1 Maximilian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Stahlberg1 1Technische Universit¨at Berlin, Germany {klimm, stahlberg}@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='de Abstract We study the complexity of computing equilibria in binary public goods games on undi- rected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In such a game, players correspond to vertices in a graph and face a binary choice of performing an action, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Each player’s decision depends only on the number of neighbors in the graph who perform the action and is encoded by a per-player binary pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that games with decreasing patterns (where players only want to act up to a threshold number of adjacent players doing so) always have a pure Nash equilibrium and that one is reached from any starting profile by following a polynomially bounded sequence of best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For non-monotonic patterns of the form 10k10∗ (where players want to act alone or alongside k +1 neighbors), we show that it is NP-hard to decide whether a pure Nash equilibrium exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We further investigate a generalization of the model that permits ties of varying strength: an edge with integral weight w behaves as w parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' While, in this model, a pure Nash equilibrium still exists for decreasing patters, we show that the task of computing one is PLS-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1 Introduction Public goods are resources that can be freely accessed and simultaneously used by many in- dividuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the physical world, they comprise abundant natural resources like sunlight and breathable air alongside artificial goods such as cultural heritage, public art, or early warning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Public goods have become ubiquitous in the information age, when data can be re- produced at a negligible cost, yet remains valuable to its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Examples from this domain are open source software, public databases, radio transmissions, and the diffusion of scientific knowledge through open channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' However, not all public goods are universal: a population warning system profits a region, herd immunity to a virus is enjoyed on the basis of human con- tact, and a scientific manuscript may be legible only to a group of peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For these scenarios, the possibility of access may be represented by a graph, where a vertex can only enjoy goods that are provided by itself or by one of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the classical setting of Bramoull´e and Kranton [2], each vertex may produce any amount of a fixed good and experiences utility based on the total amount of the good that is available in the closed neighborhood, minus a cost associated with local production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the public goods game, vertices aim to configure their own production to maximize the resulting utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Whether such strategic decisions allow for a stable arrangement and, if so, how one can be found and proposed to the actors are natural questions for policy makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Bramoull´e and Kranton show for strictly concave benefits and linear costs the existence of specialized equilibria, in which the production of each vertex is either zero or a fixed, positive amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This motivates the study of a binary variant of the game, wherein vertices have only two options: to be active and produce (one unit of) the good, or to remain inactive [6, 7, 9, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In general, costs and utility functions can be defined in a way that allows vertices to be indifferent to this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Yang and Wang [13] use this to show that it 1 Complexity Pattern class Reference O(1) 1+0+ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3† O(poly(n)) (10)∗ [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3] NP-complete 10+10∗ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='11 (10)+10∗ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='12‡ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='∗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='∗10∗ [6, Theorem 5] 10+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='∗0+ [6, Theorem 6] Table 1: Complexity of deciding the existence of equilibria in the homogeneous binary public goods games on undirected graphs, for varying classes of non-trivial (not starting with zero) best response patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The pattern class notation is defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' †The special case of 10∗ has been shown in [2], that of 110∗ in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' ‡Uses [6, Theorem 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' is NP-hard to decide whether an equilibrium exists, already for a monotone benefit function common to all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In response, Maiti and Dey [9] introduce—and Gilboa and Nisan [6] shift the focus towards—restricted games in which every vertex i always has a strict preference with respect to its options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As this preference depends only on the number of active neighbors, we may encode it by an infinite binary sequence T i, called a best response pattern [6], where T i ℓ is the activity response of i to exactly ℓ active neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' When there is a single pattern common to each vertex, this is called the fully homogeneous case [14], otherwise we call the game inhomogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1 Our results We investigate the computational hardness of deciding whether a game admits a pure Nash equilibrium, for two natural classes of patterns for which this question was unresolved [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Decreasing patterns are those that start with a positive number of ones and are zero there- after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that for this class, even the inhomogeneous binary public goods game is equiva- lent to a congestion game, which implies that best-response dynamics converge to a pure Nash equilibrium in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For an extension of the game in which edges have an integral weight, however, we show that finding an equilibrium is PLS-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call a picky pattern one where Tℓ = 1 if and only if ℓ ∈ {1, k + 1}, for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Here, vertices wish to act only alone or alongside k + 1 neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For example, when k = 1 for every vertex, then the Graph induced by the active vertices of an equilibrium consists of singletons and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that deciding the existence of such an equilibrium is an NP-complete problem, even in the fully homogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A notable corollary is the following: While deciding the infinite alternating sequence T = (1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=') is in P [11], any truncation of this sequence in which it is zero after alternating at least two times corresponds to a hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Table 1 shows a summary of our results for homogeneous patterns alongside previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 Related work Let G = (V, E) be an undirected graph and denote the open neighborhood of vertex i ∈ V by N(i) := {j ∈ V : {i, j} ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Bramoull´e and Kranton [2] study a model where each vertex is a player whose strategy is to pick a production effort si ∈ Si := R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For a strategy vector s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' , sn), the utility of each player is given by ui(s) := b(si + � j∈N(i) sj) − csi, where b: R≥0 → R≥0 is a benefit function with b(0) = 0, b′ > 0 and b′′ < 0, and where c > 0 denotes the cost of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The authors show the existence of a particular kind of Nash equilibrium, s with si ∈ {0, (b′)−1(c)}, that they call a specialized equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' They translate this result also to the more general case where bi and ci depend on the player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2 Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' [14] study a binary variant of the game with Si := {0, 1} and where they allow for arbitrary non-decreasing and player-specific bi, and for player-specific ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' They aim to show that in this setting, deciding the existence of a pure Nash equilibrium is an NP-complete problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Yang and Wang [13] point out a flaw in the proof and provide a corrected version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' They further extend this result to homogenous games where all functions bi and all parameters ci are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As noted by Gilboa and Nisan [6], the hardness result of Yang and Wang hinges on the fact that some players are indifferent to their binary decisions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=', b(k + 1) − b(k) = c for some value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Gilboa and Nisan exclude this possibility by studying best-response patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' These are binary sequences that explicitly give the best response of a player i based on the number of neighbors j ∈ N(i) with sj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The authors settle the complexity for a number of patterns, including some that start with 0, for which they consider non-trivial equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' They also show that hard patterns that start with 1 remain hard when they are prefixed with (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' See Table 1 for results on patterns starting with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Maiti and Dey [9] provide a parametrized complexity perspective wherein they study natural graph parameters such as the maximum degree and the treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' L´opez-Pintado [8] initiates the study of public goods games on directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The author analyzes a restricted binary model where each agent i prefers to play si = 1 if and only if none of their in-neighbors j plays sj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Papadimitriou and Peng [11] perform a complexity analysis of the binary variant on directed graphs for more general utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' They investigate in particular the setting where all players share a utility function that is monotone in the number of in-neighbors j who play sj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This allows for a pattern-wise analysis akin to that of Gilboa and Nisan [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Here the authors provide a complete characterization by showing that only the trivial all-zero and all-ones patterns as well as the infinite alternating sequence starting with 1 are polynomial-time decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' They further provide a PPAD-hardness result for the computation of approximate mixed Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' [7] are interested in modifying the graph in such a way that one element of a given set of strategy vectors is a pure Nash equilibrium of the binary game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The motivation behind this is to enforce a socially preferable equilibrium through the intervention of a policy maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Finally, Galeotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' [4] study a variant of the game where players have only partial knowledge of the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2 Preliminaries For n ∈ Z≥1 we write [n] short for {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For a set A, we write 1A for the indicator function of A, that is 1A(x) := 1, if x ∈ A, and 1A(x) := 0, if x ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For an undirected and simple graph G = (V, E), we write ∆(G) for its maximum vertex degree, deg(v) for the degree of v ∈ V , G[X] for the subgraph induced by X ⊆ V , NG(v) for the open neighborhood of v ∈ V , and NG(X) := {v ∈ V \\ X | ∃x ∈ X : {v, x} ∈ E} for the open neighborhood of X ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We drop suffixes when the graph in question is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' When s ∈ {0, 1}n with n = |V | denotes a strategy profile of the binary public goods game played on G = (V, E) with best-response pattern T, then we say that a vertex v ∈ V is active (in s), if sv = 1, and inactive, if sv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We further write degs(v) := |{u ∈ NG(v) | su = 1}| and call this the active degree of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Finally, we say that a vertex v ∈ V has a stable strategy assignment (under s) if it plays its best response, that is sv = Tdegs(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1 Response pattern notation We use regular expressions (REs) over the alphabet A := {0, 1} to describe classes of best response patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We first define their syntax: Every c ∈ A and the dot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' are RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If e and f are REs, then so are the concatenation ef, the k-fold concatenation of e with itself, (e)k, the zero-or-more operation (e)∗, and the one-or-more operation (e)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Parentheses may be omitted: 3 superscripts then take precedence over concatenation, for example 10∗ = 1(0)∗ ̸= (10)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The language L(e) generated by an RE e is the following: For the dot we have L(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=') := {(a) | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For c ∈ A, it is L(c) := {(c)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For REs e and f, it is L(ef) := L(e) × L(f), L(ek) := L(e)k, L(e+) := �∞ i=1 L(e) and L(e∗) := L(e+) ∪ {()}, where () is the empty sequence over A and where (c) = c is a sequence of length one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Extensible REs end in (e)∗ or (e)+ for some sub-expression e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the set of best response patterns P(e) generated by an extensible RE e, we have (x)∞ i=1 ∈ P(e) if and only if there is a threshold N such that (x)n i=1 ∈ L(e) for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For example, P(1∗0+) = P(1∗00∗) contains all infinite sequences that start with a nonnegative, finite number of ones and are zero thereafter, while P(1∗0∗) contains in addition the infinite sequence of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the following, we write T ∈ e short for T ∈ P(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We also write T = e instead of T ∈ e to signify that |P(e)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 Best-response games, strategic games, and consistent representations In the following we make explicit the relation between the binary public goods game that is specified by the players’ best-response behaviors and the classical definition where players associate a utility value with their two choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This work adopts the former representation as a best-response game that captures precisely the strict [9] public goods games on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To still leverage established notions of game equivalence that are based on utility functions, we introduce the notion of a consistent representation of a best-response game as a strategic game in which players experience utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We start the introduction of these concepts with the class of best-response games: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1 (Best-response game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A best-response game (BRG) is described by a triple (n, (Si)n i=1, (βi)n i=1) where n is the number of players, Si is the strategy set of player i ∈ [n], and βi : �i−1 j=1 Sj × �n j=i+1 Sj → Si is a function such that βi(s−i) denotes the best response of player i given that the other players play s−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For a strategy profile s ∈ �n j=1 Sj and i ∈ [n], we write s−i short for �i−1 j=1{sj}×�n j=i+1{sj} and we call s a pure Nash equilibrium (PNE) if βi(s−i) = si for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' It is straightforward to formalize the binary public goods game as a BRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 (Binary public goods game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For an undirected simple graph G = (V, E) with n vertices V = [n] and for best-response patterns T i = (τ i ℓ)∞ ℓ=1 for all i ∈ V , we call the best-response game (n, ({0, 1})n i=1, (βi)n i=1) with βi(s−i) := T i xi(s−i) and xi(s−i) := � j∈NG(i) sj a (binary) public goods game (PGG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We also refer to just G and the T i as a PGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If T i = T for all i ∈ V and for some pattern T, then we refer to the associated PGG as a homogeneous PGG with pattern T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' While in a best-response game a player’s response is defined uniquely, in strategic games players choose a strategy among all that maximize a utility function: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3 (Strategic game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A strategic game is a triple (n, (Si)n i=1, (ui)n i=1) where n is the number of players, Si is the strategy set of player i ∈ [n], and ui : �n j=1 Sj → Q is a function such that ui(s) denotes the utility experienced by player i ∈ [n] when the strategy profile s ∈ �n j=1 Sj is played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For a strategy profile s ∈ �n j=1 Sj, i ∈ [n], and a ∈ Si, we write (s−i, a) short for �i−1 j=1{sj}× {a} × �n j=i+1{sj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call a strategy profile s ∈ �n j=1 Sj with ui(s) ≥ ui(s−i, a) for all i ∈ [n] and all a ∈ Si a pure Nash equilibrium (PNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' With a suited utility function, we can represent any BRG as a strategic game with equivalent best-response dynamics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' [9]): 4 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4 (Consistent representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let A = (n, (Si)n i=1, (βi)n i=1) be a BRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call a strategic game B = (n, (Si)n i=1, (ui)n i=1) a consistent representation of A if for all s ∈ �n j=1 Sj and all i ∈ [n], it is ui(s−i, a) > ui(s−i, a′) for all a′ ∈ Si \\ {a} where a := βi(s−i) is the best response of i in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Consistent representations are closely related to the notion of a weak isomorphism between strategic games [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In particular, they also preserve the structure of pure Nash equilibria: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let A = (n, (Si)n i=1, (βi)n i=1) be a BRG and B = (n, (Si)n i=1, (ui)n i=1) a consistent representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, s ∈ �n j=1 Sj is a PNE of A if and only if s is a PNE of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let first s be a PNE of A and consider a player i ∈ [n] of A and their best response a := βi(s−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since A is a PNE, it is a = si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As further B is a consistent representation, ui(s) = ui(s−i, si) = ui(s−i, a) > ui(s−i, a′) for all a′ ∈ Si \\ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, ui(s) ≥ ui(s−i, a′′) for all a′′ ∈ Si, so s is a PNE of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let next s be a PNE of B and consider a player i ∈ [n] of B playing a := si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Assume towards a contradiction that si ̸= βi(s−i) =: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since B is a PNE, ui(s−i, a) = ui(s) ≥ ui(s−i, a′) for all a′ ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' From b ∈ Si, it follows in particular that ui(s−i, a) ≥ ui(s−i, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As B is a consistent representation and a ̸= b, ui(s−i, b) > ui(s−i, a), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Hence, si = βi(s−i), so s is a PNE of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We are further interested in the dynamics that may lead to a PNE: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='6 (Better-response sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For a best-response game (n, (Si)n i=1, (βi)n i=1), a better-response sequence is a sequence of strategy profiles (si)N i=1 such that for all i ∈ [N − 1], it is si+1 = (si −j, a) for some j ∈ [n] and a ∈ Sj with βj(si) = a and a ̸= si j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For a strategic game (n, (Si)n i=1, (ui)n i=1), a better-response sequence is a sequence of strategy profiles (si)N i=1 such that for all i ∈ [N − 1], it is si+1 = (si −j, a) for some j ∈ [n] and a ∈ Sj such that uj(si −j, a) > uj(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' It is immediate from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4 that consistent representations preserve better-response sequences: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If A is a BRG and B a consistent representation of A, then any better-response sequence in A is also a better-response sequence in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3 Game isomorphisms We will make use of the following notion of equivalence of strategic games: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='8 (Strong isomorphism [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Two strategic games A = (n, (Si)n i=1, (ui)n i=1) and B = � n, (S′ i)n i=1, (u′ i)n i=1 � are called (strongly) isomorphic when there are bijections π: [n] → [n] and φi : Si → S′ π(i) for all i ∈ [n] such that for all players i ∈ [n] (of A) and strategy profiles s ∈ �n j=1 Sj, it is ui(s) = u′ π(i) (φ(s)) where φ(s) := � φπ−1(j)(sπ−1(j)) �n j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Note that for π = id[n], it is φ(s) = (φj(sj))n j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Informally, an isomorphism describes a one-to-one mapping between the players and strategy profiles of two games that preserves the players’ utility and, by extension, the players’ strategy preferences and the PNE structure of both games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In particular: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If two games A and B are isomorphic and A admits a PNE, then so does B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If A = (n, (Si)n i=1, (ui)n i=1) has a PNE s ∈ �n i=1 Si, then ui(t) ≤ ui(s) for all i ∈ [n] and t ∈ �n j=1 Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As A and B are isomorphic, there exist bijections π and φ such that u′ π(i) (φ(t)) = ui(t) ≤ ui(s) = u′ π(i) (φ(s)) for all i and t as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As both π and φ are surjective, s′ is a PNE of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4 Encoding For the homogeneous PGG, we assume in line with previous work that the common best response pattern is part of the problem definition and not of the problem input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the inhomogeneous variant, this approach is not adequate as each player might have a distinct pattern, so here we assume that the patterns are part of the input and give the encoding explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 3 Existence of equilibria for decreasing patterns Decreasing best-response patterns are those that begin with a positive number of ones and are zero thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that for this family of patterns, formally P(1+0+), the public goods game is equivalent to a congestion game: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1 (Congestion game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let E be a set of goods equipped with a delay function de : Z≥1 → Q for every e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, we call a strategic game (n, (Si)n i=1, (ui)n i=1) with Si ⊆ E and ui(s) = − � e∈si de(xs(e)) with xs(e) := �n j=1 1sj(e) for all i ∈ [n] a congestion game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' More precisely, we show equivalence in the following sense: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The binary public goods game on undirected graphs with best-response pat- terns T i ∈ 1+0+, i ∈ V , has a consistent representation that is isomorphic to a congestion game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G = (V, E) be an instance of the PGG with patterns T i = 1ki0∗, ki ≥ 1, for all i ∈ V = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Define the strategic game Γ := (n, (Si)n i=1, (ui)n i=1) with Si := {0, 1} and ui(s) := � −(ki − 1 2), if si = 0, − degs(i), if si = 1, (1) for all i ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We first show that Γ is a consistent representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let s ∈ {0, 1}n be a strategy profile and i ∈ V a player of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let further a := βi(s−i) be the best response of i and a′ := 1 − a its unique alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that then ui(s−i, a) > ui(s−i, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If a = 0, then degs(i) ≥ ki, thus ui(s−i, a) = − � ki − 1 2 � > −ki ≥ − degs(i) = ui(s−i, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' On the other hand, if a = 1, then degs(i) ≤ ki − 1 and ui(s−i, a) = − degs(i) ≥ −(ki − 1) > − � ki − 1 2 � = ui(s−i, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Consider next the congestion game C = � n, (S′ i)n i=1, (u′ i)n i=1 � defined by the set of goods V ∪ E with delays dv(x) := kv − 1 2, for v ∈ V , and de(x) := x − 1, for e ∈ E, and by the strategies S′ v := {{v}, {e ∈ E | v ∈ e}} for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The strategy {v} corresponds to v remaining inactive, which has no effect on adjacent vertices, while {e ∈ E | v ∈ e} corresponds to v being active, which makes being active more expensive for v’s neighbors by “congesting” incident edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that C is isomorphic to Γ as witnessed by the bijections π := id[n] and φv : Sv → S′ v with φv(0) = {v} and φv(1) = {e ∈ E | v ∈ e} for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To this end, let v ∈ V be a player and s ∈ �n i=1 Si a strategy profile of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If sv = 0, then φv(sv) = {v}, and thus uv(s) = − � kv − 1 2 � = −dv(xs({v})) = u′ v(φ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 6 If sv = 1, then φv(sv) = {e ∈ E | v ∈ e}, so that uv(s) = − degs(v) = − � {v,i}∈E si = − � {v,i}∈E ��� � j ∈ [n] \\ {v} | sj = 1 ∧ {v, i} ∈ {e ∈ E | j ∈ e} � �� � j=i as j̸=v ���� = − � {v,i}∈E |{j ∈ [n] | sj = 1 ∧ {v, i} ∈ {e ∈ E | j ∈ e}}| + � {v,i}∈E ��� � j ∈ {v} | sj = 1 ∧ {v, i} ∈ {e ∈ E | j ∈ e} � �� � true as j=v ���� by the definitions of uv for sv = 1, and degs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' It follows that uv(s) = − \uf8eb \uf8ed � {v,i}∈E |{j ∈ [n] | {v, i} ∈ φj(sj)}| − degG(v) \uf8f6 \uf8f8 = − � {v,i}∈E \uf8eb \uf8ed n � j=1 1φj(sj)({v, i}) − 1 \uf8f6 \uf8f8 by the definition of φj for sj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Further, uv(s) = − � {v,i}∈E d{v,i} \uf8eb \uf8ed n � j=1 1φj(sj)({v, i}) \uf8f6 \uf8f8 = − � e∈{e∈E|v∈e} de � n � i=1 1φ(s)i(e) � follows from the definition of de for e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Finally, uv(s) = − � e∈φv(sv) de � xφ(s)(e) � = u′ v(φ(s)), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This observation allows us to state the main theorem of this section, which answers the first out of two open questions in [6]: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The binary public goods game on undirected graphs with best-response patterns T i ∈ 1+0+, i ∈ V , always admits a PNE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' the associated decision problem is thus in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As every congestion game has a PNE [12], the claim follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='9 and Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Additionally, the representation as a congestion game brings with it convergence guarantees: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the game of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3, best-response dynamics converge to a PNE after at most O(n2) improving steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let kmax := maxi∈V ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Any vertex can have at most ∆(G) active neighbors, so the game for kmax > ∆(G)+1 is equivalent to the one for kmax = ∆(G)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let thus kmax ≤ ∆(G)+1 ≤ n, without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The congestion game C in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 is an exact potential game [10] with potential function Φ(s) = � g∈V ∪E xs(g) � ℓ=1 de(ℓ) = � {u,v}∈E susv + � v∈V � kv − 1 2 � sv ≤ |E| + kmax|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' From de(ℓ) ≥ 0 for all ℓ ≥ 1 and e ∈ V ∪E, it follows that Φ(s) ≥ 0 for all s ∈ � i∈[n] S′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' On the other hand, it is Φ(s) ≤ |E| + kmax|V | ≤ 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As 2Φ is integral by construction, these bounds imply that any decreasing sequence Φ(s1) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' > Φ(sN) has length at most O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The claim follows as any better-response sequence in G corresponds to a better-response sequence in C, which has decreasing potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 4 Hardness of decreasing patterns with weighted edges We next investigate a natural extension of the public goods game, in which ties can have varying importance: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1 (Weighted binary public goods game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For an undirected simple graph G = (V, E) with n vertices V = [n], edge weights we ∈ Z≥1 for all e ∈ E, and best-response patterns T i = (τ i ℓ)∞ ℓ=1 for all i ∈ V , we call the best-response game (n, ({0, 1})n i=1, (βi)n i=1) with βi(s−i) := T i xi(s−i) and xi(s−i) := � j∈NG(i) w{i,j}sj an (edge-)weighted binary public goods game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We first notice that a straightforward adaption of the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 yields that also games with weighted edges are isomorphic to a congestion game: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The weighted binary public goods game on undirected graphs with best- response patterns T i ∈ 1+0+, i ∈ V , has a consistent representation that is isomorphic to a congestion game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof (Sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We use the same construction as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 except that the utilities in the definition of Γ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (1)) are given by ui(s) := −xi(s−1) for si = 1 and the delay functions are defined as de(x) := we(x − 1) for all e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4, we have shown that best-response dynamics converge in polynomial time to a PNE for unweighted binary public goods games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In contrast, we show that for weighted games, the computation of a PNE is PLS-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the proof, we use a straightforward reduction to the problem of computing a pure Nash equilibrium in a threshold game, a particular kind of congestion game where each player has two strategies only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The only non-trivial part in the reduction is the fact that in a threshold game, a player may be indifferent between their two strategies, while in a weighted binary public goods game this cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Intuitively, the absence of indifference makes the computation of equilibria for public goods games only harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the proof, we formalize this intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Computing a pure Nash equilibrium of a weighted binary public goods game on an undirected graph with patterns T i ∈ 1+0+ for all i ∈ [n] is PLS-complete when patterns are encoded through the number of leading ones as a binary number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1] have shown that the computation of a pure Nash equilibrium for a threshold game is PLS-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A threshold game is a congestion game where 8 the set of goods E is partitioned into two disjoint sets Ein and Eout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The set Eout := {ei | i ∈ [n]} contains a good ei for every player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The set Eout contains a good ei,j for every unordered pair of players {i, j} ⊆ [n] with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The delay function of the goods ei ∈ Eout is constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=', for every player i there is a constant θi ∈ R>0 such that dei(x) = θi for all x ∈ Z≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As explained in [1, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2], the proof of [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1] only uses delay functions of the form dei,j(x) = ai,j(x − 1) for all {i, j} ⊆ [n] with i ̸= j, where ai,j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (2) A closer examination of the proof further reveals that the values ai,j can in fact be chosen to be integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, the PLS-completeness also holds for this special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The set of strategies of each player i is given by Si = {sout i , sin i } with sout i = {ei} and sin i = {ei,j | j ∈ [n], j ̸= i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the reduction, let an instance of a threshold game with the delay functions as in (2) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We construct a corresponding instance of a weighted binary public goods game as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The game is played on a complete graph G = (V, E) with edge weights we = ai,j ∈ Z≥1 for all e = {i, j} ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We further set T i := 1⌊θi⌋0∗ for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let s be a pure Nash equilibrium of the thus defined weighted binary public goods game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We claim that ¯s defined for all i ∈ [n] as ¯si = sout i , if si = 0, and ¯si = sin i , if si = 1, is a pure Nash equilibrium of the threshold game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By the definition of ¯s and the fact that s is a pure Nash equilibrium of the weighted binary public goods game, we have ¯si = sout i whenever xi(s−i) ≥ ⌊θi⌋ + 1 and ¯si = sin i whenever xi(s−i) ≤ ⌊θi⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A player i with ¯si = sout i has utility −θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' After a deviation to sin i , the utility would be −xi(s−i) ≤ −(⌊θi⌋ + 1) ≤ −θi, so that this deviation is not profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' On the other hand, a player i with ¯si = sin i has utility −xi(s−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' After a deviation to sout i , the utility would be −θi ≤ −⌊θi⌋ ≤ −xi(s−i), so that also this deviation is not profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This concludes the PLS-reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5 Hardness of the picky pattern In a picky pattern, players want to perform the action if either none or a particular number of neighbors also does so, formally T = 10k10∗ with k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We define a number of graph gadgets that are used in a polynomial-time reduction from the NP-hard [5] POSITIVE-1IN3-SAT problem: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' An instance of the POSITIVE-1IN3-SAT problem is a collection of ℓ clauses I := � {li 1, li 2, li 3} | i ∈ [ℓ] � with li j ∈ X ∪ {⊥} for all (i, j) ∈ [ℓ] × [3], where X = {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' , ξm} is a set of boolean variables and where ⊥ denotes unconditional falsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The problem is to decide whether there is a truth assignment to the variables in X satisfying Φ(X) := � i∈[ℓ] �� li 1 ∨ li 2 ∨ li 3 � ∧ ¬ � li 1 ∧ li 2 � ∧ ¬ � li 2 ∧ li 3 � ∧ ¬ � li 3 ∧ li 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the reduction, the literals of a POSITIVE-1IN3-SAT instance (either non-negated variables or falsity) are represented by literal vertices and truth assignments to the underlying boolean variables are encoded by these vertices’ strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The gadgets represent logical operators and connect the literal vertices in such a way that the resulting graph admits a PNE if and only if the POSITIVE-1IN3-SAT formula is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To this end, every gadget has a set of operand vertices that are identified with a subset of the literal vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' no other gadget vertex has an edge leaving the gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We refer to gadget vertices that are adjacent to the operator vertices as membrane vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Three out of four gadgets are constructed in such a way that membrane vertices are inactive in any PNE on a graph containing the gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call this property safety as it rules out potential side effects when the gadget is added to a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In particular, this ensures that adding a gadget to a graph that admits no PNE will not allow a PNE to exist in the resulting graph: 9 w y y′ z z′ q x1 xℓ X (a) NEAR-OR gadget for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' w y z y1 yk zk z1 x1 xℓ X Y Z (b) NEAR-OR gadget for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Figure 1: NEAR-OR gadget: In any PNE s on a graph that contains this gadget as a subgraph in such a way that non-black vertices are connected only to other gadget vertices, it is �ℓ i=1 sxi ̸∈ {0, k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We refer to black vertices as operand vertices, to gray vertices as membrane vertices, and to white vertices as internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (a) A graph gadget comprising the 3-sun graph S3 and ℓ additional vertices X attached to its top corner, which is labeled w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The middle level of the S3 is labeled y and z, the lower level y′, q, and z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (b) A triangle with additional vertex groups X, Y , and Z attached to its corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G◦ = (V◦, E◦) be a graph gadget with operand vertices X ⊆ V◦ and membrane vertices M := NG◦(X) such that for all graphs G′ = (V ′, E′) with V ′ ∩V◦ = X, for all PNE s on G′, and for all m ∈ M, it is sm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let further G = (V, E) be a fixed graph with V ∩ V◦ = X that admits no PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, also H = (V ∪ V◦, E ∪ E◦) admits no PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G◦, X, M, G, and H as in the lemma and assume towards a contradiction that H admits a PNE s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Consider the strategy profile t obtained by limiting s to vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If degt(v) = degs(v) for all v ∈ V , then s is a PNE on G, so there is a v ∈ V with degt(v) ̸= degs(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If NH(v) ⊆ V , then degt(v) = degs(v) by construction of t, so there is further a u ∈ NH(v) \\ V with tu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since u ̸∈ V , {u, v} ̸∈ E, so {u, v} ∈ E◦, implying u, v ∈ V◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' From v ∈ V it follows that v ∈ X and from u ∈ NH(v) it follows that u ∈ NG◦(v) and thus u ∈ NG◦(X) = M, contradicting tu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1 The NEAR-OR gadget The NEAR-OR gadget (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1) is used to ensure (under the assumption that a PNE exists) that at least one literal in each clause of a POSITIVE-1IN3-SAT instance evaluates to true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The “near” in its name stems from the fact that, when used to represent an ℓ-ary logical operator with ℓ > k, the gadget cannot distinguish between a total of 0 or k + 1 operands evaluating to true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' in both cases the gadget will prevent a PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The gadget further appears as a building block in other gadgets, where we make use of this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call the NEAR-OR gadget for ℓ = 1 the TRUE gadget as it forces its single operand vertex to be active in any PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The following lemma implies that the NEAR-OR gadget forbids a PNE in which none of its operand vertices are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The NEAR-OR gadget with operand vertices removed admits no PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G be the graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1a without the vertices in X and assume towards a contradiction that G admits a PNE s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Assume further that degs(v) ≥ 4 for some v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, v is a vertex with deg(v) = 4 and v is inactive in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Consider {u, v} ∈ E with deg(u) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, u is active with degs(u) = 1 as N(u) = {v, v′} ∈ E, contradicting that s is a PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We have thus degs(v) ≤ 3 for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In particular, v ∈ V is active if and only if degs(v) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let next A ⊆ V active and I = V \\ A inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since � a∈A degs(a) + � i∈I degs(i) = � a∈A deg(a) is even, also |I| and by extension |A| = 6 − |I| are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The cases of |A| ∈ {0, 6} are easily ruled out, so either |A| = 2 or |I| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If A = {a1, a2} ∈ E, degs(a1) = 1 contradicts a1 10 active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If A = {a1, a2} ̸∈ E, diam(G) = 2 implies an i ∈ I with A ⊆ N(i) so that degs(i) = 2 contradicts i inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If I = {i1, i2} ∈ E, then G[A] is either the paw graph or the union of a P3 and a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Both have a vertex of degree one, implying a ∈ A with degs(a) = 1, which contradicts a being active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Finally, if I = {i1, i2} ̸∈ E, then degs(i1) = deg(i1) ∈ {2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' As we ruled out degs(i1) = 4 earlier, this contradicts i1 inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G be the graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1b without the vertices in X and assume towards a contradiction that G admits a PNE s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If both y and z are inactive, then w and all vertices in Y and in Z have no active neighbors and are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This contradicts y being inactive, as degs(y) = |Y | + 1 = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If both y and z are active, then degs(v) ∈ {1, 2} for all v in {w}∪Y ∪Z, so all vertices other than y and z are inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, degs(y) = 1, which contradicts y being active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If exactly one of y and z is active, say y, then all vertices in Y are inactive and all in Z are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If further w is inactive, then degs(z) = k + 1 contradicts z being inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If however w is active, then degs(y) = 1 contradicts y being active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The next lemma states that the NEAR-OR gadget also forbids a PNE in which exactly k + 1 of its operand vertices are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with V ∩ V∨ = {xi | i ∈ [ℓ]}, and H := (V ∪ V∨, E ∪ E∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, H admits no PNE s with �ℓ i=1 sxi = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Assume towards a contradiction that s is a PNE of H with �ℓ i=1 sxi = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If w is active, then both y and z are inactive as otherwise degs(w) > k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If further q is inactive, then both y′ and z′ are active as degs(y′) = degs(z′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This contradicts q inactive, as degs(q) = 2, so q is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If y′ is inactive, then degs(y) = 2 contradicts y inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' So y′ must be active, contradicting degs(y′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If w is inactive, then degs(w) > k + 1, so y or z or both are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If both y and z are active, then exactly one of y′ and q must be active, otherwise degs(y) ∈ {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If q is active, degs(y′) = 2 contradicts y′ inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If y′ is active, this contradicts degs(y′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, exactly one of y and z is active, say y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since degs(y) ∈ {0, 2} with z and w both inactive, it follows that y′ and q are either both active or both inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If both are active, this contradicts z inactive as then degs(z) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If both are inactive, it is degs(z′) = 0, so z′ is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This again implies degs(z) = 2, contradicting z inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Assume towards a contradiction that s is a PNE of H with �ℓ i=1 sxi = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Analogous to the proof for k = 1, we have that either w is active and both y and z are inactive, or that w is inactive and at least one of y and z is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the former case, all vertices in Y have no active neighbors and are active, contradicting y inactive as degs(y) = |Y | + 1 = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the latter case, if both y and z are active, then all vertices in Y are inactive, contradicting y active as degs(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If only z is active, then all vertices in Y are active, contradicting y inactive as again degs(y) = |Y | + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A symmetric argument rules out that only y is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Next, we show that the NEAR-OR gadget permits a PNE in every other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with V ∩V∨ = X, and H := (V ∪V∨, E ∪E∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, if G admits a PNE s with �ℓ i=1 sxi ̸∈ {0, k+1}, then also H admits a PNE t with tv = sv for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We claim that t with tv := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 sv, if v ∈ V, 1, if v ∈ {q}, 0, if v ∈ V∨ \\ (X ∪ {q}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 11 is a PNE on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since tv = sv and, due to tw = 0, also degt(v) = degs(v) holds for all v ∈ V by construction, it remains to show that vertices in (V ∪ V∨) \\ V = V∨ \\ X have a stable strategy assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This is the case as degt(q) = 0 and q is active, degt(w) = m ̸∈ {0, k + 1} and w is inactive, and for all v ∈ {y, z, y′, z′}, degt(v) = 1 and v is inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let Q := Y ∪ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We claim that t with tv := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 sv, if v ∈ V, 1, if v ∈ Q, 0, if v ∈ V∨ \\ (X ∪ Q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' is a PNE on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Again, we only need to show that vertices in V∨ \\ X have a stable strategy assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This is the case as for all q ∈ Q, degt(q) = 0 and q is active, degt(w) = m ̸∈ {0, k+1} and w is inactive, and degt(y) = degt(z) = k and y and z are both inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Additionally, we argue that the NEAR-OR gadget is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with V ∩ V∨ = X, and H := (V ∪ V∨, E ∪ E∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, H admits no PNE in which w, the unique vertex in NG∨(X), is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let t be a PNE of H and assume towards a contradiction that tw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4, �ℓ i=1 txi =: m ̸∈ {0, k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since w is active and degt(w) ≥ m > 0, it is degt(w) = k+1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Therefor, at least one of y and z must be active, as otherwise degt(w) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If both y and z are active, then neither y′ nor q can be active as otherwise degs(y) > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' An analogous argument rules out that z′ is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This implies deg(q) = 2, contradicting q inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, exactly one of y and z is active, say y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If further q is active, also y′ is active as degs(y′) = 2 but this contradicts y active as then degs(y) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' So q is inactive, implying that z′ is active due to degs(z′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If further y′ is inactive, then degs(q) = 2 contradicts q inactive, so also y′ is active, contradicting degs(y′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let t be a PNE of H and assume towards a contradiction that tw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By analogy with the proof for k = 1, we have degt(w) = k + 1 > m so that at least one of y and z must be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If y is active, then degt(y′) = 1 for all y′ ∈ Y , so all vertices in Y are inactive and degt(y) ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since 2 < k + 1, this contradicts y active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' An analogous argument rules out that z is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Finally, we summarize the behavior of the NEAR-OR gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G∨ = (V∨, E∨) an instance of the NEAR-OR gadget, G = (V, E) a graph with V ∩ V∨ = X, and H := (V ∪ V∨, E ∪ E∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' NEAR-OR is permissive: If G admits a PNE s with � x∈X sx ̸∈ {0, k + 1}, then also H admits a PNE t with tv = sv for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' NEAR-OR is restrictive: If H admits a PNE t, then � x∈X tx ̸∈ {0, k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' NEAR-OR is safe: In any PNE t on H and for all m ∈ NG∨(x), tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Permissiveness was shown in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='5, restrictiveness is the sum of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4, and safety follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 12 x y1 yk TRUE TRUE NEAR-OR (a) FALSE gadget: In any PNE s on a graph defined as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1, sx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' y x1 x2 z1 zk FALSE TRUE TRUE (b) EQUIV gadget: In any PNE s on a graph defined as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 1, sx1 = sx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Figure 2: FALSE and EQUIV gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Only the name and operand vertices of auxiliary gadgets are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (a) A graph gadget composed of NEAR-OR gadgets: An outer (k + 1)-ary one and k inner TRUE gadgets with common operand vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The remaining outer operand vertex is labeled x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (b) A FALSE gadget with four vertices attached: x1, x2, and two operand vertices of additional TRUE gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2 The FALSE gadget In a POSITIVE-1IN3-SAT instance, literals are either non-negated boolean variables or falsity (⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To represent the latter, we introduce a gadget that forces a vertex to be inactive in any PNE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G⊥ = (V⊥, E⊥) an instance of the FALSE gadget, G = (V, E) a graph with V ∩ V⊥ = {x}, and H := (V ∪ V⊥, E ∪ E⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' FALSE is permissive: If G admits a PNE s with sx = 0, then also H admits a PNE t with tv = sv for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' FALSE is restrictive: If H admits a PNE t, then tx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' FALSE is safe: In any PNE t on H and for all m ∈ NG⊥(x), tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Permissiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G admit a PNE s with sx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, the partial strategy profile p with pv = sv, for all v ∈ V , and pyi = 1, for all i ∈ [k], can be extended to a PNE for H by the permissiveness of the NEAR-OR gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Restrictiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Assume towards a contradiction that H admits a PNE t with tx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since TRUE is restrictive, tyi = 1 for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This contradicts NEAR-OR being restrictive, as tx + �k i=1 tyi = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Follows from the safety of the NEAR-OR gadget as x is identified with an operand vertex of the (k + 1)-ary NEAR-OR gadget in G⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='3 The EQUIV gadget Next, we introduce a gadget to identify equal variables in distinct clauses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G↔ = (V↔, E↔) an instance of the EQUIV gadget, G = (V, E) a graph with V ∩ V↔ = {x1, x2}, and H := (V ∪ V↔, E ∪ E↔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' EQUIV is permissive: If G admits a PNE s with sx1 = sx2, then also H admits a PNE t with tv = sv for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' EQUIV is restrictive: If H admits a PNE t, then tx1 = tx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' EQUIV is safe: In any PNE t on H and for all m ∈ NG⊥({x1, x2}), tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 13 t1 t2 t3 x1 y1 z1 x3 y3 z3 NEAR-OR (a) CLAUSE gadget for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' t1 t2 t3 (b) CLAUSE gadget for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Figure 3: CLAUSE gadget: Admits three symmetric PNE with st1 + st2 + st3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (a) A NEAR-OR gadget whose operand vertices t1, t2, and t3 are pairwise connected by three parallel paths with one inner vertex each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' (b) A triangle with the same operand vertex labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Permissiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G admit a PNE s with sx1 = sx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We first show that the strategy profile p with pv := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 sv, if v ∈ V, 0, if v = y, 1, if v ∈ {zi | i ∈ [k]}, is a PNE in the graph G′ that is obtained by removing all vertices in V↔\\({x1, x2, y} ∪ {zi | i ∈ [k]}) from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since py = 0, p is a stable assignment for all v ∈ V , so it remains to show that p is stable also for y and for zi with i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This is the case as for all i ∈ [k], degp(zi) = 0 and zi is active, while degp(y) ∈ {k, k + 2} with k ≥ 1 and y is inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since p is a PNE on G′, it follows from the permissiveness of the TRUE and FALSE gadgets that also H admits a PNE t with tv = pv = sv for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Restrictiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Assume towards a contradiction that H admits a PNE t with tx1 ̸= tx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Without loss of generality let tx1 = 1 and tx2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By the restrictiveness of the TRUE and FALSE gadgets, ty = 0 and tzi = 1 for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, degt(y) = k + 1, contradicting y inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' It is NG⊥({x1, x2}) = {y} and, in any PNE t on H, ty = 0 as FALSE is restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4 The CLAUSE gadget Finally, we represent each clause of a POSITIVE-1IN3-SAT instance by a CLAUSE gadget (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Here, we do not require the familiar trio of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Instead, the gadget is designed to admit three symmetric PNE, each of which has one distinct vertex from {t1, t2, t3} in active state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We prove a slightly weaker claim, which is sufficient for the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let G be an instance of the CLAUSE gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, for every t ∈ {t1, t2, t3}, G admits a PNE s with st = 1, and in any PNE s on G, st1 + st2 + st3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The case of k ≥ 2 is trivial: Exactly the profiles in which exactly one of the three vertices is active are PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let thus k = 1 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show the first claim only for t = t2 as the other cases are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We claim that the partial strategy profile s with sv := � 1, if v ∈ {t2, z1, z2, z3}, 0, if v ∈ {t1, t3, x1, x2, x3, y1, y2, y3}, can be extended to a PNE on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We first argue that s is a PNE on the graph obtained by removing all non-operand vertices of the NEAR-OR gadget from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' On this graph it is 14 degs(t2) = 0 and t2 is active while degs(t1) = degs(t3) = 3 and both t1 and t3 are inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Further, degs(z) = 0 and z is active for every z ∈ {z1, z2, z3} while degs(v) = 1 and v is inactive for all v ∈ {x1, x2, x3, y1, y2, y3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since t1 + t2 + t3 = 1 ̸∈ {0, k + 1}, it follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='5 that s can be extended to a PNE on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Next, let s be any PNE on G and assume towards a contradiction that ℓ := st1 +st2 +st3 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The cases of ℓ = 0 and ℓ = 3 are ruled out by the restrictiveness of the NEAR-OR gadget, so it remains to rule out ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Without loss of generality let st1 = st2 = 1 and st3 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' the other cases are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, degs(xi) = 2 and xi is active for all i ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Thus, degs(t1) ≥ 3, contradicting t1 active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='5 Reduction With our assortment of gadgets, we can prove the main result of this section, which answers the second open question posed in [6]: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The homogeneous binary public goods game equilibrium decision problem on undirected graphs with best-response pattern T ∈ 10+10∗ is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Containment in NP is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For NP-hardness, we describe a polynomial-time re- duction from POSITIVE-1IN3-SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the following, we consider the best-response pattern T = 10k10∗ for a fixed k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To ease notation, we write s(v) instead of sv to denote the strategy of a vertex v in a strategy profile s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let I := � {li 1, li 2, li 3} | i ∈ [ℓ] � with li j ∈ X ∪ {⊥} for all (i, j) ∈ [ℓ] × [3], ℓ ∈ Z≥1, and X = {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' , ξm} a set of boolean variables, be an instance of POSITIVE-1IN3-SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Recall that I is a yes-instance if and only if the formula Φ(X) := � i∈[ℓ] �� li 1 ∨ li 2 ∨ li 3 � ∧ ¬ � li 1 ∧ li 2 � ∧ ¬ � li 2 ∧ li 3 � ∧ ¬ � li 3 ∧ li 1 �� is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We construct an instance G = (V, E) of the binary public goods game that has a PNE if and only if this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Starting from the empty graph, we introduce a disjoint CLAUSE gadget Ci = (V i, Ei) for every {li 1, li 2, li 3} ∈ I, whose vertices we relabel with a superscript i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call the resulting graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Note that for some i ̸= i′ ∈ [ℓ] and j, j′ ∈ [3], it may be the case that li j = li′ j′ ∈ X while ti j ̸= ti′ j′ are disjoint vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For every such quadruple (i, j, i′, j′), we add an EQUIV gadget on fresh non-operand vertices whose operand vertices are identified with ti j and ti′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We call the graph at this point G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Next, for every (i, j) ∈ [ℓ] × [3] with li j = ⊥, we add a FALSE gadget on fresh non-operand vertices whose operand vertex we identify with ti j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This yields the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Clearly, the number of vertices added is polynomial in the number of clauses ℓ, so that this construction can be accomplished in time polynomial in the size of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In the following we show decision equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let first I be a yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, there is a truth assignment σ: X → {0, 1} satisfying Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We claim that the partial strategy assignment s0 with s0(ti j) := � σ(ξ), if li j = ξ ∈ X, 0, if li j = ⊥, for all (i, j) ∈ [ℓ] × [3] can be extended to a PNE on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To this end consider first the CLAUSE gadgets and the associated subgraph G′ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since Φ is satisfied, we have s0(ti 1) + s0(ti 2) + s0(ti 3) = 1 for every i ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='10, it follows that G′ admits a PNE s′ with s′(ti j) = s0(ti j) for all (i, j) ∈ [ℓ] × [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Consider next the EQUIV gadgets and the associated subgraph G′′ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let ti j and ti′ j′ be the operand vertices of an EQUIV gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, li j = li′ j′ ∈ X by construction so that s′(ti j) = s0(ti j) = s0(ti′ j′) = s′(ti′ j′) by definition of s′ and s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' From the permissiveness of the EQUIV gadgets (applied iteratively), it follows that G′′ admits a PNE 15 s′′ with s′′(ti j) = s′(ti j) = s0(ti j) for all (i, j) ∈ [ℓ] × [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Consider next the FALSE gadgets in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let ti j be the operand vertex of such a gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, by construction, li j = ⊥ and thus s′′(ti j) = s0(ti j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By the permissiveness of the FALSE gadgets (applied iteratively), we have that G admits a PNE as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let next G admit a PNE s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We show that the truth assignment σ: X → {0, 1} given by σ(li j) := s(ti j) for all (i, j) ∈ [ℓ]×[3] with li j ∈ X satisfies Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' First, we show that σ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To this end assume towards a contradiction that there are (i, j) ̸= (i′, j′) ∈ [ℓ] × [3] such that li j = li′ j′ but s(ti j) ̸= s(ti′ j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By construction, there is an EQUIV gadget in G with operand vertices ti j and ti′ j′, whose restrictiveness contradicts s being a PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Next, we show that Φ is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Let i ∈ [ℓ] index a clause {li 1, li 2, li 3} ∈ I and consider the associated CLAUSE gadget Ci with operand vertices V i = {ti 1, ti 2, ti 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since EQUIV and FALSE are safe gadgets, implying s(v) = 0 for all v ∈ NG(V i), it follows that s limited to V i is a PNE on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='10, this implies s(ti 1)+s(ti 2)+s(ti 3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Without loss of generality, assume that s(ti 1) = 1 and s(ti 2) = s(ti 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We establish that li 1 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' To this end assume towards a contradiction that li 1 = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, by construction, there is a FALSE gadget in G whose operand vertex is ti 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since FALSE is restrictive, s(ti 1) = 1 contradicts s being a PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Since li 1 ∈ X, we have σ(li 1) = s(ti 1) = 1, so � li 1 ∨ li 2 ∨ li 3 � is satisfied by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' It remains to show that both li 2 and li 3 are false under σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We do so for li 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' the argument for li 3 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The case of li 2 = ⊥ is clear, so let li 2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Then, σ(li 2) = s(ti 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' It follows that also ¬ � li 1 ∧ li 2 � ∧¬ � li 2 ∧ li 3 � ∧¬ � li 3 ∧ li 1 � and, by extension, Φ is satisfied by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Recall that we assume that the pattern of a homogeneous game is part of the problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' If we require instead that the number of intermediate zeros, k, is part of the problem input, then we only obtain a weak NP-hardness result as the reduction above produces a graph with maximum degree in Ω(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This is consistent with our argumentation in the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='4: for k > ∆(G), the precise value of k becomes irrelevant as no vertex can have k + 1 or more active neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In [6] it was shown that any NP-hard pattern starting with a 1 remains NP-hard when it is prefixed with the sequence 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This lets us identify another natural family of patterns that is NP-hard to decide, that of all truncated alternating sequences: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The homogeneous binary public goods game equilibrium decision problem on undirected graphs with best-response pattern T ∈ (10)+10∗ is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' What makes this family interesting is that its “limit case”, the alternating sequence T = (10)∗, is polynomial-time decidable already for the more general case of directed graphs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 6 Conclusions We studied equilibria of the binary public goods game from the perspective of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' We have resolved two open questions posed by Gilboa and Nisan [6] that concern the best-response patterns 1k0∗ for k ≥ 3 and 10k10∗ for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the former family, we discovered a connection to congestion games, which guarantees the existence of equilibria and yields a straightforward polynomial time algorithm to compute one: any sequence of better responses will converge to an equilibrium after at most O(n2) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' While this holds already for the inhomogeneous case where each player may follow a different pattern from this family, the problem becomes PLS-complete when we consider in addition links of varying strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' For the latter family of patterns, we proved that it is NP-hard to decide whether an equilibrium exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' The special case of 1010∗ together with a result in [6] shows that the family of truncated alternating patterns, (10)+10∗, also induces a hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' This complements nicely a positive result for (10)∗ given in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 16 References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Ackermann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' R¨oglin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' V¨ocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' On the impact of combinatorial structure on congestion games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' Journal of the ACM, 55(6):1–22, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' ISSN 1879-2294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='07.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' A refined study of the complexity of binary networked public goods games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' CoRR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='02916.' metadata={'source': 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binary networked public goods games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' In Proceedings of the 34th AAAI Conference on Artificial Intelligence (AAAI), pages 2310–2317, Palo Alto, CA, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='1609/ aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='v34i02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content='5609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFKT4oBgHgl3EQfkS62/content/2301.11849v1.pdf'} diff --git a/TtE3T4oBgHgl3EQfaQo5/content/tmp_files/2301.04504v1.pdf.txt b/TtE3T4oBgHgl3EQfaQo5/content/tmp_files/2301.04504v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fd7708dba35ce485faff11d2944ffa089ba3672 --- /dev/null +++ b/TtE3T4oBgHgl3EQfaQo5/content/tmp_files/2301.04504v1.pdf.txt @@ -0,0 +1,3649 @@ +Astronomy & Astrophysics manuscript no. ms_cocoon +©ESO 2023 +January 12, 2023 +Multiple emission components in the Cygnus cocoon +detected from Fermi-LAT observations ⋆ +X. Astiasarain1,⋆⋆, L. Tibaldo1,⋆⋆⋆, P. Martin1,⋆⋆⋆⋆, J. Knödlseder1, and Q. Remy2 +1 IRAP, Université de Toulouse, CNRS, CNES, UPS, 9 avenue Colonel Roche, 31028 Toulouse, Cedex 4, France +2 Max Planck Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany +Received 29 November 2022; Accepted 06 January 2023 +ABSTRACT +Context. Star-forming regions may play an important role in the life cycle of Galactic cosmic rays (CRs), notably as home to specific +acceleration mechanisms and transport conditions. Gamma-ray observations of Cygnus X have revealed the presence of an excess of +hard-spectrum gamma-ray emission, possibly related to a cocoon of freshly accelerated particles. +Aims. We seek an improved description of the gamma-ray emission from the cocoon using ∼13 years of observations with the Fermi- +Large Area Telescope (LAT) and use it to further constrain the processes and objects responsible for the young CR population. +Methods. We developed an emission model for a large region of interest, including a description of interstellar emission from the +background population of CRs and recent models for other gamma-ray sources in the field. Thus, we performed an improved spectro- +morphological characterisation of the residual emission including the cocoon. +Results. The best-fit model for the cocoon includes two main emission components: an extended component FCES G78.74+1.56, +described by a 2D Gaussian of extension r68 = 4.4◦ ± 0.1◦ +0.1◦ +−0.1◦ and a smooth broken power law spectrum with spectral indices +1.67 ± 0.05+0.02 +−0.01 and 2.12 ± 0.02+0.00 +−0.01 below and above 3.0 ± 0.6+0.0 +−0.2 GeV, respectively; and a central component FCES G80.00+0.50, +traced by the distribution of ionised gas within the borders of the photo-dissociation regions and with a power law spectrum of +index 2.19 ± 0.03+0.00 +−0.01 that is significantly different from the spectrum of FCES G78.74+1.56. An additional extended emission com- +ponent FCES G78.83+3.57, located on the edge of the central cavities in Cygnus X and with a spectrum compatible with that of +FCES G80.00+0.50, is likely related to the cocoon. For the two brightest components FCES G80.00+0.50 and FCES G78.74+1.56, +spectra and radial-azimuthal profiles of the emission can be accounted for in a diffusion-loss framework involving one single popula- +tion of non-thermal particles with a flat injection spectrum. Particles span the full extent of FCES G78.74+1.56 as a result of diffusion +from a central source, and give rise to source FCES G80.00+0.50 by interacting with ionised gas in the innermost region. +Conclusions. For this simple diffusion-loss model, viable setups can be very different in terms of energetics, transport conditions, and +timescales involved, and both hadronic and leptonic scenarios are possible. The solutions range from long-lasting particle acceleration, +possibly in prominent star clusters such as Cyg OB2 and NGC 6910, to a more recent and short-lived release of particles within the +last 10 − 100 kyr, likely from a supernova remnant. The observables extracted from our analysis can be used to perform detailed +comparisons with advanced models of particle acceleration and transport in star-forming regions. +Key words. Acceleration of particles – cosmic rays – open clusters and associations – Gamma rays: ISM +1. Introduction +There is firm evidence that cosmic rays (CRs) at energies be- +low 1 PeV originate from the Milky Way. Supernova remnants +(SNRs) remain the leading candidate as sources of the major- +ity of Galactic CRs, most likely through the process of diffusive +shock acceleration, while alternative source classes including +massive star-forming regions, the Galactic centre, pulsar wind +nebulae (PWNe), and compact binary systems may bring com- +plementary contributions over specific parts of the extended CR +⋆ The template used to model source FCES G80.00+0.50, the map of +excess counts in figure 5 (right), the spectral points from section 3.5, +the map of total neutral hydrogen column density in the local arm (Fig- +ure 10), and the intensity and emissivity profiles discussed in section 4.2 +are available in electronic form at the CDS via anonymous ftp to cd- +sarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi- +bin/qcat?J/A+A/. +⋆⋆ xan.astiasarain@irap.omp.eu +⋆⋆⋆ luigi.tibaldo@irap.omp.eu +⋆⋆⋆⋆ pierrick.martin@irap.omp.eu +spectrum (see, for instance Gabici et al. 2019, and references +therein). +Massive star-forming regions are of particular interest in this +context (for instance Bykov et al. 2020). The clusters of OB stars +at their centres are the progenitors of a variety of particle accel- +eration sites such as SNRs, pulsars, and PWNe, or compact bi- +nary systems. In addition, the collective action of powerful stel- +lar winds and, after a few million to a few tens of million years, +the explosion of massive stars into supernovae lead to the for- +mation of super-bubbles (SBs), which are large cavities filled by +a highly dynamical medium that, as a whole, may play a spe- +cific role in the life cycle of CRs. The isotopic abundances mea- +sured in CRs suggest that at least a fraction of the CR material +is sourced from the winds of massive stars (Binns et al. 2008; +Tatischeff et al. 2021). +Accelerated particles in distant locations can be revealed +via the gamma-ray emission produced when they interact +with interstellar gas, through inelastic collisions for nuclei or +Bremsstrahlung for leptons, and radiation fields, through the +inverse-Compton (IC) scattering by leptons. Therefore, star- +Article number, page 1 of 30 +arXiv:2301.04504v1 [astro-ph.HE] 11 Jan 2023 + +A&A proofs: manuscript no. ms_cocoon +forming regions are expected to be bright gamma-ray sources +from the interactions of particles with the large masses of in- +terstellar gas and the intense radiation fields available in these +environments. Gamma-ray emission in the GeV and TeV en- +ergy ranges is detected towards a growing number of massive +star-forming regions (for a review see for instance Tibaldo et al. +2021), and taken as evidence in favour of in situ CR accelera- +tion. However, the clustering of energetic objects and interstellar +clouds combined with the limited resolution of gamma-ray tele- +scopes makes it difficult to firmly identify the acceleration sites +and mechanisms, and to understand how particles propagate and +interact through the region and eventually escape to merge into +the large-scale CR population in the Galaxy. +Observational progress is matched by a flourishing develop- +ment of models of particle acceleration and transport by stel- +lar winds (Gupta et al. 2018; Bykov et al. 2020; Morlino et al. +2021) and SBs (Bykov 2001; Ferrand & Marcowith 2010; Tolks- +dorf et al. 2019; Vieu et al. 2022). The models show that these +objects can be efficient particle accelerators and make a contri- +bution to Galactic CRs. They predict a number of morphological +and spectral signatures that can be looked for to test the physical +processes at the origin of the observed gamma-ray signals. +Cygnus X is one of the best studied massive star-forming re- +gions in the Milky Way. Cygnus X contains Cygnus OB2 that, +with 78 confirmed O stars (Berlanas et al. 2020), is among the +largest associations of massive stars in the Milky Way. It is com- +posed of multiple substructures with a main group at ∼1.76 kpc +from the Earth and a foreground group at ∼1.35 kpc (Berlanas +et al. 2019) and at least two star-forming bursts ∼3 and ∼5 Myr +ago (Berlanas et al. 2020). A second prominent massive stel- +lar cluster in Cygnus X is NGC 6910 at a distance of ∼1.73 kpc +(Cantat-Gaudin et al. 2020), an age in the range from 5 to 10 Myr +(Delgado & Alfaro 2000; Cantat-Gaudin et al. 2020), and a flat +mass function pointing to a large number of massive stars (Kaur +et al. 2020). +The Large Area Telescope (LAT) aboard the Fermi Gamma- +ray Space Telescope (Atwood et al. 2009) unveiled a hard +gamma-ray excess towards Cygnus X with an extension1 r68 = +3.0◦ ± 0.3◦ (Ackermann et al. 2011). The excess was inter- +preted as the signature of a cocoon of freshly accelerated par- +ticles. Gamma-ray emission in the energy range from hundreds +of GeV to hundreds of TeV from the Cygnus cocoon was sub- +sequently detected using ARGO-YBJ, HAWC, and LHAASO +(Bartoli et al. 2014; Abeysekara et al. 2021; Cao et al. 2021; Li +2022). The most common interpretation involves nuclei acceler- +ated by Cygnus OB2, possibly up to PeV energies. The radial +gamma-ray emission profile above 10 GeV was taken as indica- +tion of diffusion following continuous CR injection over a few +million years (Aharonian et al. 2019). +In this paper, we present a new study of the Cygnus cocoon +based on more than 13 years of Fermi-LAT observations with +the aim of improving the morphological and spectral characteri- +sation of the emission in order to constrain particle acceleration +and propagation scenarios in the region. The characterisation of +the cocoon requires a careful modelling of the interstellar gas +distribution in the region that is presented in Section 2, while +we describe the analysis of gamma-ray data, including morpho- +logical, spectral, and spectro-morphological characterisation of +the cocoon emission in Section 3. The observables we derived +1 Throughout the paper we refer to a source extension as its 68% con- +tainment radius r68. For a 2D Gaussian intensity distribution, r68 = +1.51σ. +are then discussed and interpreted in Section 4 and our conclu- +sions are presented in Section 5. +2. Construction of interstellar gas maps +The distribution of interstellar gas towards the region of interest +is a key ingredient of our analysis for two reasons: 1) it is neces- +sary to model the strong foreground and background gamma-ray +emission from the interactions of the large-scale Galactic CR +population with interstellar gas in the direction of Cygnus, and +thus be able to extract and characterise the emission of the co- +coon; 2) it is used in the interpretation of the gamma-ray signal +in terms of the underlying CR populations. +2.1. Atomic and molecular gas +We trace atomic gas using the 21 cm emission line from the +hyperfine transition of atomic hydrogen H I. We use data from +the Canadian Galactic Plane Survey (CGPS, Taylor et al. 2003) +with an angular resolution of 1′ and a velocity resolution of 1.3 +km s−1 in the region with Galactic longitude 75.5◦ < l < 90◦ +and Galactic latitude −3◦ < b < 5◦. Outside this region we use +data from the all-sky HI4PI survey from Effelsberg and Parkes +observations (HI4PI Collaboration et al. 2016) with a lower an- +gular resolution of 0.27◦ and velocity resolution of 1.49 km s−1. +We checked the consistency of the two surveys by comparing the +data in the region covered by the CGPS. +We derived column densities N(H I) under the hypothesis of a +uniform spin temperature. All results are shown for the reference +spin temperature of 250 K suggested by emission-absorption +spectrum pairs in the CGPS area (Dickey et al. 2009) and that +was also found to best reproduce gamma-ray observations of the +Cygnus region based on an earlier analysis (Ackermann et al. +2012a). This is a highly uncertain parameter that is not expected +to be uniform along lines of sight and across the region. There- +fore, the analysis was also performed for alternative uniform spin +temperatures of 100 K (lower bound set by the brightness tem- +peratures observed in the region), 400 K, and the optically thin +case, which are used to set systematic uncertainties on relevant +quantities. +Molecular hydrogen H2 cannot be traced directly. We use the +12CO J1→0 rotational line at 2.6 mm as surrogate tracer, under the +usual hypothesis that N(H2) column densities are directly pro- +portional to the CO intensity (velocity-integrated brightness tem- +perature) WCO through a constant known as XCO ≡ N(H2)/WCO. +We use CO data from the composite survey by Dame et al. +(2001) with an angular resolution of 0.125◦ in the area consid- +ered in this paper and a velocity resolution of 1.3 km s−1. Data +were noise-filtered using the moment-masking technique (Dame +2011). +The Doppler shift of the lines can be used to infer the gas ve- +locity along the line of sight due to Galactic rotation, and there- +fore separate multiple structures. However, intrinsic velocity dis- +persion can cause biases in the estimates of gas column densi- +ties across adjacent structures. To address this problem, we used +the line profile fitting technique described in Remy et al. (2017) +to decompose emission from each line of sight into a combina- +tion of pseudo-Voigt functions. We built longitude-velocity and +latitude-velocity diagrams based on the fit results for H I and +CO, and defined in the longitude-latitude-velocity space bound- +aries that separate the gas into three structures along the line +of sight, namely the local arm (including the Cygnus complex), +the Perseus arm, and the outer arm and beyond. Figure 1 shows +Article number, page 2 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +an example of longitude-velocity diagram in the longitude range +73◦ < l < 87◦ that is used in the following analysis. +75 +80 +85 +l (deg) +−150 +−100 +−50 +0 +50 +V (km s−1) +l ∈[73.0◦, 87.0◦], b ∈[-2.0◦, 2.0◦] +100 +101 +102 +N(H I) (1020 H cm−2) +Fig. 1: H I column density as a function of Doppler-shift veloc- +ity and Galactic longitude summed over −2◦ < b < 2◦. Total +column densities from each pseudo-Voigt profile were assigned +to the velocity of the peak. The black lines show the boundaries +that we defined to separate the three structures along the line of +sight: local arm, Perseus arm, and outer Arm and beyond (from +top to bottom). +The final H I and CO maps for the three regions along the line +of sight are shown in Figure 2. Since all surveys have different +angular resolution, the maps were re-binned on a common grid +of 1.875′. +2.2. Dark neutral medium (DNM) +A significant fraction of neutral interstellar gas cannot be traced +by the H I 21 cm line nor by the 12CO J1→0 rotational line (Gre- +nier et al. 2005) and is therefore missing in the maps described +above. It can be referred to as the dark neutral medium (DNM) +and it is thought to be a combination of opaque H I and diffuse +H2 at the atomic-molecular interface of clouds, or dense H2 at +the core of molecular clouds (Remy et al. 2017). +If dust and gas in the interstellar medium (ISM) were well +mixed and the dust grains physical and chemical properties were +the same everywhere, dust thermal emission would be propor- +tional to total gas column densities along the line of sight. There- +fore, we can derive a DNM map by subtracting from the dust +thermal emission the components correlated with H I and CO. +We use a map of the dust optical depth at 353 GHz obtained from +component separation of Planck and IRAS data (Planck Collab- +oration et al. 2016b) with an effective angular resolution of 5′ in +high signal-to-noise regions. +To avoid biases from the missing DNM component in the +determination of the components correlated with H I and CO, we +used the iterative fitting procedure described in Tibaldo et al. +(2015). Briefly, the procedure consists in an iterative fitting of +the gas maps to the dust map where the positive part of the resid- +uals is re-injected in the model at each iteration to compute unbi- +ased values of the fit parameters and obtain an estimation of the +missing DNM component. A DNM map was calculated for each +uniform spin temperature considered. Figure 3 shows the DNM +map obtained for the reference spin temperature value of 250 K. +2.3. Ionised gas +We derived an ionised gas column density map from the free- +free emission measure EM(l, b) extracted from component sep- +aration of Planck, WMAP, and 408 MHz data by Planck Col- +laboration et al. (2016a). The free-free emission measure from +Cygnus X is dominated by two strong peaks that, as indicated +by 8 µm emission from dust, lie inside the cavities carved in the +ISM by the intense star-forming activity in the region. +We calculated H II column densities under the assumption +that ionised gas fills a sphere of radius 3.5◦ corresponding to +∼100 pc at a distance of 1.7 kpc and with an uniform density +along each line of sight. The sphere is meant to model the ionised +cavities at the hearth of Cygnus X. With r the radius of the sphere +and d the distance to Cygnus X the electron volume density is: +ne(l, b) = +������������ +EM(l, b) +2 +� +r2 − d2 sin2(l − l0) − d2 sin2(b − b0) +������������ +1/2 +. +(1) +Therefore, for the column density we obtain: +NH II(l, b) = ne(l, b) × 2 +� +r2 − d2 sin2(l − l0) − d2 sin2(b − b0), +(2) +where l0 and b0 are the position of the sphere’s centre and +EM(l, b) the emission measure in a given direction. +The final ionised gas column density map is displayed in Fig- +ure 4. The angular resolution of the free-free emission measure +map from Planck is 1◦. The final ionised gas column density +map was re-binned on the same grid as the MSX 8 µm map with +a grid spacing of 1”. +3. Gamma-ray analysis and results +3.1. Data selection +We analysed 13.25 years of Fermi-LAT data from the beginning +of the mission on 4 August 2008 to 3 November 2021. We used +the P8R3 data set (Atwood et al. 2013; Bruel et al. 2018) and se- +lected events in the P8R3_SOURCE class that is associated with +instrument response functions P8R3_SOURCE_V3. This event +selection has a level of background contamination sufficiently +low to study the bright extended emission from Cygnus X. Fur- +thermore, we restricted the analysis to time intervals in which +the LAT configuration and data quality is appropriate for science +analysis. +Events are separated in four independent data sets accord- +ing to their PSF event type, that is the quality of the direction +reconstruction. For each event type, we selected events above +a minimum energy so that the point spread function (PSF) 68% +containment radius is always better than 0.7◦, which roughly cor- +responds to the characteristic size of the most prominent spatial +structures in the gas maps. The minimum energy threshold used +is 0.5 GeV. Lowering the minimum energy induced instabilities +in the analysis due to bright emission from a few pulsars in the +region. To reliably characterise extended emission at energies +< 0.5 GeV event selection based on the pulsars phases would be +necessary, but this is beyond the scope of the current study. The +maximum energy is 1 TeV for all event types. +To reduce contamination from the bright gamma-ray emis- +sion from the Earth atmosphere, we selected events within a cone +from the local zenith with aperture zmax. The value of zmax was +Article number, page 3 of 30 + +A&A proofs: manuscript no. ms_cocoon +10◦ +0◦ +−10◦ +Galactic Latitude +85◦ +80◦ +75◦ +10◦ +0◦ +−10◦ +Galactic Latitude +85◦ +80◦ +75◦ +Galactic Longitude +85◦ +80◦ +75◦ +0 +101 +102 +N(H I) (1020 H cm−2) +0 +100 +101 +WCO (K km s−1) +Fig. 2: H I column densities for a spin temperature of 250 K (top row) and WCO intensities (bottom row) for the local arm, Perseus +arm, and outer arm and beyond (from left to right). +chosen from a visual inspection of the distribution of counts as +a function of zenith angle for each event type component in the +given energy ranges. Energy and zenith angle selection for the +four data sets are summarised in Table 1. +Table 1: The four data sets used in the analysis. +Event type +Energy range (GeV) +zmax +PSF3 +0.5 - 1000 +100◦ +PSF2 +1 - 1000 +100◦ +PSF1 +3.2 - 1000 +100◦ +PSF0 +5 - 1000 +105◦ +3.2. Region of interest and emission model +A major challenge in the characterisation of extended emission +from Cygnus X is to model the bright interstellar emission from +the large-scale population of CRs. Due to the large column den- +sities of the ISM in this region, emission associated with gas is +the dominant contribution at GeV energies (Ackermann et al. +2012a). Under the assumption that the large-scale CR densities +are uniform on the spatial scales of interstellar complexes, we +can model the foreground and background intensity associated +with interstellar gas Igas as a linear combination of the column +density maps for the different gas phases and structures along +Article number, page 4 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +85◦ +80◦ +75◦ +10◦ +0◦ +−10◦ +Galactic Longitude +Galactic Latitude +0 +100 +200 +300 +400 +τ353 (10−6 mag) +Fig. 3: Excess dust optical depth associated to the DNM obtained +using the procedure described in the text for the reference H I +spin temperature of 250 K. +84◦ +82◦ +80◦ +78◦ +76◦ +4◦ +2◦ +0◦ +−2◦ +Galactic Longitude +Galactic Latitude +0 +1 +2 +3 +4 +Column density (1021 H cm−2) +Fig. 4: Ionised gas column densities based on the hypothesis +of spherical geometry with a uniform density along each line +of sight. The black contours delineate the outer borders of the +photo-dissociation regions and correspond to > 1.85 10−6 W m−2 +sr−1 in the MSX 8 µm data. +the line of sight: +Igas(l, b, E) = qLIS(E) · +�������� +3 +� +ı=1 +�Aı NH I,ı(l, b) + Bı WCO,ı(l, b)� ++C τDNM(l, b) +� +, +(3) +where qLIS(E) is the local gas emissivity spectrum, that is the +gamma-ray emission rate per hydrogen atom, from Casandjian +(2015), derived from LAT data. The summation over ı describes +the combination of the three regions along the line of sight: lo- +cal arm, Perseus arm, outer arm and beyond. The free param- +eters Aı, Bı, and C account at the same time for variations of +the large-scale CR densities across the three regions, and for the +XCO ratios and the dust specific opacity σ353 = τ353/N(H I). The +spectral shape of qLIS is fixed throughout the paper. We know +that spectral variations of the emissivity along the line of sight +are small towards Cygnus (Ackermann et al. 2012a) and, in gen- +eral, towards the outer Galaxy (Acero et al. 2016). Conversely, +this implies that any spectral deviations from the local interstel- +lar spectrum (LIS) in Cygnus X are not accounted for by the +background model and characterised as part of the cocoon. +An additional diffuse component is given by IC emis- +sion from the large-scale population of CR leptons. We ac- +counted for it using the GALPROP model SYZ6R30T150C2 +(Ackermann et al. 2012b). Finally, we need to account for the +isotropic gamma-ray background, which is a combination of +extra-galactic diffuse gamma-ray emission (probably due to pop- +ulations of unresolved sources) and of residual contamination by +charged CRs. For this component, we used the tabulated spectra +provided by the Fermi-LAT collaboration and determined from +an analysis of LAT data over a large region of the sky2. +We note that the IC model is also subject to large uncertain- +ties. However, morphological variations over our limited region +of interest described below are expected to be small for con- +ventional models. Moreover, uncertainties in the spectrum are +mitigated by the fact that the isotropic background spectrum is +derived from a fit to the LAT data. +The interstellar emission model, along with the LAT re- +sponse, sets the choice of the region of interest (ROI) for the +analysis. The longitude and latitude extents should be suffi- +ciently large to separate the extended emission of the Cygnus +cocoon from the large-scale background, and so that the differ- +ent components of the background model can be reliably con- +strained by the data. We chose a ROI with Galactic longitude +73◦ ≤ l ≤ 87◦ and with Galactic latitude |b| ≤ 15◦. The longi- +tude interval leaves out complexes associated with Cygnus OB1 +at l < 73◦ and with HB 21 at l > 87◦. The wider coverage in +latitude makes it possible to better constrain emission from local +H I, IC scattering, and the isotropic background. +We modeled individual sources within the region based on +the most recent catalogue of gamma-ray sources detected by the +LAT, 4FGL-DR3 (Fermi-LAT collaboration et al. 2022). All the +sources within a square box of 40◦ side centred at l = 80◦ and +b = 0◦ were included to account for the spill-over due to the PSF. +For two extended sources with potential impact on the char- +acterisation of the cocoon emission, we replaced the 4FGL-DR3 +models with dedicated models provided by recent in-depth stud- +ies. The SNR γ Cygni is modelled according to the results from +a joint fit of Fermi-LAT and MAGIC data at energies > 5 GeV +(MAGIC Collaboration et al. 2020). The source is modelled as a +disk with a log-parabola spectrum to account for the shell, plus +a 2D Gaussian with a power law spectrum to account for an ad- +ditional component in the north of the shell. The arc component +detected by MAGIC is not included since its flux is subdominant +at energies < 1 TeV. A morphological evolution of the remnant +below 5 GeV is very challenging to characterise due to bright +2 We used files iso_P8R3_SOURCE_V3_PSFn_v1.txt, +with +n +the PSF event type, from https://fermi.gsfc.nasa.gov/ssc/ +data/access/lat/BackgroundModels.html. +Article number, page 5 of 30 + +A&A proofs: manuscript no. ms_cocoon +emission from PSR J2021+4026 (MAGIC Collaboration et al. +2020). Thus, this possibility is not considered in our study. +The SNR Cygnus Loop is modelled following the analysis by +Tutone et al. (2021) in the 0.1-100 GeV energy range. We used +two templates based on X-ray (ROSAT 0.1 − 2.4 keV) and UV +(GALEX 1771 − 2831 Å) data, each of them associated to a log- +parabola spectrum. Emission from this source above 100 GeV is +expected to be small, and we visually checked in the residuals +that there were no excess or deficit of counts at the location of +the Cygnus Loop. +Figure 5 shows the gamma-ray count map in the region of in- +terest and the excess counts associated with the cocoon obtained +by subtracting from the data counts the best-fit model presented +in Section 3.6. The cocoon, that is the excess shown in the right +85◦ +80◦ +75◦ +10◦ +0◦ +−10◦ +Galactic Latitude +102 +103 +104 +Counts +85◦ +80◦ +75◦ +0 +1000 +Excess counts +Galactic Longitude +Fig. 5: Map of data counts in the full 0.5 GeV−1 TeV en- +ergy range (left) and map of excess counts obtained by sub- +tracting from the data counts the best-fit model presented in +Section 3.6 except for the components associated to the co- +coon, namely FCES G78.74+1.56, FCES G80.00+0.50, and +FCES G78.83+3.57 (right). The dashed contours correspond to +the 8 µm emission from MSX data at 1.85 × 10−6 W m−2 sr−1. +The contours correspond to the column density of the ionised +gas template at 3.5×1021 H cm−2. The circle and the dashed cir- +cle correspond to the position and r68 of FCES G78.83+3.57 and +FCES G78.74+1.56 respectively. +panel of Figure 5, was initially modelled as in 4FGL-DR3: a 2D +Gaussian with r68 = 3◦ (Ackermann et al. 2011) and a spec- +trum described by a log-parabola function. The morphological +and spectral models were refined later during the analysis. +3.3. Analysis framework +The analysis is performed using fermitools v2.0.8 and a modi- +fied version of Fermipy v1.0.1 that enables the use of catalogue +4FGL-DR3. Models are fit to the data via a binned maximum +likelihood analysis with Poisson statistics. Events are binned on +a grid with 10 bins per decade in energy and on maps with a +pixel size of 0.1◦ in arrival direction. +Throughout the paper, we compare several models for the +region and the source of interest. In the simpler cases we use the +likelihood ratio test, that is the test statistic defined as: +TS = 2 (ln L − ln L0) , +(4) +where L0 is the maximum likelihood of a more parsimonious +emission model with fewer free parameters (null hypothesis) and +L is the maximum likelihood of the more complex model that we +want to test (test hypothesis). In the null hypothesis TS is dis- +tributed as a χ2 +n with a number of degrees of freedom n equal to +the difference of degrees of freedom between the two models. +This is only valid for nested models, that is if the model in the +null hypothesis can be obtained from the model in the test hy- +pothesis by fixing some of its parameters to values in the interior +of the allowed range (for instance Protassov et al. 2002) +For non-nested models, we use the Akaike Information Cri- +terion (AIC). The AIC of a model is defined as: +AIC = 2k − 2 ln L, +(5) +with k number of free parameters in the model and L the maxi- +mum likelihood of the model. The model providing the smallest +AIC is taken as the one best representing the data at the smaller +cost in terms of free parameters according to information theory +(for instance Burnham & Anderson 2002). +Throughout the paper, we use the method described in Bruel +(2021) to assess the goodness of fit of the different models con- +sidered. Practically, we show the deviation of the data with re- +spect to the model in units of significance based on the Poisson +statistics using the so-called PS maps. +The analysis starts with a preliminary optimisation of the +emission model via a procedure described in Appendix A.1. +In subsequent steps, unless stated otherwise, we keep free the +normalisations of the gas and IC components, as well as the +normalisations and spectral parameters of the three pulsars +PSR J2021+4026, PSR J2021+3651, and PSR J2032+4127, the +two γ Cygni extended components, and the two Cygnus Loop +extended components. The normalisation of the subdominant +isotropic background is fixed after the preliminary iterative op- +timisation of all the sources in the ROI due to the possible de- +generacy with the IC component. The best-fit normalisation ob- +tained is 0.91±0.02 (with variations ≤ 0.01 for the different spin +temperature values). +3.4. Morphological analysis +In this section, we aim at characterising the morphology of the +cocoon. As a first step, we optimised the position and extension +of the 2D Gaussian model used in Ackermann et al. (2011) and +the LAT catalogues to describe the extended cocoon emission, +following the methodology described in Appendix A.2. +The corresponding PS map is displayed in Figure 6 (panel +a, top row). An excess appears in the central region of the co- +coon, in part reminiscent of the two main peaks of ionised gas +column density within the Cygnus X cavities (Figure 4). There- +fore, we tested the addition of a central component in the cocoon +Article number, page 6 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +region using two alternative models: either the ionised gas tem- +plate clipped at the boundaries of the cavities (defined as con- +tours above 1.85 10−6 W m−2 sr−1 emission at 8 µm), or two +Gaussians with free extensions and positions, initialised at the +main peaks in the ionised gas map. All newly added sources on +top of the Gaussian model for the extended cocoon component +here and elsewhere in this section are modelled using a power- +law spectrum. +For the models described above, extended deviations are still +apparent (see Figure 6 top row). The largest excess appears in +the western part of the cocoon at l ∼ 78.8◦ and b ∼ 3.7◦. It +does not overlap with any known sources or structures in the gas +maps. A second extended region of positive deviations appears +at the edge of our ROI, at l ∼ 84.6◦ and l ∼ −5.6◦ towards the +southern arc of the Cygnus SB as imaged in soft X-rays (Cash +et al. 1980). We added two additional Gaussian components to +model those excesses, hereafter referred to as western and off- +field excesses. We initialise the Gaussian centres on the excess +peaks, and fit their positions and extensions. This results in a +significant likelihood improvement (∆ ln L ∼ 200) for all models +of the central cocoon component. +The different models for the cocoon central component are +compared in Table 2. The addition of the central cocoon com- +ponent on top of the 2D Gaussian for the extended one provides +a significant improvement in likelihood (∆ ln L > 240). Con- +versely, a model including the ionised gas map without the ex- +tended cocoon Gaussian component resulted in a marked degra- +dation of the likelihood (∆ ln L = −600). +The model including the ionised gas template for the cen- +tral cocoon component provides the largest likelihood and the +smallest AIC, and therefore it is the one favoured by our analy- +sis. It is strongly preferred over two additional Gaussian compo- +nents at the peaks in the ionised gas distribution (∆AIC < −124), +which strengthens the evidence for a correlation between part of +the gamma-ray signal and the ionised matter distribution3. This +conclusion is supported by visual inspection of the deviations in +Figure 6 (bottom row, panels b and c). +We also tested the full ionised gas map, that is not clipped +at the boundaries of the cavities, but this yielded a smaller like- +lihood. The interpretation of this result is not straightforward. +The reason may be physical, for example related to confinement +of the particles in the cavities. It could also be related to lim- +itations in the analysis, such as systematic biases in the emis- +sion measure map extracted from Planck data, approximations +in the derivation of the ionised gas column density, or degenera- +cies with other gas templates outside the two main peaks in the +map. +Spatial parameters for the multiple overlapping extended +sources may be degenerate to some level. To robustly determine +their values, we performed an iterative fit of the positions and ex- +tensions of all 2D Gaussian sources discussed in this section for +the best model. The iterations proceed until the log-likelihood +improvement between two iterations is smaller than 1. The iter- +ative fit converged after 6 iterations, with a total improvement in +ln L of 30. +Table 3 provides the best-fit morphological parameters and +TS of all the extended components discussed in this section for +the case in which the cocoon region is modelled by a broad 2D +Gaussian (extended component) plus the ionised gas map (cen- +3 The AIC criterion may not be fully appropriate in this case due to +the information entropy encoded in the geometry of the template and +not represented by any fit parameter, but the large improvement of log- +likelihood clearly favours the model with the ionised gas template. +tral component), and an additional smaller 2D Gaussian slightly +off the emission peak (western component). The extended emis- +sion components are named after their Galactic coordinates as +FCES GLL.ll ± B.bb (FCES stands for Fermi Cygnus Extended +Source). To make the paper easier to read, the sources are given a +nickname that we use throughout the paper. FCES G78.74+1.56 +is the name given to the Gaussian that describes the cocoon ex- +tended emission, nicknamed CoExt, FCES G80.00+0.50 to the +component modelled by the ionised gas map in the cocoon cen- +tral region, nicknamed CoCent, and FCES G78.83+3.57 to the +component corresponding to the excess appearing in the west- +ern part of the cocoon, nicknamed CoWest. Interestingly, the +addition of a central component for the cocoon results in a +larger r68 for the extended component with respect to previous +studies. We remark that the off-field excess, ultimately dubbed +FCES G85.00−1.78 and nicknamed OffExc, is best modelled by +a Gaussian centred at the edge of the ROI, therefore its char- +acterisation may be inaccurate. A better characterisation of this +component is left for further work. The positions and extensions +of sources in the cocoon area are shown overlaid to the excess +map in the right panel of Figure 5. +3.5. Spectral analysis +In this section, we aim at characterising the spectral properties of +the FCES sources. Based on the best morphological model de- +rived in the previous section, and for each emission component +listed in Table 3, we tested three spectral models: a simple power +law (PL), a log-parabola (LP), and a smooth broken power law +(SBPL). The expressions for these models are given in Appendix +A.3. +The models are compared in Table 4. For the components +CoCent and CoWest the best-fit model is the simple PL, with the +LP providing a negligible improvement in log-likelihood. The fit +of the SBPL for these two components did not converge, pre- +sumably due to lack of curvature in the spectrum. Conversely, +for CoExt and OffExc the models with curvature (LP or SBPL) +provide a large improvement in ln L. From the Akaike criterion, +we can conclude that the SBPL is favoured. Spectral parameters +for the best-fit models are presented in the next subsection. +The spectral indices of the components CoCent and CoWest +are compatible with each other. If we fix the spectral index of +CoWest to the best-fit value for CoCent, we obtain a decrease +in log-likelihood of 1.2 [1.2, 1.8] (1.1 [1.1, 1.3] σ, where here +and in the following variations or uncertainties refer to the dif- +ferent spin temperatures). This demonstrates that the two com- +ponents have compatible spectral shapes. However, if we model +CoWest or CoCent with the same spectral shape as CoExt and +a free normalisation, we observe a decrease in log-likelihood of +7.9 [7.9, 9.8] (∆AIC = 19.8 [19.8, 23.6]) and 70.6 [66.6, 70.6] +(∆AIC = 145.2 [137.2, 145.2]), respectively. So both CoCent +and CoWest have spectra incompatible with the spectrum of Co- +Ext, this time suggesting a different origin of the gamma-ray +emission. +We then computed the spectral energy distribution (SED) of +the four sources. To this end we performed independent analyses +over 4 energy bins per decade between 500 MeV and 1 TeV. For +this part of the analysis all spectral-shape parameters are fixed +and only normalisations are allowed to vary. However, the nor- +malisations of the gas maps are fixed to the best-fit values ob- +tained for the entire energy range to preserve the local emissiv- +ity shape. The IC normalisation is also fixed to the best-fit value +obtained for the entire energy range due to the large degeneracy +with the extended components. Finally, the normalisations of all +Article number, page 7 of 30 + +A&A proofs: manuscript no. ms_cocoon +Table 2: Comparison of different spatial models +Model +∆ ln L +∆AIC +Extended Gaussian +0 +0 ++Western + Off-field +Extended Gaussian + 2 Gaussians (IG peaks) +316 [246, 316] +−622 [−622, −492] ++ Western + Off-field +Extended Gaussian + IG template +435 [379, 435] +−868 [−868, −746] ++ Western + Off-field +Notes. ln L and AIC values are provided as differences with respect to the simplest model with only one 2D Gaussian for the extended emission +of the cocoon. The intervals correspond to the minimum and maximum spin temperatures considered. The Western and Off-field components are +named later FCES G78.83+3.57 and FCES G85.00−1.78. +Table 3: Best-fit morphological parameters and TS for the extended emission components considered in the morphological analysis. +Name +l(◦) +b(◦) +r68(◦) +TS +FCES G78.74+1.56 (CoExt) +78.7 ± 0.1+0.0 +−0.2 +1.56 ± 0.06+0.07 +−0.02 +4.4 ± 0.1+0.1 +−0.1 +2751 [2436, 2824] +FCES G80.00+0.50 (CoCent) +... +... +... +1267 [1267, 1301] +FCES G78.83+3.57 (CoWest) +78.8 ± 0.1+0.0 +−0.1 +3.6 ± 0.1+0.0 +−0.0 +0.9 ± 0.1+0.0 +−0.0 +93 [93, 106] +FCES G85.00−1.78 (OffExc) +85.0 ± 0.4+0.2 +−0.2 +−1.8 ± 0.25+0.3 +−0.2 +6.4 ± 0.2+0.1 +−0.1 +684 [680, 723] +Notes. The first uncertainties are statistical. The second uncertainties and TS variations are systematic from varying the spin temperature. +pulsars are fixed above 5 GeV because their emission fades off +rapidly. The results are shown in Figure 7. For flux densities (and +all derived quantities later) we include in the systematic uncer- +tainties those from the effective area of the LAT, combined in +quadrature with those from the spin temperature choice. +The source with the highest flux is CoExt, followed by Co- +Cent. The spectrum of CoExt extends to higher energies and con- +nects to the cocoon spectrum measured by HAWC (Abeysekara +et al. 2021), confirming earlier indications of a spectral break +between the GeV and the TeV energy ranges. Our spectrum for +CoExt is similar at energies > 1 GeV to the cocoon SED in cat- +alogue 4FGL-DR3. On the contrary, our SED lies above the one +presented in Ackermann et al. (2011), which is closer to our SED +for CoCent. Presumably, the SED determination in Ackermann +et al. (2011) was biased towards the central component, while +the extended component captured by CoExt in our analysis was +difficult to detect at that time due to a reduced amount of data (2 +years versus more than 13 years here) and a less advanced event +reconstruction scheme. +3.6. Final global fit +After the selection of the best spectral models we performed a +final optimisation of the ROI, including free normalisation and +spectral-shape parameters for the FCES sources. The final spec- +tral parameters of the FCES sources are displayed in Table 5. +Figure 8 illustrates the quality of the ROI model after all the +optimisation steps. The PS map show that we obtained a model +of the ROI with no deviation above 3σ and the fractional devia- +tion in the bottom left panel show no deviation above 10% in the +central part of the ROI. This model serves as a reference for the +following spectro-morphological analysis. The largest deviation +with a PS value close to 3σ lies at l ∼ 84◦, b ∼ 12◦ (at 1◦ from +the Geminga-like pulsar PSR J1957+5033; see Saz Parkinson +et al. 2010). The study of this excess is left for another work. +Figure 9 shows the final likelihood values for the different +spin temperature values considered. The largest likelihood is ob- +tained for a spin temperature of 100 K, but with a difference in +log-likelihood < 1 with respect to the reference value of 250 K. +The largest difference ∆ ln L ∼ 6 is found for the optically thin +case. +After the final global fit the normalisation of the H I emis- +sivity in the local arm is 0.97 ± 0.01 +0.06 +−0.17. The normalisation +is in reasonable agreement with the average value for the lo- +cal neighbourhood from Casandjian (2015). Under the hypothe- +sis that the same CR population interacts with atomic, molec- +ular and dark gas in the local arm, we can use the normali- +sations of the gas maps to infer the XCO factor, which yields +(0.75 ± 0.06 +0.15 +−0.02) × 1020 H cm−2(K km s−1)−1. This is a factor +of ∼2 lower than results from the earlier analysis of Cygnus +in Ackermann et al. (2012a), but close to gamma-ray estimates +from nearby CO clouds (for instance Remy et al. 2017), which +strengthens the hypothesis that XCO variations found between lo- +cal high-latitude clouds and the local arm may be highly sensi- +tive to the separation of DNM and CO bright molecular cloud in +the construction of gas maps and gamma-ray analyses. Other ef- +fects related to the increasing difficulty to separate the gas phases +at larger distances may also be at play. Our analysis exploiting +the PSF event types provides an improvement in this respect over +the work in Ackermann et al. (2012a). +The determination of a conversion factor for the DNM tracer +is less obvious due to the lack of knowledge on the distribu- +tion along the line of sight. However, the morphology of the +DNM in Figure 3 closely resembles the structures in the lo- +cal arm and Cygnus complex in Figure 2. Therefore, for sim- +plicity we assume that all the DNM is in the closest region. +Based on this assumption, we can follow the same procedure +used for XCO and infer a DNM dust specific opacity σ353 = +(1.370 ± 0.05 +0.059 +−0.170) × 10−26 cm2 H−1, also close to gamma-ray +results for nearby clouds (Remy et al. 2017). +We use these coefficients to build a total column density map +of neutral gas in the local arm and the Cygnus complex, which +is shown in the left panel of Figure 10 and is used for the in- +terpretation of the results in the Section 4. With respect to the +reference spin temperature of 250 K, the total column density of +neutral gas increases by ∼ 16% for a spin temperature of 100 K +and decreases by ∼ 5% for the optically thin case. +Article number, page 8 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +10◦ +0◦ +−10◦ +Galactic Latitude +85◦ +80◦ +75◦ +10◦ +0◦ +−10◦ +Galactic Latitude +85◦ +80◦ +75◦ +Galactic Longitude +85◦ +80◦ +75◦ +−3 +−2 +−1 +0 +1 +2 +3 +Sigma +(a) +(b) +(c) +Fig. 6: PS maps for different models considered in the morphological analysis. On the top panels we can see the PS maps for +different morphological models: a) one extended Gaussian, b) one extended Gaussian plus two smaller Gaussians at the peaks of the +ionised gas column density distribution, c) one extended Gaussian plus the ionised gas template. On the bottom panels, we can see +the PS maps for the same models with the addition of two Gaussians for the western and off-field excesses in the model (ultimately +labelled FCES G78.83+3.57 and FCES G85.00−1.78). The bin size is 0.1◦ and the maps were smoothed for display with a kernel +of size 0.13◦ +Table 4: Statistical comparison of different spectral models for the extended sources in Cygnus X +Component +∆ ln LLP−PL +∆ ln LSBPL−PL +∆AICSBPL−LP +FCES G78.74+1.56 (CoExt) +33 [27, 35] +42 [36, 43] +−14 [−14, −14] +FCES G80.00+0.50 (CoCent) +1 [1, 1] +... +... +FCES G78.83+3.57 (CoWest) +1 [1, 2] +... +... +FCES G85.00−1.78 (OffExc) +26 [24, 27] +38 [35, 38] +−22 [−22, −20] +Notes. The columns show the log-likelihood differences between a log-parabola and a power-law model, ∆ ln LLP−PL, or between a smooth +broken-power-law and a power-law model, ∆ ln LSBPL−PL, and the Akaike information criterion difference between a smooth broken-power-law +and a log-parabola model, ∆AICSBPL−LP. The intervals correspond to the minimum and maximum values obtained from variation of the spin +temperature. +Article number, page 9 of 30 + +A&A proofs: manuscript no. ms_cocoon +10−6 +10−5 +10−4 +E2dN +dE (MeV2 cm−2 s−1 MeV−1) +FCES G78.74+1.56 +Broadband fit +Bin per bin +LAT Ackermann et al., 2011 +LAT 4FGL-DR3 +HAWC Abeysekara et al., 2021 +FCES G80.00+0.50 +Broadband fit +Bin per bin +LAT Ackermann et al., 2011 +LAT 4FGL-DR3 +HAWC Abeysekara et al., 2021 +102 +103 +104 +105 +106 +107 +108 +E (MeV) +10−6 +10−5 +10−4 +E2dN +dE (MeV2 cm−2 s−1 MeV−1) +FCES G78.83+3.57 +Broadband fit +Bin per bin +102 +103 +104 +105 +106 +107 +108 +E (MeV) +FCES G85.00−1.78 +Broadband fit +Bin per bin +Fig. 7: Spectral energy distribution of the four extended components studied. In the top panel, we show for reference earlier de- +terminations of the cocoon spectrum. The statistical uncertainties are displayed within the error caps and the full error bar is the +quadratic sum of the statistical uncertainties, the uncertainties related to the different spin temperatures, and the uncertainties related +to the effective area of the telescope. +Table 5: Best-fit values after the final optimisation of the model. +Component +Model +Spectral parameters +N0 (cm−2 s−1 MeV−1) +γ or γ1 +γ2 +Eb (GeV) +FCES G78.74+1.56 (CoExt) +SBPL +8.6 ± 0.6+0.2 +−0.9 × 10−11 +1.67 ± 0.05+0.02 +−0.01 +2.12 ± 0.02+0.00 +−0.01 +3.0 ± 0.6+0.0 +−0.2 +FCES G80.00+0.50 (CoCent) +PL +4.7 ± 0.2+0.0 +−0.0 × 10−11 +2.19 ± 0.03+0.00 +−0.01 +... +... +FCES G78.83+3.57 (CoWest) +PL +4.8 ± 0.6+0.3 +−0.0 × 10−12 +2.3 ± 0.1+0.0 +−0.0 +... +... +FCES G85.00−1.78 (OffExc) +SBPL +1.7 ± 0.5+0.7 +−0.0 × 10−11 +0.7 ± 0.3+0.2 +−0.0 +2.12 ± 0.04+0.00 +−0.00 +3.0 ± 0.4+0.0 +−0.2 +Notes. The first uncertainties are statistical and the second uncertainties are systematic and result from varying the spin temperature. N0 is given +at the reference energy of 1 GeV. +3.7. Spectro-morphological analysis +3.7.1. Extension and position versus energy +We first tested if the best-fit spatial model of the CoExt and +CoWest components changes as a function of energy. The com- +ponent OffExc is left aside in the spectro-morphological analysis +because it is displaced regarding the Cygnus X region which we +aim to study in this paper, and it lies on the border of the ROI, +and therefore its characterisation may not be optimal. +We fitted the extension and position of CoExt and CoWest in +five energy bands: 0.5 to 1.6 GeV, 1.6 to 5 GeV, 5 to 16 GeV, +16 to 50 GeV, and 50 to 1000 GeV. As shown in Section 3.5, +CoWest has a softer spectrum: above ∼ 5 GeV the source flux +becomes very low and its TS is below 25. Therefore, fitting the +position and extension of the source above ∼ 5 GeV is not possi- +ble. In this section, the diffuse components, that is the gas maps +and the IC component, are fixed, while the two components of +Cygnus Loop are fixed above 5 GeV because of their very steep +spectrum. OffExc is fixed due to its off-centre position. +The results are shown in Figure 11. The top panel shows the +extension as a function of energy for both sources. There is no in- +dication of an evolution of the extension as a function of energy, +and the values in different energy bands are compatible with that +obtained in the broadband analysis. The lower panels show the +best-fit centroid positions for the two components. For CoExt, +all positions are compatible with each other and the broadband +Article number, page 10 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +85◦ +80◦ +75◦ +10◦ +0◦ +−10◦ +Galactic Latitude +85◦ +80◦ +75◦ +−0.25 +0.00 +0.25 +Fractional deviation +−2.5 +0.0 +2.5 +Sigma +Galactic Longitude +Fig. 8: Final deviation maps for the best-fit model over the en- +tire energy range, on the left as fractional deviation, and on the +right using the PS map. The bin size is 0.1◦ and the size of the +smoothing kernel 0.13◦. +100 250 400 +∞ +Spin temperature (K) +−6 +−4 +−2 +0 +∆ ln L +Fig. 9: Difference in log-likelihood in fits of the final model for +different spin temperatures. +fit within 1σ. For CoWest, we can see a hint of evolution of the +position in the first two bins, but the two values are compatible +within 2σ with the value from the broadband fit over the full +energy range. +3.7.2. Spectral variations across the extended sources +In this section, we search for spectral variations across the emit- +ting regions for CoExt and CoCent. We started by examining +CoExt. To this end, we replaced the Gaussian model with a com- +bination of rings and segments. We tested several combinations +but here we describe the profile obtained with a combination of: +a central disk of radius 0.7◦; five rings of external radius 1.7◦, +85◦ +80◦ +75◦ +70◦ +10◦ +5◦ +0◦ +−5◦ +Galactic Longitude +Galactic Latitude +1022 +1023 +Column density (H cm−2) +Fig. 10: Neutral gas column density in the local arm and Cygnus +region, obtained by summing N(H I), N(H2) and an estimate of +the column density for the DNM, with conversion factors cali- +brated on the gamma-ray analysis. +2.7◦, 3.7◦, 4.7◦ and 6◦, that can be decomposed into four seg- +ments spanning 90◦ in azimuth; and two large rings of external +radius 7.4◦ and 8.9◦. This somewhat arbitrary setup was chosen +to ensure a minimal TS (at least 25) in every segment and ring +(see Figure B.1 in Appendix B). Eventually, however, the wider +ring has a low TS (∼10) therefore its parameters have to be in- +terpreted with some caution. +All the components are modelled using a LP spectrum with +parameters initiated at the best-fit values found in Section 3.5. +The LP model was chosen instead of the SBPL model for this +part of the analysis because it yields more stable results when +fitting several free components at once. For this section the two +components of Cygnus Loop are fixed due to their off-centre +position. OffExc is also fixed due to its off-centre position and +proximity with the border of the ROI. +We also tested a combined description of CoExt and CoCent +via rings and segments by removing the ionised gas template +from the emission model, but the highly structured central part +was poorly described by the latter model. Thus, we decided to +proceed with the ionised gas map in the model, and we used the +combination of segments and rings to only represent the source +CoExt. The spectral shape and normalisation is left free for Co- +Cent. The decomposition of CoExt proceeded through a few sub- +sequent steps. +A We replaced the Gaussian by the aforementioned combina- +tion of seven rings and a central disk. Only the normalisation +of each template was free, while the spectral parameters (α +and β, see Appendix A.3 for a definition) were fixed to the +initial values from the analysis in Section 3.5. +B The parameters α and β for the disk and rings were free. +C The innermost five rings (beyond the central disk) were de- +composed into four azimuthal segments. Only the normali- +sations were free and the spectral parameters were fixed to +the values obtained in the step B. +D All spectral parameters were free. +For step C, a few different orientations for the segments were +tested, and we present the results for the one yielding the best +likelihood. The values of ∆ ln L and ∆AIC for the four steps are +provided in Table 6. +Article number, page 11 of 30 + +A&A proofs: manuscript no. ms_cocoon +103 +104 +105 +E (MeV) +3.8◦ +4.0◦ +4.2◦ +4.4◦ +4.6◦ +Extension r68 +FCES G78.74+1.56 +103 +104 +105 +E (MeV) +0.6◦ +0.7◦ +0.8◦ +0.9◦ +1.0◦ +1.1◦ +1.2◦ +Extension r68 +FCES G78.83+3.57 +79.0◦ +78.8◦ +78.6◦ +78.4◦ +78.2◦ +78.0◦ +Galactic Longitude +1.0◦ +1.2◦ +1.4◦ +1.6◦ +1.8◦ +2.0◦ +Galactic Latitude +0.5 - 1.6 GeV +1.6 - 5 GeV +5 - 16 GeV +16 - 50 GeV +50 - 1000 GeV +Global fit +79.25◦ +79.0◦ +78.75◦ +78.5◦ +78.25◦ +78.0◦ +Galactic Longitude +3.0◦ +3.2◦ +3.4◦ +3.6◦ +3.8◦ +4.0◦ +Galactic Latitude +0.5 - 1.6 GeV +1.6 - 5 GeV +Global fit +Fig. 11: Extension (top) and position (bottom) for FCES G78.74+1.56 (left) and FCES G78.83+3.57 (right) as a function of energy. +In the top panels the bands show the values in the global fit over the entire energy range +. In the bottom panels the grey areas show the results in the entire energy range. All uncertainties are provided at 1σ level. +Table 6: Variations of ln L and AIC for the decomposition of +FCES G78.74+1.56 in rings and segments. +Step +∆ ln L +∆AIC +A +−9 [−13, −9] +32 [32, 40] +B +7 [7, 14] +18 [4, 18] +C +58 [47, 58] +−86 [−86, −64] +D +35 [35, 42] +−10 [−24, −10] +Notes. Values are provided as difference with respect to the previous +step, and, for step A, with respect to the global analysis presented in +Section 3.5. See the text for details. +The degradation in step A is due to the approximation of +representing a 2D Gaussian with concentric rings and a central +disk. However, this is not a cause for concern as the decrease +in log-likelihood is small. Step B does not provide an improve- +ment in the description of the emitting region, that is there are no +significant variations of the spectrum as a function of distance +from the centre. Conversely, we find a model improvement in +step C, which demonstrates that the emission is not azimuthally +symmetric in intensity. The likelihood improvement is equally +shared by all the rings concerned, and it is not surprising given +the diversity of regions inside and outside the plane spanned by +each broad ring. A further improvement in the likelihood is ob- +tained in step D, showing also the presence of azimuthal spec- +tral variations, mostly driven by the two innermost rings and +the central disk with an improvement in log-likelihood of 24 +(∆AIC = −15) when only those components have the spectral +shape free. +Article number, page 12 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +However, the azimuthal variations in the first two rings could +be explained by spectral variations across CoCent. To check this +hypothesis, we sliced the ionised gas template vertically at l = +79.8◦ to separate the two main lobes of ionised gas and repeated +the step D. This results in an improvement of the log-likelihood +smaller than one, meaning that no spectral variation is detectable +between the two sides of the source CoCent. +We conclude that the best model is the one combining the +ionised gas template to describe CoCent and with CoExt decom- +posed and fitted as in step D. This is used as a basis for the in- +terpretation of the results in the next section, where the emission +profiles extracted from the data is also shown. Some additional +plots illustrating the results are provided in Appendix B. +4. Discussion +4.1. The cocoon and its landscape +Our analysis shows that the Cygnus cocoon in the LAT energy +band is best described by at least two spatial components with +different spectra: a central component, CoCent, with a power law +spectrum of index 2.19 ± 0.03+0.00 +−0.01, and an extended component, +CoExt, with a smooth broken power law spectrum with indices +1.67 ± 0.05+0.02 +−0.01 below 3.0 ± 0.6+0.0 +−0.2 GeV and 2.12 ± 0.02+0.00 +−0.01 +above. A third newly discovered extended emission component, +CoWest, overlaps in projection with the cocoon and has a spec- +trum compatible to the one of the central component, a power +law with index 2.3 ± 0.1. +Figure 12 shows the excess counts corresponding to the three +gamma-ray sources associated or potentially related to the co- +coon, that is total counts minus the best-fit model for all compo- +nents except CoExt, CoCent, and CoWest (zoomed in from Fig- +ure 5). The brightest emission in the central region of the cocoon +lies in the cavities bounded by the photo-dissociation regions +traced by 8 µm emission (right panel), as found by Ackermann +et al. (2011), and the majority of it is traced by our ionised gas +template and associated to source CoCent. The extended cocoon +component, CoExt, overlaps with the northern rim of the X-ray +structure known as Cygnus SB (Cash et al. 1980), which may +be associated with star-forming regions in Cygnus X (Uyanıker +et al. 2001), although recent data may suggest that the entire +X-ray structure is rather a hypernova remnant at a distance of +1.1-1.4 kpc (Bluem et al. 2020). Last, source CoWest is situated +along a bright arc of 8 µm emission, but does not coincide with +any over-densities in neutral or ionised gas densities (see Figs. 2, +3, and 4). Its centroid lies at approximately 1◦ from the γ Cygni +SNR, that, if we assume a distance to the Earth of 1.7 kpc, cor- +responds to a physical distance of ∼30 pc. +Under the hypothesis that the observed gamma-ray emission +is of hadronic origin we can convert the excess map into an emis- +sivity map. To this aim we divided the excess cube in the analy- +sis energy bins by the exposure cube and the total, neutral plus +ionised, gas column density map. The latter quantity is an upper +limit to the relevant gas column densities because gas could be +distributed over a larger distance along the line of sight compared +to the volume probed by the particles in the cocoon. However, we +expect most of the gas in this region to be concentrated around +the star-forming complex in Cygnus X, and we do not have an +alternative simple prescription to estimate the foreground and +background column densities to be subtracted. The results are +displayed in Figure 13. +On one hand, we can see an emissivity peak in the cocoon +central area coincident with the peaks in the ionised gas distribu- +tion (modelled by CoCent in our analysis) with broad wings ex- +tending to several degrees from the centre (CoExt). On the other +hand, we see a marked peak at the position of CoWest and around +the γ Cygni SNR and NGC 6910 stellar cluster. Although posi- +tion and spectral similarity to CoCent suggest that this source is +related to the cocoon, the interpretation is not obvious. CoWest +may be related to gas missing in our model, or else to a nearby +source or some peculiar transport configuration that results in an +accumulation of particles in this region. In the following for sim- +plicity we concentrate on the interpretation of the two brightest +sources in the cocoon area, namely CoCent and CoExt. +The striking spatial coincidence of the brightest part of the +gamma-ray signal and the contours of the cavity, and to a lesser +extent the resemblance with the extended X-ray emission struc- +ture, have suggested that both phenomena may have a common +origin: the abundant massive-star population of the region. The +most prominent stellar clusters in the regions, the Cyg OB2 asso- +ciation and NGC 6910 cluster, are natural candidates, powerful +enough to accelerate particles able to produce non-thermal emis- +sion at the observed level. +We evaluated the properties of these two objects, following +what was done in Ackermann et al. (2011). For Cygnus OB2, +we considered 78 O stars (Berlanas et al. 2020) and a power law +mass function of index 1.09 (Wright et al. 2010). For NGC 6910 +we assumed a power law mass function of index 0.74 (Kaur et al. +2020) normalised according to Figure 9 of their paper. We eval- +uated mass loss rates, cluster wind terminal velocities, and me- +chanical power of the winds by separating stars in four groups, +namely O5 to O3, O9 to O5, B5 to B0, and B8 to B5. The sample +is limited to stars heavier than B8 due to the validity range for the +reference mass-loss rate model adopted. We assumed standard +properties of O stars from Martins et al. (2005) and for B stars +from Cox (2000). We used the parametric wind model by Vink +et al. (2000). This yields a mass loss rate of 5.1 × 10−4 M⊙ yr−1 +for Cyg OB2 and of 2.7 × 10−4 M⊙ yr−1 for NGC 6910. The +mechanical power of the winds is evaluated to 8 × 1038 erg s−1 +for Cyg OB2 and 4 × 1038 erg s−1 for NGC 6910. The collective +wind terminal velocity therefore is ∼2200 km s−1 for both clus- +ters. We show in the next section that such powers are sufficient +to account for the observed signal in some scenarios. +We can estimate the physical and angular sizes of the clus- +ter wind termination shock and shocked wind bubble using the +formulae in Morlino et al. (2021), which follow the simple mod- +els in Weaver et al. (1977); Gupta et al. (2018). If we assume +ages of 5 Myr (Berlanas et al. 2020; Kaur et al. 2020) for both +clusters and interstellar gas densities of 5 H cm−3 we obtain a +size of the wind termination shock of 40 pc for Cyg OB2, and of +33 pc for NGC 6910. The total size of the wind bubble is 200 pc +for Cyg OB2, and 180 pc for NGC 6910. Figure 12 shows that +the sizes of the termination shocks are comparable to that of the +central emission component, with CoWest being located at the +edge of the termination shock from NGC 6910, while the sizes +of the wind bubbles compare well to that of the cocoon extended +emission component. +While these results tend to lend support to the idea that +Cyg OB2 and NGC 6910 may be the sources ultimately respon- +sible for the observed gamma-ray emission, we emphasise that +the modelling of the winds and bubbles is without any doubts +oversimplified. Several effects can be expected to affect the re- +sults and weaken the similarity of the gamma-ray emission and +expected SB signatures. Generally, the classical theory from +Weaver et al. (1977) is known to underestimate the radiative +losses, hence overestimate the size of the bubble. Hydrodynami- +cal simulations reveal enhanced radiative losses due to instabili- +ties at the interfaces result in a bubble being ∼ 40% smaller than +Article number, page 13 of 30 + +A&A proofs: manuscript no. ms_cocoon +82◦ +80◦ +78◦ +76◦ +4◦ +2◦ +0◦ +−2◦ +Galactic Longitude +85◦ +80◦ +75◦ +10◦ +5◦ +0◦ +−5◦ +−10◦ +Galactic Longitude +Galactic Latitude +0 +200 +400 +600 +800 +1000 +1200 +1400 +Excess counts +Fig. 12: Excess counts corresponding to the three gamma-ray sources associated or potentially related to the cocoon, namely +FCES G78.74+1.56, FCES G80.00+0.50, and FCES G78.83+3.57 (zoomed in from the left panel in Figure 5). Left: extended +region. Green contours correspond to the X-ray emission from the ROSAT all-sky survey in the 0.4 keV to 2.4 keV band. The +orange circle shows the outer radius of the third to last annulus included in the analysis. The dashed circles show the estimated +sizes of the wind bubbles for Cyg OB2 (lower left) and NGC 6910 (upper right). See text for details. Right: zoom in the central +region. Black contours correspond to the 8 µm emission from MSX data at 1.85 × 10−6 W m−2 sr−1. The stars show the positions of +PSR J2032+4127 (lower left) and PSR J2021+4026 (upper right). The green circles show the radius/68% containment radius of the +two emission components associated with the γ Cygni SNR (subtracted from the map, see Section 3.2 for details). The blue circle +shows the 68% containment radius of FCES G78.83+3.57. The continuous circles show the 50% containment radius for members +of the Cyg OB2 association (Berlanas et al. 2019) and of the NGC 6910 cluster (Cantat-Gaudin et al. 2020, NGC 6910 has a 50% +containment radius of 8.9′′which appears as a dot on this image). The dashed circles show the estimated sizes of the cluster wind +termination shock for Cyg OB2 (lower-left) and NGC 6910 (upper right). In both panels the orange diamond shows the centre of +FCES G78.74+1.56. +predicted by the classical analytical solution for well-developed +bubbles, in agreement with observations (Krause & Diehl 2014). +At earlier stages, before SB breakout from the parent molecu- +lar cloud, the dense and fractal medium surrounding the clus- +ter drives turbulent mixing and efficient cooling, resulting in +a reduction of ∼30% in bubble size for parameters relevant to +Cyg OB2 and NGC 6910 (Lancaster et al. 2021). More funda- +mentally, the simple bubble model from Weaver et al. (1977); +Gupta et al. (2018); Morlino et al. (2021) may not be straightfor- +wardly applied to Cyg OB2, which is not a compact cluster but +instead presents multiple substructures with a 50% containment +radius of stellar members spanning 0.2◦, that is 5 pc at a distance +of 1.7 kpc (Berlanas et al. 2019). Furthermore, as illustrated in +Figure 12, the bubbles from the two stellar clusters may have +interacted and it is not clear how this would have impacted the +development of the whole region and whether this should have +left specific signs that we should now see. +Therefore, the connection of the observed gamma-ray sig- +nal with gas structures imprinted by the development of a SB is +far from obvious. Actually, the separation of the cocoon emis- +sion into CoExt and CoCent, as well as the correlation of the +innermost bright signal with ionised gas, can be interpreted in +a way that weakens the link between the gamma-ray emission +and the cavity delineated by photo-dissociation regions. Indeed, +the emission could arise from the interaction of freshly acceler- +ated CRs with the ionised gas inside the cavity, the latter playing +no role. These CRs extend much beyond the limits of the cav- +ity, as was already clearly observed in Ackermann et al. (2011), +and their interactions with the ambient medium give rise to the +source CoExt. +The origin of the CRs powering the cocoon emission +can therefore be unrelated to the Cyg OB2 association and +NGC 6910 cluster (in the sense that they play no role as a whole, +but they can harbour or have harboured the actual source). The +central region actually contains a handful of extremely energetic +objects and potential particle sources. +We can list: the γ Cygni SNR (G78.2+2.1) with a proba- +ble distance of 1.7 to 2.6 kpc from association with the γ Cygni +nebula (Leahy et al. 2013) and dynamic properties estimated in +Leahy et al. (2020) as ejecta mass of 5 M⊙, age of 9.4+2.3 +−1.6 kyr, +and supernova (SN) energy of 6.3+5.8 +−3.7 × 1050 erg; the γ Cygni +pulsar, PSR J2021+4026, associated with the γ Cygni SNR and +with a spin-down power of 1.2 × 1035 erg s−1 (Ray et al. 2011); +PSR J2032+4127, a pulsar in a highly eccentric binary sys- +tem with a Be-type star (Lyne et al. 2015), probably part of +Cyg OB2, with an orbital period of 45-50 yr, a spin-down power +Article number, page 14 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +85◦ +80◦ +75◦ +5◦ +0◦ +−5◦ +Galactic Longitude +Galactic Latitude +0.2 +0.4 +0.6 +0.8 +1.0 +Emissivity (ph s−1 sr−1 H−1) +×10−27 +Fig. 13: Emissivity map calculated from the excess counts as- +sociated to the cocoon in Figure 5 (right panel). The dashed +contours correspond to the 8 µm emission from MSX data at +1.85 × 10−6 W m−2 sr−1. The contours correspond to the peak +column density of the ionised gas template at 3.5×1021 H cm−2. +The circle and the dashed circle correspond to the position and +r68 of FCES G78.83+3.57 and FCES G78.74+1.56 respectively. +of 1.5 × 1035 erg s−1, and a characteristic age of ∼200 kyr (Ho +et al. 2017). +Furthermore, the centroid of the emission from CoExt and +the peak in the emissivity map do not coincide with any of +the potential particle accelerators, stellar clusters or others (Fig- +ure 12 and 13). Therefore, we tried to account for our obser- +vations in a generic way, with a simple diffusion model based +on an unspecified source and not exclusively relevant to mas- +sive star clusters and their associated SBs. We introduce in the +following sections the model framework used and the parameter +setups yielding satisfactory fits to the data. +4.2. A simple diffusion-loss framework for the cocoon +Given the layout of the emission exposed in the previous subsec- +tion, with significantly extended radiation from a region reaching +well beyond the vicinity of potential sources, it seems reasonable +to consider that gamma rays are produced by particles that were +released by one or several sources some time ago and were trans- +ported in the surrounding medium since then. In this section, we +aim to provide a quantitative assessment of this idea. +We interpret the observations in the framework of a one-zone +diffusion-loss transport model where particles are continuously +injected at a point in space for some duration and then experi- +ence diffusive transport in a uniform and isotropic medium. This +is very likely an overly simplistic description of the processes +at stake because there may be multiple sources, not all of them +can be assumed to be of negligible size, the medium is prob- +ably not uniform over the few hundreds of parsecs probed by +the emission, and there may be other transport processes than +diffusion. Yet, our goal is to draw a few key inferences from +the observables and we defer more advanced modelling efforts +to subsequent publications. Moreover, we show later that such +a modelling with a very limited number of free parameters can +yield a fairly good representation of the observables. +The full formalism of the model framework is provided in +Appendix C. Ultimately, the main parameters of the model are: +injection luminosity Q0, power law injection spectrum slope α, +characteristic injection duration tinj, diffusion duration tdiff, and +diffusion coefficient normalisation D0. We explored a large pa- +rameter space for these four parameters and fitted the predictions +to the results of the gamma-ray analysis. +The diffuse emission from the Cygnus cocoon is very ex- +tended, with an angular size of 4.4◦ ± 0.1◦ for CoExt that trans- +lates into a ∼ 130 pc length at a distance of 1.7 kpc. A more com- +pact and central emission component CoCent is correlated with +the distribution of ionised gas within a radius of about 50 pc; the +spectrum of CoExt is flat and that of CoCent, although signifi- +cantly softer, is also pretty hard compared to interstellar emission +on larger scales. +Given these observables, we proceeded to educated guesses +for the main parameters of the model, considering first the case +of a hadronic scenario. The typical extent of the emission pro- +vides a constraint on the diffusion length, that is on the product +of diffusion coefficient and diffusion time: +rd = +� +4Dtdiff ≳ 100 pc, +(6) +D = Dism(10 GeV) = 1029 cm2 s−1 ⇒ tdiff ≃ 7.5 kyr, +(7) +D = Dsupp(10 GeV) = 1027 cm2 s−1 ⇒ tdiff ≃ 0.75 Myr. +(8) +If diffusion has the average properties inferred for transport over +large scales in the Galaxy (Trotta et al. 2011), defined by the +coefficient Dism, particles need less than 10 kyr to fill a volume +that would account for the extent of the observed emission at +a distance of 1.7 kpc. Conversely, if diffusion is for some rea- +son strongly suppressed by one to two orders of magnitudes as +inferred for a variety of sources including star-forming regions +(Aharonian et al. 2019; Abeysekara et al. 2017; Abramowski +et al. 2015), and is characterised by coefficient Dsupp, then about +1 Myr is needed. +The emissivity enhancement inferred for the cocoon is com- +parable to the local emissivity within a factor of two to three +depending on the energy range, such that the CR energy density +uCR in the region is similar to the one in the solar neighbour- +hood, uCR,local. This makes it possible to constrain the properties +of particle injection, namely its power Linj and typical duration +tinj: +4π +3 r3 +d × uCR ≃ 1 +2Linjtinj, +(9) +uCR ≃ uCR,local ≃ 1 eV cm−3, +(10) +Linjtinj ≃ 4 × 1050 erg, +(11) +Linj = 1038 erg s−1 ⇒ tinj ≃ 0.1 Myr. +(12) +As computed in the previous subsection, the mechanical lumi- +nosity of the most prominent star clusters in Cygnus is in the +range 4 − 8 × 1038 erg s−1. Such a power source can deliver par- +ticle injection at a level of 1038 erg s−1, pending efficient particle +acceleration with a yield of ∼ 10 − 30% (by some unspecified +mechanism at this stage). In that case, the inferred CR density +enhancement can be attained if injection lasts over about 100 kyr +(and particles accumulate in the volume, see the discussion in +the next paragraph). If particle acceleration is less efficient or +the source is less powerful, by about an order or magnitude, then +injection has to proceed on Myr time scales. Alternatively, a su- +pernova producing 1050 erg of accelerated particles and releasing +the majority of them over ∼ 3 − 10 kyr would provide an injec- +tion power of 3 − 10 × 1038 erg s−1 and thus allow short-lived +injection. +Article number, page 15 of 30 + +A&A proofs: manuscript no. ms_cocoon +Scenarios with tinj much smaller than tdiff are not viable be- +cause particles spread out and leave the volume too rapidly, +which results in too flat intensity profiles and too steep spectra +(because energy-dependent diffusion depletes the particle popu- +lation at the high end of the spectrum). So tinj has to be compa- +rable to or greater than tdiff, with the additional constraint that +sufficient energy should be released within a time tdiff to match +the observed level of emission. In practice, this means that: for +average interstellar diffusion, the region is filled over a ∼ 10 kyr +timescale, thus requiring a strong enough source with injection +power ∼ 1039 erg s−1 typical of a SN; alternatively, weaker +sources such as the star clusters in Cygnus with injection power +∼ 1037−38 erg s−1 require moderate to strong diffusion suppres- +sion and transport occurring over hundreds to thousands of kyr. +These considerations remain mostly valid in the case of a lep- +tonic scenario. The main difference with a hadronic scenario is +the importance of energy losses, mostly from synchrotron radia- +tion and inverse-Compton scattering. Yet, for an interstellar mag- +netic field strength B = 3 µG and optical and infrared interstel- +lar radiation fields with total energy density of about 1 eV cm−3, +such as those predicted in the large-scale model of Popescu et al. +(2017) at the Galactic position of the cocoon, particles with en- +ergy below 300−400 GeV have a cooling time of 1 Myr or more. +The transport of electrons over the distances and time scales con- +sidered above may therefore be little affected by energy losses in +many scenarios. The actual situation is however far more com- +plex because radiation densities in the innermost region of the +cocoon are much stronger than the large-scale interstellar aver- +age, by an order of magnitude, which could significantly affect +the spectral and morphological properties of the emission from +the population of propagated electrons. Unfortunately, the model +framework that we used for this work cannot handle inhomoge- +neous energy losses. +4.3. Possible diffusion scenarios for the cocoon +We tested the hypothesis that components CoExt and CoCent +are produced by a single population of non-thermal particles. +In that context, CoCent is gas-related emission (pion decay in +hadronic scenarios, and Bremsstrahlung in leptonic scenarios) +from the innermost ∼ 50 pc region of the cocoon, where a signif- +icant amount of ionised gas is present as evidenced by free-free +emission, while CoExt is additional emission on top and beyond +CoCent that is not necessarily gas-related (it can be a mix of +inverse-Compton and Bremsstrahlung in leptonic scenarios). In +the following, we relate CoCent and CoExt to so-called central +and extended regions in our model, respectively. +For computational reasons, we did not perform an overall +optimisation of all model parameters and instead investigated a +limited number of scenarios selected from the above guess for +viable parameter values. For each parameter setup, the compar- +ison of model predictions and gamma-ray analysis results goes +through the following steps that are expected to guarantee a max- +imum consistency. +Central and extended component separation: for a given run of +the model, gas-related emission from the innermost region +within 50 pc in radius of the injection point is handled sep- +arately. All related quantities (particle density, gas column +density, emission intensity,...) are not included in the prop- +erties of the complementary extended region. For instance, +gas-related emission intensity along a line of sight that passes +through the central region seen in projection is split into a +central contribution and the remaining extended contribu- +tion. Although the separation is clear-cut in the model, we +cannot exclude that there is some cross-talk between over- +lapping components in the data analysis. +Gas column density correction: : the model-predicted emission +for the extended component was divided into rings, with the +angular binning used in the spectro-morphological analysis, +and gas-related emission in each ring was rescaled by the ra- +tio of the actual average gas column density in the ring to that +corresponding to the default uniform density assumption of +the model. The same rescaling is also applied to the central +component, treated as a single region with average proper- +ties. +Fitting of total emission spectra: the total emission spectrum of +the extended component is fitted to the observed spectrum +for CoExt, which yields the injection luminosity for the +whole particle population. Then, the total emission spectrum +of the central component, re-scaled by the fitted injection lu- +minosity, is further fitted to the observed spectrum for Co- +Cent. This second fit is meant to correct for the uncertain av- +erage column density for the ionised gas in the central region. +Both fits are performed via χ2 minimisation, from significant +spectral points and using statistical uncertainties only. +There is, however, a subtlety regarding how emission from +the ionised gas should be handled. Atomic and molecular gas in +the cocoon region enter twice in the analysis: in the fit to the +Fermi-LAT data over a large ROI, where they trace emission +from the background population of CRs, and in the interpreta- +tion of the extended emission from CoExt and CoCent, where +they are associated to an additional population of particles. Con- +versely, the ionised gas template enters just once, and the associ- +ated emission may therefore comprise contributions from back- +ground CRs and from an additional population of particles. In +practice, what is fitted to the spectrum of CoCent is the spec- +trum of the central component of the model, possibly augmented +by the spectrum of the emission from the ionised gas for a local +emissivity (because the background CR population in the co- +coon region can be considered close to the local one; see Section +3.6). We tested both options and, as illustrated below, it turns out +that not including a contribution from background CRs provides +much better fits to the data. +4.3.1. Hadronic scenarios +To begin with, we present the result of complete calculations for +hadronic scenarios. Following the above discussion of the most +likely diffusion-loss model setups given the observables at hand, +we present the results of four scenarios dubbed H1, H2, H3 and +H4, the parameter sets of which are presented in Table 7. The H1 +and H2 scenarios feature constant injection of a hard spectrum +of protons and mainly differ by the diffusion time (300 kyr or +3 Myr) and level of diffusion suppression (by a factor 10 or 100 +with respect to the interstellar average). The H3 and H4 scenar- +ios corresponds to shorter-lived injection over 3 or 30 kyr and +transport over 10 or 100 kyr in a medium with no or moder- +ate diffusion suppression (by a factor 10 at most with respect +to the interstellar average). Scenario H4 actually corresponds to +a scaled version of scenario H3 (multiplying injection and dif- +fusion times and dividing diffusion normalisation and injection +power by the same amount, that is ten), such that both setups are +completely identical in terms of predictions. +Figure 14 shows the total fitted spectra for CoExt and CoCent +for model setups H1 and H2. The predicted shape for the CoExt +spectrum is in good agreement with the data, while that for the +Article number, page 16 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +Table 7: Summary of the different diffusion-loss model setups +H1 +H2 +H3 +H4 +L1 +L2 +hadronic +hadronic +hadronic +hadronic +leptonic +leptonic +tinj (yr) +108 +108 +3 × 103 +3 × 104 +108 +108 +tdiff (yr) +3 × 106 +3 × 105 +104 +105 +3 × 106 +106 +D0 (cm2 s−1) +1027 +1028 +1029 +1028 +1027 +1028 +Linj (erg/s) +3.6 × 1036 +3.2 × 1037 +2.8 × 1039 +2.8 × 1038 +6.8 × 1035 +2.4 × 1036 +α +2.0 +2.0 +2.0 +2.0 +2.0 +2.0 +CoCent spectrum is too steep. A steeper predicted spectrum for +CoCent is obtained because higher-energy particles leave the in- +nermost regions more rapidly than lower-energy particles, and +also because it contains a contribution from the background CR +population that has a steeper spectrum. +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 3.62e+36 erg/s +NIG +H = 2.70e+21 H/cm2 +2 = 25.87 + 42.16 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 3.25e+37 erg/s +NIG +H = 2.73e+21 H/cm2 +2 = 25.23 + 44.83 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. 14: Fitted model spectra for the FCES G78.74+1.56 and +FCES G80.00+0.50 emission components, for model setup H1 +(top) and H2 (bottom). A fit of the model to the spectrum of +FCES G78.74+1.56 is done first, with the injection luminosity +as fitting parameter, followed by a second fit to the spectrum of +FCES G80.00+0.50, with ionised gas column density as fitting +parameter. This last fit includes a contribution to the emission +from a background population of CRs (see text). The full ex- +tent of the error bars corresponds to the quadratic sum of the +statistical and systematic uncertainties, while the caps mark the +contribution of the statistical uncertainty only. The first χ2 cor- +responds to the fit to FCES G78.74+1.56 and the second one to +FCES G80.00+0.50. +Interestingly, a much better fit to the spectrum of CoCent is +obtained when not adding a local emissivity contribution to the +model spectrum for the central region, at the expense of higher +fitted column density for the ionised gas. This is illustrated in the +top two panels of Figure 15. Emission from the ionised gas in the +innermost regions of the cocoon would then arise only from CRs +produced in Cygnus. Background pre-existing CRs may have +been evacuated in the past during the SB growth, for instance +by advection in the stellar winds. Alternatively, the spectral sig- +nature of these pre-existing CRs may have been absorbed by an- +other component in the fit to the LAT observations (for instance +by the molecular gas or DNM templates that have a high de- +gree of correlation with ionised gas in the central region). Since +the spectral fit is so much better when not including the con- +tribution from background CRs for CoCent (this is true also in +leptonic scenarios), we present in the following only results pro- +duced with this approach. We note, however, that this has almost +no influence on the intensity and emissivity profiles presented +thereafter. +For model setup H1 (resp. H2), the fit implies a proton injec- +tion luminosity of 3.6 × 1036 erg s−1 (resp. 3.2 × 1037 erg s−1), +which would correspond to proton injection efficiencies < 0.5 − +1% (resp. < 5 − 10%) for the Cyg OB2 or NGC 6910 clus- +ters. Such low efficiencies are consistent with the assumed flat +injection spectra with α = 2.0, at least in the framework of +diffusive shock acceleration. For model setup H3 (resp. H4), +the fits implies a much higher proton injection luminosities of +2.8 × 1039 erg s−1 (resp. 2.8 × 1038 erg s−1). This would corre- +spond either to a high particle acceleration efficiency of ∼ 20% +in an SN with a 1051 erg explosion kinetic energy, with subse- +quent release of accelerated particles over a timescale of 3 kyr +(resp. 30 kyr), or to a lower acceleration efficiency in an SN more +energetic than in the canonical picture. The first option, with rel- +atively high efficiency, may be conflicting with our assumption +of a flat injection spectrum.e +Figure 16 displays the predicted intensity and emissivity pro- +files for both central and extended model components in sce- +nario H1, compared to the values inferred from the spectro- +morphological analysis in segments, in three different energy +bands: 0.5-2 GeV, 2-10 GeV, and 10 GeV-1 TeV. The comparison +for other scenarios is shown in Appendix D. To compare the dif- +ferent model setups, we provide in each panel the χ2, computed +from all significant intensity data points using statistical uncer- +tainties only. There is, however, no formal fit of the model to the +measured intensity profiles, and this χ2 is just a figure of merit +to characterise each model setup. The agreement is overall quite +good from the centre up to beyond 8◦, especially considering the +simplicity of the model and the limited number of parameters. +Most measurements are within a factor two of the predictions. +When subtracting the contribution from the central model com- +ponent, relatively flat emissivity profiles are obtained for the ex- +tended model component, in agreement with the trend inferred +from the data analysis. The results lend support to the idea that +Article number, page 17 of 30 + +A&A proofs: manuscript no. ms_cocoon +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 3.62e+36 erg/s +NIG +H = 6.01e+21 H/cm2 +2 = 25.87 + 3.60 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 3.25e+37 erg/s +NIG +H = 6.16e+21 H/cm2 +2 = 25.23 + 3.01 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 2.85e+39 erg/s +NIG +H = 6.98e+21 H/cm2 +2 = 28.51 + 5.96 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. 15: The top two panels are the same as in Figure 14, without +a contribution to the emission from a background population of +CRs in the fit to the spectrum of FCES G80.00+0.50. The bottom +panel is the corresponding figure for scenario H3. The first χ2 +corresponds to the fit to FCES G78.74+1.56 and the second one +to FCES G80.00+0.50. +CoExt and CoCent are produced by the same population of par- +ticles. All four model setups are overall equally good at account- +ing for the data, despite widely different parameter sets. +4.3.2. Leptonic scenarios +We now present the result of complete calculations for lep- +tonic scenarios, in which the emission can be produced by non- +thermal Bremsstrahlung and inverse-Compton scattering. We +considered two scenarios dubbed L1 and L2, the parameter sets +of which are presented in Table 7. Both scenarios feature con- +stant injection of a hard spectrum of electrons and mainly differ +by the diffusion time (1 or 3 Myr) and level of diffusion suppres- +sion (by a factor 10 or 100 with respect to the interstellar aver- +age). The magnetic field is assumed to have a strength B = 3 µG, +and the interstellar radiation field model is taken from the large- +scale model of Popescu et al. (2017) at the Galactic position of +the cocoon (at a 1.7 kpc distance from us). We tested the ef- +fect of a stronger interstellar radiation field model, such as the +one used in the original Cygnus cocoon paper (Ackermann et al. +2011), and obtained a much poorer fit to our measurements. This +stems mostly from the shorter propagation range and different +relative contributions of Bremsstrahlung and inverse-Compton +to the emission. This scenario is however extreme since it en- +forces very strong inverse-Compton losses over an extended vol- +ume, whereas in reality enhanced radiation fields are expected +only in the innermost regions of the cocoon. This is a caveat of +the model framework used in this work, which cannot handle in- +homogeneous environments. We also tested a leptonic version of +scenario H3, but that yields a poor fit to the data. +The fits of the model to the spectra of CoExt and CoCent +are displayed in Figure 17. They are overall pretty satisfactory +and yield χ2 similar to those obtained with the hadronic mod- +els, although slightly higher. The diffusion time range in leptonic +scenarios seems rather constrained: small ages ≲ 1 Myr tend to +produce too hard spectra for the extended component, while ages +≳ 3 Myr result in too steep spectra. The injection luminosities re- +sulting from these fits are 6.8×1035 and 2.4×1036 erg s−1 for the +L1 and L2 scenarios, respectively. This translates into electron +injection efficiencies at the sub-percent level at most if the me- +chanical luminosity from Cyg OB2 and NGC 6910 is the power +source for particle acceleration. We checked that the correspond- +ing synchrotron emission does not exceed the radio constraints +presented in Mizuno et al. (2015). +The corresponding predicted intensity profiles are compared +to the measurements in Figure 18 for model L1. As for the spec- +tra, the fit is slightly degraded compared to that obtained with +hadronic models. The best scenario is L1, with diffusion sup- +pressed by two orders of magnitude with respect to the interstel- +lar average and a diffusion time of 3 Myr. A lower level of diffu- +sion suppression results in too flat intensity profiles, undershoot- +ing the data in the inner regions and exceeding them at large dis- +tances from the injection point. As mentioned above, a smaller +diffusion time does not help because it yields a too hard spectrum +for CoExt. +We also tested the hypothesis that the observed emission ac- +tually is a pulsar halo, following the discovery of very extended +gamma-ray emission around some middle-aged pulsars (Abey- +sekara et al. 2017). In such a scenario, PSR J2032+4127 appears +as an interesting candidate because of its location close to the +peak of the emission and characteristic age of ∼200 kyr. We +used the phenomenological two-zone diffusion-loss halo model +implementation presented in Martin et al. (2022), using as base- +line key parameters: the spin-down power, estimated distance, +and characteristic age of PSR J2032+4127 from the ATNF data +base4; a broken power law injection spectrum with indices 1.8 +and 2.2 below and above a break energy of 500 GeV respec- +tively; an injection starting time of 40 kyr; diffusion suppression +by a factor of 50 within 50 pc of the pulsar, with a power law +dependence in rigidity with index 1/3; a surrounding magnetic +field with strength B = 3 µG and the interstellar radiation field +4 https://www.atnf.csiro.au/research/pulsar/psrcat/ +Article number, page 18 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +7 +10 +6 +10 +5 +10 +4 +0.5-2.2GeV intensity (ph/cm2/s/sr) +Linj = 3.62e+36 erg/s +2 = 141.80 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +29 +10 +28 +10 +27 +10 +26 +10 +25 +0.5-2.2GeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +2.2-10GeV intensity (ph/cm2/s/sr) +Linj = 3.62e+36 erg/s +2 = 310.00 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +30 +10 +29 +10 +28 +10 +27 +10 +26 +2.2-10GeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +10GeV-1TeV intensity (ph/cm2/s/sr) +Linj = 3.62e+36 erg/s +2 = 286.64 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +31 +10 +30 +10 +29 +10 +28 +10 +27 +10GeV-1TeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. 16: Intensity and emissivity radial profiles in three different gamma-ray energy bands for the FCES G78.74+1.56 and +FCES G80.00+0.50 emission components, compared to predictions for model setup H1. In the intensity plots, the intensity distri- +bution corresponding to the best-fit two-dimensional Gaussian model is displayed for comparison as a dotted line. In the emissivity +plots, the local emissivity and its uncertainty in each energy range are displayed for comparison as a dotted line and a shaded band. +The data points correspond to a decomposition of the emission into the ionised gas template, a central disk, two outer rings, and +five intermediate rings split azimuthally into four segments. For the latter, we displayed the corresponding angular range only for +one segment in each ring and introduced a small horizontal shift of the others, for readability. The full extent of the error bars cor- +responds to the quadratic sum of the statistical and systematic uncertainties, while the caps mark the contribution of the statistical +uncertainty only. +Article number, page 19 of 30 + +A&A proofs: manuscript no. ms_cocoon +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 6.84e+35 erg/s +NIG +H = 5.69e+21 H/cm2 +2 = 34.40 + 3.64 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +Linj = 2.41e+36 erg/s +NIG +H = 1.32e+22 H/cm2 +2 = 41.96 + 4.10 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. 17: Fitted model spectra for the FCES G78.74+1.56 and +FCES G80.00+0.50 emission components, for model setup L1 +(top) and L2 (bottom). A fit of the model to the spectrum of +FCES G78.74+1.56 is done first, with the injection luminosity as +fitting parameter, followed by a second fit of the Bremsstrahlung +emission only to the spectrum of FCES G80.00+0.50, with +ionised gas column density as fitting parameter. This last fit +does not include a contribution to the emission from a back- +ground population of CRs (see text). The dashed blue line is +the inverse-Compton contribution in the emission model for +FCES G78.74+1.56. The first χ2 corresponds to the fit to +FCES G78.74+1.56 and the second one to FCES G80.00+0.50. +model from the original Cygnus cocoon paper (Ackermann et al. +2011). We neglected the effect of proper motion on the emission +morphology. +The fit of the predicted emission properties to the ob- +served spectra and intensity profiles is relatively good (see Ap- +pendix D), although not at the level of those obtained in sce- +narios H1-H4 and L1-L2. Yet, the implied present-day injection +luminosity is of the order of 1036 erg s−1, an order of magnitude +larger than the spin-down power of PSR J2032+4127, which dis- +misses this pulsar as the possible source of the halo. This result +seems to be robust against variations of the main model param- +eters. One cannot exclude, however, that another currently un- +known pulsar with the right properties exists in this active star- +forming region. +4.4. The origin of the cocoon +To summarise, our extended set of observables for CoCent and +CoExt, including a radial profile for the extended component +over nearly 10◦, together with intensity measurements and emis- +sivity estimates in three energy bands from 0.5 GeV to 1 TeV, can +be accounted for reasonably well from a simple diffusion-loss +model with a small number of free parameters. Several pretty +different model setups seem to provide viable explanations of +the observations, which suggests that more developed modelling +frameworks and, more likely, additional observational data need +to be considered in future studies. +An important result is that the data can be explained from +one single population of injected particles. This population spans +the full region of extended component CoExt, and gives rise to +the central component CoCent by interacting with ionised gas in +the innermost regions. Both hadronic and leptonic scenarios are +viable, although it should be confirmed that leptonic scenarios +are still valid in a more realistic modelling framework includ- +ing non-uniform inverse-Compton losses in the strongly varying +radiation fields of the region. All solutions have in common to +require a flat particle spectrum at the source, with a power law +index 2.0, which points to very recent acceleration. +The solutions are, however, very different in terms of ener- +getics and time scales involved. Setups H1 and L1 feature con- +tinuous injection, strong diffusion suppression in the region (by +a factor 100 with respect to the large-scale interstellar average, +over a spatial extent of more than 200 pc), transport proceeding +over several Myr (in agreement with age estimates for Cyg OB2 +and NGC 6910), and low acceleration efficiencies (at the sub- +percent level in the hadronic scenario if Cyg OB2 and NGC 6910 +are the mechanical power source). Setups H2 and L2 are very +similar with continuous injection, moderate diffusion suppres- +sion (by a factor 10 with respect to the large-scale interstellar +average), a more recent injection and transport process over the +last 0.3 − 1 Myr, and five-to-ten times higher acceleration effi- +ciencies with respect to H1 and L1. +Setups H3 and H4 describe an even more recent event, with +injection lasting 3 or 30 kyr, transport proceeding over 10 or +100 kyr in a medium where diffusion is not or only moderately +suppressed, from a much more powerful source with properties +that eventually seem relevant to an SN. Support for these scenar- +ios would imply finding evidence of a middle-aged remnant in +the region. The γ Cygni SNR (G78.2+2.1), with its estimated age +∼ 10 kyr, would be an interesting candidate source for scenario +H3. Another option could be the remnant that resulted from the +explosion giving birth to PSR J2032+4127, ∼ 100−200 kyr ago. +Its age is comparable to the diffusion time involved in scenario +H4 and is high enough that the remnant has most likely gone +undetectable by now. Interestingly, the typical injection time of +30 kyr and diffusion suppression by a factor 10 in scenario H4 +are reminiscent of the results obtained in Nava et al. (2019) for +the non-linear diffusion of 0.1 − 1 TeV CRs escaping from a su- +pernova remnant in a hot ionised medium. +For comparable model setups, the difference in injection effi- +ciency between leptonic and hadronic scenarios is of an order of +magnitude at most. This means that mixed lepto-hadronic sce- +narios either imply a relatively high electron-to-proton ratio at +injection, or a predominance of the hadronic contribution to the +emission if the electron-to-proton ratio at injection is expected +to have more classical values of 10−2 at most. +All the potential sources discussed above, Cyg OB2, +NGC 6910, the γ Cygni SNR, and the unobserved SNR asso- +ciated with PSR J2032+4127 are displaced with respect to the +Article number, page 20 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +centroid of the emission (Figure 12). This could suggest an in- +homogenous transport scenario if one of these sources is indeed +responsible for the origin of the particle population. +Possible improvements to the modelling presented here in- +clude the possibility of multiple and extended sources, for in- +stance Cyg OB2 and NGC 6910 releasing accelerated particles +at their respective super-wind termination shocks, and a more +complete transport scheme, including inhomogeneous diffusion +and (or) energy losses over such a large volume, or the effect +of advection. The latter point is particularly relevant in the case +of high levels of diffusion suppression, as it may dominate the +transport of the lowest-energy particles and alter their diffusion. +The contribution from pre-existing CRs, for instance their lep- +tonic emission from inverse-Compton scattering in the dense +photon fields of the main clusters (Orlando & Strong 2007) or +their hadronic emission after reacceleration in the turbulent in- +terior of the region (Tolksdorf et al. 2019), certainly deserves a +more sophisticated treatment than done here. Meanwhile, we can +qualitatively compare our results to more sophisticated models +of particle acceleration and transport at cluster wind termination +shocks and in SBs from the literature. +Morlino et al. (2021) presents a model of particle accelera- +tion at the winds of star clusters that predicts a spectrum similar +to the cocoon central component CoCent, which spans a region +with size comparable to that of the wind termination shocks from +Cyg OB2 and NGC 6910 (see Figure 12). The spatial distribu- +tion of low-energy particles in their model can be rather flat up to +a few times the termination shock radius, which is comparable to +the extended component CoExt. However, their model predicts +that higher energy particles are more tightly confined around the +shock, which is in contrast with our results of a harder spectrum +for CoExt than CoCent. In addition, as already discussed above, +there is no clear identification of a super-wind termination shock +in the region, nor is it clear that there is a super-wind emanating +from Cyg OB2. +Alternatively, Vieu et al. (2022) have shown that their model +for particle acceleration and transport in SBs can reproduce the +overall spectrum of the Cygnus cocoon measured by the LAT +and HAWC in the case of efficient confinement in the bubble +shell. In these conditions, they show that particle densities are +rather uniform inside the SB, which might be in good agreement +with the flat radial emissivity profile of the CoExt component. +However, it is not obvious that their model can reproduce the +morphological properties of the observed emission, especially +its centrally peaked nature, if particles are efficiently trapped in +an outer shell (the location of which remains unclear in Cygnus). +Recently, Fornieri & Zhang (2022) presented a model of +gamma-ray emission from Cygnus featuring two CR sources +(Cyg OB2 and the γ Cygni SNR) and a description of particle +transport that takes into account the detection of multiple plasma +modes in the region (Zhang et al. 2020). As a consequence CR +diffusion is predicted to be inhomogenous, which results in con- +finement for a long time in the central cavities where plasma +modes are predominantly magnetosonic, and a more rapid diffu- +sion in the nearby Alfvénic-dominated regions. However, their +model does not take into account ionised gas in the central cav- +ities, which seems required to explain the emissions observed +from CoCent. Furthermore, they calculate a diffusion coefficient +for physical parameters, such as gas density and temperature, +that are not necessarily relevant for the entire gamma-ray emit- +ting region. Overall, it is not clear if their model can explain the +large extension of CoExt. +4.5. FCES G85.00−1.78 +The introduction of FCES G85.00−1.78 (OffExc), significantly +improves the likelihood of the model (Section 3.4). However, +the best-fit Gaussian model is only partially contained in our +analysis region, and therefore the results may be subject to large +uncertainties. A proper characterisation of this excess is left for +future studies, and in this section we only provide general con- +siderations on possibilities concerning its origin. There is no ob- +vious correlation between OffExc and structures in the gas maps +(Figure 2 and 3). +No SNRs are found overlapping with OffExc in SNRCat5 +(Ferrand & Safi-Harb 2012). On the other hand, the ATNF +1.67 pulsar catalogue6 (Manchester et al. 2005) lists three pul- +sars within the r68 area of OffExc with a spin-down power +> 1034 erg s−1. Among those PSR J2111+4606, the closest to +the source centroid at an offset of 3.3◦, has a spin-down power of +1.4 × 1036 erg s−1, a characteristic age of 17.5 kyr, and an uncer- +tain distance. It may be reminiscent of some middle-aged pulsars +powering large gamma-ray sources such as HESS J1825−137 +(H. E. S. S. Collaboration et al. 2019; Principe et al. 2020) or the +pulsar halos around PSR J0633+1746 or PSR B0656+14 (Abey- +sekara et al. 2017). In the latter case, the offset of the pulsar from +the centre of the emission can be explained by a combination of +proper motion and time-dependent injection (Zhang et al. 2021). +OffExc is bordered on the east by X-ray emission in the +Cygnus SB (Cash et al. 1980). The corresponding X-ray struc- +ture, dubbed as S-ARC 3 in Uyanıker et al. (2001) has been asso- +ciated to the stellar association Cyg OB4. However, the very ex- +istence of Cyg OB4 is questioned based on parallax distances (de +Zeeuw et al. 1999). Cantat-Gaudin et al. (2020) reports ten stel- +lar clusters with more than 100 members at a probability > 70% +within the r68 area of OffExc. Among those, seven are located at +distances from the Earth < 1.8 kpc, which would correspond to +a physical size < 200 pc. Most of them are quite far away from +the gamma-ray emission centroid, with the closest at 2.8◦ being +NGC 7082 at 1.339 kpc from Earth and with an estimated age +of 61 Myr, larger than typical stellar clusters with established +detections in gamma rays (for instance Tibaldo et al. 2021). +The spectrum of OffExc has a particularly strong break with +a steep slope at low energies and a hard spectrum above a few +GeV that is strikingly similar to the one of CoExt. Establish- +ing a physical connection between the two emission components +is not obvious. OffExc may be produced by particles escaping +the volume encompassed by CoExt, after an energy-dependent +transport process that depleted the particle spectrum at the low- +energy end. This is reminiscent of the illumination of neighbour- +ing gas clouds by CRs escaping from a nearby source (for in- +stance Tang 2019). Alternatively, steep slopes at low energies +has been suggested as a signature of particles reacceleration in +SBs (for instance Tolksdorf et al. 2019), and there may be radial +gradients in such a mechanism. +5. Summary and conclusions +We presented an analysis of the gamma-ray emission from the +Cygnus region based on ∼13 years of Fermi-LAT data. The ex- +traction of the emission from the so-called Cygnus cocoon was +performed from a dedicated modelling of interstellar emission +from the region. Compared to the analysis presented in Acker- +mann et al. (2011), we used almost seven times more data, pro- +duced with an improved reconstruction scheme corresponding to +5 http://www.physics.umanitoba.ca/snr/SNRcat +6 http://www.atnf.csiro.au/research/pulsar/psrcat/ +Article number, page 21 of 30 + +A&A proofs: manuscript no. ms_cocoon +enhanced instrument performance. The data analysis is based on +a much larger catalogue of gamma-ray sources and dedicated re- +sults for major sources in the fields. We also used improved gas +tracer data and an iterative procedure to derive the dark neutral +gas map. +As a result, the emission from the cocoon is now sepa- +rated into two main components: first, a central component, +FCES G80.00+0.50 (CoCent), traced by a model for the distri- +bution of ionised gas within the borders of photo-dissociation +regions, and having a power law spectrum with index 2.19 ± +0.03+0.00 +−0.01; second, an extended component, FCES G78.74+1.56 +(CoExt), that can be modelled with a 2D Gaussian intensity dis- +tribution of extension r68 = 4.4◦ ±0.1◦ +0.1◦ +−0.1◦ and a smooth broken +power law spectrum with spectral indices 1.67 ± 0.05+0.02 +−0.01 and +2.12 ± 0.02+0.00 +−0.01 below and above 3.0 ± 0.6+0.0 +−0.2 GeV, respectively. +Emission from this component is significantly detected out to +nearly 10◦ from the approximate centre of the star-forming re- +gion. Its total spectrum is significantly different from that of Co- +Cent, and it exhibits significant spectral variations in azimuth in +the innermost ≲ 3◦. +Two additional extended emission components were signifi- +cantly detected during the analysis. Source FCES G78.83+3.57 +(CoWest) overlaps with a bright arc of 8 µm emission on the +border of the central cavities in Cygnus X, and has a spectrum +statistically compatible with CoCent. Although the spectral sim- +ilarity and spatial proximity suggests a common origin, CoWest +does not show any obvious correlation with known gas struc- +tures. Another source, FCES G85.00−1.78 (OffExc), is offset by +several degrees with respect to Cygnus X and its spectrum is sig- +nificantly different from all the other extended components stud- +ied, so a common origin seems unlikely. The centroid of OffExc +lies on the edge of our analysis region, and therefore its current +characterisation may be inaccurate. A proper study of this com- +ponent is left for follow-up work. +The extended set of observables resulting from our analy- +sis for the two brightest sources making up the cocoon, CoCent +and CoExt, can be accounted for reasonably well from a simple +diffusion-loss framework with a small number of free parame- +ters, under several model setups. In all viable scenarios, one sin- +gle population of non-thermal particles with a flat injection spec- +trum at the central source is sufficient and both hadronic and lep- +tonic options are viable. Particles span the full extent of source +CoExt as a result of diffusion, and give rise to source CoCent by +interacting with ionised gas in the innermost regions. Possible +solutions are very different in terms of energetics, transport con- +ditions, and time scales involved. Some scenarios involve con- +tinuous injection during ∼ 0.3 − 3 Myr, transport in a medium +with moderately to strongly suppressed diffusion with respect +to the large-scale interstellar average, and injection luminosities +in the 1036 − 1037 erg s−1 range. They could describe a process +by which the observed gamma-ray emission is powered by par- +ticle acceleration in the prominent star clusters Cyg OB2 and +NGC 6910. Alternatively, a hadronic solution exists involving a +more recent event, with injection lasting 3 − 30 kyr and transport +proceeding over 10 kyr in a medium where diffusion is not or +only moderately suppressed, and a much more powerful source +with injection luminosity ∼ 1039 erg s−1. Such a scenario seems +more relevant to a single supernova explosion. +Possible improvements beyond this simple interpretation +framework include accounting for multiple and extended +sources, a more advanced description of particle transport in an +inhomogeneous medium and including advection, and, last but +not least, the description of physically motivated acceleration +mechanisms. The observables extracted from our analysis are +made available in machine-readable format and can be used in +the future to perform detailed comparisons with more sophisti- +cated models. +From the observational perspective, significant advances in +gamma rays can be expected from instruments with improved +sensitivity and angular resolution. The upcoming Cherenkov +Telescope Array (Cherenkov Telescope Array Consortium et al. +2019) above a few tens of GeV will provide a sensitivity an or- +der of magnitude better than previous ground-based instruments +and an angular resolution reaching a few arcmin. Proposed space +missions dedicated to the MeV to GeV domain (de Angelis et al. +2018; McEnery & Amego Team 2020) may also improve a few +times the angular resolution and by one or two orders of mag- +nitude the sensitivity compared to Fermi, and provide observa- +tions in an energy range, the sub-MeV and MeV domain, poorly +observed until now. Complementary advances in the character- +isation of interstellar gas (for instance Emig et al. 2022), and +of multi-wavelength and multi-messenger emission from the co- +coon (for instance Mizuno et al. 2015; Yoast-Hull et al. 2017) are +also key to improving our understanding of particle acceleration +and transport in this region. +Acknowledgements. The Fermi LAT Collaboration acknowledges generous on- +going support from a number of agencies and institutes that have supported both +the development and the operation of the LAT as well as scientific data analy- +sis. These include the National Aeronautics and Space Administration and the +Department of Energy in the United States, the Commissariat à l’Energie Atom- +ique and the Centre National de la Recherche Scientifique / Institut National de +Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale +Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Ed- +ucation, Culture, Sports, Science and Technology (MEXT), High Energy Accel- +erator Research Organization (KEK) and Japan Aerospace Exploration Agency +(JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research +Council and the Swedish National Space Board in Sweden. Additional support +for science analysis during the operations phase is gratefully acknowledged from +the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études +Spatiales in France. This work performed in part under DOE Contract DE-AC02- +76SF00515. +This work was supported by the "Agence Nationale de la Recherche" through +grant ANR-19-CE31-0014 (GAMALO project, PI: P. Martin). +This work makes use of NumPy (Harris et al. 2020), AstroPy (Astropy Collab- +oration et al. 2022), Matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), and +the colourmaps in the CMasher package (van der Velden 2020). +The authors would like to thank E. Orlando and M. Pesce-Rollins for their helpful +comments on the manuscript, as well as I. A. Grenier for insightful conversations +about the project. +References +Abeysekara, A. U., Albert, A., Alfaro, R., et al. 2017, Science, 358, 911 15, 18, +21 +Abeysekara, A. 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The intensity distribution corresponding to the best-fit two- +dimensional Gaussian model is displayed for comparison as a +dotted line. The χ2 value correspond to the deviation from the +2D Gaussian fit. +Article number, page 24 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +Appendix A: Data analysis +We provide in this appendix a series of technical details about +the data analysis. +Appendix A.1: Preliminary model optimisation +The preliminary optimisation of the emission model went +through the following steps: +1. a simultaneous fit of the normalisation of bright sources with +TS > 104 and predicted photons counts > 500; +2. an iterative fit of the normalisation and spectral shape of all +the sources in the ROI by order of intensity and significance +(method optimize of Fermipy); +3. a further simultaneous fit of the normalisation of sources +with TS > 104 and predicted photons counts > 500, and also +of the spectral shape of sources with the same TS condition +and predicted photons counts > 1000. +The thresholds on predicted photon counts and TS were cho- +sen to optimise all parameters for the brightest sources in the ROI +(three pulsars, Cygnus Loop, γ Cygni). We are more restrictive +for the spectral shape to reduce the number of free parameters in +the analysis, which otherwise can become unstable. +Appendix A.2: Morphological fits +In the morphological characterisation stages, the optimisation of +the related components is performed in a two-step process. +1. Extension optimisation: we perform a first scan over a coarse +grid of extension values, followed by a second finer scan +around the first optimum and then the fit of a parabola to +determine the final extension value and its uncertainty. +2. Position optimisation: we compute a map of log-likelihood +values over a position grid of 2◦ × 2◦ with a 0.2◦ binning and +determine the best-fit position and its uncertainty from the fit +of an ellipse. +Appendix A.3: Spectral models +In the spectral characterisation of the various components of the +cocoon, we consider the following models: a simple power law +(PL) of expression +dN +dE = N0 × +� E +E0 +�−γ +, +(A.1) +with N0 flux at the reference energy E0 and γ spectral index; a +log-parabola (LP) of expression +dN +dE = N0 × +� E +E0 +�− +� +α+β ln +� +E +E0 +�� +, +(A.2) +with N0 flux at the reference energy E0, α slope parameter, and +β curvature parameter; and a smooth broken power law (SBPL) +of expression +dN +dE = N0 × +� E +E0 +�−γ1 ��������1 + +� E +Eb +� γ2−γ1 +κ �������� +−κ +, +(A.3) +with N0 flux at the reference energy E0, Eb break energy, γ1 spec- +tral index at energies ≪ Eb, γ2 spectral index at energies ≫ Eb, +and κ the smoothing parameter fixed to 0.2. +Appendix B: Additional results on the +spectro-morphological analysis +Figure B.1 shows the TS values for the best decomposition of +CoExt, as in step D described in Section 3.7.2. The figure illus- +trates how the decomposition was designed to conserve a mini- +mum TS of 25 except for the largest ring where the requirement +could not be met. +85◦ +80◦ +75◦ +70◦ +10◦ +5◦ +0◦ +−5◦ +Galactic Longitude +Galactic Latitude +101 +102 +TS +Fig. B.1: TS values in rings, segments, and disk for the best de- +composition of FCES G78.74+1.56. +We used the intensities derived in Section 3.7.2 to derive +emissivities. To do so, we divided the flux associated with each +segment, ring, or disk by the total neutral gas column density +in the local arm (atomic, molecular and DNM) integrated over +solid angle for each segment, ring, or disk area (shown in Fig- +ure B.2). See Section 4.1 for a discussion on uncertainties in the +85◦ +80◦ +75◦ +70◦ +10◦ +5◦ +0◦ +−5◦ +Galactic Longitude +Galactic Latitude +1022 +4 × 1021 +6 × 1021 +2 × 1022 +3 × 1022 +4 × 1022 +Column density (H cm−2) +Fig. B.2: Total neutral gas column density in the local arm as in +Figure 10 reprojected onto the disk, rings, and segments used for +the spectro-morphological analysis in section 3.7.2. +gas column densities relevant for the emissivity computation. +Figure B.3 shows maps of intensities and emissivities for Co- +Ext as decomposed in step D in section and in three different en- +ergy bands 0.5−2.236 GeV, 2.236−10 GeV and 10−1000 GeV. +Article number, page 25 of 30 + +A&A proofs: manuscript no. ms_cocoon +Radial profiles of intensity and emissivity in the three energy +bands are shown in Section 4, where they are used for quantita- +tive interpretation of the results. +10◦ +5◦ +0◦ +−5◦ +Galactic Latitude +10−7 +10−6 +10−5 +10−7 +10−6 +Intensity (ph cm−2 s−1 sr−1) +10−8 +10−7 +10−6 +85◦ +80◦ +75◦ +70◦ +10◦ +5◦ +0◦ +−5◦ +Galactic Latitude +85◦ +80◦ +75◦ +70◦ +Galactic Longitude +85◦ +80◦ +75◦ +70◦ +10−29 +10−28 +10−29 +10−28 +Emissivity (ph s−1 sr−1 H−1) +10−30 +10−29 +0.5 - 2.2 GeV +2.2 - 10 GeV +10 GeV - 1 TeV +Fig. B.3: Intensity and emissivity maps in three different energy +ranges for component FCES G78.74+1.56, according to the de- +composition in rings and segments of step D (see Section 3.7.2 +for details). +Appendix C: Diffusion-loss model framework +In Section 4.2, we introduce a simple diffusion-loss framework +to account for the observed properties of the Cygnus cocoon. We +provide here the full formalism of this framework. +The transport equation governing the evolution of the parti- +cle distribution f in momentum p, position r, and time t is: +∂f +∂t = ∇ · (D . ∇ f) − ∂ +∂p +� ˙p . f� + Q, +(C.1) +where D is a spatial diffusion coefficient, ˙p a momentum loss +term, and Q a source term. The momentum loss term ˙p in- +cludes losses that are uniform in space and arise from radia- +tive processes, hadronic interactions for accelerated protons, and +Bremsstrahlung, inverse-Compton scattering, and synchrotron +radiation for accelerated electrons (Schlickeiser 2002). The +source term Q(p, t) is assumed to be point-like in space and +to have a constant power-law with exponential cut-off spectral +shape: +Q(p, t) = Q0 . +� +p +10 GeV/c +�−α +. e−p/pcut . e−t/tinj. +(C.2) +Particles are injected with a power law spectrum in momentum +with index α. The cut-off momentum is set to a high value of +1 PeV/c that our gamma-ray data are not sensitive to. To investi- +gate the possibility that injection is not constant in time, and may +have occurred over a finite duration some time ago followed by a +longer diffusion time, we implemented (somewhat arbitrarily) an +exponential decay of the source term. Integrating the source term +Q(p, t) over particle energies yields the time-dependent injection +luminosity Linj(t). The diffusion coefficient D(p) assumed to be +constant in space and time is defined as: +D(p) = β . D0 . +� +p +10 GeV/c +�δ +with +δ = 1/3, +(C.3) +where β = v/c with v the velocity of a particle and c the veloc- +ity of light. The power-law dependence in momentum, with an +index corresponding to a Kolmogorov scaling for the magnetic +turbulence spectrum, is adapted from models of large-scale CR +propagation in our Galaxy (Trotta et al. 2011; Orlando 2018). +We note that alternative expressions were considered recently +in the light of more accurate direct CR measurements at Earth +(Evoli et al. 2019; Génolini et al. 2019), and that the considered +deviations from a pure power law may have consequences in the +energy range we are interested in here. +The solution to the transport equation is obtained as (Atoyan +et al. 1995): +f(p, r, t) = +� tdiff +max[0, tdiff−tcool(pcut,p)] +˙p(p0) +˙p(p) . Q(p0, t0) +π3/2r3 +d +. e−r2/r2 +d . dt0, +(C.4) +with rd diffusion distance. For a present-day momentum p, the +integration runs either over the full injection and transport his- +tory, or over the recent period spanning a cooling time tcool from +the cut-off momentum down to p, with cooling time +tcool(p0, p) = +� p0 +p +−dp +˙p . +(C.5) +This three-dimensional spherically symmetric distribution of +particles is integrated along the line of sight for any angular off- +set from the centre θ and given the distance d to the source: +Φ(p, θ, t) = 2 . d2 . +� ∞ +0 +f +� +p, +√ +θ2d2 + ℓ2, t +� +dℓ. +(C.6) +The resulting angular distribution of particles is then used to +compute non-thermal emissions at any point in the region of in- +terest. To this aim, we used the naima package (Zabalza 2015) +in the approximation of isotropic radiation fields in the case of +inverse-Compton scattering and using a nuclear enhancement +factor of 1.845 in the case of pion decay (Mori 2009). +Appendix D: Possible diffusion-loss scenarios +We display here the results obtained for some of the diffusion +scenarios considered in the interpretation of the results (see Sec- +tion 4.3). The intensity and emissivity profiles for hadronic sce- +narios H2 and H3 or H4 are presented in Figs. D.1 and D.2, +and the intensity profiles for model leptonic scenario L2 are pre- +sented in Figure D.3. The full set of results for the leptonic pulsar +halo scenario is presented in Figs. D.4 and D.5. +Article number, page 26 of 30 + +3X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +0.5-2.2GeV intensity (ph/cm2/s/sr) +Linj = 3.25e+37 erg/s +2 = 145.55 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +29 +10 +28 +10 +27 +10 +26 +10 +25 +0.5-2.2GeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +2.2-10GeV intensity (ph/cm2/s/sr) +Linj = 3.25e+37 erg/s +2 = 302.86 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +30 +10 +29 +10 +28 +10 +27 +10 +26 +2.2-10GeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +10GeV-1TeV intensity (ph/cm2/s/sr) +Linj = 3.25e+37 erg/s +2 = 283.72 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +31 +10 +30 +10 +29 +10 +28 +10 +27 +10GeV-1TeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. D.1: Same as Figure 16, for model setup H2. +Article number, page 27 of 30 + +A&A proofs: manuscript no. ms_cocoon +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +0.5-2.2GeV intensity (ph/cm2/s/sr) +Linj = 2.85e+39 erg/s +2 = 199.32 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +29 +10 +28 +10 +27 +10 +26 +10 +25 +0.5-2.2GeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +2.2-10GeV intensity (ph/cm2/s/sr) +Linj = 2.85e+39 erg/s +2 = 295.93 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +30 +10 +29 +10 +28 +10 +27 +10 +26 +2.2-10GeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +10GeV-1TeV intensity (ph/cm2/s/sr) +Linj = 2.85e+39 erg/s +2 = 285.09 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +31 +10 +30 +10 +29 +10 +28 +10 +27 +10GeV-1TeV emissivity (ph/s/sr/H) +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Local emissivity +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. D.2: Same as Figure 16, for model setup H3 or H4. +Article number, page 28 of 30 + +X. Astiasarain et al.: Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +7 +10 +6 +10 +5 +10 +4 +0.5-2.2GeV intensity (ph/cm2/s/sr) +Linj = 2.41e+36 erg/s +2 = 221.36 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +2.2-10GeV intensity (ph/cm2/s/sr) +Linj = 2.41e+36 erg/s +2 = 528.59 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +10GeV-1TeV intensity (ph/cm2/s/sr) +Linj = 2.41e+36 erg/s +2 = 351.08 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. D.3: Same as Figure 18 for model setup L2. +100 +101 +102 +103 +Energy (GeV) +10 +9 +10 +8 +10 +7 +10 +6 +Spectral energy distribution (GeV/cm2/s) +D0(100TeV)=4.00e+28 cm2/s +Linj = 1.43e+36 erg/s +NIG +H = 1.03e+22 H/cm2 +2 = 23.61 + 2.89 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. D.4: Same as Figure 17 for the pulsar halo scenario. +Article number, page 29 of 30 + +A&A proofs: manuscript no. ms_cocoon +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +7 +10 +6 +10 +5 +10 +4 +0.5-2.2GeV intensity (ph/cm2/s/sr) +D0(100TeV)=4.00e+28 cm2/s +Linj = 1.43e+36 erg/s +2 = 322.50 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +7 +10 +6 +10 +5 +10 +4 +0.5-2.2GeV intensity (ph/cm2/s/sr) +D0(100TeV)=4.00e+28 cm2/s +Linj = 1.43e+36 erg/s +2 = 322.50 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Angle (deg) +10 +8 +10 +7 +10 +6 +10 +5 +2.2-10GeV intensity (ph/cm2/s/sr) +D0(100TeV)=4.00e+28 cm2/s +Linj = 1.43e+36 erg/s +2 = 524.85 +Model (FCES G78.74+1.56) +Model (FCES G80.00+0.50) +2D Gaussian fit +Data (FCES G78.74+1.56) +Data (FCES G80.00+0.50) +Fig. D.5: Same as Figure 18 for the pulsar halo scenario. +Article number, page 30 of 30 + diff --git a/TtE3T4oBgHgl3EQfaQo5/content/tmp_files/load_file.txt b/TtE3T4oBgHgl3EQfaQo5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae5ef274bdf092bae0b34adc1a86000c95eebbd4 --- /dev/null +++ b/TtE3T4oBgHgl3EQfaQo5/content/tmp_files/load_file.txt @@ -0,0 +1,2471 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf,len=2470 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon ©ESO 2023 January 12, 2023 Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations ⋆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain1,⋆⋆, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Tibaldo1,⋆⋆⋆, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Martin1,⋆⋆⋆⋆, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Knödlseder1, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Remy2 1 IRAP, Université de Toulouse, CNRS, CNES, UPS, 9 avenue Colonel Roche, 31028 Toulouse, Cedex 4, France 2 Max Planck Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Received 29 November 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Accepted 06 January 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Star-forming regions may play an important role in the life cycle of Galactic cosmic rays (CRs), notably as home to specific acceleration mechanisms and transport conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gamma-ray observations of Cygnus X have revealed the presence of an excess of hard-spectrum gamma-ray emission, possibly related to a cocoon of freshly accelerated particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We seek an improved description of the gamma-ray emission from the cocoon using ∼13 years of observations with the Fermi- Large Area Telescope (LAT) and use it to further constrain the processes and objects responsible for the young CR population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We developed an emission model for a large region of interest, including a description of interstellar emission from the background population of CRs and recent models for other gamma-ray sources in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Thus, we performed an improved spectro- morphological characterisation of the residual emission including the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The best-fit model for the cocoon includes two main emission components: an extended component FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56, described by a 2D Gaussian of extension r68 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ and a smooth broken power law spectrum with spectral indices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 below and above 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 GeV, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' and a central component FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, traced by the distribution of ionised gas within the borders of the photo-dissociation regions and with a power law spectrum of index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 that is significantly different from the spectrum of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' An additional extended emission com- ponent FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57, located on the edge of the central cavities in Cygnus X and with a spectrum compatible with that of FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, is likely related to the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For the two brightest components FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56, spectra and radial-azimuthal profiles of the emission can be accounted for in a diffusion-loss framework involving one single popula- tion of non-thermal particles with a flat injection spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Particles span the full extent of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 as a result of diffusion from a central source, and give rise to source FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 by interacting with ionised gas in the innermost region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For this simple diffusion-loss model, viable setups can be very different in terms of energetics, transport conditions, and timescales involved, and both hadronic and leptonic scenarios are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The solutions range from long-lasting particle acceleration, possibly in prominent star clusters such as Cyg OB2 and NGC 6910, to a more recent and short-lived release of particles within the last 10 − 100 kyr, likely from a supernova remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The observables extracted from our analysis can be used to perform detailed comparisons with advanced models of particle acceleration and transport in star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Acceleration of particles – cosmic rays – open clusters and associations – Gamma rays: ISM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Introduction There is firm evidence that cosmic rays (CRs) at energies be- low 1 PeV originate from the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Supernova remnants (SNRs) remain the leading candidate as sources of the major- ity of Galactic CRs, most likely through the process of diffusive shock acceleration, while alternative source classes including massive star-forming regions, the Galactic centre, pulsar wind nebulae (PWNe), and compact binary systems may bring com- plementary contributions over specific parts of the extended CR ⋆ The template used to model source FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, the map of excess counts in figure 5 (right), the spectral points from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5, the map of total neutral hydrogen column density in the local arm (Fig- ure 10), and the intensity and emissivity profiles discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 are available in electronic form at the CDS via anonymous ftp to cd- sarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='fr (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5) or via https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='fr/cgi- bin/qcat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='J/A+A/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ⋆⋆ xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='astiasarain@irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='eu ⋆⋆⋆ luigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='tibaldo@irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='eu ⋆⋆⋆⋆ pierrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='martin@irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='eu spectrum (see, for instance Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Massive star-forming regions are of particular interest in this context (for instance Bykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The clusters of OB stars at their centres are the progenitors of a variety of particle accel- eration sites such as SNRs, pulsars, and PWNe, or compact bi- nary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In addition, the collective action of powerful stel- lar winds and, after a few million to a few tens of million years, the explosion of massive stars into supernovae lead to the for- mation of super-bubbles (SBs), which are large cavities filled by a highly dynamical medium that, as a whole, may play a spe- cific role in the life cycle of CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The isotopic abundances mea- sured in CRs suggest that at least a fraction of the CR material is sourced from the winds of massive stars (Binns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Tatischeff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Accelerated particles in distant locations can be revealed via the gamma-ray emission produced when they interact with interstellar gas, through inelastic collisions for nuclei or Bremsstrahlung for leptons, and radiation fields, through the inverse-Compton (IC) scattering by leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Therefore, star- Article number, page 1 of 30 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='04504v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='HE] 11 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon forming regions are expected to be bright gamma-ray sources from the interactions of particles with the large masses of in- terstellar gas and the intense radiation fields available in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gamma-ray emission in the GeV and TeV en- ergy ranges is detected towards a growing number of massive star-forming regions (for a review see for instance Tibaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021), and taken as evidence in favour of in situ CR accelera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, the clustering of energetic objects and interstellar clouds combined with the limited resolution of gamma-ray tele- scopes makes it difficult to firmly identify the acceleration sites and mechanisms, and to understand how particles propagate and interact through the region and eventually escape to merge into the large-scale CR population in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Observational progress is matched by a flourishing develop- ment of models of particle acceleration and transport by stel- lar winds (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Bykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021) and SBs (Bykov 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Ferrand & Marcowith 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Tolks- dorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Vieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The models show that these objects can be efficient particle accelerators and make a contri- bution to Galactic CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' They predict a number of morphological and spectral signatures that can be looked for to test the physical processes at the origin of the observed gamma-ray signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Cygnus X is one of the best studied massive star-forming re- gions in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Cygnus X contains Cygnus OB2 that, with 78 confirmed O stars (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020), is among the largest associations of massive stars in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' It is com- posed of multiple substructures with a main group at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='76 kpc from the Earth and a foreground group at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='35 kpc (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019) and at least two star-forming bursts ∼3 and ∼5 Myr ago (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A second prominent massive stel- lar cluster in Cygnus X is NGC 6910 at a distance of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='73 kpc (Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020), an age in the range from 5 to 10 Myr (Delgado & Alfaro 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020), and a flat mass function pointing to a large number of massive stars (Kaur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The Large Area Telescope (LAT) aboard the Fermi Gamma- ray Space Telescope (Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2009) unveiled a hard gamma-ray excess towards Cygnus X with an extension1 r68 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3◦ (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The excess was inter- preted as the signature of a cocoon of freshly accelerated par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gamma-ray emission in the energy range from hundreds of GeV to hundreds of TeV from the Cygnus cocoon was sub- sequently detected using ARGO-YBJ, HAWC, and LHAASO (Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Li 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The most common interpretation involves nuclei acceler- ated by Cygnus OB2, possibly up to PeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The radial gamma-ray emission profile above 10 GeV was taken as indica- tion of diffusion following continuous CR injection over a few million years (Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In this paper, we present a new study of the Cygnus cocoon based on more than 13 years of Fermi-LAT observations with the aim of improving the morphological and spectral characteri- sation of the emission in order to constrain particle acceleration and propagation scenarios in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The characterisation of the cocoon requires a careful modelling of the interstellar gas distribution in the region that is presented in Section 2, while we describe the analysis of gamma-ray data, including morpho- logical, spectral, and spectro-morphological characterisation of the cocoon emission in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The observables we derived 1 Throughout the paper we refer to a source extension as its 68% con- tainment radius r68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For a 2D Gaussian intensity distribution, r68 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='51σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' are then discussed and interpreted in Section 4 and our conclu- sions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Construction of interstellar gas maps The distribution of interstellar gas towards the region of interest is a key ingredient of our analysis for two reasons: 1) it is neces- sary to model the strong foreground and background gamma-ray emission from the interactions of the large-scale Galactic CR population with interstellar gas in the direction of Cygnus, and thus be able to extract and characterise the emission of the co- coon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2) it is used in the interpretation of the gamma-ray signal in terms of the underlying CR populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Atomic and molecular gas We trace atomic gas using the 21 cm emission line from the hyperfine transition of atomic hydrogen H I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We use data from the Canadian Galactic Plane Survey (CGPS, Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2003) with an angular resolution of 1′ and a velocity resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 km s−1 in the region with Galactic longitude 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5◦ < l < 90◦ and Galactic latitude −3◦ < b < 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Outside this region we use data from the all-sky HI4PI survey from Effelsberg and Parkes observations (HI4PI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2016) with a lower an- gular resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='27◦ and velocity resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='49 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We checked the consistency of the two surveys by comparing the data in the region covered by the CGPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We derived column densities N(H I) under the hypothesis of a uniform spin temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All results are shown for the reference spin temperature of 250 K suggested by emission-absorption spectrum pairs in the CGPS area (Dickey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2009) and that was also found to best reproduce gamma-ray observations of the Cygnus region based on an earlier analysis (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is a highly uncertain parameter that is not expected to be uniform along lines of sight and across the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There- fore, the analysis was also performed for alternative uniform spin temperatures of 100 K (lower bound set by the brightness tem- peratures observed in the region), 400 K, and the optically thin case, which are used to set systematic uncertainties on relevant quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Molecular hydrogen H2 cannot be traced directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We use the 12CO J1→0 rotational line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 mm as surrogate tracer, under the usual hypothesis that N(H2) column densities are directly pro- portional to the CO intensity (velocity-integrated brightness tem- perature) WCO through a constant known as XCO ≡ N(H2)/WCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We use CO data from the composite survey by Dame et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2001) with an angular resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='125◦ in the area consid- ered in this paper and a velocity resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Data were noise-filtered using the moment-masking technique (Dame 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The Doppler shift of the lines can be used to infer the gas ve- locity along the line of sight due to Galactic rotation, and there- fore separate multiple structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, intrinsic velocity dis- persion can cause biases in the estimates of gas column densi- ties across adjacent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To address this problem, we used the line profile fitting technique described in Remy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2017) to decompose emission from each line of sight into a combina- tion of pseudo-Voigt functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We built longitude-velocity and latitude-velocity diagrams based on the fit results for H I and CO, and defined in the longitude-latitude-velocity space bound- aries that separate the gas into three structures along the line of sight, namely the local arm (including the Cygnus complex), the Perseus arm, and the outer arm and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 1 shows Article number, page 2 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations an example of longitude-velocity diagram in the longitude range 73◦ < l < 87◦ that is used in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 75 80 85 l (deg) −150 −100 −50 0 50 V (km s−1) l ∈[73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦], b ∈[-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦] 100 101 102 N(H I) (1020 H cm−2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1: H I column density as a function of Doppler-shift veloc- ity and Galactic longitude summed over −2◦ < b < 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Total column densities from each pseudo-Voigt profile were assigned to the velocity of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The black lines show the boundaries that we defined to separate the three structures along the line of sight: local arm, Perseus arm, and outer Arm and beyond (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The final H I and CO maps for the three regions along the line of sight are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Since all surveys have different angular resolution, the maps were re-binned on a common grid of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='875′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Dark neutral medium (DNM) A significant fraction of neutral interstellar gas cannot be traced by the H I 21 cm line nor by the 12CO J1→0 rotational line (Gre- nier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2005) and is therefore missing in the maps described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' It can be referred to as the dark neutral medium (DNM) and it is thought to be a combination of opaque H I and diffuse H2 at the atomic-molecular interface of clouds, or dense H2 at the core of molecular clouds (Remy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' If dust and gas in the interstellar medium (ISM) were well mixed and the dust grains physical and chemical properties were the same everywhere, dust thermal emission would be propor- tional to total gas column densities along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There- fore, we can derive a DNM map by subtracting from the dust thermal emission the components correlated with H I and CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We use a map of the dust optical depth at 353 GHz obtained from component separation of Planck and IRAS data (Planck Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2016b) with an effective angular resolution of 5′ in high signal-to-noise regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To avoid biases from the missing DNM component in the determination of the components correlated with H I and CO, we used the iterative fitting procedure described in Tibaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Briefly, the procedure consists in an iterative fitting of the gas maps to the dust map where the positive part of the resid- uals is re-injected in the model at each iteration to compute unbi- ased values of the fit parameters and obtain an estimation of the missing DNM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A DNM map was calculated for each uniform spin temperature considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 3 shows the DNM map obtained for the reference spin temperature value of 250 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Ionised gas We derived an ionised gas column density map from the free- free emission measure EM(l, b) extracted from component sep- aration of Planck, WMAP, and 408 MHz data by Planck Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The free-free emission measure from Cygnus X is dominated by two strong peaks that, as indicated by 8 µm emission from dust, lie inside the cavities carved in the ISM by the intense star-forming activity in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We calculated H II column densities under the assumption that ionised gas fills a sphere of radius 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5◦ corresponding to ∼100 pc at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 kpc and with an uniform density along each line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The sphere is meant to model the ionised cavities at the hearth of Cygnus X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' With r the radius of the sphere and d the distance to Cygnus X the electron volume density is: ne(l, b) = ������������ EM(l, b) 2 � r2 − d2 sin2(l − l0) − d2 sin2(b − b0) ������������ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (1) Therefore, for the column density we obtain: NH II(l, b) = ne(l, b) × 2 � r2 − d2 sin2(l − l0) − d2 sin2(b − b0), (2) where l0 and b0 are the position of the sphere’s centre and EM(l, b) the emission measure in a given direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The final ionised gas column density map is displayed in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The angular resolution of the free-free emission measure map from Planck is 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The final ionised gas column density map was re-binned on the same grid as the MSX 8 µm map with a grid spacing of 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gamma-ray analysis and results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Data selection We analysed 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25 years of Fermi-LAT data from the beginning of the mission on 4 August 2008 to 3 November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We used the P8R3 data set (Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2018) and se- lected events in the P8R3_SOURCE class that is associated with instrument response functions P8R3_SOURCE_V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This event selection has a level of background contamination sufficiently low to study the bright extended emission from Cygnus X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Fur- thermore, we restricted the analysis to time intervals in which the LAT configuration and data quality is appropriate for science analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Events are separated in four independent data sets accord- ing to their PSF event type, that is the quality of the direction reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For each event type, we selected events above a minimum energy so that the point spread function (PSF) 68% containment radius is always better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦, which roughly cor- responds to the characteristic size of the most prominent spatial structures in the gas maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The minimum energy threshold used is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Lowering the minimum energy induced instabilities in the analysis due to bright emission from a few pulsars in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To reliably characterise extended emission at energies < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 GeV event selection based on the pulsars phases would be necessary, but this is beyond the scope of the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The maximum energy is 1 TeV for all event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To reduce contamination from the bright gamma-ray emis- sion from the Earth atmosphere, we selected events within a cone from the local zenith with aperture zmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The value of zmax was Article number, page 3 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 10◦ 0◦ −10◦ Galactic Latitude 85◦ 80◦ 75◦ 10◦ 0◦ −10◦ Galactic Latitude 85◦ 80◦ 75◦ Galactic Longitude 85◦ 80◦ 75◦ 0 101 102 N(H I) (1020 H cm−2) 0 100 101 WCO (K km s−1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2: H I column densities for a spin temperature of 250 K (top row) and WCO intensities (bottom row) for the local arm, Perseus arm, and outer arm and beyond (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' chosen from a visual inspection of the distribution of counts as a function of zenith angle for each event type component in the given energy ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Energy and zenith angle selection for the four data sets are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Table 1: The four data sets used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Event type Energy range (GeV) zmax PSF3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 - 1000 100◦ PSF2 1 - 1000 100◦ PSF1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 - 1000 100◦ PSF0 5 - 1000 105◦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Region of interest and emission model A major challenge in the characterisation of extended emission from Cygnus X is to model the bright interstellar emission from the large-scale population of CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Due to the large column den- sities of the ISM in this region, emission associated with gas is the dominant contribution at GeV energies (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Under the assumption that the large-scale CR densities are uniform on the spatial scales of interstellar complexes, we can model the foreground and background intensity associated with interstellar gas Igas as a linear combination of the column density maps for the different gas phases and structures along Article number, page 4 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 85◦ 80◦ 75◦ 10◦ 0◦ −10◦ Galactic Longitude Galactic Latitude 0 100 200 300 400 τ353 (10−6 mag) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3: Excess dust optical depth associated to the DNM obtained using the procedure described in the text for the reference H I spin temperature of 250 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 84◦ 82◦ 80◦ 78◦ 76◦ 4◦ 2◦ 0◦ −2◦ Galactic Longitude Galactic Latitude 0 1 2 3 4 Column density (1021 H cm−2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4: Ionised gas column densities based on the hypothesis of spherical geometry with a uniform density along each line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The black contours delineate the outer borders of the photo-dissociation regions and correspond to > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85 10−6 W m−2 sr−1 in the MSX 8 µm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' the line of sight: Igas(l, b, E) = qLIS(E) · �������� 3 � ı=1 �Aı NH I,ı(l, b) + Bı WCO,ı(l, b)� +C τDNM(l, b) � , (3) where qLIS(E) is the local gas emissivity spectrum, that is the gamma-ray emission rate per hydrogen atom, from Casandjian (2015), derived from LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The summation over ı describes the combination of the three regions along the line of sight: lo- cal arm, Perseus arm, outer arm and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The free param- eters Aı, Bı, and C account at the same time for variations of the large-scale CR densities across the three regions, and for the XCO ratios and the dust specific opacity σ353 = τ353/N(H I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The spectral shape of qLIS is fixed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We know that spectral variations of the emissivity along the line of sight are small towards Cygnus (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2012a) and, in gen- eral, towards the outer Galaxy (Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Conversely, this implies that any spectral deviations from the local interstel- lar spectrum (LIS) in Cygnus X are not accounted for by the background model and characterised as part of the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' An additional diffuse component is given by IC emis- sion from the large-scale population of CR leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We ac- counted for it using the GALPROP model SYZ6R30T150C2 (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Finally, we need to account for the isotropic gamma-ray background, which is a combination of extra-galactic diffuse gamma-ray emission (probably due to pop- ulations of unresolved sources) and of residual contamination by charged CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For this component, we used the tabulated spectra provided by the Fermi-LAT collaboration and determined from an analysis of LAT data over a large region of the sky2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We note that the IC model is also subject to large uncertain- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, morphological variations over our limited region of interest described below are expected to be small for con- ventional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Moreover, uncertainties in the spectrum are mitigated by the fact that the isotropic background spectrum is derived from a fit to the LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The interstellar emission model, along with the LAT re- sponse, sets the choice of the region of interest (ROI) for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The longitude and latitude extents should be suffi- ciently large to separate the extended emission of the Cygnus cocoon from the large-scale background, and so that the differ- ent components of the background model can be reliably con- strained by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We chose a ROI with Galactic longitude 73◦ ≤ l ≤ 87◦ and with Galactic latitude |b| ≤ 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The longi- tude interval leaves out complexes associated with Cygnus OB1 at l < 73◦ and with HB 21 at l > 87◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The wider coverage in latitude makes it possible to better constrain emission from local H I, IC scattering, and the isotropic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We modeled individual sources within the region based on the most recent catalogue of gamma-ray sources detected by the LAT, 4FGL-DR3 (Fermi-LAT collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All the sources within a square box of 40◦ side centred at l = 80◦ and b = 0◦ were included to account for the spill-over due to the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For two extended sources with potential impact on the char- acterisation of the cocoon emission, we replaced the 4FGL-DR3 models with dedicated models provided by recent in-depth stud- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The SNR γ Cygni is modelled according to the results from a joint fit of Fermi-LAT and MAGIC data at energies > 5 GeV (MAGIC Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The source is modelled as a disk with a log-parabola spectrum to account for the shell, plus a 2D Gaussian with a power law spectrum to account for an ad- ditional component in the north of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The arc component detected by MAGIC is not included since its flux is subdominant at energies < 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A morphological evolution of the remnant below 5 GeV is very challenging to characterise due to bright 2 We used files iso_P8R3_SOURCE_V3_PSFn_v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='txt, with n the PSF event type, from https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='gov/ssc/ data/access/lat/BackgroundModels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 5 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon emission from PSR J2021+4026 (MAGIC Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Thus, this possibility is not considered in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The SNR Cygnus Loop is modelled following the analysis by Tutone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2021) in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1-100 GeV energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We used two templates based on X-ray (ROSAT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 keV) and UV (GALEX 1771 − 2831 Å) data, each of them associated to a log- parabola spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Emission from this source above 100 GeV is expected to be small, and we visually checked in the residuals that there were no excess or deficit of counts at the location of the Cygnus Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 5 shows the gamma-ray count map in the region of in- terest and the excess counts associated with the cocoon obtained by subtracting from the data counts the best-fit model presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The cocoon, that is the excess shown in the right 85◦ 80◦ 75◦ 10◦ 0◦ −10◦ Galactic Latitude 102 103 104 Counts 85◦ 80◦ 75◦ 0 1000 Excess counts Galactic Longitude Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 5: Map of data counts in the full 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 GeV−1 TeV en- ergy range (left) and map of excess counts obtained by sub- tracting from the data counts the best-fit model presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 except for the components associated to the co- coon, namely FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56, FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The dashed contours correspond to the 8 µm emission from MSX data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85 × 10−6 W m−2 sr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The contours correspond to the column density of the ionised gas template at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5×1021 H cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The circle and the dashed cir- cle correspond to the position and r68 of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' panel of Figure 5, was initially modelled as in 4FGL-DR3: a 2D Gaussian with r68 = 3◦ (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011) and a spec- trum described by a log-parabola function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The morphological and spectral models were refined later during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Analysis framework The analysis is performed using fermitools v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 and a modi- fied version of Fermipy v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 that enables the use of catalogue 4FGL-DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Models are fit to the data via a binned maximum likelihood analysis with Poisson statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Events are binned on a grid with 10 bins per decade in energy and on maps with a pixel size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ in arrival direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Throughout the paper, we compare several models for the region and the source of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the simpler cases we use the likelihood ratio test, that is the test statistic defined as: TS = 2 (ln L − ln L0) , (4) where L0 is the maximum likelihood of a more parsimonious emission model with fewer free parameters (null hypothesis) and L is the maximum likelihood of the more complex model that we want to test (test hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the null hypothesis TS is dis- tributed as a χ2 n with a number of degrees of freedom n equal to the difference of degrees of freedom between the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is only valid for nested models, that is if the model in the null hypothesis can be obtained from the model in the test hy- pothesis by fixing some of its parameters to values in the interior of the allowed range (for instance Protassov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2002) For non-nested models, we use the Akaike Information Cri- terion (AIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The AIC of a model is defined as: AIC = 2k − 2 ln L, (5) with k number of free parameters in the model and L the maxi- mum likelihood of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The model providing the smallest AIC is taken as the one best representing the data at the smaller cost in terms of free parameters according to information theory (for instance Burnham & Anderson 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Throughout the paper, we use the method described in Bruel (2021) to assess the goodness of fit of the different models con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Practically, we show the deviation of the data with re- spect to the model in units of significance based on the Poisson statistics using the so-called PS maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The analysis starts with a preliminary optimisation of the emission model via a procedure described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In subsequent steps, unless stated otherwise, we keep free the normalisations of the gas and IC components, as well as the normalisations and spectral parameters of the three pulsars PSR J2021+4026, PSR J2021+3651, and PSR J2032+4127, the two γ Cygni extended components, and the two Cygnus Loop extended components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The normalisation of the subdominant isotropic background is fixed after the preliminary iterative op- timisation of all the sources in the ROI due to the possible de- generacy with the IC component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The best-fit normalisation ob- tained is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02 (with variations ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 for the different spin temperature values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Morphological analysis In this section, we aim at characterising the morphology of the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' As a first step, we optimised the position and extension of the 2D Gaussian model used in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011) and the LAT catalogues to describe the extended cocoon emission, following the methodology described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The corresponding PS map is displayed in Figure 6 (panel a, top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' An excess appears in the central region of the co- coon, in part reminiscent of the two main peaks of ionised gas column density within the Cygnus X cavities (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There- fore, we tested the addition of a central component in the cocoon Article number, page 6 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations region using two alternative models: either the ionised gas tem- plate clipped at the boundaries of the cavities (defined as con- tours above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85 10−6 W m−2 sr−1 emission at 8 µm), or two Gaussians with free extensions and positions, initialised at the main peaks in the ionised gas map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All newly added sources on top of the Gaussian model for the extended cocoon component here and elsewhere in this section are modelled using a power- law spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For the models described above, extended deviations are still apparent (see Figure 6 top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The largest excess appears in the western part of the cocoon at l ∼ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ and b ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' It does not overlap with any known sources or structures in the gas maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A second extended region of positive deviations appears at the edge of our ROI, at l ∼ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ and l ∼ −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ towards the southern arc of the Cygnus SB as imaged in soft X-rays (Cash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We added two additional Gaussian components to model those excesses, hereafter referred to as western and off- field excesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We initialise the Gaussian centres on the excess peaks, and fit their positions and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This results in a significant likelihood improvement (∆ ln L ∼ 200) for all models of the central cocoon component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The different models for the cocoon central component are compared in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The addition of the central cocoon com- ponent on top of the 2D Gaussian for the extended one provides a significant improvement in likelihood (∆ ln L > 240).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Con- versely, a model including the ionised gas map without the ex- tended cocoon Gaussian component resulted in a marked degra- dation of the likelihood (∆ ln L = −600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The model including the ionised gas template for the cen- tral cocoon component provides the largest likelihood and the smallest AIC, and therefore it is the one favoured by our analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' It is strongly preferred over two additional Gaussian compo- nents at the peaks in the ionised gas distribution (∆AIC < −124), which strengthens the evidence for a correlation between part of the gamma-ray signal and the ionised matter distribution3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This conclusion is supported by visual inspection of the deviations in Figure 6 (bottom row, panels b and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We also tested the full ionised gas map, that is not clipped at the boundaries of the cavities, but this yielded a smaller like- lihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The interpretation of this result is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The reason may be physical, for example related to confinement of the particles in the cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' It could also be related to lim- itations in the analysis, such as systematic biases in the emis- sion measure map extracted from Planck data, approximations in the derivation of the ionised gas column density, or degenera- cies with other gas templates outside the two main peaks in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Spatial parameters for the multiple overlapping extended sources may be degenerate to some level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To robustly determine their values, we performed an iterative fit of the positions and ex- tensions of all 2D Gaussian sources discussed in this section for the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The iterations proceed until the log-likelihood improvement between two iterations is smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The iter- ative fit converged after 6 iterations, with a total improvement in ln L of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Table 3 provides the best-fit morphological parameters and TS of all the extended components discussed in this section for the case in which the cocoon region is modelled by a broad 2D Gaussian (extended component) plus the ionised gas map (cen- 3 The AIC criterion may not be fully appropriate in this case due to the information entropy encoded in the geometry of the template and not represented by any fit parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' but the large improvement of log- likelihood clearly favours the model with the ionised gas template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' tral component), and an additional smaller 2D Gaussian slightly off the emission peak (western component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The extended emis- sion components are named after their Galactic coordinates as FCES GLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='ll ± B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='bb (FCES stands for Fermi Cygnus Extended Source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To make the paper easier to read, the sources are given a nickname that we use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 is the name given to the Gaussian that describes the cocoon ex- tended emission, nicknamed CoExt, FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 to the component modelled by the ionised gas map in the cocoon cen- tral region, nicknamed CoCent, and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 to the component corresponding to the excess appearing in the west- ern part of the cocoon, nicknamed CoWest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Interestingly, the addition of a central component for the cocoon results in a larger r68 for the extended component with respect to previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We remark that the off-field excess, ultimately dubbed FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 and nicknamed OffExc, is best modelled by a Gaussian centred at the edge of the ROI, therefore its char- acterisation may be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A better characterisation of this component is left for further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The positions and extensions of sources in the cocoon area are shown overlaid to the excess map in the right panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Spectral analysis In this section, we aim at characterising the spectral properties of the FCES sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Based on the best morphological model de- rived in the previous section, and for each emission component listed in Table 3, we tested three spectral models: a simple power law (PL), a log-parabola (LP), and a smooth broken power law (SBPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The expressions for these models are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The models are compared in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For the components CoCent and CoWest the best-fit model is the simple PL, with the LP providing a negligible improvement in log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The fit of the SBPL for these two components did not converge, pre- sumably due to lack of curvature in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Conversely, for CoExt and OffExc the models with curvature (LP or SBPL) provide a large improvement in ln L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' From the Akaike criterion, we can conclude that the SBPL is favoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Spectral parameters for the best-fit models are presented in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The spectral indices of the components CoCent and CoWest are compatible with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' If we fix the spectral index of CoWest to the best-fit value for CoCent, we obtain a decrease in log-likelihood of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3] σ, where here and in the following variations or uncertainties refer to the dif- ferent spin temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This demonstrates that the two com- ponents have compatible spectral shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, if we model CoWest or CoCent with the same spectral shape as CoExt and a free normalisation, we observe a decrease in log-likelihood of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9 [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8] (∆AIC = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 [19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6]) and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 [66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6] (∆AIC = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 [137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2, 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' So both CoCent and CoWest have spectra incompatible with the spectrum of Co- Ext, this time suggesting a different origin of the gamma-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We then computed the spectral energy distribution (SED) of the four sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To this end we performed independent analyses over 4 energy bins per decade between 500 MeV and 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For this part of the analysis all spectral-shape parameters are fixed and only normalisations are allowed to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, the nor- malisations of the gas maps are fixed to the best-fit values ob- tained for the entire energy range to preserve the local emissiv- ity shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The IC normalisation is also fixed to the best-fit value obtained for the entire energy range due to the large degeneracy with the extended components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Finally, the normalisations of all Article number, page 7 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon Table 2: Comparison of different spatial models Model ∆ ln L ∆AIC Extended Gaussian 0 0 +Western + Off-field Extended Gaussian + 2 Gaussians (IG peaks) 316 [246, 316] −622 [−622, −492] + Western + Off-field Extended Gaussian + IG template 435 [379, 435] −868 [−868, −746] + Western + Off-field Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ln L and AIC values are provided as differences with respect to the simplest model with only one 2D Gaussian for the extended emission of the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The intervals correspond to the minimum and maximum spin temperatures considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The Western and Off-field components are named later FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 and FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Table 3: Best-fit morphological parameters and TS for the extended emission components considered in the morphological analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Name l(◦) b(◦) r68(◦) TS FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 (CoExt) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 2751 [2436, 2824] FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 (CoCent) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1267 [1267, 1301] FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (CoWest) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 93 [93, 106] FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 (OffExc) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 684 [680, 723] Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The first uncertainties are statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The second uncertainties and TS variations are systematic from varying the spin temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' pulsars are fixed above 5 GeV because their emission fades off rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The results are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For flux densities (and all derived quantities later) we include in the systematic uncer- tainties those from the effective area of the LAT, combined in quadrature with those from the spin temperature choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The source with the highest flux is CoExt, followed by Co- Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The spectrum of CoExt extends to higher energies and con- nects to the cocoon spectrum measured by HAWC (Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021), confirming earlier indications of a spectral break between the GeV and the TeV energy ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Our spectrum for CoExt is similar at energies > 1 GeV to the cocoon SED in cat- alogue 4FGL-DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' On the contrary, our SED lies above the one presented in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011), which is closer to our SED for CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Presumably, the SED determination in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011) was biased towards the central component, while the extended component captured by CoExt in our analysis was difficult to detect at that time due to a reduced amount of data (2 years versus more than 13 years here) and a less advanced event reconstruction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Final global fit After the selection of the best spectral models we performed a final optimisation of the ROI, including free normalisation and spectral-shape parameters for the FCES sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The final spec- tral parameters of the FCES sources are displayed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 8 illustrates the quality of the ROI model after all the optimisation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The PS map show that we obtained a model of the ROI with no deviation above 3σ and the fractional devia- tion in the bottom left panel show no deviation above 10% in the central part of the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This model serves as a reference for the following spectro-morphological analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The largest deviation with a PS value close to 3σ lies at l ∼ 84◦, b ∼ 12◦ (at 1◦ from the Geminga-like pulsar PSR J1957+5033;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' see Saz Parkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The study of this excess is left for another work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 9 shows the final likelihood values for the different spin temperature values considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The largest likelihood is ob- tained for a spin temperature of 100 K, but with a difference in log-likelihood < 1 with respect to the reference value of 250 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The largest difference ∆ ln L ∼ 6 is found for the optically thin case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' After the final global fit the normalisation of the H I emis- sivity in the local arm is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The normalisation is in reasonable agreement with the average value for the lo- cal neighbourhood from Casandjian (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Under the hypothe- sis that the same CR population interacts with atomic, molec- ular and dark gas in the local arm, we can use the normali- sations of the gas maps to infer the XCO factor, which yields (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='06 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02) × 1020 H cm−2(K km s−1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is a factor of ∼2 lower than results from the earlier analysis of Cygnus in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2012a), but close to gamma-ray estimates from nearby CO clouds (for instance Remy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017), which strengthens the hypothesis that XCO variations found between lo- cal high-latitude clouds and the local arm may be highly sensi- tive to the separation of DNM and CO bright molecular cloud in the construction of gas maps and gamma-ray analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Other ef- fects related to the increasing difficulty to separate the gas phases at larger distances may also be at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Our analysis exploiting the PSF event types provides an improvement in this respect over the work in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The determination of a conversion factor for the DNM tracer is less obvious due to the lack of knowledge on the distribu- tion along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, the morphology of the DNM in Figure 3 closely resembles the structures in the lo- cal arm and Cygnus complex in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Therefore, for sim- plicity we assume that all the DNM is in the closest region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Based on this assumption, we can follow the same procedure used for XCO and infer a DNM dust specific opacity σ353 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='370 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='059 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='170) × 10−26 cm2 H−1, also close to gamma-ray results for nearby clouds (Remy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We use these coefficients to build a total column density map of neutral gas in the local arm and the Cygnus complex, which is shown in the left panel of Figure 10 and is used for the in- terpretation of the results in the Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' With respect to the reference spin temperature of 250 K, the total column density of neutral gas increases by ∼ 16% for a spin temperature of 100 K and decreases by ∼ 5% for the optically thin case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 8 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 10◦ 0◦ −10◦ Galactic Latitude 85◦ 80◦ 75◦ 10◦ 0◦ −10◦ Galactic Latitude 85◦ 80◦ 75◦ Galactic Longitude 85◦ 80◦ 75◦ −3 −2 −1 0 1 2 3 Sigma (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 6: PS maps for different models considered in the morphological analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' On the top panels we can see the PS maps for different morphological models: a) one extended Gaussian, b) one extended Gaussian plus two smaller Gaussians at the peaks of the ionised gas column density distribution, c) one extended Gaussian plus the ionised gas template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' On the bottom panels, we can see the PS maps for the same models with the addition of two Gaussians for the western and off-field excesses in the model (ultimately labelled FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 and FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The bin size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ and the maps were smoothed for display with a kernel of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='13◦ Table 4: Statistical comparison of different spectral models for the extended sources in Cygnus X Component ∆ ln LLP−PL ∆ ln LSBPL−PL ∆AICSBPL−LP FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 (CoExt) 33 [27, 35] 42 [36, 43] −14 [−14, −14] FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 (CoCent) 1 [1, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (CoWest) 1 [1, 2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 (OffExc) 26 [24, 27] 38 [35, 38] −22 [−22, −20] Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The columns show the log-likelihood differences between a log-parabola and a power-law model, ∆ ln LLP−PL, or between a smooth broken-power-law and a power-law model, ∆ ln LSBPL−PL, and the Akaike information criterion difference between a smooth broken-power-law and a log-parabola model, ∆AICSBPL−LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The intervals correspond to the minimum and maximum values obtained from variation of the spin temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 9 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 10−6 10−5 10−4 E2dN dE (MeV2 cm−2 s−1 MeV−1) FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 Broadband fit Bin per bin LAT Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', 2011 LAT 4FGL-DR3 HAWC Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', 2021 FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 Broadband fit Bin per bin LAT Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', 2011 LAT 4FGL-DR3 HAWC Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', 2021 102 103 104 105 106 107 108 E (MeV) 10−6 10−5 10−4 E2dN dE (MeV2 cm−2 s−1 MeV−1) FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 Broadband fit Bin per bin 102 103 104 105 106 107 108 E (MeV) FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 Broadband fit Bin per bin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 7: Spectral energy distribution of the four extended components studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the top panel, we show for reference earlier de- terminations of the cocoon spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The statistical uncertainties are displayed within the error caps and the full error bar is the quadratic sum of the statistical uncertainties, the uncertainties related to the different spin temperatures, and the uncertainties related to the effective area of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Table 5: Best-fit values after the final optimisation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Component Model Spectral parameters N0 (cm−2 s−1 MeV−1) γ or γ1 γ2 Eb (GeV) FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 (CoExt) SBPL 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9 × 10−11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 (CoCent) PL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 × 10−11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (CoWest) PL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 × 10−12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 (OffExc) SBPL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 × 10−11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='04+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The first uncertainties are statistical and the second uncertainties are systematic and result from varying the spin temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' N0 is given at the reference energy of 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Spectro-morphological analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Extension and position versus energy We first tested if the best-fit spatial model of the CoExt and CoWest components changes as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The com- ponent OffExc is left aside in the spectro-morphological analysis because it is displaced regarding the Cygnus X region which we aim to study in this paper, and it lies on the border of the ROI, and therefore its characterisation may not be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We fitted the extension and position of CoExt and CoWest in five energy bands: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 GeV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 to 5 GeV, 5 to 16 GeV, 16 to 50 GeV, and 50 to 1000 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' As shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5, CoWest has a softer spectrum: above ∼ 5 GeV the source flux becomes very low and its TS is below 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Therefore, fitting the position and extension of the source above ∼ 5 GeV is not possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In this section, the diffuse components, that is the gas maps and the IC component, are fixed, while the two components of Cygnus Loop are fixed above 5 GeV because of their very steep spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' OffExc is fixed due to its off-centre position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The results are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The top panel shows the extension as a function of energy for both sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There is no in- dication of an evolution of the extension as a function of energy, and the values in different energy bands are compatible with that obtained in the broadband analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The lower panels show the best-fit centroid positions for the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For CoExt, all positions are compatible with each other and the broadband Article number, page 10 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 85◦ 80◦ 75◦ 10◦ 0◦ −10◦ Galactic Latitude 85◦ 80◦ 75◦ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25 Fractional deviation −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 Sigma Galactic Longitude Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 8: Final deviation maps for the best-fit model over the en- tire energy range, on the left as fractional deviation, and on the right using the PS map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The bin size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ and the size of the smoothing kernel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='13◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 100 250 400 ∞ Spin temperature (K) −6 −4 −2 0 ∆ ln L Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 9: Difference in log-likelihood in fits of the final model for different spin temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' fit within 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For CoWest, we can see a hint of evolution of the position in the first two bins, but the two values are compatible within 2σ with the value from the broadband fit over the full energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Spectral variations across the extended sources In this section, we search for spectral variations across the emit- ting regions for CoExt and CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We started by examining CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To this end, we replaced the Gaussian model with a com- bination of rings and segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We tested several combinations but here we describe the profile obtained with a combination of: a central disk of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' five rings of external radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦, 85◦ 80◦ 75◦ 70◦ 10◦ 5◦ 0◦ −5◦ Galactic Longitude Galactic Latitude 1022 1023 Column density (H cm−2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 10: Neutral gas column density in the local arm and Cygnus region, obtained by summing N(H I), N(H2) and an estimate of the column density for the DNM, with conversion factors cali- brated on the gamma-ray analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦ and 6◦, that can be decomposed into four seg- ments spanning 90◦ in azimuth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' and two large rings of external radius 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This somewhat arbitrary setup was chosen to ensure a minimal TS (at least 25) in every segment and ring (see Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Eventually, however, the wider ring has a low TS (∼10) therefore its parameters have to be in- terpreted with some caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All the components are modelled using a LP spectrum with parameters initiated at the best-fit values found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The LP model was chosen instead of the SBPL model for this part of the analysis because it yields more stable results when fitting several free components at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For this section the two components of Cygnus Loop are fixed due to their off-centre position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' OffExc is also fixed due to its off-centre position and proximity with the border of the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We also tested a combined description of CoExt and CoCent via rings and segments by removing the ionised gas template from the emission model, but the highly structured central part was poorly described by the latter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Thus, we decided to proceed with the ionised gas map in the model, and we used the combination of segments and rings to only represent the source CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The spectral shape and normalisation is left free for Co- Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The decomposition of CoExt proceeded through a few sub- sequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A We replaced the Gaussian by the aforementioned combina- tion of seven rings and a central disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Only the normalisation of each template was free, while the spectral parameters (α and β, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 for a definition) were fixed to the initial values from the analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' B The parameters α and β for the disk and rings were free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' C The innermost five rings (beyond the central disk) were de- composed into four azimuthal segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Only the normali- sations were free and the spectral parameters were fixed to the values obtained in the step B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D All spectral parameters were free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For step C, a few different orientations for the segments were tested, and we present the results for the one yielding the best likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The values of ∆ ln L and ∆AIC for the four steps are provided in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 11 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 103 104 105 E (MeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ Extension r68 FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 103 104 105 E (MeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦ Extension r68 FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ Galactic Longitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ Galactic Latitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 GeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 - 5 GeV 5 - 16 GeV 16 - 50 GeV 50 - 1000 GeV Global fit 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25◦ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='75◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25◦ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ Galactic Longitude 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6◦ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0◦ Galactic Latitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 GeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 - 5 GeV Global fit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 11: Extension (top) and position (bottom) for FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 (left) and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (right) as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the top panels the bands show the values in the global fit over the entire energy range .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the bottom panels the grey areas show the results in the entire energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All uncertainties are provided at 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Table 6: Variations of ln L and AIC for the decomposition of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 in rings and segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Step ∆ ln L ∆AIC A −9 [−13, −9] 32 [32, 40] B 7 [7, 14] 18 [4, 18] C 58 [47, 58] −86 [−86, −64] D 35 [35, 42] −10 [−24, −10] Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Values are provided as difference with respect to the previous step, and, for step A, with respect to the global analysis presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' See the text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The degradation in step A is due to the approximation of representing a 2D Gaussian with concentric rings and a central disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, this is not a cause for concern as the decrease in log-likelihood is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Step B does not provide an improve- ment in the description of the emitting region, that is there are no significant variations of the spectrum as a function of distance from the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Conversely, we find a model improvement in step C, which demonstrates that the emission is not azimuthally symmetric in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The likelihood improvement is equally shared by all the rings concerned, and it is not surprising given the diversity of regions inside and outside the plane spanned by each broad ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A further improvement in the likelihood is ob- tained in step D, showing also the presence of azimuthal spec- tral variations, mostly driven by the two innermost rings and the central disk with an improvement in log-likelihood of 24 (∆AIC = −15) when only those components have the spectral shape free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 12 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations However, the azimuthal variations in the first two rings could be explained by spectral variations across CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To check this hypothesis, we sliced the ionised gas template vertically at l = 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ to separate the two main lobes of ionised gas and repeated the step D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This results in an improvement of the log-likelihood smaller than one, meaning that no spectral variation is detectable between the two sides of the source CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We conclude that the best model is the one combining the ionised gas template to describe CoCent and with CoExt decom- posed and fitted as in step D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is used as a basis for the in- terpretation of the results in the next section, where the emission profiles extracted from the data is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Some additional plots illustrating the results are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The cocoon and its landscape Our analysis shows that the Cygnus cocoon in the LAT energy band is best described by at least two spatial components with different spectra: a central component, CoCent, with a power law spectrum of index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01, and an extended component, CoExt, with a smooth broken power law spectrum with indices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 GeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A third newly discovered extended emission component, CoWest, overlaps in projection with the cocoon and has a spec- trum compatible to the one of the central component, a power law with index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 12 shows the excess counts corresponding to the three gamma-ray sources associated or potentially related to the co- coon, that is total counts minus the best-fit model for all compo- nents except CoExt, CoCent, and CoWest (zoomed in from Fig- ure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The brightest emission in the central region of the cocoon lies in the cavities bounded by the photo-dissociation regions traced by 8 µm emission (right panel), as found by Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011), and the majority of it is traced by our ionised gas template and associated to source CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The extended cocoon component, CoExt, overlaps with the northern rim of the X-ray structure known as Cygnus SB (Cash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1980), which may be associated with star-forming regions in Cygnus X (Uyanıker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2001), although recent data may suggest that the entire X-ray structure is rather a hypernova remnant at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 kpc (Bluem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Last, source CoWest is situated along a bright arc of 8 µm emission, but does not coincide with any over-densities in neutral or ionised gas densities (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2, 3, and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Its centroid lies at approximately 1◦ from the γ Cygni SNR, that, if we assume a distance to the Earth of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 kpc, cor- responds to a physical distance of ∼30 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Under the hypothesis that the observed gamma-ray emission is of hadronic origin we can convert the excess map into an emis- sivity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To this aim we divided the excess cube in the analy- sis energy bins by the exposure cube and the total, neutral plus ionised, gas column density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The latter quantity is an upper limit to the relevant gas column densities because gas could be distributed over a larger distance along the line of sight compared to the volume probed by the particles in the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, we expect most of the gas in this region to be concentrated around the star-forming complex in Cygnus X, and we do not have an alternative simple prescription to estimate the foreground and background column densities to be subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The results are displayed in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' On one hand, we can see an emissivity peak in the cocoon central area coincident with the peaks in the ionised gas distribu- tion (modelled by CoCent in our analysis) with broad wings ex- tending to several degrees from the centre (CoExt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' On the other hand, we see a marked peak at the position of CoWest and around the γ Cygni SNR and NGC 6910 stellar cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Although posi- tion and spectral similarity to CoCent suggest that this source is related to the cocoon, the interpretation is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' CoWest may be related to gas missing in our model, or else to a nearby source or some peculiar transport configuration that results in an accumulation of particles in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the following for sim- plicity we concentrate on the interpretation of the two brightest sources in the cocoon area, namely CoCent and CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The striking spatial coincidence of the brightest part of the gamma-ray signal and the contours of the cavity, and to a lesser extent the resemblance with the extended X-ray emission struc- ture, have suggested that both phenomena may have a common origin: the abundant massive-star population of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The most prominent stellar clusters in the regions, the Cyg OB2 asso- ciation and NGC 6910 cluster, are natural candidates, powerful enough to accelerate particles able to produce non-thermal emis- sion at the observed level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We evaluated the properties of these two objects, following what was done in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For Cygnus OB2, we considered 78 O stars (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020) and a power law mass function of index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='09 (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For NGC 6910 we assumed a power law mass function of index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74 (Kaur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020) normalised according to Figure 9 of their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We eval- uated mass loss rates, cluster wind terminal velocities, and me- chanical power of the winds by separating stars in four groups, namely O5 to O3, O9 to O5, B5 to B0, and B8 to B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The sample is limited to stars heavier than B8 due to the validity range for the reference mass-loss rate model adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We assumed standard properties of O stars from Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2005) and for B stars from Cox (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We used the parametric wind model by Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This yields a mass loss rate of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 × 10−4 M⊙ yr−1 for Cyg OB2 and of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 × 10−4 M⊙ yr−1 for NGC 6910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The mechanical power of the winds is evaluated to 8 × 1038 erg s−1 for Cyg OB2 and 4 × 1038 erg s−1 for NGC 6910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The collective wind terminal velocity therefore is ∼2200 km s−1 for both clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We show in the next section that such powers are sufficient to account for the observed signal in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We can estimate the physical and angular sizes of the clus- ter wind termination shock and shocked wind bubble using the formulae in Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2021), which follow the simple mod- els in Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (1977);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' If we assume ages of 5 Myr (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Kaur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020) for both clusters and interstellar gas densities of 5 H cm−3 we obtain a size of the wind termination shock of 40 pc for Cyg OB2, and of 33 pc for NGC 6910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The total size of the wind bubble is 200 pc for Cyg OB2, and 180 pc for NGC 6910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 12 shows that the sizes of the termination shocks are comparable to that of the central emission component, with CoWest being located at the edge of the termination shock from NGC 6910, while the sizes of the wind bubbles compare well to that of the cocoon extended emission component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' While these results tend to lend support to the idea that Cyg OB2 and NGC 6910 may be the sources ultimately respon- sible for the observed gamma-ray emission, we emphasise that the modelling of the winds and bubbles is without any doubts oversimplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Several effects can be expected to affect the re- sults and weaken the similarity of the gamma-ray emission and expected SB signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Generally, the classical theory from Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (1977) is known to underestimate the radiative losses, hence overestimate the size of the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Hydrodynami- cal simulations reveal enhanced radiative losses due to instabili- ties at the interfaces result in a bubble being ∼ 40% smaller than Article number, page 13 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 82◦ 80◦ 78◦ 76◦ 4◦ 2◦ 0◦ −2◦ Galactic Longitude 85◦ 80◦ 75◦ 10◦ 5◦ 0◦ −5◦ −10◦ Galactic Longitude Galactic Latitude 0 200 400 600 800 1000 1200 1400 Excess counts Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 12: Excess counts corresponding to the three gamma-ray sources associated or potentially related to the cocoon, namely FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56, FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (zoomed in from the left panel in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Left: extended region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Green contours correspond to the X-ray emission from the ROSAT all-sky survey in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 keV to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The orange circle shows the outer radius of the third to last annulus included in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The dashed circles show the estimated sizes of the wind bubbles for Cyg OB2 (lower left) and NGC 6910 (upper right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Right: zoom in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Black contours correspond to the 8 µm emission from MSX data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85 × 10−6 W m−2 sr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The stars show the positions of PSR J2032+4127 (lower left) and PSR J2021+4026 (upper right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The green circles show the radius/68% containment radius of the two emission components associated with the γ Cygni SNR (subtracted from the map, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The blue circle shows the 68% containment radius of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The continuous circles show the 50% containment radius for members of the Cyg OB2 association (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019) and of the NGC 6910 cluster (Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020, NGC 6910 has a 50% containment radius of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='9′′which appears as a dot on this image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The dashed circles show the estimated sizes of the cluster wind termination shock for Cyg OB2 (lower-left) and NGC 6910 (upper right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In both panels the orange diamond shows the centre of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' predicted by the classical analytical solution for well-developed bubbles, in agreement with observations (Krause & Diehl 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' At earlier stages, before SB breakout from the parent molecu- lar cloud, the dense and fractal medium surrounding the clus- ter drives turbulent mixing and efficient cooling, resulting in a reduction of ∼30% in bubble size for parameters relevant to Cyg OB2 and NGC 6910 (Lancaster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' More funda- mentally, the simple bubble model from Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (1977);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2021) may not be straightfor- wardly applied to Cyg OB2, which is not a compact cluster but instead presents multiple substructures with a 50% containment radius of stellar members spanning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦, that is 5 pc at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 kpc (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Furthermore, as illustrated in Figure 12, the bubbles from the two stellar clusters may have interacted and it is not clear how this would have impacted the development of the whole region and whether this should have left specific signs that we should now see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Therefore, the connection of the observed gamma-ray sig- nal with gas structures imprinted by the development of a SB is far from obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Actually, the separation of the cocoon emis- sion into CoExt and CoCent, as well as the correlation of the innermost bright signal with ionised gas, can be interpreted in a way that weakens the link between the gamma-ray emission and the cavity delineated by photo-dissociation regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Indeed, the emission could arise from the interaction of freshly acceler- ated CRs with the ionised gas inside the cavity, the latter playing no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' These CRs extend much beyond the limits of the cav- ity, as was already clearly observed in Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011), and their interactions with the ambient medium give rise to the source CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The origin of the CRs powering the cocoon emission can therefore be unrelated to the Cyg OB2 association and NGC 6910 cluster (in the sense that they play no role as a whole, but they can harbour or have harboured the actual source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The central region actually contains a handful of extremely energetic objects and potential particle sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We can list: the γ Cygni SNR (G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1) with a proba- ble distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 kpc from association with the γ Cygni nebula (Leahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2013) and dynamic properties estimated in Leahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2020) as ejecta mass of 5 M⊙, age of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 kyr, and supernova (SN) energy of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 × 1050 erg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' the γ Cygni pulsar, PSR J2021+4026, associated with the γ Cygni SNR and with a spin-down power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 × 1035 erg s−1 (Ray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' PSR J2032+4127, a pulsar in a highly eccentric binary sys- tem with a Be-type star (Lyne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2015), probably part of Cyg OB2, with an orbital period of 45-50 yr, a spin-down power Article number, page 14 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 85◦ 80◦ 75◦ 5◦ 0◦ −5◦ Galactic Longitude Galactic Latitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 Emissivity (ph s−1 sr−1 H−1) ×10−27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 13: Emissivity map calculated from the excess counts as- sociated to the cocoon in Figure 5 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The dashed contours correspond to the 8 µm emission from MSX data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85 × 10−6 W m−2 sr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The contours correspond to the peak column density of the ionised gas template at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5×1021 H cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The circle and the dashed circle correspond to the position and r68 of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 and FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 × 1035 erg s−1, and a characteristic age of ∼200 kyr (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Furthermore, the centroid of the emission from CoExt and the peak in the emissivity map do not coincide with any of the potential particle accelerators, stellar clusters or others (Fig- ure 12 and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Therefore, we tried to account for our obser- vations in a generic way, with a simple diffusion model based on an unspecified source and not exclusively relevant to mas- sive star clusters and their associated SBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We introduce in the following sections the model framework used and the parameter setups yielding satisfactory fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A simple diffusion-loss framework for the cocoon Given the layout of the emission exposed in the previous subsec- tion, with significantly extended radiation from a region reaching well beyond the vicinity of potential sources, it seems reasonable to consider that gamma rays are produced by particles that were released by one or several sources some time ago and were trans- ported in the surrounding medium since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In this section, we aim to provide a quantitative assessment of this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We interpret the observations in the framework of a one-zone diffusion-loss transport model where particles are continuously injected at a point in space for some duration and then experi- ence diffusive transport in a uniform and isotropic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is very likely an overly simplistic description of the processes at stake because there may be multiple sources, not all of them can be assumed to be of negligible size, the medium is prob- ably not uniform over the few hundreds of parsecs probed by the emission, and there may be other transport processes than diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Yet, our goal is to draw a few key inferences from the observables and we defer more advanced modelling efforts to subsequent publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Moreover, we show later that such a modelling with a very limited number of free parameters can yield a fairly good representation of the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The full formalism of the model framework is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Ultimately, the main parameters of the model are: injection luminosity Q0, power law injection spectrum slope α, characteristic injection duration tinj, diffusion duration tdiff, and diffusion coefficient normalisation D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We explored a large pa- rameter space for these four parameters and fitted the predictions to the results of the gamma-ray analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The diffuse emission from the Cygnus cocoon is very ex- tended, with an angular size of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ for CoExt that trans- lates into a ∼ 130 pc length at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A more com- pact and central emission component CoCent is correlated with the distribution of ionised gas within a radius of about 50 pc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' the spectrum of CoExt is flat and that of CoCent, although signifi- cantly softer, is also pretty hard compared to interstellar emission on larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Given these observables, we proceeded to educated guesses for the main parameters of the model, considering first the case of a hadronic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The typical extent of the emission pro- vides a constraint on the diffusion length, that is on the product of diffusion coefficient and diffusion time: rd = � 4Dtdiff ≳ 100 pc, (6) D = Dism(10 GeV) = 1029 cm2 s−1 ⇒ tdiff ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 kyr, (7) D = Dsupp(10 GeV) = 1027 cm2 s−1 ⇒ tdiff ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='75 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (8) If diffusion has the average properties inferred for transport over large scales in the Galaxy (Trotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011), defined by the coefficient Dism, particles need less than 10 kyr to fill a volume that would account for the extent of the observed emission at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Conversely, if diffusion is for some rea- son strongly suppressed by one to two orders of magnitudes as inferred for a variety of sources including star-forming regions (Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Abramowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2015), and is characterised by coefficient Dsupp, then about 1 Myr is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The emissivity enhancement inferred for the cocoon is com- parable to the local emissivity within a factor of two to three depending on the energy range, such that the CR energy density uCR in the region is similar to the one in the solar neighbour- hood, uCR,local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This makes it possible to constrain the properties of particle injection, namely its power Linj and typical duration tinj: 4π 3 r3 d × uCR ≃ 1 2Linjtinj, (9) uCR ≃ uCR,local ≃ 1 eV cm−3, (10) Linjtinj ≃ 4 × 1050 erg, (11) Linj = 1038 erg s−1 ⇒ tinj ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (12) As computed in the previous subsection, the mechanical lumi- nosity of the most prominent star clusters in Cygnus is in the range 4 − 8 × 1038 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Such a power source can deliver par- ticle injection at a level of 1038 erg s−1, pending efficient particle acceleration with a yield of ∼ 10 − 30% (by some unspecified mechanism at this stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In that case, the inferred CR density enhancement can be attained if injection lasts over about 100 kyr (and particles accumulate in the volume, see the discussion in the next paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' If particle acceleration is less efficient or the source is less powerful, by about an order or magnitude, then injection has to proceed on Myr time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Alternatively, a su- pernova producing 1050 erg of accelerated particles and releasing the majority of them over ∼ 3 − 10 kyr would provide an injec- tion power of 3 − 10 × 1038 erg s−1 and thus allow short-lived injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 15 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon Scenarios with tinj much smaller than tdiff are not viable be- cause particles spread out and leave the volume too rapidly, which results in too flat intensity profiles and too steep spectra (because energy-dependent diffusion depletes the particle popu- lation at the high end of the spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' So tinj has to be compa- rable to or greater than tdiff, with the additional constraint that sufficient energy should be released within a time tdiff to match the observed level of emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In practice, this means that: for average interstellar diffusion, the region is filled over a ∼ 10 kyr timescale, thus requiring a strong enough source with injection power ∼ 1039 erg s−1 typical of a SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' alternatively, weaker sources such as the star clusters in Cygnus with injection power ∼ 1037−38 erg s−1 require moderate to strong diffusion suppres- sion and transport occurring over hundreds to thousands of kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' These considerations remain mostly valid in the case of a lep- tonic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The main difference with a hadronic scenario is the importance of energy losses, mostly from synchrotron radia- tion and inverse-Compton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Yet, for an interstellar mag- netic field strength B = 3 µG and optical and infrared interstel- lar radiation fields with total energy density of about 1 eV cm−3, such as those predicted in the large-scale model of Popescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2017) at the Galactic position of the cocoon, particles with en- ergy below 300−400 GeV have a cooling time of 1 Myr or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The transport of electrons over the distances and time scales con- sidered above may therefore be little affected by energy losses in many scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The actual situation is however far more com- plex because radiation densities in the innermost region of the cocoon are much stronger than the large-scale interstellar aver- age, by an order of magnitude, which could significantly affect the spectral and morphological properties of the emission from the population of propagated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Unfortunately, the model framework that we used for this work cannot handle inhomoge- neous energy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Possible diffusion scenarios for the cocoon We tested the hypothesis that components CoExt and CoCent are produced by a single population of non-thermal particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In that context, CoCent is gas-related emission (pion decay in hadronic scenarios, and Bremsstrahlung in leptonic scenarios) from the innermost ∼ 50 pc region of the cocoon, where a signif- icant amount of ionised gas is present as evidenced by free-free emission, while CoExt is additional emission on top and beyond CoCent that is not necessarily gas-related (it can be a mix of inverse-Compton and Bremsstrahlung in leptonic scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the following, we relate CoCent and CoExt to so-called central and extended regions in our model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For computational reasons, we did not perform an overall optimisation of all model parameters and instead investigated a limited number of scenarios selected from the above guess for viable parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For each parameter setup, the compar- ison of model predictions and gamma-ray analysis results goes through the following steps that are expected to guarantee a max- imum consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Central and extended component separation: for a given run of the model, gas-related emission from the innermost region within 50 pc in radius of the injection point is handled sep- arately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All related quantities (particle density, gas column density, emission intensity,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=') are not included in the prop- erties of the complementary extended region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For instance, gas-related emission intensity along a line of sight that passes through the central region seen in projection is split into a central contribution and the remaining extended contribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Although the separation is clear-cut in the model, we cannot exclude that there is some cross-talk between over- lapping components in the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Gas column density correction: : the model-predicted emission for the extended component was divided into rings, with the angular binning used in the spectro-morphological analysis, and gas-related emission in each ring was rescaled by the ra- tio of the actual average gas column density in the ring to that corresponding to the default uniform density assumption of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The same rescaling is also applied to the central component, treated as a single region with average proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Fitting of total emission spectra: the total emission spectrum of the extended component is fitted to the observed spectrum for CoExt, which yields the injection luminosity for the whole particle population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Then, the total emission spectrum of the central component, re-scaled by the fitted injection lu- minosity, is further fitted to the observed spectrum for Co- Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This second fit is meant to correct for the uncertain av- erage column density for the ionised gas in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Both fits are performed via χ2 minimisation, from significant spectral points and using statistical uncertainties only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There is, however, a subtlety regarding how emission from the ionised gas should be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Atomic and molecular gas in the cocoon region enter twice in the analysis: in the fit to the Fermi-LAT data over a large ROI, where they trace emission from the background population of CRs, and in the interpreta- tion of the extended emission from CoExt and CoCent, where they are associated to an additional population of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Con- versely, the ionised gas template enters just once, and the associ- ated emission may therefore comprise contributions from back- ground CRs and from an additional population of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In practice, what is fitted to the spectrum of CoCent is the spec- trum of the central component of the model, possibly augmented by the spectrum of the emission from the ionised gas for a local emissivity (because the background CR population in the co- coon region can be considered close to the local one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We tested both options and, as illustrated below, it turns out that not including a contribution from background CRs provides much better fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Hadronic scenarios To begin with, we present the result of complete calculations for hadronic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Following the above discussion of the most likely diffusion-loss model setups given the observables at hand, we present the results of four scenarios dubbed H1, H2, H3 and H4, the parameter sets of which are presented in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The H1 and H2 scenarios feature constant injection of a hard spectrum of protons and mainly differ by the diffusion time (300 kyr or 3 Myr) and level of diffusion suppression (by a factor 10 or 100 with respect to the interstellar average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The H3 and H4 scenar- ios corresponds to shorter-lived injection over 3 or 30 kyr and transport over 10 or 100 kyr in a medium with no or moder- ate diffusion suppression (by a factor 10 at most with respect to the interstellar average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Scenario H4 actually corresponds to a scaled version of scenario H3 (multiplying injection and dif- fusion times and dividing diffusion normalisation and injection power by the same amount, that is ten), such that both setups are completely identical in terms of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure 14 shows the total fitted spectra for CoExt and CoCent for model setups H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The predicted shape for the CoExt spectrum is in good agreement with the data, while that for the Article number, page 16 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations Table 7: Summary of the different diffusion-loss model setups H1 H2 H3 H4 L1 L2 hadronic hadronic hadronic hadronic leptonic leptonic tinj (yr) 108 108 3 × 103 3 × 104 108 108 tdiff (yr) 3 × 106 3 × 105 104 105 3 × 106 106 D0 (cm2 s−1) 1027 1028 1029 1028 1027 1028 Linj (erg/s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 × 1036 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 × 1037 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 × 1039 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 × 1038 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 × 1035 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 × 1036 α 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 CoCent spectrum is too steep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A steeper predicted spectrum for CoCent is obtained because higher-energy particles leave the in- nermost regions more rapidly than lower-energy particles, and also because it contains a contribution from the background CR population that has a steeper spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='62e+36 erg/s NIG H = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='70e+21 H/cm2 2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='87 + 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='16 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25e+37 erg/s NIG H = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='73e+21 H/cm2 2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='23 + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 14: Fitted model spectra for the FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 emission components, for model setup H1 (top) and H2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A fit of the model to the spectrum of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 is done first, with the injection luminosity as fitting parameter, followed by a second fit to the spectrum of FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, with ionised gas column density as fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This last fit includes a contribution to the emission from a background population of CRs (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The full ex- tent of the error bars corresponds to the quadratic sum of the statistical and systematic uncertainties, while the caps mark the contribution of the statistical uncertainty only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The first χ2 cor- responds to the fit to FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and the second one to FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Interestingly, a much better fit to the spectrum of CoCent is obtained when not adding a local emissivity contribution to the model spectrum for the central region, at the expense of higher fitted column density for the ionised gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is illustrated in the top two panels of Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Emission from the ionised gas in the innermost regions of the cocoon would then arise only from CRs produced in Cygnus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Background pre-existing CRs may have been evacuated in the past during the SB growth, for instance by advection in the stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Alternatively, the spectral sig- nature of these pre-existing CRs may have been absorbed by an- other component in the fit to the LAT observations (for instance by the molecular gas or DNM templates that have a high de- gree of correlation with ionised gas in the central region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Since the spectral fit is so much better when not including the con- tribution from background CRs for CoCent (this is true also in leptonic scenarios), we present in the following only results pro- duced with this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We note, however, that this has almost no influence on the intensity and emissivity profiles presented thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For model setup H1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' H2), the fit implies a proton injec- tion luminosity of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6 × 1036 erg s−1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 × 1037 erg s−1), which would correspond to proton injection efficiencies < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 − 1% (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' < 5 − 10%) for the Cyg OB2 or NGC 6910 clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Such low efficiencies are consistent with the assumed flat injection spectra with α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0, at least in the framework of diffusive shock acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For model setup H3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' H4), the fits implies a much higher proton injection luminosities of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 × 1039 erg s−1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 × 1038 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This would corre- spond either to a high particle acceleration efficiency of ∼ 20% in an SN with a 1051 erg explosion kinetic energy, with subse- quent release of accelerated particles over a timescale of 3 kyr (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 30 kyr), or to a lower acceleration efficiency in an SN more energetic than in the canonical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The first option, with rel- atively high efficiency, may be conflicting with our assumption of a flat injection spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='e Figure 16 displays the predicted intensity and emissivity pro- files for both central and extended model components in sce- nario H1, compared to the values inferred from the spectro- morphological analysis in segments, in three different energy bands: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2 GeV, 2-10 GeV, and 10 GeV-1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The comparison for other scenarios is shown in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To compare the dif- ferent model setups, we provide in each panel the χ2, computed from all significant intensity data points using statistical uncer- tainties only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There is, however, no formal fit of the model to the measured intensity profiles, and this χ2 is just a figure of merit to characterise each model setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The agreement is overall quite good from the centre up to beyond 8◦, especially considering the simplicity of the model and the limited number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Most measurements are within a factor two of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' When subtracting the contribution from the central model com- ponent, relatively flat emissivity profiles are obtained for the ex- tended model component, in agreement with the trend inferred from the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The results lend support to the idea that Article number, page 17 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='62e+36 erg/s NIG H = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01e+21 H/cm2 2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='87 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='60 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25e+37 erg/s NIG H = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='16e+21 H/cm2 2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='23 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85e+39 erg/s NIG H = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='98e+21 H/cm2 2 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='51 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='96 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 15: The top two panels are the same as in Figure 14, without a contribution to the emission from a background population of CRs in the fit to the spectrum of FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The bottom panel is the corresponding figure for scenario H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The first χ2 corresponds to the fit to FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and the second one to FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' CoExt and CoCent are produced by the same population of par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All four model setups are overall equally good at account- ing for the data, despite widely different parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Leptonic scenarios We now present the result of complete calculations for lep- tonic scenarios, in which the emission can be produced by non- thermal Bremsstrahlung and inverse-Compton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We considered two scenarios dubbed L1 and L2, the parameter sets of which are presented in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Both scenarios feature con- stant injection of a hard spectrum of electrons and mainly differ by the diffusion time (1 or 3 Myr) and level of diffusion suppres- sion (by a factor 10 or 100 with respect to the interstellar aver- age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The magnetic field is assumed to have a strength B = 3 µG, and the interstellar radiation field model is taken from the large- scale model of Popescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2017) at the Galactic position of the cocoon (at a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7 kpc distance from us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We tested the ef- fect of a stronger interstellar radiation field model, such as the one used in the original Cygnus cocoon paper (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011), and obtained a much poorer fit to our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This stems mostly from the shorter propagation range and different relative contributions of Bremsstrahlung and inverse-Compton to the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This scenario is however extreme since it en- forces very strong inverse-Compton losses over an extended vol- ume, whereas in reality enhanced radiation fields are expected only in the innermost regions of the cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is a caveat of the model framework used in this work, which cannot handle in- homogeneous environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We also tested a leptonic version of scenario H3, but that yields a poor fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The fits of the model to the spectra of CoExt and CoCent are displayed in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' They are overall pretty satisfactory and yield χ2 similar to those obtained with the hadronic mod- els, although slightly higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The diffusion time range in leptonic scenarios seems rather constrained: small ages ≲ 1 Myr tend to produce too hard spectra for the extended component, while ages ≳ 3 Myr result in too steep spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The injection luminosities re- sulting from these fits are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8×1035 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4×1036 erg s−1 for the L1 and L2 scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This translates into electron injection efficiencies at the sub-percent level at most if the me- chanical luminosity from Cyg OB2 and NGC 6910 is the power source for particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We checked that the correspond- ing synchrotron emission does not exceed the radio constraints presented in Mizuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The corresponding predicted intensity profiles are compared to the measurements in Figure 18 for model L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' As for the spec- tra, the fit is slightly degraded compared to that obtained with hadronic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The best scenario is L1, with diffusion sup- pressed by two orders of magnitude with respect to the interstel- lar average and a diffusion time of 3 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A lower level of diffu- sion suppression results in too flat intensity profiles, undershoot- ing the data in the inner regions and exceeding them at large dis- tances from the injection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' As mentioned above, a smaller diffusion time does not help because it yields a too hard spectrum for CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We also tested the hypothesis that the observed emission ac- tually is a pulsar halo, following the discovery of very extended gamma-ray emission around some middle-aged pulsars (Abey- sekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In such a scenario, PSR J2032+4127 appears as an interesting candidate because of its location close to the peak of the emission and characteristic age of ∼200 kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We used the phenomenological two-zone diffusion-loss halo model implementation presented in Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2022), using as base- line key parameters: the spin-down power, estimated distance, and characteristic age of PSR J2032+4127 from the ATNF data base4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' a broken power law injection spectrum with indices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 below and above a break energy of 500 GeV respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' an injection starting time of 40 kyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' diffusion suppression by a factor of 50 within 50 pc of the pulsar, with a power law dependence in rigidity with index 1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' a surrounding magnetic field with strength B = 3 µG and the interstellar radiation field 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='atnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='au/research/pulsar/psrcat/ Article number, page 18 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 0 1 2 3 4 5 6 7 8 Angle (deg) 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='62e+36 erg/s 2 = 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='80 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 29 10 28 10 27 10 26 10 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV intensity (ph/cm2/s/sr) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='62e+36 erg/s 2 = 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 30 10 29 10 28 10 27 10 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 10GeV-1TeV intensity (ph/cm2/s/sr) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='62e+36 erg/s 2 = 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='64 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 31 10 30 10 29 10 28 10 27 10GeV-1TeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 16: Intensity and emissivity radial profiles in three different gamma-ray energy bands for the FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 emission components, compared to predictions for model setup H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the intensity plots, the intensity distri- bution corresponding to the best-fit two-dimensional Gaussian model is displayed for comparison as a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the emissivity plots, the local emissivity and its uncertainty in each energy range are displayed for comparison as a dotted line and a shaded band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The data points correspond to a decomposition of the emission into the ionised gas template, a central disk, two outer rings, and five intermediate rings split azimuthally into four segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For the latter, we displayed the corresponding angular range only for one segment in each ring and introduced a small horizontal shift of the others, for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The full extent of the error bars cor- responds to the quadratic sum of the statistical and systematic uncertainties, while the caps mark the contribution of the statistical uncertainty only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 19 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='84e+35 erg/s NIG H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='69e+21 H/cm2 2 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='40 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='64 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='41e+36 erg/s NIG H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='32e+22 H/cm2 2 = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='96 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='10 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 17: Fitted model spectra for the FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 emission components, for model setup L1 (top) and L2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A fit of the model to the spectrum of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 is done first, with the injection luminosity as fitting parameter, followed by a second fit of the Bremsstrahlung emission only to the spectrum of FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50, with ionised gas column density as fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This last fit does not include a contribution to the emission from a back- ground population of CRs (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The dashed blue line is the inverse-Compton contribution in the emission model for FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The first χ2 corresponds to the fit to FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and the second one to FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' model from the original Cygnus cocoon paper (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We neglected the effect of proper motion on the emission morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The fit of the predicted emission properties to the ob- served spectra and intensity profiles is relatively good (see Ap- pendix D), although not at the level of those obtained in sce- narios H1-H4 and L1-L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Yet, the implied present-day injection luminosity is of the order of 1036 erg s−1, an order of magnitude larger than the spin-down power of PSR J2032+4127, which dis- misses this pulsar as the possible source of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This result seems to be robust against variations of the main model param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' One cannot exclude, however, that another currently un- known pulsar with the right properties exists in this active star- forming region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The origin of the cocoon To summarise, our extended set of observables for CoCent and CoExt, including a radial profile for the extended component over nearly 10◦, together with intensity measurements and emis- sivity estimates in three energy bands from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 GeV to 1 TeV, can be accounted for reasonably well from a simple diffusion-loss model with a small number of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Several pretty different model setups seem to provide viable explanations of the observations, which suggests that more developed modelling frameworks and, more likely, additional observational data need to be considered in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' An important result is that the data can be explained from one single population of injected particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This population spans the full region of extended component CoExt, and gives rise to the central component CoCent by interacting with ionised gas in the innermost regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Both hadronic and leptonic scenarios are viable, although it should be confirmed that leptonic scenarios are still valid in a more realistic modelling framework includ- ing non-uniform inverse-Compton losses in the strongly varying radiation fields of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All solutions have in common to require a flat particle spectrum at the source, with a power law index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0, which points to very recent acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The solutions are, however, very different in terms of ener- getics and time scales involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Setups H1 and L1 feature con- tinuous injection, strong diffusion suppression in the region (by a factor 100 with respect to the large-scale interstellar average, over a spatial extent of more than 200 pc), transport proceeding over several Myr (in agreement with age estimates for Cyg OB2 and NGC 6910), and low acceleration efficiencies (at the sub- percent level in the hadronic scenario if Cyg OB2 and NGC 6910 are the mechanical power source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Setups H2 and L2 are very similar with continuous injection, moderate diffusion suppres- sion (by a factor 10 with respect to the large-scale interstellar average), a more recent injection and transport process over the last 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 − 1 Myr, and five-to-ten times higher acceleration effi- ciencies with respect to H1 and L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Setups H3 and H4 describe an even more recent event, with injection lasting 3 or 30 kyr, transport proceeding over 10 or 100 kyr in a medium where diffusion is not or only moderately suppressed, from a much more powerful source with properties that eventually seem relevant to an SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Support for these scenar- ios would imply finding evidence of a middle-aged remnant in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The γ Cygni SNR (G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1), with its estimated age ∼ 10 kyr, would be an interesting candidate source for scenario H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Another option could be the remnant that resulted from the explosion giving birth to PSR J2032+4127, ∼ 100−200 kyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Its age is comparable to the diffusion time involved in scenario H4 and is high enough that the remnant has most likely gone undetectable by now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Interestingly, the typical injection time of 30 kyr and diffusion suppression by a factor 10 in scenario H4 are reminiscent of the results obtained in Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2019) for the non-linear diffusion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 − 1 TeV CRs escaping from a su- pernova remnant in a hot ionised medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For comparable model setups, the difference in injection effi- ciency between leptonic and hadronic scenarios is of an order of magnitude at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This means that mixed lepto-hadronic sce- narios either imply a relatively high electron-to-proton ratio at injection, or a predominance of the hadronic contribution to the emission if the electron-to-proton ratio at injection is expected to have more classical values of 10−2 at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' All the potential sources discussed above, Cyg OB2, NGC 6910, the γ Cygni SNR, and the unobserved SNR asso- ciated with PSR J2032+4127 are displaced with respect to the Article number, page 20 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations centroid of the emission (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This could suggest an in- homogenous transport scenario if one of these sources is indeed responsible for the origin of the particle population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Possible improvements to the modelling presented here in- clude the possibility of multiple and extended sources, for in- stance Cyg OB2 and NGC 6910 releasing accelerated particles at their respective super-wind termination shocks, and a more complete transport scheme, including inhomogeneous diffusion and (or) energy losses over such a large volume, or the effect of advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The latter point is particularly relevant in the case of high levels of diffusion suppression, as it may dominate the transport of the lowest-energy particles and alter their diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The contribution from pre-existing CRs, for instance their lep- tonic emission from inverse-Compton scattering in the dense photon fields of the main clusters (Orlando & Strong 2007) or their hadronic emission after reacceleration in the turbulent in- terior of the region (Tolksdorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019), certainly deserves a more sophisticated treatment than done here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Meanwhile, we can qualitatively compare our results to more sophisticated models of particle acceleration and transport at cluster wind termination shocks and in SBs from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2021) presents a model of particle accelera- tion at the winds of star clusters that predicts a spectrum similar to the cocoon central component CoCent, which spans a region with size comparable to that of the wind termination shocks from Cyg OB2 and NGC 6910 (see Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The spatial distribu- tion of low-energy particles in their model can be rather flat up to a few times the termination shock radius, which is comparable to the extended component CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, their model predicts that higher energy particles are more tightly confined around the shock, which is in contrast with our results of a harder spectrum for CoExt than CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In addition, as already discussed above, there is no clear identification of a super-wind termination shock in the region, nor is it clear that there is a super-wind emanating from Cyg OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Alternatively, Vieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2022) have shown that their model for particle acceleration and transport in SBs can reproduce the overall spectrum of the Cygnus cocoon measured by the LAT and HAWC in the case of efficient confinement in the bubble shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In these conditions, they show that particle densities are rather uniform inside the SB, which might be in good agreement with the flat radial emissivity profile of the CoExt component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, it is not obvious that their model can reproduce the morphological properties of the observed emission, especially its centrally peaked nature, if particles are efficiently trapped in an outer shell (the location of which remains unclear in Cygnus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Recently, Fornieri & Zhang (2022) presented a model of gamma-ray emission from Cygnus featuring two CR sources (Cyg OB2 and the γ Cygni SNR) and a description of particle transport that takes into account the detection of multiple plasma modes in the region (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' As a consequence CR diffusion is predicted to be inhomogenous, which results in con- finement for a long time in the central cavities where plasma modes are predominantly magnetosonic, and a more rapid diffu- sion in the nearby Alfvénic-dominated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, their model does not take into account ionised gas in the central cav- ities, which seems required to explain the emissions observed from CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Furthermore, they calculate a diffusion coefficient for physical parameters, such as gas density and temperature, that are not necessarily relevant for the entire gamma-ray emit- ting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Overall, it is not clear if their model can explain the large extension of CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 The introduction of FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 (OffExc), significantly improves the likelihood of the model (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, the best-fit Gaussian model is only partially contained in our analysis region, and therefore the results may be subject to large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A proper characterisation of this excess is left for future studies, and in this section we only provide general con- siderations on possibilities concerning its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' There is no ob- vious correlation between OffExc and structures in the gas maps (Figure 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' No SNRs are found overlapping with OffExc in SNRCat5 (Ferrand & Safi-Harb 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' On the other hand, the ATNF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='67 pulsar catalogue6 (Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2005) lists three pul- sars within the r68 area of OffExc with a spin-down power > 1034 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Among those PSR J2111+4606, the closest to the source centroid at an offset of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3◦, has a spin-down power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 × 1036 erg s−1, a characteristic age of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 kyr, and an uncer- tain distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' It may be reminiscent of some middle-aged pulsars powering large gamma-ray sources such as HESS J1825−137 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Principe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020) or the pulsar halos around PSR J0633+1746 or PSR B0656+14 (Abey- sekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In the latter case, the offset of the pulsar from the centre of the emission can be explained by a combination of proper motion and time-dependent injection (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' OffExc is bordered on the east by X-ray emission in the Cygnus SB (Cash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The corresponding X-ray struc- ture, dubbed as S-ARC 3 in Uyanıker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2001) has been asso- ciated to the stellar association Cyg OB4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' However, the very ex- istence of Cyg OB4 is questioned based on parallax distances (de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2020) reports ten stel- lar clusters with more than 100 members at a probability > 70% within the r68 area of OffExc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Among those, seven are located at distances from the Earth < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8 kpc, which would correspond to a physical size < 200 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Most of them are quite far away from the gamma-ray emission centroid, with the closest at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='8◦ being NGC 7082 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='339 kpc from Earth and with an estimated age of 61 Myr, larger than typical stellar clusters with established detections in gamma rays (for instance Tibaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The spectrum of OffExc has a particularly strong break with a steep slope at low energies and a hard spectrum above a few GeV that is strikingly similar to the one of CoExt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Establish- ing a physical connection between the two emission components is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' OffExc may be produced by particles escaping the volume encompassed by CoExt, after an energy-dependent transport process that depleted the particle spectrum at the low- energy end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This is reminiscent of the illumination of neighbour- ing gas clouds by CRs escaping from a nearby source (for in- stance Tang 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Alternatively, steep slopes at low energies has been suggested as a signature of particles reacceleration in SBs (for instance Tolksdorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019), and there may be radial gradients in such a mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Summary and conclusions We presented an analysis of the gamma-ray emission from the Cygnus region based on ∼13 years of Fermi-LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The ex- traction of the emission from the so-called Cygnus cocoon was performed from a dedicated modelling of interstellar emission from the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Compared to the analysis presented in Acker- mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (2011), we used almost seven times more data, pro- duced with an improved reconstruction scheme corresponding to 5 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='umanitoba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='ca/snr/SNRcat 6 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='atnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='au/research/pulsar/psrcat/ Article number, page 21 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon enhanced instrument performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The data analysis is based on a much larger catalogue of gamma-ray sources and dedicated re- sults for major sources in the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We also used improved gas tracer data and an iterative procedure to derive the dark neutral gas map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' As a result, the emission from the cocoon is now sepa- rated into two main components: first, a central component, FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 (CoCent), traced by a model for the distri- bution of ionised gas within the borders of photo-dissociation regions, and having a power law spectrum with index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' second, an extended component, FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 (CoExt), that can be modelled with a 2D Gaussian intensity dis- tribution of extension r68 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4◦ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1◦ and a smooth broken power law spectrum with spectral indices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='01 below and above 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Emission from this component is significantly detected out to nearly 10◦ from the approximate centre of the star-forming re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Its total spectrum is significantly different from that of Co- Cent, and it exhibits significant spectral variations in azimuth in the innermost ≲ 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Two additional extended emission components were signifi- cantly detected during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Source FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='83+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='57 (CoWest) overlaps with a bright arc of 8 µm emission on the border of the central cavities in Cygnus X, and has a spectrum statistically compatible with CoCent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Although the spectral sim- ilarity and spatial proximity suggests a common origin, CoWest does not show any obvious correlation with known gas struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Another source, FCES G85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='78 (OffExc), is offset by several degrees with respect to Cygnus X and its spectrum is sig- nificantly different from all the other extended components stud- ied, so a common origin seems unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The centroid of OffExc lies on the edge of our analysis region, and therefore its current characterisation may be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A proper study of this com- ponent is left for follow-up work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The extended set of observables resulting from our analy- sis for the two brightest sources making up the cocoon, CoCent and CoExt, can be accounted for reasonably well from a simple diffusion-loss framework with a small number of free parame- ters, under several model setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' In all viable scenarios, one sin- gle population of non-thermal particles with a flat injection spec- trum at the central source is sufficient and both hadronic and lep- tonic options are viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Particles span the full extent of source CoExt as a result of diffusion, and give rise to source CoCent by interacting with ionised gas in the innermost regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Possible solutions are very different in terms of energetics, transport con- ditions, and time scales involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Some scenarios involve con- tinuous injection during ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 − 3 Myr, transport in a medium with moderately to strongly suppressed diffusion with respect to the large-scale interstellar average, and injection luminosities in the 1036 − 1037 erg s−1 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' They could describe a process by which the observed gamma-ray emission is powered by par- ticle acceleration in the prominent star clusters Cyg OB2 and NGC 6910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Alternatively, a hadronic solution exists involving a more recent event, with injection lasting 3 − 30 kyr and transport proceeding over 10 kyr in a medium where diffusion is not or only moderately suppressed, and a much more powerful source with injection luminosity ∼ 1039 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Such a scenario seems more relevant to a single supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Possible improvements beyond this simple interpretation framework include accounting for multiple and extended sources, a more advanced description of particle transport in an inhomogeneous medium and including advection, and, last but not least, the description of physically motivated acceleration mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The observables extracted from our analysis are made available in machine-readable format and can be used in the future to perform detailed comparisons with more sophisti- cated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' From the observational perspective, significant advances in gamma rays can be expected from instruments with improved sensitivity and angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The upcoming Cherenkov Telescope Array (Cherenkov Telescope Array Consortium et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019) above a few tens of GeV will provide a sensitivity an or- der of magnitude better than previous ground-based instruments and an angular resolution reaching a few arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Proposed space missions dedicated to the MeV to GeV domain (de Angelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' McEnery & Amego Team 2020) may also improve a few times the angular resolution and by one or two orders of mag- nitude the sensitivity compared to Fermi, and provide observa- tions in an energy range, the sub-MeV and MeV domain, poorly observed until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Complementary advances in the character- isation of interstellar gas (for instance Emig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2022), and of multi-wavelength and multi-messenger emission from the co- coon (for instance Mizuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Yoast-Hull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2017) are also key to improving our understanding of particle acceleration and transport in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The Fermi LAT Collaboration acknowledges generous on- going support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' These include the National Aeronautics and Space Administration and the Department of Energy in the United States,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' the Commissariat à l’Energie Atom- ique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' the Ministry of Ed- ucation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Culture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Sports,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Science and Technology (MEXT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' High Energy Accel- erator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' and the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This work performed in part under DOE Contract DE-AC02- 76SF00515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This work was supported by the "Agence Nationale de la Recherche" through grant ANR-19-CE31-0014 (GAMALO project, PI: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Martin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' This work makes use of NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020), AstroPy (Astropy Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2022), Matplotlib (Hunter 2007), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020), and the colourmaps in the CMasher package (van der Velden 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The authors would like to thank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Orlando and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Pesce-Rollins for their helpful comments on the manuscript, as well as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Grenier for insightful conversations about the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' References Abeysekara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' U.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', Yan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2020, Nature Astronomy, 4, 1001 21 Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=', & Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2021, ApJ, 922, 130 21 Article number, page 23 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 0 1 2 3 4 5 6 7 8 Angle (deg) 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) Linj = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='84e+35 erg/s 2 = 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='89 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV intensity (ph/cm2/s/sr) Linj = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='84e+35 erg/s 2 = 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='36 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 9 10 8 10 7 10 6 10 5 10GeV-1TeV intensity (ph/cm2/s/sr) Linj = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='84e+35 erg/s 2 = 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='71 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 18: Intensity radial profiles in three different gamma-ray en- ergy bands for the FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56 and FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 emission components, compared to predictions for model setup L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The intensity distribution corresponding to the best-fit two- dimensional Gaussian model is displayed for comparison as a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The χ2 value correspond to the deviation from the 2D Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 24 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations Appendix A: Data analysis We provide in this appendix a series of technical details about the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1: Preliminary model optimisation The preliminary optimisation of the emission model went through the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' a simultaneous fit of the normalisation of bright sources with TS > 104 and predicted photons counts > 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' an iterative fit of the normalisation and spectral shape of all the sources in the ROI by order of intensity and significance (method optimize of Fermipy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' a further simultaneous fit of the normalisation of sources with TS > 104 and predicted photons counts > 500, and also of the spectral shape of sources with the same TS condition and predicted photons counts > 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The thresholds on predicted photon counts and TS were cho- sen to optimise all parameters for the brightest sources in the ROI (three pulsars, Cygnus Loop, γ Cygni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We are more restrictive for the spectral shape to reduce the number of free parameters in the analysis, which otherwise can become unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2: Morphological fits In the morphological characterisation stages, the optimisation of the related components is performed in a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Extension optimisation: we perform a first scan over a coarse grid of extension values, followed by a second finer scan around the first optimum and then the fit of a parabola to determine the final extension value and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Position optimisation: we compute a map of log-likelihood values over a position grid of 2◦ × 2◦ with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2◦ binning and determine the best-fit position and its uncertainty from the fit of an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3: Spectral models In the spectral characterisation of the various components of the cocoon, we consider the following models: a simple power law (PL) of expression dN dE = N0 × � E E0 �−γ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1) with N0 flux at the reference energy E0 and γ spectral index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' a log-parabola (LP) of expression dN dE = N0 × � E E0 �− � α+β ln � E E0 �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2) with N0 flux at the reference energy E0, α slope parameter, and β curvature parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' and a smooth broken power law (SBPL) of expression dN dE = N0 × � E E0 �−γ1 ��������1 + � E Eb � γ2−γ1 κ �������� −κ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3) with N0 flux at the reference energy E0, Eb break energy, γ1 spec- tral index at energies ≪ Eb, γ2 spectral index at energies ≫ Eb, and κ the smoothing parameter fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Appendix B: Additional results on the spectro-morphological analysis Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 shows the TS values for the best decomposition of CoExt, as in step D described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The figure illus- trates how the decomposition was designed to conserve a mini- mum TS of 25 except for the largest ring where the requirement could not be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 85◦ 80◦ 75◦ 70◦ 10◦ 5◦ 0◦ −5◦ Galactic Longitude Galactic Latitude 101 102 TS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1: TS values in rings, segments, and disk for the best de- composition of FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We used the intensities derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 to derive emissivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To do so, we divided the flux associated with each segment, ring, or disk by the total neutral gas column density in the local arm (atomic, molecular and DNM) integrated over solid angle for each segment, ring, or disk area (shown in Fig- ure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 for a discussion on uncertainties in the 85◦ 80◦ 75◦ 70◦ 10◦ 5◦ 0◦ −5◦ Galactic Longitude Galactic Latitude 1022 4 × 1021 6 × 1021 2 × 1022 3 × 1022 4 × 1022 Column density (H cm−2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2: Total neutral gas column density in the local arm as in Figure 10 reprojected onto the disk, rings, and segments used for the spectro-morphological analysis in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' gas column densities relevant for the emissivity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3 shows maps of intensities and emissivities for Co- Ext as decomposed in step D in section and in three different en- ergy bands 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='236 GeV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='236−10 GeV and 10−1000 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 25 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon Radial profiles of intensity and emissivity in the three energy bands are shown in Section 4, where they are used for quantita- tive interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 10◦ 5◦ 0◦ −5◦ Galactic Latitude 10−7 10−6 10−5 10−7 10−6 Intensity (ph cm−2 s−1 sr−1) 10−8 10−7 10−6 85◦ 80◦ 75◦ 70◦ 10◦ 5◦ 0◦ −5◦ Galactic Latitude 85◦ 80◦ 75◦ 70◦ Galactic Longitude 85◦ 80◦ 75◦ 70◦ 10−29 10−28 10−29 10−28 Emissivity (ph s−1 sr−1 H−1) 10−30 10−29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 - 10 GeV 10 GeV - 1 TeV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3: Intensity and emissivity maps in three different energy ranges for component FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56, according to the de- composition in rings and segments of step D (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Appendix C: Diffusion-loss model framework In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2, we introduce a simple diffusion-loss framework to account for the observed properties of the Cygnus cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We provide here the full formalism of this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The transport equation governing the evolution of the parti- cle distribution f in momentum p, position r, and time t is: ∂f ∂t = ∇ · (D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ∇ f) − ∂ ∂p � ˙p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' f� + Q, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1) where D is a spatial diffusion coefficient, ˙p a momentum loss term, and Q a source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The momentum loss term ˙p in- cludes losses that are uniform in space and arise from radia- tive processes, hadronic interactions for accelerated protons, and Bremsstrahlung, inverse-Compton scattering, and synchrotron radiation for accelerated electrons (Schlickeiser 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The source term Q(p, t) is assumed to be point-like in space and to have a constant power-law with exponential cut-off spectral shape: Q(p, t) = Q0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' � p 10 GeV/c �−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' e−p/pcut .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' e−t/tinj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2) Particles are injected with a power law spectrum in momentum with index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The cut-off momentum is set to a high value of 1 PeV/c that our gamma-ray data are not sensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To investi- gate the possibility that injection is not constant in time, and may have occurred over a finite duration some time ago followed by a longer diffusion time, we implemented (somewhat arbitrarily) an exponential decay of the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Integrating the source term Q(p, t) over particle energies yields the time-dependent injection luminosity Linj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The diffusion coefficient D(p) assumed to be constant in space and time is defined as: D(p) = β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' � p 10 GeV/c �δ with δ = 1/3, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3) where β = v/c with v the velocity of a particle and c the veloc- ity of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The power-law dependence in momentum, with an index corresponding to a Kolmogorov scaling for the magnetic turbulence spectrum, is adapted from models of large-scale CR propagation in our Galaxy (Trotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Orlando 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' We note that alternative expressions were considered recently in the light of more accurate direct CR measurements at Earth (Evoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Génolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 2019), and that the considered deviations from a pure power law may have consequences in the energy range we are interested in here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The solution to the transport equation is obtained as (Atoyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 1995): f(p, r, t) = � tdiff max[0, tdiff−tcool(pcut,p)] ˙p(p0) ˙p(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Q(p0, t0) π3/2r3 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' e−r2/r2 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' dt0, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4) with rd diffusion distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' For a present-day momentum p, the integration runs either over the full injection and transport his- tory, or over the recent period spanning a cooling time tcool from the cut-off momentum down to p, with cooling time tcool(p0, p) = � p0 p −dp ˙p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5) This three-dimensional spherically symmetric distribution of particles is integrated along the line of sight for any angular off- set from the centre θ and given the distance d to the source: Φ(p, θ, t) = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' � ∞ 0 f � p, √ θ2d2 + ℓ2, t � dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='6) The resulting angular distribution of particles is then used to compute non-thermal emissions at any point in the region of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' To this aim, we used the naima package (Zabalza 2015) in the approximation of isotropic radiation fields in the case of inverse-Compton scattering and using a nuclear enhancement factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='845 in the case of pion decay (Mori 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Appendix D: Possible diffusion-loss scenarios We display here the results obtained for some of the diffusion scenarios considered in the interpretation of the results (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The intensity and emissivity profiles for hadronic sce- narios H2 and H3 or H4 are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2, and the intensity profiles for model leptonic scenario L2 are pre- sented in Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' The full set of results for the leptonic pulsar halo scenario is presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 26 of 30 3X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25e+37 erg/s 2 = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='55 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 29 10 28 10 27 10 26 10 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV intensity (ph/cm2/s/sr) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25e+37 erg/s 2 = 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='86 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 30 10 29 10 28 10 27 10 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 10GeV-1TeV intensity (ph/cm2/s/sr) Linj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='25e+37 erg/s 2 = 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='72 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 31 10 30 10 29 10 28 10 27 10GeV-1TeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='1: Same as Figure 16, for model setup H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 27 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85e+39 erg/s 2 = 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='32 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 29 10 28 10 27 10 26 10 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV intensity (ph/cm2/s/sr) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85e+39 erg/s 2 = 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='93 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 30 10 29 10 28 10 27 10 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 10GeV-1TeV intensity (ph/cm2/s/sr) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85e+39 erg/s 2 = 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='09 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 31 10 30 10 29 10 28 10 27 10GeV-1TeV emissivity (ph/s/sr/H) Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Local emissivity Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2: Same as Figure 16, for model setup H3 or H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 28 of 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Astiasarain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' : Multiple emission components in the Cygnus cocoon detected from Fermi-LAT observations 0 1 2 3 4 5 6 7 8 Angle (deg) 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='41e+36 erg/s 2 = 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='36 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV intensity (ph/cm2/s/sr) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='41e+36 erg/s 2 = 528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='59 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 10GeV-1TeV intensity (ph/cm2/s/sr) Linj = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='41e+36 erg/s 2 = 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='08 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='3: Same as Figure 18 for model setup L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' 100 101 102 103 Energy (GeV) 10 9 10 8 10 7 10 6 Spectral energy distribution (GeV/cm2/s) D0(100TeV)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00e+28 cm2/s Linj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='43e+36 erg/s NIG H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='03e+22 H/cm2 2 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='61 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='89 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='4: Same as Figure 17 for the pulsar halo scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 29 of 30 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' ms_cocoon 0 1 2 3 4 5 6 7 8 Angle (deg) 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) D0(100TeV)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00e+28 cm2/s Linj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='43e+36 erg/s 2 = 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 7 10 6 10 5 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2GeV intensity (ph/cm2/s/sr) D0(100TeV)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00e+28 cm2/s Linj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='43e+36 erg/s 2 = 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 0 1 2 3 4 5 6 7 8 Angle (deg) 10 8 10 7 10 6 10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='2-10GeV intensity (ph/cm2/s/sr) D0(100TeV)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00e+28 cm2/s Linj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='43e+36 erg/s 2 = 524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='85 Model (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Model (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) 2D Gaussian fit Data (FCES G78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='56) Data (FCES G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='50) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content='5: Same as Figure 18 for the pulsar halo scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} +page_content=' Article number, page 30 of 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfaQo5/content/2301.04504v1.pdf'} diff --git a/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf b/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a5dbf65517d2dee77b3f72237f64faf9e9a631cd --- /dev/null +++ b/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +1 +FedICT: Federated Multi-task Distillation for +Multi-access Edge Computing +Zhiyuan Wu, Member, IEEE, Sheng Sun, Yuwei Wang, Member, IEEE, +Min Liu, Senior Member, IEEE, Xuefeng Jiang, and Bo Gao, Member, IEEE +Abstract—The growing interest in intelligent services and privacy protection for mobile devices has given rise to the widespread +application of federated learning in Multi-access Edge Computing (MEC). Diverse user behaviors call for personalized services with +heterogeneous Machine Learning (ML) models on different devices. Federated Multi-task Learning (FMTL) is proposed to train related +but personalized ML models for different devices, whereas previous works suffer from excessive communication overhead during training +and neglect the model heterogeneity among devices in MEC. Introducing knowledge distillation into FMTL can simultaneously enable +efficient communication and model heterogeneity among clients, whereas existing methods rely on a public dataset, which is impractical +in reality. To tackle this dilemma, Federated MultI-task Distillation for Multi-access Edge CompuTing (FedICT) is proposed. FedICT +direct local-global knowledge aloof during bi-directional distillation processes between clients and the server, aiming to enable multi-task +clients while alleviating client drift derived from divergent optimization directions of client-side local models. Specifically, FedICT includes +Federated Prior Knowledge Distillation (FPKD) and Local Knowledge Adjustment (LKA). FPKD is proposed to reinforce the clients’ fitting +of local data by introducing prior knowledge of local data distributions. Moreover, LKA is proposed to correct the distillation loss of the +server, making the transferred local knowledge better match the generalized representation. Extensive experiments on three datasets +demonstrate that FedICT significantly outperforms all compared benchmarks in various data heterogeneous and model architecture +settings, achieving improved accuracy with less than 1.2% training communication overhead compared with FedAvg and no more than +75% training communication round compared with FedGKT in all considered scenarios. +Index Terms—Federated learning, multi-task learning, knowledge distillation, multi-access edge computing, distributed optimization +! +1 +INTRODUCTION +M +ULTI-ACCESS Edge Computing (MEC) pushes com- +putation and memory resources to the network edge, +enabling low communication latency and convenient ser- +vices for accessed devices [1]. Along with the development +of wireless network technology and the proliferation of +mobile devices, increasing amounts of distributed data gen- +erated in diverse devices are processed in MEC scenarios. +Besides, the growing interest in edge intelligence services +motivates the prominent demands for deploying Machine +Learning (ML) models on devices. Whereas for privacy +concerns, collecting data from devices to the remote server +for model training is often prohibited [2]. +Federated Learning (FL) [3] opens a new horizon for +• +Zhiyuan Wu and Xuefeng Jiang are with the Institute of Computing +Technology, Chinese Academy of Sciences, Beijing, China, and also with +the University of Chinese Academy of Sciences, Beijing, China. +E-mail: {wuzhiyuan22s, jiangxuefeng21b}@ict.ac.cn. +• +Sheng Sun and Yuwei Wang are with the Institute of Computing Tech- +nology, Chinese Academy of Sciences, Beijing, China. +E-mail: {sunsheng, ywwang}@ict.ac.cn. +• +Min Liu is with the Institute of Computing Technology, Chinese Academy +of Sciences, Beijing, China, and also with the Zhongguancun Laboratory, +Beijing, China. +E-mail: {liumin}@ict.ac.cn. +• +Bo Gao is with the School of Computer and Information Technology, and +the Engineering Research Center of Network Management Technology for +High-Speed Railway of Ministry of Education, Beijing Jiaotong Univer- +sity, Beijing, China. +E-mail: {bogao}@bjtu.edu.cn. +Corresponding author: Yuwei Wang. +This work was supported by the National Key Research and Development +Program of China (2021YFB2900102) and the National Natural Science +Foundation of China (No. 61732017, No. 62072436, No. 62002346 and No. +61872028). +training ML models in a distributed manner while keeping +private data locally, and is well suited for privacy-sensitive +applications in MEC, such as the internet of vehicles [4], +[5], healthcare [6], [7], etc. However, local data distribu- +tions across devices usually exhibit discrepant characteris- +tics and evident skews in MEC due to diversified individual +behaviours [8]. This phenomenon poses requirements to +inconsistent update targets among client-side local mod- +els, and thus the shared server-side global model trained +through conventional FL methods generalizes poorly on +heterogeneous local data [9], [10], [11], [12]. +To collaboratively train separate models with different +update targets, Federated Multi-task Learning (FMTL) [13] +regards local model training on each device as a learning +task to fit personalized requirements. However, most exist- +ing FMTL methods face two challenges to tackle in MEC. +On the one hand, exchanging large-scale model parameters +or gradients during training is unaffordable for devices +with inferior communication capabilities [14], [15]. On the +other hand, personalized models with heterogeneous model +architectures are required to be deployed on clients since +differentiated computational capabilities, energy states and +data distributions are ubiquitous among clients [2], [16], +[17]. Whereas existing FMTL methods [18], [19], [20], [21] +require large-scale parameters transmission as well as only +support adopting the same model architecture on the server +and clients, hence are unavailable when local models are +heterogeneous in MEC with constrained resources. +One prospective way to avoid large-scale parameters +transmission and enable heterogeneous models in FMTL +is to introduce Knowledge Distillation (KD) [22], [23] as +arXiv:2301.00389v1 [cs.LG] 1 Jan 2023 + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +2 +an exchange protocol across model representations (called +Federated Distillation, FD), transferring knowledge or inter- +mediate features instead of model parameters between the +server and clients. However, all existing FD methods that +support multi-task clients [10], [24] are built on frameworks +that rely on public datasets whose data distribution should +be close to private data on clients [25]. Since collected +public data needs to be compared with the clients’ private +data on data distributions, all FD methods rely on public +datasets will undoubtedly lead to privacy leakage of clients +and are impractical in MEC [17], [26]. Although few FD +approaches can achieve client-server co-distillation without +public datasets [27], [28], they are only appropriate to the +single-task setting because of neglecting data discrepancy +among clients. However, directly imposing individualized +parameters update on local models in the above FD ap- +proaches without public datasets [27], [28] is commonly +ineffective, since it aggravates local optimization directions +deviating from that of the global model, i.e., client drift, +which causes unsatisfactory global convergence and dra- +matically limits the individual performance of clients in +turn [8], [10], [24]. How to overcome the adverse impact of +client drift and well achieve local distillation differentiation +becomes the primary issue in FD-based FMTL without the +assistance of public datasets. +In this paper, we propose an FD-based FMTL frame- +work for MEC without a public dataset, named Federated +MultI-task Distillation for Multi-access Edge CompuTing +(FedICT). FedICT enables differentiated learning on client- +side local models via distillation-based personalized opti- +mization while disaffecting the knowledge transferred be- +tween the server and clients, so as to mitigate the impact +of client drift on model convergence while enabling per- +sonalized local models. Specifically, FedICT consists of two +parts, Federated Prior Knowledge Distillation (FPKD) for +personalizing client-side distillation and Local Knowledge +Adjustment (LKA) for correcting server-side distillation. +The former enhances clients’ multi-task capability based on +prior knowledge of local data distributions and reinforces +the fitting degree of local models to their local data by +controlling class attention during local distillation. The latter +is proposed to correct the loss of global distillation on the +server, which prevents the global optimization direction +from being skewed by local updates. To our best knowl- +edge, this paper is the first work to investigate federated +multi-task distillation without additional public datasets +in multi-access edge computing, which realizes multi- +task training requirements in a communication-efficient and +model-heterogeneity-allowable manner, and is practical for +MEC. +In general, our contributions can be summarized as +follows: +• +We propose a novel FD-based FMTL framework in +MEC (namely FedICT), which can realize distillation- +based personalized optimization on clients while +reducing the impact of client drift from a novel per- +spective of alienating local-global knowledge with- +out public datasets. +• +We propose FPKD to enhance fitting degrees of +client-side local models on discrepant data via intro- +ducing prior knowledge of local data distributions. +Further, LKA is proposed to correct the distillation +loss of the server-side global model, aiming to alle- +viate client drift derived from knowledge mismatch +between clients and the server. +• +We conduct extensive experiments on CIFAR-10, +CINIC-10 and TMD datasets. Results show that our +proposed FedICT can improve average User model +Accuracy (UA) [18] of all compared benchmarks. +Besides, FedICT enables efficient communication and +faster convergence, achieving the same average UA +with less than 1.2% of training communication over- +head compared with FedAvg and no more than 75% +of communication rounds compared with FedGKT in +all experimental settings. +2 +RELATED WORK +2.1 +Federated Multi-task Learning +FMTL [13] is proposed to fit related but personalized models +over FL, which enables clients to collaboratively explore a +shared generalized representation while allowing person- +alized objectives on local models. Motivated by this idea, +a series of approaches are proposed, such as introduc- +ing non-federated network layers [18], adopting diversified +optimization objectives [20], [29], or leveraging ensemble +models to fit client-side data distributions [19]. Specifically, +[18] allows clients to separately optimize personalization +layers. [19] adopt linear combinations of multiple shared +component models, assuming that data distributions of +clients are a mixture of multiple unknown underlying distri- +butions. Some approaches utilize Laplacian Regularization +to constrain local models [20] or adopt dynamic weights +on local model gradients [29]. However, none of the above +approaches enables local training on clients with heteroge- +neous models, and they all require exchanging large-scale +model parameters between the server and clients. +2.2 +Federated Learning in Multi-access Edge Comput- +ing +FL performs collaborative model training on distributed de- +vices at the network termination, whereas these devices of- +ten possess heterogeneous system configurations and train- +ing goals with constrained resources [2], [16]. A series of ap- +proaches are proposed to reduce the computational or com- +munication on devices through transferring computation +burden from devices to the edge server [30], adopting model +pruning methods to lighten model sizes on devices [31], or +establishing computing- and communication-friendly train- +ing paradigm [27]. Another line of research is to fit different +requirements among devices: adopting adaptive learning +rates to fit the personalized accuracy goals of clients [32], +transferring historical information from previous person- +alized models to maintain local models’ well performance +on individual clients [33], or leveraging memory-efficient +source-free unsupervised domain adaptation to make local +models adapt their respective data [8]. However, none of the +above approaches can simultaneously meet communication +constraints and enable model heterogeneity among clients, +which is inapplicable to MEC scenarios in practice. + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +3 +2.3 +Knowledge Distillation in Federated Learning +KD enables knowledge to be transferred from one ML +model to another to facilitate constructive optimization of +the latter model. KD has been utilized in various fields +up to date, such as model compression [22], [34], domain +adaptation [35], [36], [37] and distributed training [38], [39]. +Jeong et al. [40] first introduce KD to FL as an exchange +protocol for cross-clients model representations, and such +distillation-based FL methods are called federated distilla- +tion (FD). +One of the most representative FD methods is proposed +in [41], where the server iteratively generates consensus +based on client logits and then distributes consensus to +clients for local distillation. Subsequent approaches are im- +proved in terms of data dependency [42], [43], knowledge +distribution [42], [44], knowledge filtering or weighting +[10], [24], [45], [46], etc. Several works [42], [43] extend +conventional supervised FD methods to semi-supervised +paradigms. Besides, some approaches adjust the knowl- +edge distribution during distillation to accelerate client- +side convergence [42] or counteract poisoning attacks [44]. +More recent works are proposed to filter, weight, or cluster +knowledge from clients with similar local data distributions +[10], [24], [45], [46]. However, all the above approaches rely +on public datasets whose data distribution should be similar +to local training data [25], but such datasets are hard to +access in reality [17], [26]. Although few approaches can +realize FD without public datasets [27], [28], [47], [48], they +either neglect knowledge deviation of local models derived +in multi-task setting [27], [28], or confront with tremendous +communication overhead for exchanging model parameters +[47], [48]. Therefore, existing FD methods are not suitable +for FMTL in MEC. +3 +NOTATIONS AND PRELIMINARY +3.1 +Formulation of Federated Multi-task Learning +This paper investigates the cross-device FMTL in which +heterogeneous clients jointly train ML models coordinated +by the server, with the goal of training personalized lo- +cal models that can adapt to local data distributions. The +main notations in this paper are summarized in TABLE 1. +Without loss of generality, we study C class classification in +FMTL. Assuming that K clients participate in FL training +and K := {1, 2, ......, K}. Each client k ∈ K possesses a +local dataset ˆDk := +N k +� +i=1 +{( ˆXk +i , ˆyk +i )} with N k samples.The +local dataset ˆDk is sampled from the local data distribution +Dk := +∞ +� +i=1 +{(Xk +i , yk +i )}, where ˆDk ⊂ Dk. Different from the +optimization objectives of conventional FL methods [49], +[50], [51] where all clients share the same model, we expect +that client k obtains a local model Fk(·) that can maximize +the localized evaluation metric M(·) for its personalized +local data, i.e., +arg max +W k +E +(Xk +i ,yk +i )∼Dk[M(Fk(Xk +i ; W k), yk +i )], +(1) +where W k is the parameter of the local model at client k. +Generally, FMTL guides local models to accommodate uni- +versal representations integrated from all clients during the +TABLE 1 +Main notations and descriptions. +Notation +Description +K +Number of clients +R +Maximum number of communication rounds +ˆDk +Local dataset of client k +N k +Number of samples in ˆDk +ˆXk +i +The i-th sample of ˆDk +ˆyk +i +The label of ˆXk +i +W S +The global model parameters of the server +W k +The local model parameters of client k +zS +ˆ +Xk +i +The global knowledge of ˆXk +i +zk +ˆ +Xk +i +The local knowledge of ˆXk +i +ˆHk +i +The extracted features of ˆXk +i +dk +The local data distribution vector of client k +dS +The global data distribution vector +JS +ICT +The optimization objective of global model +when adopting FedICT +Jk +ICT +The optimization objective of local model +on client k when adopting FedICT +training process, so as to improve local models’ performance +on local data. +3.2 +Basic Process of Federated Distillation +This paper follows the framework of proxy-data-free FD +[27], [28], where the model of arbitrary client k is divided +into two parts, the feature extractor and the predictor +with corresponding parameters W k +e and W k +p respectively. +Hence, the model parameters of client k are denoted as +W k := {W k +e , W k +p }. The server adopts a global model with +only the predictor to synthesize local knowledge, whose +parameters are denoted as W S. It is worth noting that +the inputs of all feature extractors and the outputs of all +predictors share the same shape. +Proxy-data-free FD relaxes the requirements of model +homogeneity and decreases the communication overhead +through exchanging knowledge or features in replacement +of model parameters between the server and clients. The +overall training procedure consists of multiple communica- +tion rounds, and each round adopts a stage-wise training +paradigm, successively updating global and local model +parameters in a co-distillation manner [38]. Specifically, let +f(·; W ∗) denotes the non-linear mapping determined by +the parameters W ∗ ∈ { +K� +k=1 +W k ∪ W S}, and R denotes the +maximum number of communication rounds. τ(·) is the +softmax mapping, LCE(·) is the cross-entropy loss func- +tion, and Lsim(·) is the customized knowledge similarity +loss function, which takes KL divergence loss by default. +Throughout the training process, we refer to the logits from +clients as local knowledge and the logits from the server as +global knowledge. +The basic process of FD can be divided into two stages +as follows: +• +Local Distillation. Client k updates its local model +parameters W k based on the local labels ˆyk +i and + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +4 +(a) Most FL +Task1 +Task2 +Task3 +Global +(b) FMTL +Task1 +Task2 +Task3 +Local1 +Local2 +Local3 +Local +Adaptation +Task1 +Task2 +Task3 +Local1 +Local2 +Local3 +Distillation +(c) Most FD +Task1 +Task2 +Task3 +Local1 +Local2 +Local3 +FPKD +(d) FedICT +LKA +Fig. 1. Comparison of different FL methods in MEC. Grey circles indicate the parameter requirements for different training tasks on devices, and +the blue circles indicate the trained model parameters. Each circle’s size represents the scale of model parameters, and the distance between two +arbitrary circles implies the degree of differences between their corresponding parameters. +the downloaded global knowledge zS +ˆ +Xk +i . The basic +objective of local model optimization on client k +Jk(·) can be expressed as follows: +arg min +W k Jk(W k) += arg min +W k +E +( ˆ +Xk +i ,ˆyk +i )∼ ˆ +Dk[LCE(τ(f( ˆXk +i ; W k)), ˆyk +i ) ++β · Lsim(τ(f( ˆXk +i ; W k)), τ(zS +ˆ +Xk +i ))], +(2) +where zS +ˆ +Xk +i is the global knowledge extracted from +the local features ˆHk +i in the previous communication +round, which is derived by: +zS +ˆ +Xk +i = f( ˆHk +i ; W S). +(3) +• +Global Distillation. The server updates the global +model parameters W S based on the uploaded local +knowledge zk +ˆ +Xk +i , the uploaded local features ˆHk +i and +labels ˆyk +i . The basic objective of global model opti- +mization JS(·) can be expressed as follows: +arg min +W S JS(W S) += arg min +W S +E +( ˆ +Xk +i ,ˆyk +i )∼ � +k∈K +ˆ +Dk[LCE(τ(f( ˆHk +i ; W S)), ˆyk +i ) ++β · Lsim(τ(f( ˆHk +i ; W S)), τ(zk +ˆ +Xk +i ))], +(4) +where ˆHk +i and zk +ˆ +Xk +i are the local features and knowl- +edge of client k generated in the last local distillation +process. They can be derived by: +ˆHk +i = f( ˆXk +i ; W k +e ), +(5) +zk +ˆ +Xk +i = f( ˆXk +i ; W k). +(6) +Local and global distillation stages are alternately executed +until model convergence. As only embedded features, logits, +and labels are exchanged between the server and clients +and their sizes are much smaller than model parameters +[27], [28], FD can naturally guarantee communication effec- +tiveness. Furthermore, FD does not require homogeneous +model architectures on clients and thus can support various +devices with different system configurations. +4 +FEDERATED +MULTI-TASK +DISTILLATION +FOR +MULTI-ACCESS EDGE COMPUTING +4.1 +Motivation +4.1.1 +Superiority of FD for FMTL in MEC +The core challenges of FMTL in MEC are twofold: limited +communication capabilities and heterogeneous models. +• +Limited Communication Capabilities. Devices pos- +sess poor communication capabilities and are unable +to communicate at scale [2], [14], [15], [16]. +• +Heterogeneous Models. Each client call for inde- +pendently designed models with differentiated pa- +rameters to satisfy personalized requirements since +devices vary in computational capabilities, energy +states and data distributions [2], [16], [17]. +Most FMTL methods require to exchange large-scale model +parameters during training. Hence, tremendous communi- +cation overhead is a key trouble when deploying to MEC. +In addition, model heterogeneity combined with multi- +tasking is also a big issue in MEC, as shown in Fig 1. As +displayed in Fig. 1 (b), although existing FMTL methods can +capture common representations between interrelated tasks +and generalize well to different tasks via local adaptation, +they fail to deploy models with suitable parameters size for +each client. +We claim that adopting FD for FMTL in MEC has the +following advantages: +• +Communication Efficiency. The size of knowledge +or embedded features exchanged between the server +and clients are much smaller than that of model +parameters. As a result, FD-based FMTL methods +are effective in MEC scenario, where communication +resources among clients are strictly limited. +• +Heterogeneous +Models +Supportability. Even if +clients adopt independent models with various ar- +chitectures, FD-based FMTL can be deployed and +trained as long as few preconditions are met (e.g. +agreement on the size of knowledge or features), +which is applicable to MEC. +• +Multi-task Feasibility. Local distillation can be tai- +lored to adapt local data distributions, meeting +client-side local task requirements. + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +5 +TABLE 2 +Comparisons of FedICT with other FL methods in terms of four conditions to represent the practicality of deployment in MEC. +Method +Task Hetero. +Among Clients +Model Hetero. +Among Clients +Efficient +Communication +Do Not Require +Public Data +FedAvg [49] /FedProx [50]/FedAdam [51] +� +� +� +� +pFedMe [20]/FedEM [19]/MTFL [18] +� +� +� +� +FedMD [41]/DS-FL [52]/FedGEMS [46] +� +� +� +� +PERFED-CKT [10]/KT-pFL [24]/CoFED [53] +� +� +� +� +FedGKT [27]/FedDKC [28] +� +� +� +� +FedICT +� +� +� +� +In general, adopting FD for FMTL is a feasible choice for +MEC: it not only meets the communication limitation and +model heterogeneity requirements of MEC, but also enables +collaborative training among clients with different tasks. +4.1.2 +Insight of Aloof Local-Global Knowledge in FD +Since FD requires local models to mimic the global model +partially, local models tend to learn an isomorphic represen- +tation of the global model, somewhat inhibiting the ability +to accommodate multiple tasks on clients. As shown in Fig. +1 (c), all clients tend to learn a common representation that +is similar to the server in existing FD methods, and fail +to perform well on different local tasks due to ignoring +adapt local models to local data [27], [28]. Furthermore, as +FMTL expects to train local models with a high degree of +personalization, it raises a question of how the global model +learns a uniform generalizable representation from highly +biased local knowledge: local models need to perform well +on heterogeneous local data distributions, and their induc- +tive preferences necessarily deviate from that of the global +model, which in turn increases the difficulty of distillation- +based fusion of local knowledge. +Based on the above analysis, we suggest that knowledge +correction is necessary during local and global distillation. +Therefore, we expect to inject localized prior knowledge +in local distillation and de-localize local knowledge in +global distillation, i.e., keeping local-global knowledge +aloof. Based on the customized local distillation objective, +each local model can better adapt to the local task. Based +on the de-localized global distillation objective, the global +model can converge stably towards global generalization. +Through adopting this idea, the server can learn gener- +alizable knowledge while clients possess satisfactory ca- +pabilities of learning discrepant local tasks, with different +representations between the server and clients. +Based on the above insight, FedICT is proposed, whose +optimization sketch map in MEC is shown in Fig. 1 (d), and +comparisons with other FL methods are listed in TABLE +2. Compared with the state-of-the-art methods, our pro- +posed FedICT not only allows task and model heterogeneity +among clients, but also enables efficient communication +without the assistance of a public dataset, which can be +deemed as the first FD work on multi-task setting to be +practically deployed in MEC. +4.2 +Framework Formulation +Different from previous methods [27], [28], we perform +knowledge adaptation processes in both local and global +distillation stages. Specifically, prior knowledge of local +data distributions is introduced to personalize local models +during local distillation; the discordance of global versus +local data distributions is considered to reduce global-local +knowledge divergence during global distillation. +To be specific, we define dk := dist( ˆDk) as the local data +distribution vector of client k and dS := dist( +K� +k=1 +ˆDk) as +the global data distribution vector, where dist(·) maps the +input dataset to its corresponding data distribution vector +for estimating the data distribution of a given dataset. In this +paper, we adopt data category frequencies tepresent data +distributions. For any dataset ˆD∗ := +N ∗ +� +i=1 +{( ˆX∗ +i , ˆy∗ +i )} with N ∗ +samples, the i-th dimension of its data distribution vector +dist( ˆD∗)i depends on the frequency of its i-th class f ∗ +i , that +is: +dist( ˆD∗)i = f ∗ +i = +� +y∗ +i ∈D∗ δ(y∗ +i = i) +N ∗ +, +(7) +where δ(·) is an indicator function that returns 1 when the +input equation holds and 0 otherwise. +During local distillation, local models are updated with +reference to local data distribution information, aiming to +achieve superior performance on local tasks. Specifically, +we formulate the new local distillation objective Jk +ICT (·) for +client k as follows: +arg min +W k Jk +ICT (W k) += arg min +W k [Jk(W k) + λ · Jk +F P KD(W k; dk)], +(8) +where Jk +F P KD(·) is the optimization component of client k +based on the distribution vector of local data dk. +During global distillation, the global model is updated +considering the discordance of the global versus local data +distributions, realizing the global knowledge de-localization +to maintain the global model’s global-to-local perspective +rather than a narrow local perspective. Specifically, we for- +mulate the new global distillation objective JS +ICT (W S) as +follows: +arg min +W S JS +ICT (W S) += arg min +W S [JS(W S) + µ · JS +LKA(W S; dS, dk)], +(9) + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +6 +where JS +LKA(·) is the optimization component based on the +de-localized local knowledge. +In general, we anticipate that the transferred knowledge +from both global and local models will be biased toward +the data distribution associated with their respective target +models, i.e. inducing aloof local-global knowledge. Such in- +duction during bi-directional distillation processes enables +local models to sufficiently fit local tasks while facilitating +the global model to integrate personalized local knowledge +for achieving faster convergence. Specifically, we propose +Federated Prior Knowledge Distillation (FPKD, related to +Jk +F P KD) and Local Knowledge Adjustment (LKA, related +to JS +LKA) to jointly achieve aloof local-global knowledge. +The details of our proposed techniques are described in the +following sections. +4.3 +Federated Prior Knowledge Distillation +Existing FD methods [27], [28] without public datasets +simply let local models fit downloaded global knowledge +during local distillation, during which all local models learn +a uniform global representation, which is commonly gen- +eralized and relatively class-balanced. However, in FMTL, +the training tasks of local models are highly correlated +with local data distributions, and more biased local repre- +sentation is preferred. Thus, we optimize client-side local +models utilizing local data distributions and concentrate +on classes with high frequencies to adapt to skewed local +data. Specifically, for the i-th sample of client k denoted as +ˆXk +i , the r-th dimension of its global knowledge is denoted +as globalr := (zS +ˆ +Xk +i )r, and the r-th dimension of its local +knowledge is denoted as localr := (zk +ˆ +Xk +i )r. In addition, wk +i +is defined to weight the i-th component of KL-divergence +between the local knowledge of client k and the global +knowledge. Accordingly, the optimization objective of client +k is defined as follows: +Jk +F P KD(W k; dk) = +E +( ˆ +Xk +i ,ˆyk +i )∼ ˆ +Dk[ +C +� +r=1 +wk +r · globalr·log globalr +localr +], +(10) +where wk +r is positively correlated to local class frequencies +and is controlled by a hyperparameter T, that is: +wk +r = +exp( f k +r +T ) +C +� +j=1 +exp( +f k +j +T ) +, +(11) +where f k +i denotes the sample frequency of category i in ˆDk +i . +4.4 +Local Knowledge Adjustment +An essential issue of noteworthy divergence among local +models needs to be solved during global distillation in +FMTL, deriving from data heterogeneity and personalized +local distillation (e.g., FPKD discussed in section 4.3). Recent +works have demonstrated that local divergence is detrimen- +tal to the overall FL training, as client-side local models +tend to gradually forget representations of global mod- +els and drift towards their local objectives [54], [55]. This +phenomenon inevitably poses inconsistent updates and un- +stable convergence when aggregating highly-differentiated +local models, i.e. client drift [54], [55], [56], [57]. To this +end, we expect to tackle the above-mentioned problem by +assigning importance to local knowledge. Specifically, we +consider two levels: +• +Client level. The global model optimization pays +more attention to local knowledge from clients +whose local data distributions are similar to the +overall data distribution. +• +Class level. The class importance in global distil- +lation is positively correlated with the residuals of +global-local class frequencies. +Based on the above-mentioned two insights, we propose +similarity-based and class-balanced LKA respectively. They +will be elaborated on in the following subsections. +4.4.1 +Similarity-based Local Knowledge Adjustment +The training performance of FD can be improved through +knowledge collaboration among clients with similar data +distributions, as pointed out in [10], [24]. Likewise, global +distillation can be enhanced with the collaboration of clients +whose data distributions are similar to overall data dis- +tribution. Hence, we design distribution-wise weights on +local knowledge, aiming to reduce the negative effects of +inconsistent knowledge on the global model. Precisely, the +similarity difference between global and local knowledge is +measured by the cosine similarity of global and local data +distribution vectors. Then, the weights of local knowledge +from clients are proportional to the resulting knowledge +similarity during global distillation. The global distillation +objective based on data distribution similarity is defined as +follows: +JS +LKA(W k; dS, dk) += E +k∈K{ +(dS)⊤·dk +∥dS∥2·∥dk∥2 · +E +( ˆ +Xk +i ,ˆyk +i )∼ ˆ +Dk[Lsim(global, local)]}. +(12) +4.4.2 +Class-balanced Local Knowledge Adjustment +Due to different user behaviours, local data is often class- +unbalanced in FL scenarios [58]. As a result, local model +training on each client is strongly correlated with local +class distributions and naturally pays more attention to +high-frequency categories. Not only because high-frequency +categories are assigned higher probabilities to reduce the +local loss, but also because FPKD enhances local data fitting +degrees of local models. This phenomenon hampers global +distillation and slows down model convergence. To this +end, we propose a soft-label weighting technique based on +class frequency residuals, which assigns lower weights to +classes whose local class frequencies on clients are higher +than global class frequencies during global distillation. This +technique can narrow global-local knowledge discrepancy +by balancing the transferred local knowledge among classes, +preventing the global model from learning skewed local +class representations. The global distillation objective based +on class importance is defined as follows: +JS +LKA(W k; dS, dk) += E +k∈K{ +E +( ˆ +Xk +i ,ˆyk +i )∼ ˆ +Dk[ +C +� +r=1 +vk +r · localr · log localr +globalr ]}, +(13) + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +7 +where vk +r is positively related to the residuals between the +global and local class frequencies and is controlled by a +hyperparameter U, that is: +vk +r = +exp( f S +r −f k +r +U +) +C +� +j=1 +exp( +f S +j −f k +j +U +) +, +(14) +where f S +i +denotes the sample frequency of category i in +� +k∈K +ˆDk +i . +4.5 +Formal Description of FedICT +The proposed FedICT on clients and the server are illus- +trated in algorithms 1 and 2 respectively, where Hk := +N k +� +i=1 +ˆHk +i , Y k := +N k +� +i=1 +ˆyk +i , Zk +ˆ +Xk := +N k +� +i=1 +zk +ˆ +Xk +i , ZS +ˆ +Xk := +N k +� +i=1 +zS +ˆ +Xk +i and +other notations are listed in TABLE 1. To start with, K clients +and the server simultaneously execute their corresponding +algorithms, where clients start execution by calling FedICT- +CLIENT (Algorithm 1, line 1), and the server starts by +calling FedICT-SERVER (Algorithm 2, line 1). +All clients first perform local initialization (Algorithm +1, line 2) as follows: clients parallelly compute their local +data distribution vectors based on Eq. (7) (Algorithm 1, +line 7). After that, the local data distribution vectors, local +sample numbers and local labels are sent to the server +(Algorithm 1, line 8), followed by iteratively conducting +local distillation (Algorithm 1, line 4). Meanwhile, the server +first performs global initialization (Algorithm 2, line 2), +which includes receiving the local data information from +all clients (Algorithm 2, line 7) and then calculating the +global data distribution vector (Algorithm 2, line 8). After +that, the server sets the global knowledge to zeros and +distributes the initialized values to all clients (Algorithm +2, lines 9-11). Subsequently, the server iteratively performs +global distillation until training stops (Algorithm 2, line 4). +At the beginning of each training round, all clients +parallelly receive the global knowledge generated by the +Algorithm 1: FedICT on Client k. +1: procedure FEDICT-CLIENT( ˆDk, W k, N k) +2: +dk= LOCALINIT( ˆDk, N k) +3: +repeat +4: +W k=LOCALDISTILL( ˆDk, W k, dk) +until Reaches communication rounds R; +5: +return Trained W k +6: procedure LOCALINIT( ˆDk, N k) +7: +Compute dk according to Eq. (7) +8: +Upload dk, N k and Y k to the server +9: +return dk +10: procedure LOCALDISTILL( ˆDk, W k, dk) +11: +Receive ZS +ˆ +XS from the server +12: +Optimize Jk +ICT according to Eq. (8) +13: +Extract Hk according to Eq. (5) +14: +Extract Zk +ˆ +Xk according to Eq. (6) +15: +Upload Hk and Zk +ˆ +Xk to the server +16: +return Trained W k +Algorithm 2: FedICT on the Server. +1: procedure FEDICT-SERVER(W S) +2: +dS, +K� +k=1 +dk, +K� +k=1 +Y k=GLOBALINIT() +3: +repeat +4: +W S= GLOBALDISTILL(W S,dS, +K� +k=1 +dk, +K� +k=1 +Y k) +until Reaches communication rounds R; +5: +return Trained W S +6: procedure GLOBALINIT() +7: +Receive all dk, N k and Y k from clients +8: +Compute dS = +K +� +k=1 +N k · dk +� K +� +k=1 +N k +9: +forall Client k do +10: +Initialize ZS +ˆ +Xk with zeros +11: +Distribute ZS +ˆ +Xk to client k +end +12: +return dk, +K� +k=1 +dk, +K� +k=1 +Y k +13: procedure GLOBALDISTILL(W S,dS, +K� +k=1 +dk, +K� +k=1 +Y k) +14: +forall Client k do +15: +Receive Hk and Zk +ˆ +Xk from client k +16: +Optimize JS +ICT according to Eq. (9) +17: +Generate ZS +ˆ +Xk according to Eq. (3) +18: +Distribute ZS +ˆ +Xk to client k +end +19: +return Trained W S +server in the previous round (Algorithm 1, line 11). The local +model parameters are then optimized according to Eq. (8), +during which the prior knowledge about clients’ local data +distributions is injected to guide local models to accommo- +date their local data (Algorithm 1, line 12). Subsequently, +local knowledge is extracted and uploaded to the server +(Algorithm 1, lines 13-15). The server then accepts the local +knowledge uploaded by each client (Algorithm 2, line 15) +and optimizes the global model parameters according to Eq. +(9) (Algorithm 2, line 16). Noting that this operation benefits +global distillation via similarity-based LKA according to Eq. +(12) or class-balanced LKA according to Eq. (13). Further, +the server extracts the global knowledge based on the +updated global model parameters and distributes them to +corresponding clients (Algorithm 2, lines 17-19). The whole +training process is completed until model convergence. +5 +EXPERIMENTS +5.1 +Experimental Setup +5.1.1 +Datasets and Preprocessing +Datasets. We conduct experiments on image datasets +CIFAR-10 [59], CINIC-10 [60] for classification, and one mo- +bile sensor data mining dataset TMD [61] for transportation +mode detection. CIFAR-10 and CINIC-10 are 10-class image +classification datasets with common objects. TMD is a 5- +class transportation mode detection dataset that categorizes +heterogeneous users’ transportation modes by mining em- + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +8 +Training Data Distribution +Testing Data Distribution +0 +10% +5% +10% +5% +α=0.5 +α=3.0 +Fig. 2. Data distributions with different α on CIFAR-10. Each heat map represents the training/testing data distributions for all clients. Each row of +heat maps represents the class distributions of a single client, where the column label gives the category. Each cell represents the sample number +of corresponding classes for a given client’s training/testing dataset, and the shade of the color indicates the proportion to the total. +bedded sensor data from smartphones. All datasets are pre- +split into training and testing datasets. +Data Partition. For all of our experiments, data partitioning +strategy in [62] is adopted, where the hyper-parameter α +(α > 0) controls the degree of data heterogeneity, with a +smaller α indicating a stronger degree of heterogeneity. In +the FMTL setup, the testing dataset of each client satisfies +a similar distribution with its training dataset. Fig. 2 shows +the data distributions of training/testing datasets on CIFAR- +10 when 10 clients participate in FMTL. As displayed, +the heat map with the smaller α exhibits more uneven +color distributions, i.e., more unbalanced data partition. +Moreover, the color distributions of training and testing +datasets for each client are almost identical, i.e., isomorphic +training/testing data distribution for individual clients. For +experiments on image classification, we conduct two groups +of experiments under conditions of homogeneous and het- +erogeneous models, each with 10 and 5 clients, respectively. +Each experiment group validates on three different degrees +of data heterogeneity, α ∈ {0.5, 1.0, 3.0}. For experiments +on transportation mode detection, we respectively set the +numbers of participated devices to 120 and 150 under two +data heterogeneity settings, α ∈ {1.0, 3.0}. +Data Augmentation and Normalization. For experiments +on image classification, we conduct random crop, random +horizontal flip and mean-variance standardization before +feeding images into models. For experiments on transporta- +tion mode detection, we normalize the sensor data to have +a mean of 0 and a variance of 1. +5.1.2 +Models +In our experiments, a total of eight local model architec- +tures {AC +1 , ......, AC +8 } are adopted, wherein {AC +1 , ......, AC +5 } +are convolutional neural networks for image classification, +and {AC +6 , AC +7 , AC +8 } are fully connected neural networks for +transportation mode detection. In particular, global model +TABLE 3 +Main configuration of models. H and W are the height and width of +input images, respectively. +Notation +Type +Feat. Shape +Params +AC +1 +Convolutional +Neural +Network +H × W × 16 +0.7K +AC +2 +5.2K +AC +3 +10.5K +AC +4 /AC +5 +9.8K +AS +1 +588.2K +AC +6 +Fully Connected +Neural Network +13 +1109 +AC +7 +1335 +AC +8 +1877 +AS +2 +2053 +architectures AS +0 and AS +1 are adopted for image classification +and transportation mode detection, respectively. Details of +model configurations are provided in TABLE 3. For image +classification experiments with homogeneous models, all +clients adopt the same model architecture AC +1 . For image +classification experiments with heterogeneous models, each +of the five clients adopts a different model architecture +{AC +1 , ......, AC +5 }. In transportation mode detection experi- +ments, we randomly choose AC +8 architecture with a 10% +probability, AC +7 architecture with a 30% probability, and AC +6 +architecture for the rest when adopting FD methods. For +clients adopting non-FD methods, we conduct three groups +of experiments with different model architectures, in which +AC +6 , AC +7 and AC +8 are respectively adopted for all clients in +each group. +5.1.3 +Benchmarks +We compare FedICT combined with FPKD and LKA with +state-of-the-art methods as follows: +• +Classical FL method, FedAvg [49] and FedAdam [51]. + +8185 +JJJ +J08 +toe +J083 +c8 +0 +J +23Q +J02J +JQ寸3 +1 +2Q3 +err +52a3 +0 +0 +J +4 +JJSS +855 +0 +345 +J4 +JSO +230 +3 +520 +48 +J3 +5002 +502 +230 +202 +30J2 +SSJ +J008 +52 +@t0 +J0 +Qe +JJJ3 +JOQO +28寸 +aee +200 +2 +30寸 +JJe +82 +1 +40 +JJ8 +soe +JS3 +52 +30 +1000 +104 +385 +JS +JJ8 +20 +523J +Ses +35J +J00 +SQ8 +52e +0 +54 +J3J +J80 +20 +Jae +ars +SOJO +0 +SOJ +J0Q2 +455 +482 +JQ2S +Q3 +JS3J +0 +0 +02Q +53 +2 +JI +SJ2 +Je +0 +J +Qe +0 +0 +82 +J42 +425 +0 +0 +J +0 +3寸 +JQ3 +0 +4Q +J0 +J2 +52 +10寸 +J +4e +J30 +5 +330 +32 +03 +JOQ +210 +32 +J +0 +JJ3 +54 +JJ +J82 +80 +J20 +84 +21 +J0 +J2 +8 +31 +4 +JJ +41 +J32 +10 +3 +SJ +se +寸8J +41 +2 +J +J8 +25 +41 +0 +44 +rs +ree +JS +J42 +23 +320 +0 +40 +5J4 +82 +350 +J3 +5te +0 +0 +0GJ!GUI a +CI!SUI 8 +CJ!GUI +CI!GUI Q +CI!GI 2 +CI!GI 4 +CJ!GI 3 +CI!GI S +CI!GI J +0 trsilb452 +Ja +Jae +J3J +200 +353 +100 +a20 +QQE +28 +421 +STT +eso +58寸 +J25 +J41 +S8e +452 +358 +312 +31 +8J +T8S +28Q +532 +4寸寸 +80 +245 +rss +J3J2 +240 +802 +Q寸2 +Q寸0 +SS +JSao +Tee +0 +0 +寸寸3 +541 +res +JO +e +JJ寸 +258 +J25 +3寸Q +350 +J03J +300 +30J +400 +512 +Qe0 +2Q0 +J5 +0 +J18 +J03 +323 +350 +8rr +43 +Jas +Q03 +J8 +J +J28 +ree +sel +Jod +40J +cee +JJJ +418 +4寸3 +518 +252 +102 +240 +ces +4寸 +812 +ser +414 +1OSJ +234 +Q2] +Q8S +0 +082 +38 +8es +52 +re +Q2 +J38 +Jao +18 +J3寸 +80 +J22 +J54 +21 +J20 +30 +2e +8e +ee +512 +JQ +2Q +4Q +82 +02 +JO +44 +J25 +JOQ +J18 +J58 +Jse +Jse +40 +523 +0 +0 +82 +41 +21 +35 +JOJ +Q2 +Q +50寸 +20 +er +er +22 +JSJ +JJS +542 +0 +33 +86 +Qd +QS +J20 +8J +30 +J3J +3寸 +ra +JJJ +2Q +30 +QQ +23 +35寸 +er +SJ +rr +81 +20 +JJ3 +82 +J33 +8r +JOS +42 +J12 +82 +J12 +J2 +02 +JO1 +J35 +28 +28 +0 +0THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +9 +TABLE 4 +Average UA (%) [18] on homogeneous local models. Bold values represent the best performance, and underlined values represent the +second-best performance. The same as below. +Method +Model +CIFAR-10 +CINIC-10 +α=3.0 +α=1.0 +α=0.5 +α=3.0 +α=1.0 +α=0.5 +FedAvg +AC +1 +45.73 +39.97 +38.28 +45.76 +42.06 +39.30 +FedAdam +49.09 +53.03 +40.13 +55.71 +54.03 +49.72 +pFedMe +37.53 +34.78 +32.73 +41.03 +38.33 +34.59 +MTFL +42.59 +38.99 +36.96 +42.60 +39.32 +35.67 +FedGKT +59.34 +63.83 +71.26 +46.96 +48.58 +57.56 +FedDKC +60.30 +62.70 +71.53 +50.92 +51.35 +61.09 +FedICT (sim) +60.96 +65.42 +73.54 +56.49 +57.05 +65.46 +FedICT (balance) +61.28 +65.15 +73.37 +56.34 +57.12 +65.72 +• +Personalized FL method, pFedMe [63]. +• +FMTL method, MTFL [18]. +• +FD methods, FedGKT [27] and FedDKC [28]. +Of all the above methods, FD methods support heteroge- +neous local models, while non-FD methods only support +homogeneous local models. Hence, in image classification +experiments, we compare FedICT with all the above state- +of-the-art methods on homogeneous models, while only +compare FedICT with FD methods on heterogeneous mod- +els. In experiments on transportation mode detection, we +simultaneously compare our proposed methods with all the +above benchmarks, where FD-based methods adopt hetero- +geneous models with random model architectures, and non- +FD methods respectively adopt three different model archi- +tectures, as discussed in section 5.1.2. Moreover, we adopt +average User model Accuracy (UA) as the evaluation metric +referred to [18], where UA denotes the training accuracy of +client-side local models through validating on local testing +datasets. +5.1.4 +Hyper-parameter Settings +We adopt stochastic gradient descent to optimize all models. +For experiments on image classification, we set the learning +rate to 1 × 10−2, the l2 weight decay value to 5 × 10−4, +and the batch size to 256. For experiments on transportation +mode detection, the learning rate, weight decay value, and +batch size are set as 3 × 10−4, 5 × 10−4 and 2, respec- +tively. For all the compared methods, each client optimizes +its local model for an epoch before conducting parameter +aggregation or global distillation. Some methods require +individualized hyper-parameters, which are set as follows: +• +We set β1 = 0.9, β2 = 0.99 and τ = 0.001 in +FedAdam referencing to [51]. +• +We set η = 0.005, λ = 15, β = 1 in pFedMe, +referencing to [63]. +• +We adopt implementation based on FedAvg in +MTFL, with other hyper-parameters kept as default +in [64]. +• +We adopt the empirically more effective scheme, +KKR-FedDKC, with β = 1.5 and T = 0.12 refer- +encing to [28]. +• +We set β = λ = µ = 1.5, T = 3.0 and U = 7.0 in our +proposed FedICT. +5.2 +Results on Image Classification +5.2.1 +Performance on Homogeneous Models +TABLE 4 compares our proposed FedICT with existing state- +of-the-art methods on two image classification datasets, +where all clients adopt the same model architecture AC +1 . For +the last two lines in the table, we adopt similarity-based +LKA in FedICT (sim) and class-balanced LKA in FedICT +(balance) , the same as in the following sections. As shown +in TABLE 4, FedICTs both outperform all other baselines +on both CIFAR-10 and CINIC-10 in all data heterogeneity +settings. Specifically, FedICT (sim) increases the average +UA by up to 1.41% and 2.72% on CIFAR-10 and CINIC-10 +compared with the best performances on six benchmarks +respectively, and the improvements are with 1.38% and +2.78% for FedICT (balance). Hence, we can conclude that +our proposed methods are effective in challenging federated +multi-task classification with clients’ local data exhibiting +heterogeneity among each other. +5.2.2 +Performance on Heterogeneous Models +TABLE 5 compares the performance of FedICTs with +FedGKT and FedDKC, including results on two datasets +with three degrees of data heterogeneity and five inde- +pendently designed models. We can see that both Fe- +dICT (sim) and FedICT (balance) outperform the compared +benchmarks in all image classification datasets, all data +heterogeneity settings, and all adopted model architectures +in terms of the average UA, with more than 3.06% im- +provement in average on FedICT (sim), and more than +3.23% improvement in average on FedICT (balance). No- +tably, in the total of 30 client settings, both FedICT (sim) +and FedICT (balance) outperform the best performances in +FedGKT and FedDKC on 29 clients, i.e., UA’s improvement +covering 96.67% of clients. This result demonstrates that +our proposed methods not only improve the average UA of +clients, but also are robust to model architectures, which are +satisfactory for clients with different data distributions and +model architectures. This property motivates diversified +devices with heterogeneous data to participate in FMTL +training, and significantly promotes the availability in real +MEC scenarios. + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +10 +TABLE 5 +UA (%) on heterogeneous local models. +Method +Model +CIFAR-10 +CINIC-10 +α=3.0 +α=1.0 +α=0.5 +α=3.0 +α=1.0 +α=0.5 +FedGKT +AC +1 +35.55 +44.62 +49.90 +39.95 +48.82 +52.21 +AC +2 +52.97 +59.09 +56.67 +43.14 +49.84 +56.97 +AC +3 +61.04 +67.15 +70.16 +62.75 +59.40 +65.84 +AC +4 +50.30 +54.20 +68.89 +45.15 +43.24 +62.21 +AC +5 +57.98 +58.79 +55.49 +55.05 +53.21 +63.35 +Clients Avg. +51.57 +56.77 +60.22 +49.21 +50.90 +60.12 +FedDKC +AC +1 +39.63 +46.83 +51.90 +42.47 +52.06 +52.07 +AC +2 +56.48 +66.43 +61.61 +46.66 +56.43 +59.41 +AC +3 +66.68 +70.33 +70.20 +65.35 +67.07 +66.51 +AC +4 +56.37 +56.86 +71.23 +52.72 +50.13 +62.44 +AC +5 +64.86 +62.41 +61.77 +62.67 +59.73 +64.09 +Clients Avg. +56.08 +60.57 +63.34 +53.97 +57.08 +60.90 +FedICT (sim) +AC +1 +42.40 +49.77 +54.44 +42.62 +54.03 +55.42 +AC +2 +59.85 +68.62 +70.01 +48.18 +57.42 +67.74 +AC +3 +66.56 +72.63 +74.37 +65.92 +67.65 +67.32 +AC +4 +59.18 +60.74 +73.57 +56.13 +52.81 +69.58 +AC +5 +69.99 +63.54 +66.49 +66.27 +61.51 +66.79 +Clients Avg. +59.60 +63.06 +67.78 +55.82 +58.68 +65.37 +FedICT (balance) +AC +1 +42.98 +50.04 +55.06 +42.76 +53.00 +55.15 +AC +2 +57.51 +68.33 +70.20 +48.10 +60.15 +69.13 +AC +3 +66.63 +72.46 +74.66 +66.97 +68.61 +67.96 +AC +4 +61.19 +63.02 +71.27 +55.70 +53.76 +68.56 +AC +5 +71.59 +62.97 +66.83 +65.80 +59.70 +66.74 +Clients Avg. +59.98 +63.36 +67.60 +55.87 +59.04 +65.51 +(a) CIFAR-10 α=3.0 +10 +30 +50 +70 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +FedAvg +FedAdam +pFedMe +MTFL +10 +30 +50 +70 +0 +50 +100 +150 +200 +250 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +FedAvg +FedAdam +pFedMe +MTFL +10 +30 +50 +70 +0 +50 +100 +150 +200 +250 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +FedAvg +FedAdam +pFedMe +MTFL +(b) CIFAR-10 α=1.0 +(c) CIFAR-10 α=0.5 +(d) CINIC-10 α=1.0 +Comm. Rounds +Comm. Rounds +Comm. Rounds +0 +20 +40 +60 +0 +20 +40 +60 +FedGKT +FedAvg +FedAdam +pFedMe +MTFL +FedICT (sim) +FedICT (balance) +FedDKC +Comm. Rounds +Avg. +UA + +(%) +Fig. 3. Learning curves of local models measured by average UA on different degrees of data heterogeneity and datasets. +5.2.3 +Convergence Analysis +We first suggest that FD methods generally converge much +faster than non-FD methods, as displayed in Fig. 3. Since +knowledge and features exchanged in each communica- +tion round contain information about multiple rounds of +model optimization, FD methods always converge to a +higher average UA than non-FD methods under the same +number of communication rounds regardless of datasets, +model architecture setups, and degrees of data heterogene- +ity. Therefore, we only compare the convergence speed of +our proposed FedICTs with existing FD methods by com- +paring the number of communication rounds required to +reach a given average UA. As displayed in TABLE 6, the +required number of communication rounds to converge to +all given average UAs for FedICTs are smaller than that of +existing FD methods in all settings. Specifically, the number +of communication rounds required by FedICTs is no more +than 75% of FedGKT to achieve all given average UAs. +Thus, we can draw that FedICTs achieve convergence accel- +eration, and their training performance suits various data +distributions and model architectures. This is because LKA +mitigates client drift derived by local knowledge divergence +during global distillation, so the server can capture a more +generalizable representation and facilitate local distillation +with the assistance of FPKD in turn. +We further confirm the effectiveness of FedICTs in im- +proving the convergence of individual clients. Fig. 4 dis- +plays the learning curves of selected models under both + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +11 +TABLE 6 +Communication rounds of different FD methods when reaching a given average UA. +Model Homo. +Method +CIFAR-10 +α=3.0 +α=1.0 +α=0.5 +50% +60% +50% +60% +60% +70% +FedGKT +101 +432 +48 +161 +28 +203 +FedDKC +72 +366 +37 +136 +22 +189 +FedICT (sim) +42 +212 +23 +92 +18 +95 +FedICT (balance) +42 +208 +23 +92 +19 +95 +Method +CINIC-10 +α=3.0 +α=1.0 +α=0.5 +40% +50% +40% +50% +50% +60% +FedGKT +15 +- +4 +- +3 +- +FedDKC +13 +76 +3 +41 +2 +54 +FedICT (sim) +6 +40 +1 +24 +2 +26 +FedICT (balance) +6 +40 +1 +19 +2 +26 +Model Hetero. +Method +CIFAR-10 +α=3.0 +α=1.0 +α=0.5 +50% +55% +50% +55% +55% +60% +FedGKT +84 +- +42 +94 +28 +96 +FedDKC +71 +112 +30 +57 +22 +70 +FedICT (sim) +42 +80 +18 +43 +13 +43 +FedICT (balance) +45 +80 +18 +42 +13 +41 +Method +CINIC-10 +α=3.0 +α=1.0 +α=0.5 +40% +50% +50% +55% +55% +60% +FedGKT +8 +59 +57 +- +37 +- +FedDKC +8 +61 +35 +84 +15 +54 +FedICT (sim) +6 +30 +30 +47 +11 +38 +FedICT (balance) +6 +33 +27 +46 +11 +36 +UA +(%) +20 +40 +60 +0 +40 +80 +120 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +40 +60 +80 +0 +40 +80 +120 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +20 +40 +60 +0 +40 +80 +120 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +UA +(%) +(b) Client 3 +(a) Client 1 +(a) Client 1 +(c) Client 10 +(c) Client 10 +(b) Client 3 +(a) Client 1 +(c) Client 10 +30 +50 +70 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +30 +50 +70 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +20 +40 +60 +80 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +UA +(%) +(b) Client 3 +(a) Client 1 +(c) Client 10 +30 +50 +70 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +30 +50 +70 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +20 +40 +60 +80 +0 +50 +100 +150 +200 +LGA-FD (sim) +LGA-FD (balance) +FedDKC +FedGKT +Model Homogeneity +Model Heterogeneity +(b) 3 +C +A +(a) +2 +C +A +(c) +4 +C +A +(b) 3 +C +A +(a) +2 +C +A +(c) +4 +C +A +FedICT (sim) +FedICT (sim) +FedICT (balance) +FedICT (balance) +FedDKC +FedDKC +FedGKT +FedGKT +FedICT (sim) +FedICT (balance) +FedDKC +FedGKT +Fig. 4. Learning curves on selected local models, where the horizontal +coordinates indicate the number of communication rounds. Results are +derived from CIFAR-10, taking α=1.0. +homogeneous and heterogeneous local model settings. We +can figure out that FedICTs consistently exhibit faster con- +vergence compared to FedGKT and FedDKC and can con- +verge to higher UA in all selected clients. This confirms that +our proposed methods can improve the convergence perfor- +mance of heterogeneous individual clients, which supports +the fairness of FedICTs for clients under various conditions. +5.3 +Results on Transportation Mode Detection +TABLE 7 shows the comparison of FedICTs with all con- +sidered state-of-the-art methods on TMD dataset under +different model architecture settings. We can see that our +proposed methods achieve the highest communication ef- +ficiency than all benchmarks on both 120 and 150 clients +settings, regardless of the degrees of data heterogeneity and +model architectures. Specifically, benefiting from exchang- +ing only compact features and knowledge between the +server and clients, FedICTs require less than 1.2% and 0.6% +of communication overheads to achieve 37% average UA in +settings of 120 and 150 clients compared with FedAvg. This +demonstrates that our proposed methods simultaneously +achieve efficient communication, allow heterogeneous local +models, and enable performance on task-diverse clients +superior to state-of-the-art methods, which are not only +practical for MEC but also can remarkably improve client- +side training accuracy in multi-task settings. + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +12 +TABLE 7 +Average UA (%) and communication overheads on TMD dataset, taking α=1.0. +Method +Model +120 Clients +150 Clients +Maximum +Average UA +Comm. Overhead when +Reaching Average UA +Maximum +Average UA +Comm. Overhead when +Reaching Average UA +37% +60% +37% +60% +FedAvg +AC +6 +39.06 +113.24M +- +44.60 +96.36M +- +FedAdam +27.48 +- +- +39.26 +356.46M +- +pFedMe +36.00 +- +- +42.10 +237.19M +- +MTFL +39.20 +111.21M +- +44.98 +101.75M +- +FedAvg +AC +7 +40.75 +45.24M +- +45.06 +117.99M +- +FedAdam +37.35 +176.98M +- +39.18 +444.46M +- +pFedMe +37.81 +97.51M +- +38.58 +277.98M +- +MTFL +40.15 +47.38M +- +45.16 +110.35M +- +FedAvg +AC +8 +42.80 +64.45M +- +45.46 +137.50M +- +FedAdam +40.42 +249.22M +- +36.00 +- +- +pFedMe +37.69 +151.25M +- +36.39 +- +- +MTFL +42.52 +65.74M +- +45.20 +137.50M +- +FedGKT +AC +6 , AC +7 , AC +8 +61.00 +0.70M +4.97M +64.41 +0.54M +3.72M +FedDKC +60.83 +0.70M +4.60M +66.89 +0.54M +2.89M +FedICT (sim) +61.53 +0.54M +3.45M +66.98 +0.54M +1.99M +FedICT (balance) +62.85 +0.54M +2.83M +67.41 +0.54M +2.89M +6 +ABLATION STUDY +6.1 +Ablation Settings +To verify that our proposed methods actually benefit from +leveraging local/global data distribution information, we +conduct the ablation operation Dmeta@ where the randomly +generated data distribution vectors instead of the actual lo- +cal data distribution vectors are used in FedICT. Specifically, +random local data distribution vectors dk ∼ τ(Dmeta), so as +to simulate dk that is independent of local data distributions. +According to algorithm 2, line 8, dS is calculated from dk, so +it is also set as random. In this paper, we try several common +Dmeta to generate dk, which are U(0, 3), N(0, 3) and E(3). +On this basis, we conduct ablation experiments with oper- +ation Dmeta@ on both FedICT (sim) and FedICT (balance). +Specifically, both homogeneous and heterogeneous model +settings are considered, with the same experimental config- +urations as provided in section 5. +6.2 +Results +TABLE 8 displays the experimental results with different +ablation operations and model architectures. We can figure +out that the average UAs of FedICTs with operation Dmeta@ +are all degraded, regardless of adopted LKA techniques +and model architecture settings. This result confirms that +our methods indeed improve average user performance by +transferring the knowledge of local/global data distribu- +tions. +7 +ANALYSIS ON COMPUTATION COST +We compare the computation complexity of FedICT with +existing FD methods without public datasets [27], [28], as +TABLE 8 +Average UA (%) with different ablation operations. Results are derived +on CIFAR-10 dataset, taking α=1.0. +Model +Homo. +Operation +FedICT +(sim) +FedICT +(balance) +U(0, 3)@ +64.86 +64.63 +N(0, 3)@ +63.34 +64.35 +E(3)@ +63.19 +63.88 +None +65.42 +65.15 +Model +Hetero. +Operation +FedICT +(sim) +FedICT +(balance) +U(0, 3)@ +62.82 +62.46 +N(0, 3)@ +60.67 +61.75 +E(3)@ +62.12 +62.47 +None +63.06 +63.36 +shown in TABLE 9. Compared with FedGKT, FedICT in- +troduces additional computational overhead twofold: train- +ing initialization and loss computation. At the client side, +FedICT requires to compute data distribution vectors dur- +ing local initialization, which introduces O(N k + C) ex- +tra computation cost on client k compared with previous +works [27], [28]. Besides, the newly introduced optimiza- +tion component Jk +F P KD(·) requires additional RN k·O(C) +computation cost. At the server side, local data distribution +vectors should be utilized to compute the global data distri- +bution vector during global initialization, where additional +K·O(C) computational cost is required. Likewise, Jk +LKA(·) +introduced by LKA needs extra R +K +� +k=1 +N k·O(C) computa- +tion in the server, regardless of similarity-based or class- + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. +13 +TABLE 9 +Computation complexity of existing FD methods without public datasets. Backward propagation, forward propagation, and stochastic gradient +descent are denoted as BP., FP., SGD., respectively. +Network +Termination +Method +Initialization +BP./FP./SGD. +Loss Computation +Total +FedGKT +- +RN k·O(W k) +RN k·O(C) +RN k·O(W k) +KKR-FedDKC +SKR-FedDKC +FedICT (sim) +O(N k + C) +FedICT (balance) +Network +Edge +Method +Initialization +BP./FP./SGD. +Loss Computation +Total +FedGKT +K +� +k=1 +N k·O(C) +R +K +� +k=1 +N k·O(W S) +R +K +� +k=1 +N k·O(C) +R +K +� +k=1 +N k·O(W S) +KKR-FedDKC +SKR-FedDKC +R +K +� +k=1 +N k·O(C log |ϵ1−ϵ2| +ε +) +FedICT (sim) +(K + +K +� +k=1 +N k)·O(C) +R +K +� +k=1 +N k·O(C) +FedICT (balance) +balanced technique is adopted. +Although extra computation cost is introduced during +initialization and each training round, we still suggest that +FedICT is a computation-efficient FD paradigm compared +with prior works [27], [28]. On the one hand, the ad- +ditional computation cost introduced during initialization +and loss computation is orders of magnitude less than +forward/backward propagation or gradient descent, i.e. +O(N k + C) ≪ N k·O(W k), K·O(C) ≪ +K +� +k=1 +N k·O(W S) +during initialization and RN k·O(C) +≪ +RN k·O(W k), +RK·O(C) ≪ R +K +� +k=1 +N k·O(W S) during model training. On +the other hand, the overall computational overhead is pro- +portional to the number of training rounds, and FedICT can +effectively accelerate model convergence with at least 25% +and 14% fewer training rounds to achieve the same average +UA compared with FedGKT and FedDKC, respectively, as +discussed in section 5.2.3. Therefore, we can conclude that +FedICT generally requires less computation cost than state- +of-the-art methods. +8 +CONCLUSION +This paper proposes a federated multi-task distillation +framework for multi-access edge computing (FedICT). In +our framework, local and global knowledge is disaffected +to achieve client-side adaptation to multiple tasks while +alleviating client drift derived from divergent client-side +optimization directions. Specifically, we propose FPKD and +LKA techniques to reinforce the clients’ fitting to local data +or to match the transferred local knowledge to better suit +generalized representation. To our best knowledge, this pa- +per is the first work that enables federated multi-task learn- +ing to be deployed practically in multi-access edge com- +puting. Extensive experiments on both image classification +and transportation mode detection demonstrate that our +proposed methods achieve superior performance than the +state-of-the-art while improving communication efficiency +and convergence speed by a large margin without requiring +additional public datasets. +ACKNOWLEDGMENTS +We thank Hui Jiang, Qingxiang Liu and Xujing Li from +Institute of Computing Technology, Chinese Academy of +Sciences, Jinda Lu from University of Science and Technol- +ogy of China, Zhiqi Ge from Zhejiang University, Zixuan +Li from Sun Yat-sen University and Yiming Cheng from +University of the Arts London for inspiring suggestions. +ACKNOWLEDGMENT +REFERENCES +[1] +P. Cruz, N. Achir, and A. C. Viana, “On the edge of the deploy- +ment: A survey on multi-access edge computing,” ACM Computing +Surveys (CSUR), 2022. +[2] +A. Tak and S. Cherkaoui, “Federated edge learning: Design issues +and challenges,” IEEE Network, vol. 35, no. 2, pp. 252–258, 2020. +[3] +Q. 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Carpineti, V. Lomonaco, L. Bedogni, M. Di Felice, and +L. Bononi, “Custom dual transportation mode detection by smart- +phone devices exploiting sensor diversity,” in 2018 IEEE Inter- +national Conference on Pervasive Computing and Communications +Workshops (PerCom Workshops). +IEEE, 2018, pp. 367–372. +[62] C. He, S. Li, J. So, X. Zeng, M. Zhang, H. Wang, X. Wang, +P. Vepakomma, A. Singh, H. Qiu et al., “Fedml: A research library +and benchmark for federated machine learning,” arXiv preprint +arXiv:2007.13518, 2020. +[63] C. T Dinh, N. Tran, and J. Nguyen, “Personalized federated +learning with moreau envelopes,” Advances in Neural Information +Processing Systems, vol. 33, pp. 21 394–21 405, 2020. +[64] J. Mills, J. Hu, and G. Min, https://github.com/JedMills/ +MTFL-For-Personalised-DNNs, 2022. +Zhiyuan Wu (Member, IEEE) is currently a re- +search assistant with the Institute of Computing +Technology, Chinese Academy of Sciences. He +is also a member of Distributed Computing and +Systems Committee as well as the Artificial In- +telligence and Pattern Recognition Committee in +China Computer Federation (CCF). His research +interests include mobile computing, federated +learning, knowledge distillation, and distributed +optimization. +Sheng Sun received her B.S. and Ph.D degrees +in computer science from Beihang University, +China, and the University of Chinese Academy +of Sciences, China, respectively. She is currently +an assistant professor at the Institute of Comput- +ing Technology, Chinese Academy of Sciences, +Beijing, China. Her current research interests in- +clude federated learning, mobile computing and +edge intelligence. +Yuwei Wang (Member, IEEE) received his Ph.D. +degree in computer science from the Univer- +sity of Chinese Academy of Sciences, Beijing, +China. He is currently an associate professor +at the Institute of Computing Technology, Chi- +nese Academy of Sciences. He has been re- +sponsible for setting over 30 international and +national standards, and also holds various posi- +tions in both international and national industrial +standards development organizations (SDOs) +as well as local research institutions, including +the associate rapporteur at the ITU-T SG16 Q5, and the deputy director +of China Communications Standards Association (CCSA) TC1 WG1. +His current research interests include federated learning, mobile edge +computing, and next-generation network architecture. +Min Liu (Senior Member, IEEE) received her +Ph.D degree in computer science from the Grad- +uate University of the Chinese Academy of Sci- +ences, China. Before that, she received her B.S. +and M.S. degrees in computer science from +Xi’an Jiaotong University, China. She is currently +a professor at the Institute of Computing Tech- +nology, Chinese Academy of Sciences, and also +holds a position at the Zhongguancun Labora- +tory. Her current research interests include mo- +bile computing and edge intelligence. +Xuefeng Jiang is currently a Ph.D candidate +with the Institute of Computing Technology, Chi- +nese Academy of Sciences. Before that, he re- +ceived his bachelor degree with honor at Beijing +University of Posts and Telecommunications. His +research interests include distributed optimiza- +tion and machine learning. +Bo Gao (Member, IEEE) received his M.S. de- +gree in electrical engineering from the School of +Electronic Information and Electrical Engineer- +ing at Shanghai Jiaotong University, Shanghai, +China in 2009, and his Ph.D. degree in com- +puter engineering from the Bradley Department +of Electrical and Computer Engineering at Vir- +ginia Tech, Blacksburg, USA in 2014. He was +an Assistant Professor with the Institute of Com- +puting Technology at Chinese Academy of Sci- +ences, Beijing, China from 2014 to 2017. He was +a Visiting Researcher with the School of Computing and Communica- +tions at Lancaster University, Lancaster, UK from 2018 to 2019. He +is currently an Associate Professor with the School of Computer and +Information Technology at Beijing Jiaotong University, Beijing, China. He +has directed a number of research projects sponsored by the National +Natural Science Foundation of China (NSFC) or other funding agencies. +He is a member of IEEE, ACM, and China Computer Federation (CCF). +His research interests include wireless networking, mobile/edge com- +puting, multiagent systems, and machine learning. + diff --git a/XtAyT4oBgHgl3EQfh_jm/content/tmp_files/load_file.txt b/XtAyT4oBgHgl3EQfh_jm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d2a4873c64972698cb49da24570f71eafd7a239 --- /dev/null +++ b/XtAyT4oBgHgl3EQfh_jm/content/tmp_files/load_file.txt @@ -0,0 +1,1991 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf,len=1990 +page_content='THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1 FedICT: Federated Multi-task Distillation for Multi-access Edge Computing Zhiyuan Wu, Member, IEEE, Sheng Sun, Yuwei Wang, Member, IEEE, Min Liu, Senior Member, IEEE, Xuefeng Jiang, and Bo Gao, Member, IEEE Abstract—The growing interest in intelligent services and privacy protection for mobile devices has given rise to the widespread application of federated learning in Multi-access Edge Computing (MEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Diverse user behaviors call for personalized services with heterogeneous Machine Learning (ML) models on different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Federated Multi-task Learning (FMTL) is proposed to train related but personalized ML models for different devices, whereas previous works suffer from excessive communication overhead during training and neglect the model heterogeneity among devices in MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Introducing knowledge distillation into FMTL can simultaneously enable efficient communication and model heterogeneity among clients, whereas existing methods rely on a public dataset, which is impractical in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To tackle this dilemma, Federated MultI-task Distillation for Multi-access Edge CompuTing (FedICT) is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' FedICT direct local-global knowledge aloof during bi-directional distillation processes between clients and the server, aiming to enable multi-task clients while alleviating client drift derived from divergent optimization directions of client-side local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, FedICT includes Federated Prior Knowledge Distillation (FPKD) and Local Knowledge Adjustment (LKA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' FPKD is proposed to reinforce the clients’ fitting of local data by introducing prior knowledge of local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Moreover, LKA is proposed to correct the distillation loss of the server, making the transferred local knowledge better match the generalized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Extensive experiments on three datasets demonstrate that FedICT significantly outperforms all compared benchmarks in various data heterogeneous and model architecture settings, achieving improved accuracy with less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2% training communication overhead compared with FedAvg and no more than 75% training communication round compared with FedGKT in all considered scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Index Terms—Federated learning, multi-task learning, knowledge distillation, multi-access edge computing, distributed optimization !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1 INTRODUCTION M ULTI-ACCESS Edge Computing (MEC) pushes com- putation and memory resources to the network edge, enabling low communication latency and convenient ser- vices for accessed devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Along with the development of wireless network technology and the proliferation of mobile devices, increasing amounts of distributed data gen- erated in diverse devices are processed in MEC scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Besides, the growing interest in edge intelligence services motivates the prominent demands for deploying Machine Learning (ML) models on devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Whereas for privacy concerns, collecting data from devices to the remote server for model training is often prohibited [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Federated Learning (FL) [3] opens a new horizon for Zhiyuan Wu and Xuefeng Jiang are with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China, and also with the University of Chinese Academy of Sciences, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' E-mail: {wuzhiyuan22s, jiangxuefeng21b}@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Sheng Sun and Yuwei Wang are with the Institute of Computing Tech- nology, Chinese Academy of Sciences, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' E-mail: {sunsheng, ywwang}@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Min Liu is with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China, and also with the Zhongguancun Laboratory, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' E-mail: {liumin}@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Bo Gao is with the School of Computer and Information Technology, and the Engineering Research Center of Network Management Technology for High-Speed Railway of Ministry of Education, Beijing Jiaotong Univer- sity, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' E-mail: {bogao}@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Corresponding author: Yuwei Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This work was supported by the National Key Research and Development Program of China (2021YFB2900102) and the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 61732017, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 62072436, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 62002346 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 61872028).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' training ML models in a distributed manner while keeping private data locally, and is well suited for privacy-sensitive applications in MEC, such as the internet of vehicles [4], [5], healthcare [6], [7], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, local data distribu- tions across devices usually exhibit discrepant characteris- tics and evident skews in MEC due to diversified individual behaviours [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This phenomenon poses requirements to inconsistent update targets among client-side local mod- els, and thus the shared server-side global model trained through conventional FL methods generalizes poorly on heterogeneous local data [9], [10], [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To collaboratively train separate models with different update targets, Federated Multi-task Learning (FMTL) [13] regards local model training on each device as a learning task to fit personalized requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, most exist- ing FMTL methods face two challenges to tackle in MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' On the one hand, exchanging large-scale model parameters or gradients during training is unaffordable for devices with inferior communication capabilities [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' On the other hand, personalized models with heterogeneous model architectures are required to be deployed on clients since differentiated computational capabilities, energy states and data distributions are ubiquitous among clients [2], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Whereas existing FMTL methods [18], [19], [20], [21] require large-scale parameters transmission as well as only support adopting the same model architecture on the server and clients, hence are unavailable when local models are heterogeneous in MEC with constrained resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' One prospective way to avoid large-scale parameters transmission and enable heterogeneous models in FMTL is to introduce Knowledge Distillation (KD) [22], [23] as arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='00389v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='LG] 1 Jan 2023 THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 2 an exchange protocol across model representations (called Federated Distillation, FD), transferring knowledge or inter- mediate features instead of model parameters between the server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, all existing FD methods that support multi-task clients [10], [24] are built on frameworks that rely on public datasets whose data distribution should be close to private data on clients [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Since collected public data needs to be compared with the clients’ private data on data distributions, all FD methods rely on public datasets will undoubtedly lead to privacy leakage of clients and are impractical in MEC [17], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Although few FD approaches can achieve client-server co-distillation without public datasets [27], [28], they are only appropriate to the single-task setting because of neglecting data discrepancy among clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, directly imposing individualized parameters update on local models in the above FD ap- proaches without public datasets [27], [28] is commonly ineffective, since it aggravates local optimization directions deviating from that of the global model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', client drift, which causes unsatisfactory global convergence and dra- matically limits the individual performance of clients in turn [8], [10], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' How to overcome the adverse impact of client drift and well achieve local distillation differentiation becomes the primary issue in FD-based FMTL without the assistance of public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In this paper, we propose an FD-based FMTL frame- work for MEC without a public dataset, named Federated MultI-task Distillation for Multi-access Edge CompuTing (FedICT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' FedICT enables differentiated learning on client- side local models via distillation-based personalized opti- mization while disaffecting the knowledge transferred be- tween the server and clients, so as to mitigate the impact of client drift on model convergence while enabling per- sonalized local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, FedICT consists of two parts, Federated Prior Knowledge Distillation (FPKD) for personalizing client-side distillation and Local Knowledge Adjustment (LKA) for correcting server-side distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The former enhances clients’ multi-task capability based on prior knowledge of local data distributions and reinforces the fitting degree of local models to their local data by controlling class attention during local distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The latter is proposed to correct the loss of global distillation on the server, which prevents the global optimization direction from being skewed by local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To our best knowl- edge, this paper is the first work to investigate federated multi-task distillation without additional public datasets in multi-access edge computing, which realizes multi- task training requirements in a communication-efficient and model-heterogeneity-allowable manner, and is practical for MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In general, our contributions can be summarized as follows: We propose a novel FD-based FMTL framework in MEC (namely FedICT), which can realize distillation- based personalized optimization on clients while reducing the impact of client drift from a novel per- spective of alienating local-global knowledge with- out public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We propose FPKD to enhance fitting degrees of client-side local models on discrepant data via intro- ducing prior knowledge of local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Further, LKA is proposed to correct the distillation loss of the server-side global model, aiming to alle- viate client drift derived from knowledge mismatch between clients and the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We conduct extensive experiments on CIFAR-10, CINIC-10 and TMD datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Results show that our proposed FedICT can improve average User model Accuracy (UA) [18] of all compared benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Besides, FedICT enables efficient communication and faster convergence, achieving the same average UA with less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2% of training communication over- head compared with FedAvg and no more than 75% of communication rounds compared with FedGKT in all experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Federated Multi-task Learning FMTL [13] is proposed to fit related but personalized models over FL, which enables clients to collaboratively explore a shared generalized representation while allowing person- alized objectives on local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Motivated by this idea, a series of approaches are proposed, such as introduc- ing non-federated network layers [18], adopting diversified optimization objectives [20], [29], or leveraging ensemble models to fit client-side data distributions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, [18] allows clients to separately optimize personalization layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' [19] adopt linear combinations of multiple shared component models, assuming that data distributions of clients are a mixture of multiple unknown underlying distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Some approaches utilize Laplacian Regularization to constrain local models [20] or adopt dynamic weights on local model gradients [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, none of the above approaches enables local training on clients with heteroge- neous models, and they all require exchanging large-scale model parameters between the server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Federated Learning in Multi-access Edge Comput- ing FL performs collaborative model training on distributed de- vices at the network termination, whereas these devices of- ten possess heterogeneous system configurations and train- ing goals with constrained resources [2], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' A series of ap- proaches are proposed to reduce the computational or com- munication on devices through transferring computation burden from devices to the edge server [30], adopting model pruning methods to lighten model sizes on devices [31], or establishing computing- and communication-friendly train- ing paradigm [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Another line of research is to fit different requirements among devices: adopting adaptive learning rates to fit the personalized accuracy goals of clients [32], transferring historical information from previous person- alized models to maintain local models’ well performance on individual clients [33], or leveraging memory-efficient source-free unsupervised domain adaptation to make local models adapt their respective data [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, none of the above approaches can simultaneously meet communication constraints and enable model heterogeneity among clients, which is inapplicable to MEC scenarios in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='3 Knowledge Distillation in Federated Learning KD enables knowledge to be transferred from one ML model to another to facilitate constructive optimization of the latter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' KD has been utilized in various fields up to date, such as model compression [22], [34], domain adaptation [35], [36], [37] and distributed training [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' [40] first introduce KD to FL as an exchange protocol for cross-clients model representations, and such distillation-based FL methods are called federated distilla- tion (FD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' One of the most representative FD methods is proposed in [41], where the server iteratively generates consensus based on client logits and then distributes consensus to clients for local distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Subsequent approaches are im- proved in terms of data dependency [42], [43], knowledge distribution [42], [44], knowledge filtering or weighting [10], [24], [45], [46], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Several works [42], [43] extend conventional supervised FD methods to semi-supervised paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Besides, some approaches adjust the knowl- edge distribution during distillation to accelerate client- side convergence [42] or counteract poisoning attacks [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' More recent works are proposed to filter, weight, or cluster knowledge from clients with similar local data distributions [10], [24], [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, all the above approaches rely on public datasets whose data distribution should be similar to local training data [25], but such datasets are hard to access in reality [17], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Although few approaches can realize FD without public datasets [27], [28], [47], [48], they either neglect knowledge deviation of local models derived in multi-task setting [27], [28], or confront with tremendous communication overhead for exchanging model parameters [47], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Therefore, existing FD methods are not suitable for FMTL in MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 3 NOTATIONS AND PRELIMINARY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Formulation of Federated Multi-task Learning This paper investigates the cross-device FMTL in which heterogeneous clients jointly train ML models coordinated by the server, with the goal of training personalized lo- cal models that can adapt to local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The main notations in this paper are summarized in TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Without loss of generality, we study C class classification in FMTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Assuming that K clients participate in FL training and K := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='., K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each client k ∈ K possesses a local dataset ˆDk := N k � i=1 {( ˆXk i , ˆyk i )} with N k samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The local dataset ˆDk is sampled from the local data distribution Dk := ∞ � i=1 {(Xk i , yk i )}, where ˆDk ⊂ Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Different from the optimization objectives of conventional FL methods [49], [50], [51] where all clients share the same model, we expect that client k obtains a local model Fk(·) that can maximize the localized evaluation metric M(·) for its personalized local data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', arg max W k E (Xk i ,yk i )∼Dk[M(Fk(Xk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W k), yk i )], (1) where W k is the parameter of the local model at client k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Generally, FMTL guides local models to accommodate uni- versal representations integrated from all clients during the TABLE 1 Main notations and descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Number of clients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Maximum number of communication rounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ˆDk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Local dataset of client k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='N k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Number of samples in ˆDk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ˆXk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The i-th sample of ˆDk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ˆyk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The label of ˆXk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='W S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The global model parameters of the server ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='W k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The local model parameters of client k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='zS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The global knowledge of ˆXk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='zk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The local knowledge of ˆXk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ˆHk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The extracted features of ˆXk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='dk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The local data distribution vector of client k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The global data distribution vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='JS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ICT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The optimization objective of global model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='when adopting FedICT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Jk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='ICT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='The optimization objective of local model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='on client k when adopting FedICT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='training process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' so as to improve local models’ performance on local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Basic Process of Federated Distillation This paper follows the framework of proxy-data-free FD [27], [28], where the model of arbitrary client k is divided into two parts, the feature extractor and the predictor with corresponding parameters W k e and W k p respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Hence, the model parameters of client k are denoted as W k := {W k e , W k p }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The server adopts a global model with only the predictor to synthesize local knowledge, whose parameters are denoted as W S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' It is worth noting that the inputs of all feature extractors and the outputs of all predictors share the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Proxy-data-free FD relaxes the requirements of model homogeneity and decreases the communication overhead through exchanging knowledge or features in replacement of model parameters between the server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The overall training procedure consists of multiple communica- tion rounds, and each round adopts a stage-wise training paradigm, successively updating global and local model parameters in a co-distillation manner [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, let f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W ∗) denotes the non-linear mapping determined by the parameters W ∗ ∈ { K� k=1 W k ∪ W S}, and R denotes the maximum number of communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' τ(·) is the softmax mapping, LCE(·) is the cross-entropy loss func- tion, and Lsim(·) is the customized knowledge similarity loss function, which takes KL divergence loss by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Throughout the training process, we refer to the logits from clients as local knowledge and the logits from the server as global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The basic process of FD can be divided into two stages as follows: Local Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Client k updates its local model parameters W k based on the local labels ˆyk i and THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4 (a) Most FL Task1 Task2 Task3 Global (b) FMTL Task1 Task2 Task3 Local1 Local2 Local3 Local Adaptation Task1 Task2 Task3 Local1 Local2 Local3 Distillation (c) Most FD Task1 Task2 Task3 Local1 Local2 Local3 FPKD (d) FedICT LKA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Comparison of different FL methods in MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Grey circles indicate the parameter requirements for different training tasks on devices, and the blue circles indicate the trained model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each circle’s size represents the scale of model parameters, and the distance between two arbitrary circles implies the degree of differences between their corresponding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' the downloaded global knowledge zS ˆ Xk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The basic objective of local model optimization on client k Jk(·) can be expressed as follows: arg min W k Jk(W k) = arg min W k E ( ˆ Xk i ,ˆyk i )∼ ˆ Dk[LCE(τ(f( ˆXk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W k)), ˆyk i ) +β · Lsim(τ(f( ˆXk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W k)), τ(zS ˆ Xk i ))], (2) where zS ˆ Xk i is the global knowledge extracted from the local features ˆHk i in the previous communication round, which is derived by: zS ˆ Xk i = f( ˆHk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (3) Global Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The server updates the global model parameters W S based on the uploaded local knowledge zk ˆ Xk i , the uploaded local features ˆHk i and labels ˆyk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The basic objective of global model opti- mization JS(·) can be expressed as follows: arg min W S JS(W S) = arg min W S E ( ˆ Xk i ,ˆyk i )∼ � k∈K ˆ Dk[LCE(τ(f( ˆHk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W S)), ˆyk i ) +β · Lsim(τ(f( ˆHk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W S)), τ(zk ˆ Xk i ))], (4) where ˆHk i and zk ˆ Xk i are the local features and knowl- edge of client k generated in the last local distillation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' They can be derived by: ˆHk i = f( ˆXk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W k e ), (5) zk ˆ Xk i = f( ˆXk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' W k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (6) Local and global distillation stages are alternately executed until model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As only embedded features, logits, and labels are exchanged between the server and clients and their sizes are much smaller than model parameters [27], [28], FD can naturally guarantee communication effec- tiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Furthermore, FD does not require homogeneous model architectures on clients and thus can support various devices with different system configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4 FEDERATED MULTI-TASK DISTILLATION FOR MULTI-ACCESS EDGE COMPUTING 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Motivation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Superiority of FD for FMTL in MEC The core challenges of FMTL in MEC are twofold: limited communication capabilities and heterogeneous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Limited Communication Capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Devices pos- sess poor communication capabilities and are unable to communicate at scale [2], [14], [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Heterogeneous Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each client call for inde- pendently designed models with differentiated pa- rameters to satisfy personalized requirements since devices vary in computational capabilities, energy states and data distributions [2], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Most FMTL methods require to exchange large-scale model parameters during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Hence, tremendous communi- cation overhead is a key trouble when deploying to MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In addition, model heterogeneity combined with multi- tasking is also a big issue in MEC, as shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1 (b), although existing FMTL methods can capture common representations between interrelated tasks and generalize well to different tasks via local adaptation, they fail to deploy models with suitable parameters size for each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We claim that adopting FD for FMTL in MEC has the following advantages: Communication Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The size of knowledge or embedded features exchanged between the server and clients are much smaller than that of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As a result, FD-based FMTL methods are effective in MEC scenario, where communication resources among clients are strictly limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Heterogeneous Models Supportability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Even if clients adopt independent models with various ar- chitectures, FD-based FMTL can be deployed and trained as long as few preconditions are met (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' agreement on the size of knowledge or features), which is applicable to MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Multi-task Feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Local distillation can be tai- lored to adapt local data distributions, meeting client-side local task requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5 TABLE 2 Comparisons of FedICT with other FL methods in terms of four conditions to represent the practicality of deployment in MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Method Task Hetero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Among Clients Model Hetero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Among Clients Efficient Communication Do Not Require Public Data FedAvg [49] /FedProx [50]/FedAdam [51] � � � � pFedMe [20]/FedEM [19]/MTFL [18] � � � � FedMD [41]/DS-FL [52]/FedGEMS [46] � � � � PERFED-CKT [10]/KT-pFL [24]/CoFED [53] � � � � FedGKT [27]/FedDKC [28] � � � � FedICT � � � � In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' adopting FD for FMTL is a feasible choice for MEC: it not only meets the communication limitation and model heterogeneity requirements of MEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' but also enables collaborative training among clients with different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Insight of Aloof Local-Global Knowledge in FD Since FD requires local models to mimic the global model partially, local models tend to learn an isomorphic represen- tation of the global model, somewhat inhibiting the ability to accommodate multiple tasks on clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1 (c), all clients tend to learn a common representation that is similar to the server in existing FD methods, and fail to perform well on different local tasks due to ignoring adapt local models to local data [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Furthermore, as FMTL expects to train local models with a high degree of personalization, it raises a question of how the global model learns a uniform generalizable representation from highly biased local knowledge: local models need to perform well on heterogeneous local data distributions, and their induc- tive preferences necessarily deviate from that of the global model, which in turn increases the difficulty of distillation- based fusion of local knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Based on the above analysis, we suggest that knowledge correction is necessary during local and global distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Therefore, we expect to inject localized prior knowledge in local distillation and de-localize local knowledge in global distillation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', keeping local-global knowledge aloof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Based on the customized local distillation objective, each local model can better adapt to the local task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Based on the de-localized global distillation objective, the global model can converge stably towards global generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Through adopting this idea, the server can learn gener- alizable knowledge while clients possess satisfactory ca- pabilities of learning discrepant local tasks, with different representations between the server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Based on the above insight, FedICT is proposed, whose optimization sketch map in MEC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1 (d), and comparisons with other FL methods are listed in TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Compared with the state-of-the-art methods, our pro- posed FedICT not only allows task and model heterogeneity among clients, but also enables efficient communication without the assistance of a public dataset, which can be deemed as the first FD work on multi-task setting to be practically deployed in MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Framework Formulation Different from previous methods [27], [28], we perform knowledge adaptation processes in both local and global distillation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, prior knowledge of local data distributions is introduced to personalize local models during local distillation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' the discordance of global versus local data distributions is considered to reduce global-local knowledge divergence during global distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To be specific, we define dk := dist( ˆDk) as the local data distribution vector of client k and dS := dist( K� k=1 ˆDk) as the global data distribution vector, where dist(·) maps the input dataset to its corresponding data distribution vector for estimating the data distribution of a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In this paper, we adopt data category frequencies tepresent data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For any dataset ˆD∗ := N ∗ � i=1 {( ˆX∗ i , ˆy∗ i )} with N ∗ samples, the i-th dimension of its data distribution vector dist( ˆD∗)i depends on the frequency of its i-th class f ∗ i , that is: dist( ˆD∗)i = f ∗ i = � y∗ i ∈D∗ δ(y∗ i = i) N ∗ , (7) where δ(·) is an indicator function that returns 1 when the input equation holds and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' During local distillation, local models are updated with reference to local data distribution information, aiming to achieve superior performance on local tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, we formulate the new local distillation objective Jk ICT (·) for client k as follows: arg min W k Jk ICT (W k) = arg min W k [Jk(W k) + λ · Jk F P KD(W k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' dk)], (8) where Jk F P KD(·) is the optimization component of client k based on the distribution vector of local data dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' During global distillation, the global model is updated considering the discordance of the global versus local data distributions, realizing the global knowledge de-localization to maintain the global model’s global-to-local perspective rather than a narrow local perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, we for- mulate the new global distillation objective JS ICT (W S) as follows: arg min W S JS ICT (W S) = arg min W S [JS(W S) + µ · JS LKA(W S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' dS, dk)], (9) THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 6 where JS LKA(·) is the optimization component based on the de-localized local knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In general, we anticipate that the transferred knowledge from both global and local models will be biased toward the data distribution associated with their respective target models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' inducing aloof local-global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Such in- duction during bi-directional distillation processes enables local models to sufficiently fit local tasks while facilitating the global model to integrate personalized local knowledge for achieving faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, we propose Federated Prior Knowledge Distillation (FPKD, related to Jk F P KD) and Local Knowledge Adjustment (LKA, related to JS LKA) to jointly achieve aloof local-global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The details of our proposed techniques are described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='3 Federated Prior Knowledge Distillation Existing FD methods [27], [28] without public datasets simply let local models fit downloaded global knowledge during local distillation, during which all local models learn a uniform global representation, which is commonly gen- eralized and relatively class-balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' However, in FMTL, the training tasks of local models are highly correlated with local data distributions, and more biased local repre- sentation is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Thus, we optimize client-side local models utilizing local data distributions and concentrate on classes with high frequencies to adapt to skewed local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, for the i-th sample of client k denoted as ˆXk i , the r-th dimension of its global knowledge is denoted as globalr := (zS ˆ Xk i )r, and the r-th dimension of its local knowledge is denoted as localr := (zk ˆ Xk i )r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In addition, wk i is defined to weight the i-th component of KL-divergence between the local knowledge of client k and the global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Accordingly, the optimization objective of client k is defined as follows: Jk F P KD(W k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' dk) = E ( ˆ Xk i ,ˆyk i )∼ ˆ Dk[ C � r=1 wk r · globalr·log globalr localr ], (10) where wk r is positively correlated to local class frequencies and is controlled by a hyperparameter T, that is: wk r = exp( f k r T ) C � j=1 exp( f k j T ) , (11) where f k i denotes the sample frequency of category i in ˆDk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='4 Local Knowledge Adjustment An essential issue of noteworthy divergence among local models needs to be solved during global distillation in FMTL, deriving from data heterogeneity and personalized local distillation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', FPKD discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Recent works have demonstrated that local divergence is detrimen- tal to the overall FL training, as client-side local models tend to gradually forget representations of global mod- els and drift towards their local objectives [54], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This phenomenon inevitably poses inconsistent updates and un- stable convergence when aggregating highly-differentiated local models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' client drift [54], [55], [56], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To this end, we expect to tackle the above-mentioned problem by assigning importance to local knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, we consider two levels: Client level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The global model optimization pays more attention to local knowledge from clients whose local data distributions are similar to the overall data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Class level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The class importance in global distil- lation is positively correlated with the residuals of global-local class frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Based on the above-mentioned two insights, we propose similarity-based and class-balanced LKA respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' They will be elaborated on in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Similarity-based Local Knowledge Adjustment The training performance of FD can be improved through knowledge collaboration among clients with similar data distributions, as pointed out in [10], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Likewise, global distillation can be enhanced with the collaboration of clients whose data distributions are similar to overall data dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Hence, we design distribution-wise weights on local knowledge, aiming to reduce the negative effects of inconsistent knowledge on the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Precisely, the similarity difference between global and local knowledge is measured by the cosine similarity of global and local data distribution vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Then, the weights of local knowledge from clients are proportional to the resulting knowledge similarity during global distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The global distillation objective based on data distribution similarity is defined as follows: JS LKA(W k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' dS, dk) = E k∈K{ (dS)⊤·dk ∥dS∥2·∥dk∥2 · E ( ˆ Xk i ,ˆyk i )∼ ˆ Dk[Lsim(global, local)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (12) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Class-balanced Local Knowledge Adjustment Due to different user behaviours, local data is often class- unbalanced in FL scenarios [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As a result, local model training on each client is strongly correlated with local class distributions and naturally pays more attention to high-frequency categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Not only because high-frequency categories are assigned higher probabilities to reduce the local loss, but also because FPKD enhances local data fitting degrees of local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This phenomenon hampers global distillation and slows down model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To this end, we propose a soft-label weighting technique based on class frequency residuals, which assigns lower weights to classes whose local class frequencies on clients are higher than global class frequencies during global distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This technique can narrow global-local knowledge discrepancy by balancing the transferred local knowledge among classes, preventing the global model from learning skewed local class representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The global distillation objective based on class importance is defined as follows: JS LKA(W k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' dS, dk) = E k∈K{ E ( ˆ Xk i ,ˆyk i )∼ ˆ Dk[ C � r=1 vk r · localr · log localr globalr ]}, (13) THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 7 where vk r is positively related to the residuals between the global and local class frequencies and is controlled by a hyperparameter U, that is: vk r = exp( f S r −f k r U ) C � j=1 exp( f S j −f k j U ) , (14) where f S i denotes the sample frequency of category i in � k∈K ˆDk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 Formal Description of FedICT The proposed FedICT on clients and the server are illus- trated in algorithms 1 and 2 respectively, where Hk := N k � i=1 ˆHk i , Y k := N k � i=1 ˆyk i , Zk ˆ Xk := N k � i=1 zk ˆ Xk i , ZS ˆ Xk := N k � i=1 zS ˆ Xk i and other notations are listed in TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To start with, K clients and the server simultaneously execute their corresponding algorithms, where clients start execution by calling FedICT- CLIENT (Algorithm 1, line 1), and the server starts by calling FedICT-SERVER (Algorithm 2, line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' All clients first perform local initialization (Algorithm 1, line 2) as follows: clients parallelly compute their local data distribution vectors based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (7) (Algorithm 1, line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' After that, the local data distribution vectors, local sample numbers and local labels are sent to the server (Algorithm 1, line 8), followed by iteratively conducting local distillation (Algorithm 1, line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Meanwhile, the server first performs global initialization (Algorithm 2, line 2), which includes receiving the local data information from all clients (Algorithm 2, line 7) and then calculating the global data distribution vector (Algorithm 2, line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' After that, the server sets the global knowledge to zeros and distributes the initialized values to all clients (Algorithm 2, lines 9-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Subsequently, the server iteratively performs global distillation until training stops (Algorithm 2, line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' At the beginning of each training round, all clients parallelly receive the global knowledge generated by the Algorithm 1: FedICT on Client k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1: procedure FEDICT-CLIENT( ˆDk, W k, N k) 2: dk= LOCALINIT( ˆDk, N k) 3: repeat 4: W k=LOCALDISTILL( ˆDk, W k, dk) until Reaches communication rounds R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5: return Trained W k 6: procedure LOCALINIT( ˆDk, N k) 7: Compute dk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (7) 8: Upload dk, N k and Y k to the server 9: return dk 10: procedure LOCALDISTILL( ˆDk, W k, dk) 11: Receive ZS ˆ XS from the server 12: Optimize Jk ICT according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (8) 13: Extract Hk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (5) 14: Extract Zk ˆ Xk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (6) 15: Upload Hk and Zk ˆ Xk to the server 16: return Trained W k Algorithm 2: FedICT on the Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 1: procedure FEDICT-SERVER(W S) 2: dS, K� k=1 dk, K� k=1 Y k=GLOBALINIT() 3: repeat 4: W S= GLOBALDISTILL(W S,dS, K� k=1 dk, K� k=1 Y k) until Reaches communication rounds R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5: return Trained W S 6: procedure GLOBALINIT() 7: Receive all dk, N k and Y k from clients 8: Compute dS = K � k=1 N k · dk � K � k=1 N k 9: forall Client k do 10: Initialize ZS ˆ Xk with zeros 11: Distribute ZS ˆ Xk to client k end 12: return dk, K� k=1 dk, K� k=1 Y k 13: procedure GLOBALDISTILL(W S,dS, K� k=1 dk, K� k=1 Y k) 14: forall Client k do 15: Receive Hk and Zk ˆ Xk from client k 16: Optimize JS ICT according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (9) 17: Generate ZS ˆ Xk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (3) 18: Distribute ZS ˆ Xk to client k end 19: return Trained W S server in the previous round (Algorithm 1, line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The local model parameters are then optimized according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (8), during which the prior knowledge about clients’ local data distributions is injected to guide local models to accommo- date their local data (Algorithm 1, line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Subsequently, local knowledge is extracted and uploaded to the server (Algorithm 1, lines 13-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The server then accepts the local knowledge uploaded by each client (Algorithm 2, line 15) and optimizes the global model parameters according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (9) (Algorithm 2, line 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Noting that this operation benefits global distillation via similarity-based LKA according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (12) or class-balanced LKA according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Further, the server extracts the global knowledge based on the updated global model parameters and distributes them to corresponding clients (Algorithm 2, lines 17-19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The whole training process is completed until model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5 EXPERIMENTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Experimental Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Datasets and Preprocessing Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We conduct experiments on image datasets CIFAR-10 [59], CINIC-10 [60] for classification, and one mo- bile sensor data mining dataset TMD [61] for transportation mode detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' CIFAR-10 and CINIC-10 are 10-class image classification datasets with common objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' TMD is a 5- class transportation mode detection dataset that categorizes heterogeneous users’ transportation modes by mining em- THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 8 Training Data Distribution Testing Data Distribution 0 10% 5% 10% 5% α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Data distributions with different α on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each heat map represents the training/testing data distributions for all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each row of heat maps represents the class distributions of a single client, where the column label gives the category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each cell represents the sample number of corresponding classes for a given client’s training/testing dataset, and the shade of the color indicates the proportion to the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' bedded sensor data from smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' All datasets are pre- split into training and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Data Partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For all of our experiments, data partitioning strategy in [62] is adopted, where the hyper-parameter α (α > 0) controls the degree of data heterogeneity, with a smaller α indicating a stronger degree of heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In the FMTL setup, the testing dataset of each client satisfies a similar distribution with its training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 2 shows the data distributions of training/testing datasets on CIFAR- 10 when 10 clients participate in FMTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As displayed, the heat map with the smaller α exhibits more uneven color distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', more unbalanced data partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Moreover, the color distributions of training and testing datasets for each client are almost identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', isomorphic training/testing data distribution for individual clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For experiments on image classification, we conduct two groups of experiments under conditions of homogeneous and het- erogeneous models, each with 10 and 5 clients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Each experiment group validates on three different degrees of data heterogeneity, α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For experiments on transportation mode detection, we respectively set the numbers of participated devices to 120 and 150 under two data heterogeneity settings, α ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Data Augmentation and Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For experiments on image classification, we conduct random crop, random horizontal flip and mean-variance standardization before feeding images into models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For experiments on transporta- tion mode detection, we normalize the sensor data to have a mean of 0 and a variance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Models In our experiments, a total of eight local model architec- tures {AC 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='., AC 8 } are adopted, wherein {AC 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='., AC 5 } are convolutional neural networks for image classification, and {AC 6 , AC 7 , AC 8 } are fully connected neural networks for transportation mode detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In particular, global model TABLE 3 Main configuration of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' H and W are the height and width of input images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Notation Type Feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Shape Params AC 1 Convolutional Neural Network H × W × 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='7K AC 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2K AC 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5K AC 4 /AC 5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='8K AS 1 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2K AC 6 Fully Connected Neural Network 13 1109 AC 7 1335 AC 8 1877 AS 2 2053 architectures AS 0 and AS 1 are adopted for image classification and transportation mode detection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Details of model configurations are provided in TABLE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For image classification experiments with homogeneous models, all clients adopt the same model architecture AC 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For image classification experiments with heterogeneous models, each of the five clients adopts a different model architecture {AC 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='., AC 5 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In transportation mode detection experi- ments, we randomly choose AC 8 architecture with a 10% probability, AC 7 architecture with a 30% probability, and AC 6 architecture for the rest when adopting FD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For clients adopting non-FD methods, we conduct three groups of experiments with different model architectures, in which AC 6 , AC 7 and AC 8 are respectively adopted for all clients in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': 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PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 9 TABLE 4 Average UA (%) [18] on homogeneous local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Bold values represent the best performance, and underlined values represent the second-best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' The same as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Method Model CIFAR-10 CINIC-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 FedAvg AC 1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='73 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='28 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='30 FedAdam 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='09 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='03 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='13 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='71 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='03 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='72 pFedMe 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='53 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='78 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='73 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='03 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='33 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='59 MTFL 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='59 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='99 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='96 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='60 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='32 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='67 FedGKT 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='34 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='83 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='26 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='96 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='58 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='56 FedDKC 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='30 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='70 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='53 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='92 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='35 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='09 FedICT (sim) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='96 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='42 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='49 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='05 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='46 FedICT (balance) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='28 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='37 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='34 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='12 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='72 Personalized FL method, pFedMe [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' FMTL method, MTFL [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' FD methods, FedGKT [27] and FedDKC [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Of all the above methods, FD methods support heteroge- neous local models, while non-FD methods only support homogeneous local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Hence, in image classification experiments, we compare FedICT with all the above state- of-the-art methods on homogeneous models, while only compare FedICT with FD methods on heterogeneous mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In experiments on transportation mode detection, we simultaneously compare our proposed methods with all the above benchmarks, where FD-based methods adopt hetero- geneous models with random model architectures, and non- FD methods respectively adopt three different model archi- tectures, as discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Moreover, we adopt average User model Accuracy (UA) as the evaluation metric referred to [18], where UA denotes the training accuracy of client-side local models through validating on local testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='4 Hyper-parameter Settings We adopt stochastic gradient descent to optimize all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For experiments on image classification, we set the learning rate to 1 × 10−2, the l2 weight decay value to 5 × 10−4, and the batch size to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For experiments on transportation mode detection, the learning rate, weight decay value, and batch size are set as 3 × 10−4, 5 × 10−4 and 2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For all the compared methods, each client optimizes its local model for an epoch before conducting parameter aggregation or global distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Some methods require individualized hyper-parameters, which are set as follows: We set β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='99 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='001 in FedAdam referencing to [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We set η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='005, λ = 15, β = 1 in pFedMe, referencing to [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We adopt implementation based on FedAvg in MTFL, with other hyper-parameters kept as default in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We adopt the empirically more effective scheme, KKR-FedDKC, with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 and T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='12 refer- encing to [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We set β = λ = µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5, T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 and U = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 in our proposed FedICT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Results on Image Classification 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Performance on Homogeneous Models TABLE 4 compares our proposed FedICT with existing state- of-the-art methods on two image classification datasets, where all clients adopt the same model architecture AC 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' For the last two lines in the table, we adopt similarity-based LKA in FedICT (sim) and class-balanced LKA in FedICT (balance) , the same as in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As shown in TABLE 4, FedICTs both outperform all other baselines on both CIFAR-10 and CINIC-10 in all data heterogeneity settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, FedICT (sim) increases the average UA by up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='41% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='72% on CIFAR-10 and CINIC-10 compared with the best performances on six benchmarks respectively, and the improvements are with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='38% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='78% for FedICT (balance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Hence, we can conclude that our proposed methods are effective in challenging federated multi-task classification with clients’ local data exhibiting heterogeneity among each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Performance on Heterogeneous Models TABLE 5 compares the performance of FedICTs with FedGKT and FedDKC, including results on two datasets with three degrees of data heterogeneity and five inde- pendently designed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We can see that both Fe- dICT (sim) and FedICT (balance) outperform the compared benchmarks in all image classification datasets, all data heterogeneity settings, and all adopted model architectures in terms of the average UA, with more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06% im- provement in average on FedICT (sim), and more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='23% improvement in average on FedICT (balance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' No- tably, in the total of 30 client settings, both FedICT (sim) and FedICT (balance) outperform the best performances in FedGKT and FedDKC on 29 clients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', UA’s improvement covering 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='67% of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This result demonstrates that our proposed methods not only improve the average UA of clients, but also are robust to model architectures, which are satisfactory for clients with different data distributions and model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This property motivates diversified devices with heterogeneous data to participate in FMTL training, and significantly promotes the availability in real MEC scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 10 TABLE 5 UA (%) on heterogeneous local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Method Model CIFAR-10 CINIC-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 FedGKT AC 1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='55 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='62 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='90 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='95 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='82 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='21 AC 2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='09 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='67 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='14 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='84 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97 AC 3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='04 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='16 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='75 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='40 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='84 AC 4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='30 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='20 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='89 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='24 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='21 AC 5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='98 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='79 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='49 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='05 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='21 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='35 Clients Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='57 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='77 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='22 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='21 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='90 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='12 FedDKC AC 1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='63 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='83 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='90 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='47 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='07 AC 2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='48 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='43 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='61 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='66 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='43 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='41 AC 3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='68 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='33 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='20 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='35 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='07 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='51 AC 4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='37 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='86 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='67 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='73 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='09 Clients Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='08 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='57 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='34 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='08 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='90 FedICT (sim) AC 1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='40 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='77 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='44 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='62 54.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='49 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='27 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='51 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='79 Clients Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='60 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='78 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='82 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='68 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='37 FedICT (balance) AC 1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='98 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='04 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='76 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='00 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 AC 2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='51 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='33 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='20 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='10 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='13 AC 3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='63 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='46 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='66 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='61 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='96 AC 4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='19 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='02 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='27 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='70 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='76 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='56 AC 5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='59 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='83 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='80 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='70 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='74 Clients Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='98 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='36 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='60 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='87 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='04 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='51 (a) CIFAR-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 10 30 50 70 0 50 100 150 200 LGA-FD (sim) LGA-FD (balance) FedDKC FedGKT FedAvg FedAdam pFedMe MTFL 10 30 50 70 0 50 100 150 200 250 LGA-FD (sim) LGA-FD (balance) FedDKC FedGKT FedAvg FedAdam pFedMe MTFL 10 30 50 70 0 50 100 150 200 250 LGA-FD (sim) LGA-FD (balance) FedDKC FedGKT FedAvg FedAdam pFedMe MTFL (b) CIFAR-10 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 (c) CIFAR-10 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 (d) CINIC-10 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Rounds Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Rounds Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Rounds 0 20 40 60 0 20 40 60 FedGKT FedAvg FedAdam pFedMe MTFL FedICT (sim) FedICT (balance) FedDKC Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Rounds Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' UA (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Learning curves of local models measured by average UA on different degrees of data heterogeneity and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='3 Convergence Analysis We first suggest that FD methods generally converge much faster than non-FD methods, as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Since knowledge and features exchanged in each communica- tion round contain information about multiple rounds of model optimization, FD methods always converge to a higher average UA than non-FD methods under the same number of communication rounds regardless of datasets, model architecture setups, and degrees of data heterogene- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Therefore, we only compare the convergence speed of our proposed FedICTs with existing FD methods by com- paring the number of communication rounds required to reach a given average UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' As displayed in TABLE 6, the required number of communication rounds to converge to all given average UAs for FedICTs are smaller than that of existing FD methods in all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, the number of communication rounds required by FedICTs is no more than 75% of FedGKT to achieve all given average UAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Thus, we can draw that FedICTs achieve convergence accel- eration, and their training performance suits various data distributions and model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This is because LKA mitigates client drift derived by local knowledge divergence during global distillation, so the server can capture a more generalizable representation and facilitate local distillation with the assistance of FPKD in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We further confirm the effectiveness of FedICTs in im- proving the convergence of individual clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4 dis- plays the learning curves of selected models under both THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 11 TABLE 6 Communication rounds of different FD methods when reaching a given average UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Model Homo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Method CIFAR-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 50% 60% 50% 60% 60% 70% FedGKT 101 432 48 161 28 203 FedDKC 72 366 37 136 22 189 FedICT (sim) 42 212 23 92 18 95 FedICT (balance) 42 208 23 92 19 95 Method CINIC-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 40% 50% 40% 50% 50% 60% FedGKT 15 4 3 FedDKC 13 76 3 41 2 54 FedICT (sim) 6 40 1 24 2 26 FedICT (balance) 6 40 1 19 2 26 Model Hetero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Method CIFAR-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 50% 55% 50% 55% 55% 60% FedGKT 84 42 94 28 96 FedDKC 71 112 30 57 22 70 FedICT (sim) 42 80 18 43 13 43 FedICT (balance) 45 80 18 42 13 41 Method CINIC-10 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='40% ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedICT (sim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedICT (sim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedICT (balance) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedICT (balance) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedDKC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedDKC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedGKT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedGKT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedICT (sim) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedICT (balance) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedDKC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='FedGKT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Learning curves on selected local models, where the horizontal coordinates indicate the number of communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Results are derived from CIFAR-10, taking α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' homogeneous and heterogeneous local model settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We can figure out that FedICTs consistently exhibit faster con- vergence compared to FedGKT and FedDKC and can con- verge to higher UA in all selected clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This confirms that our proposed methods can improve the convergence perfor- mance of heterogeneous individual clients, which supports the fairness of FedICTs for clients under various conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='3 Results on Transportation Mode Detection TABLE 7 shows the comparison of FedICTs with all con- sidered state-of-the-art methods on TMD dataset under different model architecture settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We can see that our proposed methods achieve the highest communication ef- ficiency than all benchmarks on both 120 and 150 clients settings, regardless of the degrees of data heterogeneity and model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, benefiting from exchang- ing only compact features and knowledge between the server and clients, FedICTs require less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='6% of communication overheads to achieve 37% average UA in settings of 120 and 150 clients compared with FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This demonstrates that our proposed methods simultaneously achieve efficient communication, allow heterogeneous local models, and enable performance on task-diverse clients superior to state-of-the-art methods, which are not only practical for MEC but also can remarkably improve client- side training accuracy in multi-task settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 12 TABLE 7 Average UA (%) and communication overheads on TMD dataset, taking α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Method Model 120 Clients 150 Clients Maximum Average UA Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Overhead when Reaching Average UA Maximum Average UA Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Overhead when Reaching Average UA 37% 60% 37% 60% FedAvg AC 6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='24M 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='60 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='36M FedAdam 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='48 39.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='38M 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='16 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='35M FedAvg AC 8 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='80 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='45M 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='46 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='50M FedAdam 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='42 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='22M 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='00 pFedMe 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='69 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='25M 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='39 MTFL 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='52 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='74M 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='20 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='50M FedGKT AC 6 , AC 7 , AC 8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='70M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='97M 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='72M FedDKC 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='70M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='60M 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='89M FedICT (sim) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='45M 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='99M FedICT (balance) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='83M 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='54M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='89M 6 ABLATION STUDY 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='1 Ablation Settings To verify that our proposed methods actually benefit from leveraging local/global data distribution information, we conduct the ablation operation Dmeta@ where the randomly generated data distribution vectors instead of the actual lo- cal data distribution vectors are used in FedICT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, random local data distribution vectors dk ∼ τ(Dmeta), so as to simulate dk that is independent of local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' According to algorithm 2, line 8, dS is calculated from dk, so it is also set as random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In this paper, we try several common Dmeta to generate dk, which are U(0, 3), N(0, 3) and E(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' On this basis, we conduct ablation experiments with oper- ation Dmeta@ on both FedICT (sim) and FedICT (balance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, both homogeneous and heterogeneous model settings are considered, with the same experimental config- urations as provided in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2 Results TABLE 8 displays the experimental results with different ablation operations and model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' We can figure out that the average UAs of FedICTs with operation Dmeta@ are all degraded, regardless of adopted LKA techniques and model architecture settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' This result confirms that our methods indeed improve average user performance by transferring the knowledge of local/global data distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 7 ANALYSIS ON COMPUTATION COST We compare the computation complexity of FedICT with existing FD methods without public datasets [27], [28], as TABLE 8 Average UA (%) with different ablation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Results are derived on CIFAR-10 dataset, taking α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Model Homo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Operation FedICT (sim) FedICT (balance) U(0, 3)@ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='86 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='63 N(0, 3)@ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='34 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='35 E(3)@ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='19 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='88 None 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='42 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='15 Model Hetero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Operation FedICT (sim) FedICT (balance) U(0, 3)@ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='82 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='46 N(0, 3)@ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='67 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='75 E(3)@ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='12 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='47 None 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='06 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='36 shown in TABLE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Compared with FedGKT, FedICT in- troduces additional computational overhead twofold: train- ing initialization and loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' At the client side, FedICT requires to compute data distribution vectors dur- ing local initialization, which introduces O(N k + C) ex- tra computation cost on client k compared with previous works [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Besides, the newly introduced optimiza- tion component Jk F P KD(·) requires additional RN k·O(C) computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' At the server side, local data distribution vectors should be utilized to compute the global data distri- bution vector during global initialization, where additional K·O(C) computational cost is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Likewise, Jk LKA(·) introduced by LKA needs extra R K � k=1 N k·O(C) computa- tion in the server, regardless of similarity-based or class- THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 13 TABLE 9 Computation complexity of existing FD methods without public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Backward propagation, forward propagation, and stochastic gradient descent are denoted as BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Network Termination Method Initialization BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='/FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='/SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Loss Computation Total FedGKT RN k·O(W k) RN k·O(C) RN k·O(W k) KKR-FedDKC SKR-FedDKC FedICT (sim) O(N k + C) FedICT (balance) Network Edge Method Initialization BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='/FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='/SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Loss Computation Total FedGKT K � k=1 N k·O(C) R K � k=1 N k·O(W S) R K � k=1 N k·O(C) R K � k=1 N k·O(W S) KKR-FedDKC SKR-FedDKC R K � k=1 N k·O(C log |ϵ1−ϵ2| ε ) FedICT (sim) (K + K � k=1 N k)·O(C) R K � k=1 N k·O(C) FedICT (balance) balanced technique is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Although extra computation cost is introduced during initialization and each training round, we still suggest that FedICT is a computation-efficient FD paradigm compared with prior works [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' On the one hand, the ad- ditional computation cost introduced during initialization and loss computation is orders of magnitude less than forward/backward propagation or gradient descent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' O(N k + C) ≪ N k·O(W k), K·O(C) ≪ K � k=1 N k·O(W S) during initialization and RN k·O(C) ≪ RN k·O(W k), RK·O(C) ≪ R K � k=1 N k·O(W S) during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' On the other hand, the overall computational overhead is pro- portional to the number of training rounds, and FedICT can effectively accelerate model convergence with at least 25% and 14% fewer training rounds to achieve the same average UA compared with FedGKT and FedDKC, respectively, as discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Therefore, we can conclude that FedICT generally requires less computation cost than state- of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' 8 CONCLUSION This paper proposes a federated multi-task distillation framework for multi-access edge computing (FedICT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' In our framework, local and global knowledge is disaffected to achieve client-side adaptation to multiple tasks while alleviating client drift derived from divergent client-side optimization directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Specifically, we propose FPKD and LKA techniques to reinforce the clients’ fitting to local data or to match the transferred local knowledge to better suit generalized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' To our best knowledge, this pa- per is the first work that enables federated multi-task learn- ing to be deployed practically in multi-access edge com- puting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Extensive experiments on both image classification and transportation mode detection demonstrate that our proposed methods achieve superior performance than the state-of-the-art while improving communication efficiency and convergence speed by a large margin without requiring additional public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Hui Jiang, Qingxiang Liu and Xujing Li from Institute of Computing Technology, Chinese Academy of Sciences, Jinda Lu from University of Science and Technol- ogy of China, Zhiqi Ge from Zhejiang University, Zixuan Li from Sun Yat-sen University and Yiming Cheng from University of the Arts London for inspiring suggestions.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Hu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Min, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='com/JedMills/ MTFL-For-Personalised-DNNs, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Zhiyuan Wu (Member, IEEE) is currently a re- search assistant with the Institute of Computing Technology, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He is also a member of Distributed Computing and Systems Committee as well as the Artificial In- telligence and Pattern Recognition Committee in China Computer Federation (CCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' His research interests include mobile computing, federated learning, knowledge distillation, and distributed optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Sheng Sun received her B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='D degrees in computer science from Beihang University, China, and the University of Chinese Academy of Sciences, China, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' She is currently an assistant professor at the Institute of Comput- ing Technology, Chinese Academy of Sciences, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Her current research interests in- clude federated learning, mobile computing and edge intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Yuwei Wang (Member, IEEE) received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' degree in computer science from the Univer- sity of Chinese Academy of Sciences, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He is currently an associate professor at the Institute of Computing Technology, Chi- nese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He has been re- sponsible for setting over 30 international and national standards, and also holds various posi- tions in both international and national industrial standards development organizations (SDOs) as well as local research institutions, including the associate rapporteur at the ITU-T SG16 Q5, and the deputy director of China Communications Standards Association (CCSA) TC1 WG1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' His current research interests include federated learning, mobile edge computing, and next-generation network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Min Liu (Senior Member, IEEE) received her Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='D degree in computer science from the Grad- uate University of the Chinese Academy of Sci- ences, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Before that, she received her B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' degrees in computer science from Xi’an Jiaotong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' She is currently a professor at the Institute of Computing Tech- nology, Chinese Academy of Sciences, and also holds a position at the Zhongguancun Labora- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Her current research interests include mo- bile computing and edge intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Xuefeng Jiang is currently a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='D candidate with the Institute of Computing Technology, Chi- nese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Before that, he re- ceived his bachelor degree with honor at Beijing University of Posts and Telecommunications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' His research interests include distributed optimiza- tion and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' Bo Gao (Member, IEEE) received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' de- gree in electrical engineering from the School of Electronic Information and Electrical Engineer- ing at Shanghai Jiaotong University, Shanghai, China in 2009, and his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' degree in com- puter engineering from the Bradley Department of Electrical and Computer Engineering at Vir- ginia Tech, Blacksburg, USA in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He was an Assistant Professor with the Institute of Com- puting Technology at Chinese Academy of Sci- ences, Beijing, China from 2014 to 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He was a Visiting Researcher with the School of Computing and Communica- tions at Lancaster University, Lancaster, UK from 2018 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He is currently an Associate Professor with the School of Computer and Information Technology at Beijing Jiaotong University, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He has directed a number of research projects sponsored by the National Natural Science Foundation of China (NSFC) or other funding agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' He is a member of IEEE, ACM, and China Computer Federation (CCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} +page_content=' His research interests include wireless networking, mobile/edge com- puting, multiagent systems, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfh_jm/content/2301.00389v1.pdf'} diff --git a/XtE0T4oBgHgl3EQfWADS/content/tmp_files/2301.02273v1.pdf.txt b/XtE0T4oBgHgl3EQfWADS/content/tmp_files/2301.02273v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aae129dcdf7096d1bb0df68cb336f173f9b077ab --- /dev/null +++ b/XtE0T4oBgHgl3EQfWADS/content/tmp_files/2301.02273v1.pdf.txt @@ -0,0 +1,2945 @@ +DRAFT +1 +Microphysically modified magnetosonic +modes in collisionless, high-β plasmas +S. Majeski 1†, M. W. Kunz1,2, and J. Squire3 +1Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ +08544, USA +2Princeton Plasma Physics Laboratory, PO Box 451, Princeton, NJ 08543, USA +3Department of Physics, University of Otago, 730 Cumberland St, North Dunedin, Dunedin +9016, New Zealand +(compiled on 9 January 2023) +With the support of hybrid-kinetic simulations and analytic theory, we describe the +nonlinear behaviour of long-wavelength non-propagating (NP) modes and fast magne- +tosonic waves in high-β collisionless plasmas, with particular attention to their excitation +of, and reaction to, kinetic micro-instabilities. The perpendicularly pressure balanced +polarization of NP modes produces an excess of perpendicular pressure over parallel +pressure in regions where the plasma β is increased. For mode amplitudes δB/B0 ≳ +0.3, this excess excites the mirror instability. Particle scattering off these micro-scale +mirrors frustrates the nonlinear saturation of transit-time damping, ensuring that large- +amplitude NP modes continue their decay to small amplitudes. At asymptotically large +wavelengths, we predict that the mirror-induced scattering will be large enough to inter- +rupt transit-time damping entirely, isotropizing the pressure perturbations and morphing +the collisionless NP mode into the magnetohydrodynamic (MHD) entropy mode. In fast +waves, a fluctuating pressure anisotropy drives both mirror and firehose instabilities +when the wave amplitude satisfies δB/B0 ≳ 2β−1. The induced particle scattering leads +to delayed shock formation and MHD-like wave dynamics. Taken alongside prior work on +self-interrupting Alfv´en waves and self-sustaining ion-acoustic waves, our results establish +a foundation for new theories of electromagnetic turbulence in low-collisionality, high-β +plasmas such as the intracluster medium, radiatively inefficient accretion flows, and the +near-Earth solar wind. +1. Introduction +1.1. Context and motivation +Nearly half of all the baryonic matter in the Universe resides in a hot and dilute +plasma state, in which Coulomb collisions are relatively rare and cosmic magnetic fields +greatly influence the trajectories of the constituent particles. Examples include the warm- +hot intergalactic medium, having number densities n ≳ 10−6 cm−3 and temperatures +T ∼ 105–107 K, and the intracluster medium of galaxy clusters, with n ≳ 10−3 cm−3 +and T +∼ 107–108 K. Radiatively inefficient accretion flows such as that onto the +supermassive black hole at the Galactic centre, as well as the Solar wind that pervades +interplanetary space, provide smaller-scale examples of systems characterized by large +collisional mean free paths and small particle gyro-radii. A key feature of these systems +is that the transport of momentum and heat are anisotropic with respect to the magnetic- +field direction, even when the magnetic energy is much less than the thermal pressure, +† Email address for correspondence: smajeski@princeton.edu +arXiv:2301.02273v1 [astro-ph.HE] 5 Jan 2023 + +2 +S. Majeski, M. W. Kunz, and J. Squire +viz. β .= 8πnT/B2 ≫ 1. This spatial anisotropy is a direct result of the velocity-space +anisotropy in the particle distribution function, which is allowed by the rarity of particle- +particle collisions and shaped by the particles’ primary allegiance to the local magnetic- +field direction. In high-β plasmas, such field-biased deviations from local thermodynamic +equilibrium can have important dynamical consequences on both the large ‘fluid’ scales +and the small plasma-kinetic ‘micro’ scales. It is this multi-scale connection between a +high-β plasma’s thermodynamics and its fluid dynamics that is the focus of this paper. +In particular, by elucidating the non-linear behaviour of long-wavelength magnetosonic +modes, and placing our findings in the company of complementary work on Alfv´enic and +acoustic fluctuations, we demonstrate that even textbook examples of plasma dynamics, +such as basic waves, can be fundamentally different in weakly collisional, high-β plasmas. +1.2. Pressure anisotropy, micro-instabilities, and collisionless damping +Collisionless and weakly collisional plasmas possess particles whose motions are bound +by adiabatic invariants that are otherwise broken in highly collisional MHD plasmas. +While there are three adiabatic invariants most commonly considered in plasma physics, +two of them – the magnetic moment µ for cross-field gyro-motion and the bounce +invariant J for field-parallel bounce motion – are associated with frequencies that are +generally large enough for these invariants to be approximately conserved even when +some collisions are present. For describing collective behaviour, these invariants are +often adapted into the form of the double adiabats p⊥/nB and p∥B2/n3, which are +conserved in time along the flow of the plasma if the density n and magnetic-field +strength B change slowly relative to the periodic (gyro- or bounce) motion. In this +case, the thermal pressure p is split up into components along and across the magnetic- +field direction, p∥ and p⊥ respectively, a result of the invariants each being associated +with different components of the particles’ motions. In essence, the random thermal +motions of a collisionless or weakly collisional plasma are restricted differently depending +on whether they are along or across the magnetic field. Their dynamical importance +with respect to the magnetic field can also be defined separately, as β⊥ .= 8πp⊥/B2 and +β∥ .= 8πp∥/B2. Natural variations in the plasma density and magnetic-field strength +that are present, coupled with approximate double-adiabatic invariance, lead to the +development of pressure anisotropy ∆ .= p⊥/p∥ − 1 ̸= 0. Often, the magnitude of ∆ +is small and may have little effect on a plasma’s evolution. However, in high-β plasmas +where the thermal pressure is much larger than the magnetic energy, even small deviations +from thermal isotropy may be significant enough to grant the pressure anisotropy a role +comparable to that of the magnetic field. +Two mechanisms by which the pressure anisotropy plays this elevated role are the +modification of magnetic-field-line tension and the triggering of rapidly growing, kinetic +micro-instabilities. An illustration of the former mechanism is a process named ‘Alfv´en +wave interruption’ (Squire et al. 2016, 2017a,b), in which a linearly polarized Alfv´en wave +whose amplitude satisfies (δB⊥/B)2 ≳ 2/β adiabatically generates a pressure anisotropy +large enough to nullify the restoring magnetic tension and prevent the wave’s propagation. +In this paper, we are focused primarily on large-scale compressive fluctuations, for which +magnetic tension ends up being of little importance at high β. Our focus is therefore +primarily on the connection that pressure anisotropy has with ion-Larmor-scale kinetic +instabilities, specifically the firehose and mirror instabilities. +The firehose instability is triggered in pressure-anisotropic plasmas satisfying β∥∆ ≲ +−2. This threshold is commonly referred to as the ‘fluid firehose’ threshold, and corre- +sponds to an exact balance between the restoring magnetic tension force and the desta- + +High-β collisionless magnetosonic modes +3 +bilizing viscous stress from the negative pressure anisotropy.1 In this case, when small +perpendicular fluctuations in the magnetic field are present, the excess parallel pressure +leads to a centrifugal force that acts in the bends of the magnetic-field lines. When the +pressure anisotropy is sufficiently negative, this force cannot be stably balanced by the +magnetic tension and the bends grow very rapidly (Parker 1958; Vedenov & Sagdeev +1958), increasingly so on smaller lengthscales (down to the ion-Larmor scale, where +they are stabilized by finite-Larmor-radius effects; Kennel & Sagdeev 1967; Davidson +& V¨olk 1968; Yoon et al. 1993; Hellinger & Matsumoto 2000). In a driven system, the +unstable pressure anisotropy is regulated through a combination of the particles pitch- +angle scattering off of these bends and the compensating positive pressure anisotropy +associated with the growing magnetic perturbations (Schekochihin et al. 2008; Rosin +et al. 2011; Kunz et al. 2014a). Conversely, the mirror instability is triggered when +an excessively positive pressure anisotropy satisfies β⊥∆ ≳ 1 (Barnes 1966; Hasegawa +1969). In this case, the enhanced perpendicular pressure is able to push out against local +decrements in the magnetic-field strength, causing ion-Larmor-scale ‘magnetic mirrors’ +to form. These mirrors resonantly confine particles with large pitch angles (v⊥ > v∥) +through their conservation of µ (e.g., Southwood & Kivelson 1993). The anisotropic +thermal energy of these resonant particles reinforces the outward push against the field +lines, further growing the fluctuations (and thus the confining mirror force) until the +ends of the mirrors become so kinked that the particles can pitch-angle scatter off of +their sharp edges and regulate the pressure anisotropy (Kunz et al. 2014a; Riquelme +et al. 2015; Rincon et al. 2015). +Kunz et al. (2020) demonstrated that these kinetic instabilities interfere with the +collisionless damping of long-wavelength, parallel-propagating ion-acoustic waves (IAWs). +Namely, IAW amplitudes satisfying |δn/n| ≳ 2/β generate a pressure anisotropy large +enough to drive firehose and mirror instabilities, whose associated scattering and trapping +impede the maintenance of Landau resonances that enable such waves’ otherwise potent +decay. The result is self-sustaining wave dynamics that evince a weakly collisional plasma: +the ion distribution function is near-Maxwellian, the field-parallel flow of heat resembles +its Braginskii form (except in regions where large-amplitude magnetic mirrors strongly +suppress particle transport), and the relations between various thermodynamic quantities +are more ‘fluid-like’ than kinetic. +1.3. Non-propagating modes, fast waves, and oblique IAWs +In this work, a combination of elements from both Alfv´en waves and IAWs is in- +vestigated in the study of collisionless magnetosonic modes – namely, non-propagating +(NP) modes (in §2), fast waves (in §3), and to a more limited extent oblique IAWs +(in appendix C). We investigate fast waves in the limit of perpendicular propagation, +in which magnetic tension and collisionless damping play no role, but the associated +fluctuations in B and n drive unstable pressure anisotropies. The NP modes, on the +other hand, are highly oblique, perpendicular-pressure-balanced structures, in which +collisionless transit-time damping (or ‘Barnes damping’; Barnes 1966) is responsible +for the entirety of the modes’ dynamics. Barnes damping is a form of Landau (1946) +damping in which sinusoidal fluctuations in magnetic-field strength caused by an oblique +1Certain conditions can lead to the dominance of a resonant oblique firehose instability having +a less stringent threshold of β∥∆ ≲ −1.4 (Hellinger & Matsumoto 2000; A.F.A. Bott et al., in +preparation). As none of the magnetosonic fluctuations investigated in this paper are subject to +self-interruption, the difference between −2 and −1.4 is of little consequence, and we generically +refer to the ‘firehose threshold’ as being at −2. + +4 +S. Majeski, M. W. Kunz, and J. Squire +perturbation (magnetic ‘mirrors’) resonantly confine µ-conserving particles and perform +work on their guiding centres, thereby transferring free energy from the electromagnetic +perturbations to the particles and damping the mode. For large values of β, the damping +rate of the NP mode is relatively slow, and nonlinear saturation of the damping process +can occur before the mode decays by a significant fraction. In this case, trapped particles +in near resonance with the mode are rearranged in phase space, flattening the velocity +distribution function of the particles f(v∥) in the vicinity of the phase velocity (in this +case, v∥ ∼ 0). Once (∂f/∂v∥)|0 ∼ 0, there is no more free energy left to be gained by the +distribution from rearranging particles, and the damping process stalls. This swapping +of phase-space positions occurs on the order of a bounce time, ∼Ω−1 +b , which is the +time it takes for a (just barely) trapped particle to make a full orbit of its confining +magnetic mirror. The larger amplitude a mode, the shorter its bounce time, so the +nonlinear saturation ensures that large-amplitude NP modes are longer lived than their +small-amplitude counterparts. The principal question here is to what extent the pressure +anisotropy associated with these modes affects their character and longevity. +2. Non-propagating modes: Suppression of nonlinear saturation +2.1. Theory +2.1.1. Model equations and assumptions +The linear evolution of the NP mode at long wavelengths can be treated analytically +in the drift-kinetic approximation, in which all relevant time- and lengthscales are much +larger than those associated with the particles’ gyromotion and the velocity distribution +function of the particles is gyrotropic. We adopt this framework, and further simplify +the calculation by treating the electrons as a massless, neutralizing, isothermal fluid +having constant temperature Te.2 In this model the velocity of magnetic-field lines, and +equivalently the perpendicular fluid flow, is captured by the E × B drift velocity u⊥. +The perpendicular velocity peculiar to this drift, denoted by w⊥, then describes the +perpendicular particle motion relative to the field lines and the fluid flow, under the +constraint that the magnetic moment µ .= miw2 +⊥/2B is conserved. The component of the +particle velocity directed along the local magnetic-field direction is denoted by v∥. +In what follows, we solve for the evolution of small perturbations δf(t, r, v∥, w⊥) to a +spatially uniform ‘background’ ion velocity distribution function F0(v∥, w⊥). The parallel +(∥) and perpendicular (⊥) coordinate directions are fixed with respect to a uniform +background magnetic field, B0. Assuming that spatial variations in the plasma are due +only to a sinusoidal perturbation having wavenumbers k∥ and k⊥, the relevant equations +in their linearized forms are the drift-kinetic Vlasov equation, +� ∂ +∂t + ik∥v∥ +� � +δf + δB∥ +B0 +w⊥ +2 +∂F0 +∂w⊥ +� ++ e +mi +δE∥ +∂F0 +∂v∥ +− ik∥ +δB∥ +B0 +w2 +⊥ +2 +∂F0 +∂v∥ += 0; +(2.1a) +the force equation for the evolution of the drift velocity, +du⊥ +dt += − ik⊥ +min0 +� +δp⊥i + Teδn +� +− ik⊥v2 +A +δB∥ +B0 ++ ik∥v2 +A +δB⊥ +B0 +; +(2.1b) +2The choice of isothermal electrons is for consistency with the simulations performed using the +Pegasus++ hybrid-kinetic particle-in-cell code (see §2.2), though it can be justified physically +in some weakly collisional plasmas such as the ICM, where the electrons are collisional enough +to remain near-Maxwellian and fast enough to be approximately isothermal along perturbed +magnetic-field lines (e.g., Kunz 2011). + +High-β collisionless magnetosonic modes +5 +the ideal induction equation governing the parallel and perpendicular components of the +perturbed magnetic field δB, +d +dt +δB∥ +B0 += −ik⊥u⊥ +and +d +dt +δB⊥ +B0 += ik∥u⊥; +(2.1c) +and a generalized Ohm’s law for the parallel electric field, +δE∥ = −ik∥ +Te +e +δn +n0 +. +(2.1d) +The perturbed number density and perpendicular ion pressure are given by +δn .= +� +d3v δf +and +δp⊥i .= +� +d3v 1 +2miw2 +⊥δf, +(2.2) +respectively, with d3v = 2πw⊥dw⊥dv∥. The other symbols have their usual meanings: e +is the elementary charge, mi is the ion mass, and vA .= B0/(4πmin0)1/2 is the Alfv´en +speed given B0 and a uniform background density n0 (the zeroth moment of F0). Note +that u⊥ is not an explicit moment of the perturbed distribution function, and must be +evolved independently using (2.1b). This combination of the drift-kinetic equation with +a fluid equation for the drift velocity and a frozen-in magnetic field is commonly referred +to as ‘kinetic MHD’ (Kulsrud 1964, 1983). +At this point we take F0 to be a stationary, isotropic, Maxwell–Boltzmann distribution, +F0 = FM(v), with +� +d3v FM(v) = n0 and +� +d3v miv2FM(v) = 3n0Ti0 .= 3pi0. This not +only simplifies the analysis, but also ensures that the background distribution function +itself is not kinetically unstable. Equation (2.1a) can then be readily integrated in time +to obtain +δf(t, w⊥, v∥) = δf(0, w⊥, v∥) e−ik∥v∥t +− +� t +0 +dt′ FM(v) e−ik∥v∥(t−t′) +� +ik∥v∥ +Te +Ti0 +δn(t′) +n0 +− w2 +⊥ +v2 +th,i +d +dt′ +δB∥(t′) +B0 +� +, +(2.3) +where vth,i .= (2Ti0/mi)1/2 is the ion thermal speed. The first term on the right-hand +side of (2.3) represents the parallel phase mixing of the initial perturbation by the free +streaming of particles along the (unperturbed) magnetic field. If δf(0, w⊥, v∥) ∝ FM(v), +then any velocity-space moment of this term will decay as exp[−(k∥vth,it/2)2]. The second +term in (2.3) captures the self-consistent response of the plasma to the induced parallel +electric field (∝δn/n0) and the magnetic mirror force (∝δB∥/B0). It is this eigenmode +response that we first calculate and discuss, before moving on to take the second moments +of (2.3) and compute the time-dependent pressure anisotropy in §2.1.3. +2.1.2. Eigenmode response for the NP mode +If we take the fluctuation amplitudes to be proportional to exp(−iωt) with complex +frequency ω, the dispersion relation that results after combining (2.1) may be written as +D(ζ) .= +� +ω2 − k2v2 +A +�� +1 + Ti0 +Te ++ ζZ(ζ) +� ++k2 +⊥v2 +th,i ζZ(ζ) +� +1 + Ti0 +Te ++ 1 +2ζZ(ζ) +� += 0, +(2.4) +where k2 = k2 +∥ + k2 +⊥, ζ +.= ω/|k∥|vth,i is the dimensionless phase speed, and Z(ζ) is +the plasma dispersion function. The first term in parentheses captures the combined +restoring force of the magnetic pressure and tension, and indicates that we are examining +magnetosonic modes. Indeed, setting the accompanying multiplicative term in square +brackets to zero provides the dispersion relation for a Landau-damped IAW in the + +6 +S. Majeski, M. W. Kunz, and J. Squire +limit (me/mi)1/2 ≪ 1. The final term in (2.4), proportional to k2 +⊥v2 +th,i, couples these +Alfv´enic and acoustic responses; its presence can be traced back to the final term in (2.3) +representing the mirror force, and thus introduces collisionless damping of the mode +through transit-time damping. +In order to isolate the NP mode, we focus specifically on highly oblique wavenumbers +(k⊥ ≫ k∥) and low frequencies (ζ ≪ 1). In this limit, the plasma dispersion function +in (2.4) can be approximated as Z(ζ) ≈ i√π, and we may simplify the dispersion relation +further by neglecting terms of order ζ2. The result is an approximate expression for the +decay rate of the NP mode: +ζ ≃ − +i +√πβi0 +k2 +k2 +⊥ +, +where +βi0 = 8πpi0 +B2 +0 += +v2 +th,i +v2 +A +. +(2.5) +For ζ ≪ 1 to be satisfied by (2.5), we require that βi0 ≫ k∥/k⊥, which is easily satisfied +by our obliqueness assumption and aligns well with our interest in high-β plasmas. Other +useful properties of the NP mode, such as the proportionalities between δn, δp⊥,i, and +δB∥, can be found by taking moments of the kinetic equation (2.1a): +δn +n0 += −ζZ(ζ) +� +1 + Te +Ti0 +� +1 + ζZ(ζ) +��−1 δB∥ +B0 +≃ − 1 +β0 +k2 +k2 +⊥ +δB∥ +B0 +, +(2.6a) +δp⊥i +pi0 += − Te +Ti0 +δn +n0 ++ 2 ω2 − k2v2 +A +k2 +⊥v2 +th,i +δB∥ +B0 +≃ +� +2 + Te +Ti0 +�δn +n0 +, +(2.6b) +where β0 .= βi0(1 + Te/Ti0). Equation (2.6b) implies approximate perpendicular pressure +balance when k∥ ≪ k⊥, since then +δp⊥i + δpe + δB2 +8π ≃ − +k2 +∥ +k2 +⊥ +δB2 +4π ≪ δB2 +4π . +(2.6c) +Additionally, the parallel ion pressure perturbation is given by +δp∥i +pi0 += − Te +Ti0 +δn +n0 +− 2ζ2� +1 + ζZ(ζ) +��δB∥ +B0 ++ Te +Ti0 +δn +n0 +� +≃ − Te +Ti0 +δn +n0 +, +(2.6d) +so that δp∥i+δpe ≃ 0. Equations (2.5) and (2.6) highlight some of the essential properties +of the NP mode, namely, that it does not oscillate but rather decays slowly at high β, and +that its perturbations to the magnetic-field strength and the density are anti-correlated. +The physical mechanism behind the damping rate is primarily transit-time magnetic +pumping, in which Landau-resonant particles (technically, their guiding centres) that +are trapped between large-scale magnetic mirrors formed by an oblique perturbation in +the magnetic field extract energy from the mirror force. They experience net heating +by betatron acceleration because the number of particles heated in regions where |B| +increases (lower v∥ particles) is greater than the number of particles cooled where |B| +decreases (higher v∥ particles). At higher plasma β this difference is smaller, hence the +β−1 dependence of the damping rate. +This type of collisionless damping is susceptible to nonlinear saturation, whereby +the particles in the well explore the phase space available to them by µ conservation, +phase-mixing out the original Maxwellian according to their differing bounce times and +flattening the distribution function in the magnetic well to create a plateau around v∥ ∼ 0. +This effectively increases the plasma β of the resonant particles, and the damping rate +weakens dramatically. Because of the slow nature of the NP mode’s decay rate at high β, +nonlinear saturation occurs comparatively rapidly, at a rate comparable to the bounce + +High-β collisionless magnetosonic modes +7 +frequency of a thermal particle, +Ωb .= 1 +2k∥vth,i +���� +δB∥ +B0 +���� +1/2 +. +(2.7) +For |δB∥/B0| ≳ β−2 +i0 , the bounce frequency will be larger than the decay rate (2.5), and +thus nonlinear saturation will be important. Because of our interest in plasmas with +β ≫ 1, even modes that may often be considered ‘linear’ in amplitude will thus decay +by only a small amount before experiencing nonlinear saturation, the implication being +that these structures should be long lived. That is, unless some process is able to erode +the resonant plateau in the perturbed distribution function on a timescale ≲Ω−1 +b . +2.1.3. Generation of pressure anisotropy and triggering of the mirror instability +The eigenmode (2.6) implies a dimensionless pressure anisotropy in the ions given by +∆NP ≃ 2 +� +1 + Te +Ti0 +�δn +n0 +≃ − 2 +βi0 +k2 +k2 +⊥ +δB∥ +B0 +. +(2.8) +This suggests that, for δB∥/B0 ∼ 1, the pressure anisotropy associated with the NP mode +is sufficient to excite both the firehose and mirror instabilities, the former occurring in +regions where δB∥ > 0, the latter occurring in regions where δB∥ < 0. There are two +considerations that complicate this conclusion. +The first complication concerns the additional pressure anisotropy that is generated +when the initial perturbation to the distribution function is anisotropically phase mixed +by particles streaming freely along, but not across, the field lines. To see this effect, +let us return to the time-dependent solution for the perturbed distribution function, +equation (2.3), and suppose that, at t = 0, the plasma hosts an isothermal, pressure- +balanced perturbation with +δf(0, w⊥, v∥) = δn(0) +n0 +FM(v) = − 2 +β0 +δB∥(0) +B0 +FM(v). +(2.9) +This initial condition guarantees that the pressure anisotropy that develops as the +particles free stream and the plasma settles into the NP eigenmode is generated self- +consistently and not put in by hand. Calculating the difference of the (1/2)miw2 +⊥ and +miv2 +∥ moments of (2.3) with the initial condition (2.9) yields the following expression for +the time-dependent pressure anisotropy: +∆NP(t) = 2 +�k∥vth,it +2 +�2 +e−(k∥vth,it/2)2� +1 + Te +Ti0 +�δn(0) +n0 ++ +� t +0 +dt′ e−[k∥vth,i(t−t′)/2]2 d +dt′ +δB∥(t′) +B0 ++ 2 +� t +0 +dt′ +�k∥vth,i(t − t′) +2 +�2 +e−[k∥vth,i(t−t′)/2]2 d +dt′ +� Te +Ti0 +δn(t′) +n0 ++ δB∥(t′) +B0 +� +. (2.10) +All terms involving the combination k∥vth,it/2 describe the damping effect of phase +mixing on the moments of the perturbed distribution function due to the production +of fine-scale structure along v∥. As discussed by Kunz et al. (2020, their equation (3.7)), +the first term on the right-hand side of (2.10) captures a transiently produced pressure +anisotropy resulting from the anisotropy of particle motion: as the magnetized particles +free stream along, but not across, the field, the w2 +⊥ and v2 +∥ moments of δf(0) phase +mix differently. The integral terms in (2.10) capture the pressure anisotropy driven by + +8 +S. Majeski, M. W. Kunz, and J. Squire +(a) +(b) +Figure 1: (a) Solution of (2.10) using the method presented in appendix A for the time- +dependent root-mean-square pressure anisotropy of a linear NP mode with wavenumber +k∥ and dimensionless initial amplitude α .= δB∥(0)/B0 for βi0 = 16 and various Te/Ti0. +The small oscillations present after the initial adjustment are due to fast waves generated +as the isothermal, pressure-balanced initial condition settles into the NP eigenmode. The +approximate analytic solution (2.12) is shown with the dashed line. (b) Maximum pressure +anisotropy (divided by α) vs. Te/Ti0; its values at Te/Ti0 = 1/2, 1, and 2 are indicated. +adiabatic invariance as the mode is excited and then decays in time. It is this contribution +to ∆NP(t) that includes the pressure anisotropy of the eigenmode, equation (2.8). +The integrals in (2.10) can be computed numerically (see appendix A) and the pressure +anisotropy ∆NP(t) determined for a given initial mode amplitude +α .= +���� +δB∥(0) +B0 +���� . +(2.11) +The result is shown in figure 1(a) at a selection of values of Te/Ti0. The initial rise in +∆NP is due to a combination of the anisotropic phase mixing of the initially perturbed +density and the pressure anisotropy adiabatically produced as the system settles into +the NP eigenmode. After approximately one thermal-crossing time of the mode’s parallel +wavelength, the eigenmode is established and the slow exponential decay of ∆NP seen in +the figure reflects the Barnes damping of the mode. (The higher-frequency oscillations +seen on top of this slow decay are caused by fast modes excited by the initial conditions +and represent rapid oscillations about perpendicular pressure balance.) An approximate +analytic solution for ∆NP(t) may be obtained in the limit of βi0 ≫ 1, (k∥/k⊥)2 ≪ 1, and +Te/Ti0 ∼ 1 upon substituting the damped eigenmode (2.6a) into the time integrals in +equation (2.10). The result is that +∆NP(t) ≃ 2τ 2e−τ 2� +1 + Te +Ti0 +�δn(0) +n0 +− +� +erf(τ) − τ +√πe−τ 2� 2 +βi0 +δB∥(t) +B0 +(2.12a) += − +� +2τ 2e−τ 2 + e−2iζτ +� +erf(τ) − τ +√πe−τ 2�� 2 +βi0 +δB∥(0) +B0 +, +(2.12b) +where τ .= k∥vth,it/2. The term in square brackets goes as ∼2τ 2 + τ/√π for early times, +suggesting that the plasma would become mirror-unstable at a time tm ∼ (√αk∥vth,i)−1, +comparable to the inverse of the bounce frequency (2.7). With the mode then slowly + +High-β collisionless magnetosonic modes +9 +decaying exponentially, the maximum value of the pressure anisotropy may be estimated +by setting exp(−2iζτ) ≃ 1 and maximizing (2.12b) with respect to τ. The result is +a maximum pressure anisotropy ≃2.6αβ−1 +i0 +(cf. (2.8)) occurring at k∥vth,it ≃ 2.3. The +approximate solution (2.12) is traced by the dashed line in figure 1(a), and is a manifestly +good description of the full solution. +The second complication when using (2.8) to determine the kinetic stability of the +NP mode is related to how the mode perturbs the perpendicular and parallel plasma β +parameters that feature in the firehose and mirror instability thresholds. Using (2.6) and +that δB⊥ = −(k∥/k⊥)δB∥, one obtains +β∥i ≃ βi0 +� +1 + 2δB∥ +B0 ++ k2 +k2 +⊥ +δB2 +∥ +B2 +0 +�−1� +1 − k2 +k2 +⊥ +1 +βi0 +Te +Ti0 +� +1 + Te +Ti0 +�−1 δB∥ +B0 +� +, +(2.13a) +β⊥i ≃ βi0 +� +1 + 2δB∥ +B0 ++ k2 +k2 +⊥ +δB2 +∥ +B2 +0 +�−1� +1 − k2 +k2 +⊥ +1 +βi0 +� +2 + Te +Ti0 +�� +1 + Te +Ti0 +�−1 δB∥ +B0 +� +. (2.13b) +The final terms in both of these expressions may be dropped in the limit of βi0 ≫ 1. +Combining the result with (2.8) yields +β∥i∆NP ≈ β⊥i∆NP ≈ −2δB∥ +B0 +� +1 + 2δB∥ +B0 ++ k2 +k2 +⊥ +δB2 +∥ +B2 +0 +�−1 +. +(2.14) +Equation (2.14) indicates that is impossible to produce a pressure anisotropy that is +sufficiently negative to destabilize the plasma to the firehose. Regions in which ∆NP < 0 +also have a reduced plasma β, and so the more negative the anisotropy becomes (for +larger δB∥ > 0), the further the firehose threshold (≈ − 2/β∥i) moves away. In contrast, +the plasma in regions where δB∥ < 0 that acquire a positive pressure anisotropy have an +easier time of reaching the reduced mirror threshold (≈1/β⊥i). Setting the right-hand side +of (2.14) to unity and solving for δB∥ = −|δB∥| then provides the following amplitude +threshold for the NP mode to trigger the mirror instability: +���� +δB∥ +B0 +���� ≳ 0.3 +(NP mode amplitude threshold). +(2.15) +When this criterion is satisfied, we anticipate regions of kinetically unstable plasma to +be localized to where δB∥ < 0 and to host ion-Larmor-scale mirrors. +With these predictions borne in mind, we now determine the spatial extent of these +mirror-unstable regions and discuss how the mirrors growing within them evolve to +regulate the pressure anisotropy. +2.1.4. Regulation of pressure anisotropy by the mirror instability +In §2.1.3, we showed that the plasma where δB∥ < 0 becomes mirror-unstable at tm ∼ +(√αk∥vth,i)−1 if initialized from isothermal pressure balance. With α ≳ 0.3 (i.e., when +instability is possible), this time is comparable to the timescale over which the NP mode’s +pressure anisotropy is set up (see figure 1). We may then view the mirror instability as +growing on top of an otherwise weakly decaying positive pressure anisotropy satisfying +(2.14) with δB∥ < 0. The maximum growth rate of the instability depends on how far the +local pressure anisotropy ventures beyond the instability threshold, Λm .= ∆ − β−1 +⊥i > 0. +In the asymptotic limit β⊥iΛm ≪ 1, the maximum mirror growth rate and associated +wavenumber are given by (Hellinger 2007; A.F.A. Bott et al., in preparation) +γm/Ωi ≈ 0.06β⊥iΛ2 +m, +k∥,mρi ≈ 0.2β⊥iΛm, +k⊥,mρi ≈ 0.6(β⊥iΛm)1/2. +(2.16) + +10 +S. Majeski, M. W. Kunz, and J. Squire +(a) +(b) +Figure 2: (a) Perpendicular (k⊥,m) and parallel (k∥,m) wavenumbers of the fastest-growing +mirror mode having growth rate γm, all computed from linear Vlasov–Maxwell theory +using the instability parameter Λm corresponding to a NP mode with α .= |δB∥/B0| and +k⊥/k∥ = 4 in a βi0 = 16 plasma (see (2.14); these values are weakly dependent upon +βi0 and k⊥/k∥ so long as βi0 ≳ 10 and k ≃ k⊥). The dotted lines trace the asymptotic +expressions from (2.16), valid when β⊥iΛm ≪ 1. (b) The predicted number of mirrors +Nm within the δB∥ < 0 region of a NP mode having wavelength λ∥ and amplitude α +(see (2.20)). +However, because of the sensitive dependence of the instability parameter β⊥iΛm on +the NP mode amplitude (see (2.14)), with its value ranging from ∼1 to ∼100 for +α ∈ [0.4, 0.9], only very marginally unstable NP modes (viz., α ≃ 0.3) satisfy the +ordering used to derive (2.16). For arbitrary NP mode amplitude α, the growth rate and +wavenumber of the fastest-growing mirrors can be calculated numerically by solving the +linearized Vlasov–Maxwell equations for a bi-Maxwellian plasma (A.F.A. Bott, private +communication) with (2.14) specifying the pressure anisotropy; their values are shown +versus α in figure 2(a). For this figure we used βi0 = 16 and k⊥/k∥ = 4, although +the values shown are fairly insensitive to either parameter as long as βi0 ≳ 10 and +(k/k⊥)2 ≈ 1. The asymptotic expressions from (2.16) are shown by the dotted lines, and +appear to be accurate only for α ≲ 0.4. As the NP mode amplitude approaches unity, +the maximal mirror instability growth rate and associated wavenumber tend towards +γm/Ωi ≈ 0.2Λm, +k∥,mρi ≈ 0.6, +k⊥,mρi ≈ 1.2, +(2.17) +respectively. +In order for the mirror instability to be relevant to the linear evolution of the NP +mode, two criteria must be satisfied. First, the mirror growth rate must be much larger +than the rate at which the NP mode decays (2.5), i.e., γm +√πβi ≫ k∥vth,i. This condition +appears to be trivially satisfied in high-β plasmas for unstable NP modes with parallel +wavelengths λ∥ ≳ 103ρi. The second criterion is that the mirror modes must actually fit +inside the length of the region that is mirror unstable, viz. 2π/k∥,m ≲ ℓmirror. We estimate +ℓmirror by asking where in the NP mode the quantity (2.14) is larger than unity: +Λm ≃ 1 +βi0 +� +−1 − 4δB∥ +B0 +− k2 +k2 +⊥ +δB2 +∥ +B2 +0 +� +≳ 0. +(2.18) +Because the leading-order eigenvector components are all real, we can take δB∥ = +−αB0 cos(k∥x + k⊥y) (as used in our simulations; see §2.2). Courtesy of our assump- + +High-β collisionless magnetosonic modes +11 +tion that k⊥ ≫ k∥, we have that δB⊥ ≪ δB∥, so the field lines are approximately +straight everywhere and the paths taken by the trapped particles as they bounce are +approximately parallel to B0. Then, taking the appropriate root of (2.18) to ensure that +the inverse cosine is defined for mirror-unstable amplitudes, we find that the length of +the mirror-unstable portion of the wave satisfies +ℓmirror ≈ λ∥ +π cos−1 +�2 − +� +4 − k2/k2 +⊥ +α +� +.= fmλ∥. +(2.19) +For α ≈ 0.3–0.9 and k∥ ≪ k⊥, fm ≈ 0.1–0.4. The number of maximally growing mirrors +that can fit within ℓmirror is then +Nm ≈ fm +2 +�k∥,mρi +2π +��λ∥ +ρi +� +. +(2.20) +In writing (2.20), we have included an additional factor of ≈1/2 to account for the fact +that the pressure anisotropy is not expected to be uniform within the mirror-unstable +region and so the full extent of ℓmirror is unlikely to be filled with mirrors of identical +wavelengths; the bespoke factor of ≈1/2 was obtained empirically from examining the +hybrid-kinetic simulations of unstable NP modes presented in §2.2. A further, and final, +adjustment to Nm accounts for the fact that the ion-Larmor radius ρi ∝ √T⊥i/B in the +mirror-unstable region is larger than ρi0, primarily because of the decrease in the local +magnetic-field strength. Using (2.6) to express δT⊥i in terms of δB∥, we find that +ρi +ρi0 +≈ +� +1 − α +βi0 +k2 +k2 +⊥ +�1/2� +1 − 2α + k2 +k2 +⊥ +α2 +�−1/2 +. +(2.21) +With k∥,mρi taken from figure 2(a), we can assemble (2.18)–(2.21) to predict Nm for a +given λ∥/ρi0, α, and k∥/k⊥ of the NP mode at βi0 ≫ 1. +The result of this procedure is shown in figure 2(b) as the open circles. Note that the +number of mirrors Nm is fairly independent of the NP mode amplitude for α ≳ 0.4, with +the consequence that several mirrors can fit within the mirror-unstable region of a NP +mode with λ∥ ∼ 103ρi0. However, at the critical amplitude α ≈ 0.3, only one or two +mirrors are predicted to fit if λ∥ ∼ 103ρi0. In this case, the mirror instability might be +ineffective at regulating the pressure anisotropy. +In summary, we predict that a NP mode with α ≳ 0.4 and λ∥ ≳ 103ρi0 should be able +to support a robust collection of mirror-unstable fluctuations. +2.1.5. Effective collisionality induced by the mirror instability +We now seek an estimate for the effective collision frequency instigated by these mirror- +unstable distortions in the magnetic-field lines. For this, we follow the arguments of +Newman (2020) for the pitch-angle diffusion of charged particles in regions of Larmor- +scale magnetic irregularities. First, we conjecture that each encounter of an ion with the +edges of a single mirror depletes the plasma’s temperature anisotropy A .= w2 +⊥/2 − v2 +∥ by +a fraction χ (here, the overline indicates an average over the ion distribution function). +Following Newman (2020), we identify χ with (3/2) sin2 ϑ, where ϑ is the local deflection +angle of the perturbed magnetic-field lines. We estimate sin2 ϑ ∼ |δBm/B|2, and argue +that the energy of the mirror modes will be comparable to the free energy available to + +12 +S. Majeski, M. W. Kunz, and J. Squire +them in the unstable distribution function, viz. |δBm/B|2 ∼ Λm (Kunz et al. 2015).3 The +result is that +χ ∼ Λm. +(2.22) +In words, larger pressure anisotropies produce larger mirror fluctuations, which in turn +are able to decrease by larger amounts the pitch angles of trapped particles. +To obtain the effective collision frequency νeff, we then multiply χ by the number of +Larmor-scale mirrors per unit time encountered by a typical particle. For a NP mode +with amplitude α ≳ 0.4, the criterion for a particle to be able to pass through the +NP mode’s enhancement in |B| is v∥/w⊥ ≳ +� +4/3. In other words, for a near-Maxwellian +distribution of particle velocities, a majority of the particles will be confined to the trough +of the NP mode where ion-Larmor-scale mirrors should be present, passing through this +mirror-unstable region twice per bounce time. In this case, Nm scattering mirrors are +encountered by each trapped particle every transit time ∆t ≈ πΩ−1 +b . The average rate +of change of the ion anisotropy is then +∆A +∆t ≈ −χ +πNmΩbA .= −νeffA, +(2.23) +where in the last equality we have introduced the effective collision frequency νeff. +Assembling (2.7) and (2.18)–(2.23), we find that +νeff ≈ Gβ−1 +i0 Ωi0, +(2.24a) +where +G .= k∥,mρi +4π2 +� +α−2α2 + k2 +k2 +⊥ +α3 +�1/2� +−1+4α− k2 +k2 +⊥ +α2 +� +cos−1 +�2 − +� +4 − k2/k2 +⊥ +α +� +(2.24b) +is a function of only the amplitude and wavenumber obliquity of the NP mode. +Equation (2.24) states that the predicted νeff is independent of the wavelength of the +NP mode and increases with increasing α, key features that are tested (and confirmed) +in §2.2.6. The predicted dependence of νeff upon α at βi0 = 16 and k⊥/k∥ = 4 is shown in +figure 3(a); νeff may be obtained for any βi0 by multiplying the plotted values by 16/βi0. +The predicted collision frequency drops gradually between α = 0.9 and 0.5, and then falls +sharply by more than an order of magnitude to νeff ∼ 10−4β−1 +i0 Ωi0 at α = 0.3. In panel (b), +we plot the minimum parallel wavelength λ∥ of a NP mode for which νeff∆t ⩾ 1, where +∆t = πΩ−1 +b . Such modes should host mirrors whose scattering frequency is comparable +to the transit time. Note that, for α = 0.3 and βi0 = 16, λ∥/ρi0 must be ≳105 for the +scattering frequency to be larger than the inverse transit time. It is worth bearing these +numbers in mind when interpreting the simulation results presented in §§2.2.3 and 2.2.6. +2.1.6. Suppression of nonlinear saturation of the NP mode +Once νeff becomes competitive with the bounce frequency, the induced scattering will +isotropize the ion distribution function faster than the nonlinear saturation can maintain +the plateau in δf(v∥) around v∥ ∼ 0. In this case, the nonlinear saturation is suppressed +and the NP mode should resume its decay at a rate comparable to (2.5). At some +point during this decay, the mode amplitude will pass below its critical threshold for +triggering the mirror instability (2.15), and the mirror modes themselves will become +3For plasmas in which the pressure anisotropy is persistently driven (e.g., by a background shear +flow or double-adiabatic compression) rather than supplied as an initial condition, the mirror +instability can grow to amplitudes δBm/B ≈ 0.3 before saturating through strong pitch-angle +scattering (Kunz et al. 2014b; Riquelme et al. 2015; Sironi & Narayan 2015). + +High-β collisionless magnetosonic modes +13 +(a) +(b) +Figure 3: (a) Predicted scattering frequency νeff (see (2.24)) caused by the mirror +instability for a NP mode with amplitude α, using the values of k∥,mρi in figure 2(a). +(b) Minimum parallel wavelength λ∥ of a NP mode for which νeff∆t ⩾ 1, where +∆t = πΩ−1 +b . Such modes should host mirrors whose scattering frequency is comparable +to the transit time. The data in both panels correspond to βi0 = 16; to re-scale them for +any βi0, multiply νeff by 16/βi0 and λ∥ by βi0/16. +short-wavelength decaying NP modes. Near the mirror-instability threshold, these short- +wavelength NP modes decay very slowly, and so the associated magnetic-field-strength +fluctuations will remain nonlinear for some time after the large-scale NP mode is no longer +formally mirror unstable. We therefore conjecture that the NP mode will continue to +decay until the mirror fluctuations (and their induced scattering) have had sufficient time +to dissipate. Excepting perhaps the case of asymptotically long NP mode wavelengths, +then, there should be some delay between when the NP mode passes below threshold +and when its nonlinear saturation is re-established. +The preceding arguments imply that four distinct regimes exist for collisionless NP +modes in high-β plasmas: (i) When the mode amplitude satisfies |δB∥/B0| ≲ β−2 +i0 , +its behaviour is nearly linear. The rate of Barnes damping is faster than the bounce +frequency, therefore allowing a substantial fraction of the initial mode amplitude to +decay prior to the onset of nonlinear saturation. (ii) If β−2 +i0 +≲ |δB∥/B0| ≲ 0.3, the +pressure anisotropy associated with the mode is too small to trigger the mirror instability, +but the rate at which nonlinear saturation flattens the distribution function is greater +than the Barnes damping rate. At these amplitudes then, nonlinear saturation occurs +before the mode can decay appreciably, implying these pressure-balanced structures are +thus long-lived. (iii) When |δB∥/B0| ≳ 0.3, the pressure anisotropy of the NP mode +triggers the mirror instability in regions where δB∥ < 0 and eventually introduces an +effective collisionality that, for sufficiently large NP mode wavelengths, suppresses the +maintenance of a nonlinear plateau. As a result, linear decay resumes until the NP mode +decays back well below its amplitude threshold. (iv) Because the induced scattering +rate (2.24) does not scale with the wavelength of the NP mode, one might expect a +fourth fluid-like regime at very long wavelengths, when νeff ≫ k∥vth,i and the collisionless +damping is arrested altogether. We discuss the realizability of this fourth regime and +speculate on its behaviour in §2.2.6. + +14 +S. Majeski, M. W. Kunz, and J. Squire +2.2. Numerical results +2.2.1. Method of solution and initial conditions +To test the theory presented in §2.1 and explore the nonlinear evolution of a mirror- +infested NP mode, we employ the hybrid-kinetic particle-in-cell code Pegasus++ (Kunz +et al. 2014b; Arzamasskiy et al., in preparation). Pegasus++ evolves the ion distribution +function f(t, r, v) using a collection of positively charged macro-particles that interact +with the self-consistent electromagnetic fields E(t, r) and B(t, r), which are in turn +evolved on a discrete mesh using Faraday’s law and a generalized Ohm’s law that +includes the inductive electric field, the Hall effect, and a thermoelectric field caused +by pressure gradients in the (assumed massless) electron fluid. The latter ensures quasi- +neutrality. For simplicity, we adopt an isothermal equation of state for the electrons with +temperature Te = Ti0. Both the interpolation of fields to the macro-particle locations, +and the deposition of the macro-particles’ phase-space information on the mesh, are +performed using second-order-accurate triangle-shaped stencils. +All simulations of the NP mode are performed on a two-dimensional mesh that is +elongated in the direction of a mean magnetic field B0 = B0ˆx and spans one full NP +mode wavelength, Lx × Ly = λ∥ × λ⊥. The latter ranges from λ∥ = 1000ρi0 to 4000ρi0, +with aspect ratios of either λ∥/λ⊥ = 4 or 8. When varying these two dimensions, the +transverse dimension is never smaller than 250ρi0, thereby guaranteeing sufficient scale +separation between the NP mode and any ion-Larmor-scale instabilities. In all runs, the +spatial resolution is ∆x = ∆y ≃ 0.3ρi0 and the number of macro-particles per cell is +either 104 or 5 × 103 (the latter used only in our largest simulations); these values are +similar to those used in previously published Pegasus simulations of collisionless Alfv´en +waves (Squire et al. 2017a) and IAWs (Kunz et al. 2020) in firehose/mirror-susceptible +plasmas. +At t = 0 we perturb the magnetic field using the vector potential +A(x, y) = −αB0 +|k| sin(k∥x + k⊥y)ˆz, +(2.25) +where k∥ = 2π/λ∥, k⊥ = 2π/λ⊥, and α is a dimensionless number quantifying the mode +amplitude. To excite the NP mode, the associated change in the magnetic pressure, +δB2 +8π = −αB2 +0 +8π cos(k∥x + k⊥y) +�2k⊥ +|k| − α cos(k∥x + k⊥y) +� +, +(2.26) +must be exactly balanced by a perturbation to the perpendicular pressure of the plasma +(cf. (2.6c)). In order to keep the initialization of the latter relatively simple, we choose +to begin not from an exact NP eigenmode but rather from an isothermal perturbation +to the plasma density δn, in which case the perturbed perpendicular pressure is simply +δp⊥ = δn(Ti0 + Te). Balancing this expression by (2.26) and solving for δn leads to the +initial ion distribution function +f(0, x, y, v) = FM(v) +� +1 + α +β0 +cos(k∥x + k⊥y) +�2k⊥ +|k| − α cos(k∥x + k⊥y) +�� +. +(2.27) +In this case, the initial total (magnetic plus thermal) pressure in the simulation domain +is constant and equal to (B2 +0/8π)(1 + β0); recall that β0 .= βi0(1 + Te/Ti0). Starting +from a pressure-isotropic plasma has the advantage that any pressure anisotropy that +develops is generated self-consistently and not put in by hand. It is also consistent with +the assumptions made to obtain the analytic solution for ∆NP(t), equation (2.10). +In all of our simulations, we set βi0 = 16 so that it is large enough to agree with +asymptotic expressions derived in §2.1, but not so large that we cannot capture a full + +High-β collisionless magnetosonic modes +15 +decay time of the linear NP decay rate. We vary α ∈ [0.1, 0.8], spanning the predicted NP +amplitude threshold for triggering the mirror instability (2.15). Special attention is paid +to the case with λ∥ = 2000ρi0, λ⊥ = 500ρi0, and α = 0.8; we refer to this as our fiducial +case. Hereafter, ⟨ · ⟩ denotes a spatial average taken over the entire domain, and ⟨ · ⟩k +denotes a spatial average taken over the y-direction while accounting for the changing +position of the wavefront (so as to align all of the perturbed and unperturbed regions +within the domain). The latter is referred to as a ‘wavefront average’; note that it leaves +the x-coordinate (in the direction of B0) unchanged. +2.2.2. Overall evolution of the fiducial run +We begin our discussion of the Pegasus++ results by using the fiducial run to make +contact with some of the predictions laid out in §2.1. These predictions include the +excitation and subsequent linear collisionless damping of the NP mode, its nonlinear +saturation, the simultaneous generation of mirror-unstable pressure anisotropy in the +regions of the mode where δB∥ < 0, and the resumption of linear damping following +the pitch-angle scattering of trapped ions by the saturated Larmor-scale mirrors at a +rate larger than the bounce frequency. Figure 4 illustrates these evolutionary phases by +depicting the amplitude of the NP mode versus time. After a rapid adjustment from the +isothermal pressure-balanced initial condition, the NP mode emerges and decays at the +linear rate (black line) for approximately one bounce time, Ω−1 +b . Immediately thereafter, +the decay stalls (blue line) as nonlinear saturation sets in. Figure 5 demonstrates that, +meanwhile, the NP mode has produced a large, positive pressure anisotropy in the +regions where δB∥ < 0 and almost zero pressure anisotropy elsewhere, consistent with +the prediction (2.14) (dashed line). The mirror-unstable region (with ⟨β⊥i∆⟩k above the +dotted line in figure 5) is seen to occupy ∼40% of the NP mode wavelength, consistent +with (2.19) for α = 0.8. It is in this mirror-unstable region that the magnetic field acquires +moderate-amplitude, oblique fluctuations in its strength on ion-Larmor scales, which are +clearly apparent in figure 6. The strongest fluctuations occupy roughly a quarter of the +box length and acquire amplitudes comparable to that of the mean field. The associated +distortions in the field lines ultimately scatter particles at a rate comparable to the +bounce frequency (see figure 7 and the accompanying discussion in §2.2.3). As a result, +the NP mode amplitude enters a ‘suppressed saturation’ phase (figure 4, red line), during +which the nonlinear plateau is eroded by the mirror-induced collisionality and the Barnes +damping resumes.4 +In the remainder of §2.2, we examine these phases in more detail and their dependence +on mode amplitude and scale separation, starting with the mirror-induced scattering and +its impact on the NP mode’s pressure anisotropy. +2.2.3. Effective collisionality: particle scattering and trapping +Figure 7 displays the evolution of the mirror-induced effective collisionality νeff in the +fiducial run, calculated following the method used in Kunz et al. (2014a, 2020), Melville +et al. (2016), and Squire et al. (2017a). Namely, the individual magnetic moments of +∼104 particles are tracked and monitored for (both abrupt and accumulated) changes +by at least a factor of κ = 1.2 (as used by Kunz et al. 2020 to measure firehose/mirror- +induced scattering in unstable IAWs). The times at which these changes are registered +are stored, along with the locations at which they occurred, and a spatially dependent +effective collision frequency νeff is calculated from the mean scattering time τ using νeff .= +4We were not able to discern any fluctuations above the noise floor in the out-of-plane component +Bz, which would be indicative of the ion-cyclotron instability (e.g., Gary & Lee 1994). + +16 +S. Majeski, M. W. Kunz, and J. Squire +0 +2 +4 +6 +8 +10 +12 +14 +k||vth,it +0.6 +0.7 +0.8 +δB∥(k|| = 2π/Lx)/B0 +γ = k2/(k2 +⊥ +√πβi0) +decay +saturation +suppressed saturation +Ω−1 +b +Figure 4: Amplitude of the magnetic-field-strength perturbation of the NP mode vs. time +from the fiducial run, with the different phases of the predicted evolution labelled and +colour-coded. The dashed line indicates the linear decay rate (2.5) of the NP mode in a +pressure-isotropic plasma with βi0 ≫ 1. See §2.2.2 for discussion. +0 +5 +10 +15 +⟨β⊥i∆⟩k +mirror +k||vth,it = 3.1 +λ|| = 2000ρi0 = 4λ⊥ +λ|| = 1000ρi0 = 4λ⊥ +λ|| = 2000ρi0 = 8λ⊥ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x/λ|| +−5 +0 +5 +10 +15 +⟨β||i∆⟩k +firehose +Figure 5: Wavefront-averaged profiles of β⊥i∆ and β∥i∆ at k∥vth,it = 3.1, when +the pressure anisotropy is near its maximum value, compared against the theoretical +predictions from the linear eigenmode (2.14), for α = 0.8 and different NP mode +wavelengths λ∥ and λ⊥. The fiducial run corresponds to the solid black line. Positive +values of βi∆ far exceeding the mirror threshold occur in the regions where δB∥ < 0. +Elsewhere, negative pressure anisotropy is compensated by a decrease in βi to avoid +exciting the firehose instability. +(ln κ)2/τ. This calculation was also performed using κ ∈ [1.1, 1.5], with no significant +dependence of νeff on κ. +In the bottom panel of figure 7, the box-averaged effective collisionality (black line) +and maximum value of the wavefront-averaged effective collisionality (red line) are shown + +High-β collisionless magnetosonic modes +17 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +x (ρi0) +0 +100 +200 +300 +400 +500 +y (ρi0) +mirror δBx/B0, k||vth,it = 25 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +Figure 6: The x-component of the magnetic-field perturbation, filtered to remove +wavenumbers associated with the α = 0.8 NP mode, at k∥vth,it = 25 in the fiducial run. +By this time, the mirror instability is fully nonlinear, causing large-amplitude, small- +wavelength deflections in the magnetic-field direction that pitch-angle scatter particles. +as functions of time. Both exhibit rapid growth during the initial phase of the mirror +instability and then reach a quasi-steady state, with max(⟨νeff⟩k) ≈ 0.0035Ωi0 ≈ 2.5Ωb. +We have found the timescale for the scattering rate to reach this steady state to be largely +independent of the wavelength of the NP mode, although it increases somewhat with +decreasing α because of the slower linear growth rate of the mirror instability. The space- +time diagram of the wavefront-averaged collisionality shown in the top panel indicates +that the maximum value of νeff is localized to the centre of the mirror-unstable region, +with slightly smaller values occurring near this region’s boundaries where the mirror +amplitudes are smaller (cf. figure 6). A large fraction of the thermal plasma is subject to +this collisionality, because the mode amplitude is large enough that most of the plasma +particles are confined in the regions where δB∥ < 0 (i.e., large δn > 0). For example, when +α = 0.8, particles whose pitch angles satisfy v∥/w⊥ ⩽ +� +max(B)/min(B) − 1 ≈ 2.8 would +be mirror-confined in the absence of collisions. Outside of these regions, where the plasma +is stable, the collisionality is very low; as a result, the box-averaged collisionality is more +than a factor of 5 smaller than the maximum value. The top panel also shows the path +of a single tracked particle as a grey line. The initial evolution demonstrates bouncing +within the δB∥ < 0 region. Once the mirror fluctuations reach nonlinear amplitudes, the +particle is temporarily trapped within a growing mirror. Eventually, it scatters enough +in pitch angle to become de-trapped and traverses the δB∥ > 0 region, breaking its +resonance with the NP mode. +2.2.4. Evolution of pressure anisotropy +The top panel of figure 8 shows the evolution of the maximum of the wavefront-averaged +∆ and β⊥i∆ in the fiducial run. The bottom panel depicts the growth of the root-mean- +square amplitude of the mirror fluctuations, averaged over the mirror-unstable region +where δB∥ < 0. These fluctuations grow large enough to scatter particles and restore the +linear decay of the NP mode, through which the pressure anisotropy decays. Indeed, ⟨∆⟩k +is similar to the linear prediction (2.12), denoted here by the blue dashed line. Likewise, +⟨β⊥i∆⟩k is modeled well by (2.14) with the substitution δB∥/B0 = α exp(−iζk∥vth,it) +where ζ is the linear eigenvalue (2.5). This expression is traced by the dashed red line + +18 +S. Majeski, M. W. Kunz, and J. Squire +Figure 7: Effective collisionality νeff caused by the mirror instability in the fiducial run +with α = 0.8 and λ∥ = 2000ρi0 = 4λ⊥. Top: Space-time diagram of ⟨νeff⟩k (colour). A +illustrative particle trajectory is shown with the grey line, exhibiting resonant bouncing, +followed by trapping within a mirror fluctuation, and eventual scattering out of resonance +with the NP mode. Bottom: Box-averaged (black) and maximum wavefront-averaged +(red) collision frequencies vs. time. +in figure 8, where we have started the decay at k∥vth,it = 6 and set α = 0.75 in order +to account for the delay due to the (temporary) nonlinear saturation. At larger scale +separations, we anticipate that faster pitch-angle scattering induced by the mirrors will +be able to regulate the pressure anisotropy more efficiently than its linear decay, at which +point the mode will no longer resemble the collisionless linear NP eigenmode (see §2.2.6). +The growth of mirrors leads to modifications in the shape of the NP mode profile, as +shown in figure 9. The evolution of the wavefront-averaged profile of β⊥i∆ in the fiducial +run at k∥vth,it = 3, 6, 11, and 27 is shown. The profile in the region where the mirror +instability is active has flattened, although the mode seems to remain close to the linear +eigenmode, as evidenced by figure 8. The reduction in β⊥i∆ occurs considerably faster +than the linear decay of ∆ by itself, which highlights the importance of β⊥i in achieving +marginal stability. This reinforces the idea that the mirror fluctuations do not so much +act directly on the anisotropy to achieve β⊥i∆ = 1, but rather they enable the NP mode +to decay and reduce both ∆ and β⊥i to achieve marginal stability more rapidly than +would otherwise occur. + +High-β collisionless magnetosonic modes +19 +0.00 +0.05 +0.10 +max(⟨∆⟩k) +Eq. (3.11) +Eq. (3.13) +0 +5 +10 +15 +20 +25 +k∥vth,it +0.0 +0.1 +0.2 +⟨δBrms +m ⟩m/B0 +0 +5 +10 +15 +max(⟨β⊥i∆⟩k) +Figure 8: Top: Maximum of the wavefront-averaged ∆ (solid blue line) and β⊥i∆ (solid +red line) versus time in the fiducial run. The evolution of ⟨∆⟩k matches well the +predicted linear evolution (blue dashed line), suggesting that the rapid reduction of +β⊥i∆ is due mostly to the resumed decay of the NP mode and the decrease in β⊥i +caused by the growing mirror fluctuations. Bottom: Root-mean-square amplitude of the +mirror fluctuations, averaged over the mirror unstable region. The growth of the mirror +instability coincides with a drop in ⟨β⊥i∆⟩k. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x/λ|| +0 +5 +10 +15 +⟨β⊥i∆⟩k +mirror +k||vth,it = 3 +k||vth,it = 6 +k||vth,it = 11 +k||vth,it = 27 +Figure 9: Temporal evolution of the wavefront-averaged profile of β⊥i∆. Four times are +shown: just after the adjustment into the NP eigenmode during the initial decay phase +(black line); an intermediate time during which the NP mode decay is saturated (blue +line); after the mirrors become nonlinear and scatter particles fast enough to suppress +the NP mode’s saturation (red line); and later once β⊥i∆ has been reduced enough that +the mirrors are marginally stable (grey line). +2.2.5. Suppression of nonlinear saturation and resumption of transit-time damping +The effects of nonlinear saturation and mirror-induced collisionality across a variety +of NP mode amplitudes can be seen in figure 10. For reasons of computational cost, for +these runs we used λ∥ = 1000ρi rather than the fiducial 2000ρi. A Fourier transform is +used to select the magnitude of the box-wavelength perturbation to the background field +(i.e., the amplitude of the NP mode); this quantity is plotted as a function of time. In + +20 +S. Majeski, M. W. Kunz, and J. Squire +(a) +(b) +0 +2 +4 +6 +8 +k∥vth,it +1.0 +0.8 +0.9 +α−1δB∥(k∥ = 2π/Lx) +α = 0.1 +α = 0.2 +α = 0.4 +α = 0.6 +α = 0.8 +0 +10 +20 +30 +40 +k∥vth,it +1.0 +0.5 +0.6 +0.7 +0.8 +0.9 +α = 0.1 +α = 0.2 +α = 0.4 +α = 0.6 +α = 0.8 +Figure 10: Amplitude of the magnetic-field-strength perturbation of the NP mode, +normalized to its initial value, vs. time for λ∥ = 1000ρi0 = 4λ⊥ and different α. (a) Early +times, during which the NP mode nonlinearly saturates after approximately one bounce +time ∼Ω−1 +b +(vertical dotted lines; see (2.7)). The dashed line indicates the linear decay +rate (2.5). (b) Late times, showing suppression of nonlinear saturation and resumption +of linear damping for amplitudes α ⩾ 0.6. +panel (a), the initial phase of evolution is featured, at first demonstrating linear decay +at a rate similar to the prediction (2.5) (shown by a black dashed line), approximately +independent of α. After roughly one bounce time (marked by dotted lines of matching +colour), the decay begins to stall and the mode amplitude tends towards a constant value. +This nonlinear saturation occurs at earlier times for larger mode amplitudes, trending +with the α−1/2 scaling of the bounce time (see (2.7)). At amplitudes α ≳ 0.4, more than +90% of the original mode amplitude is preserved by the nonlinear saturation, suggesting +that large-amplitude collisionless NP modes at high β can be rather long lived. +Figure 10(b) shows the behaviour of these modes over longer timescales. For amplitudes +α ⩽ 0.4, nonlinear saturation remains and the linear decay rate is never again realized. +By contrast, the larger values of pressure anisotropy in α = 0.6 and 0.8 NP modes +produce mirror-unstable fluctuations with amplitude large enough to interfere with the +maintenance of the nonlinear plateau. As a result, the linear decay rate is almost re- +established at α = 0.6 and is restored fully at α = 0.8. These modes are then able +to decay further and convert magnetic energy into particle energy through a balance +between plateau generation and pitch-angle scattering. With the value of λ∥ used in +these runs being twice smaller than that in the fiducial run, it is notable that the time +at which near-linear decay is restored by mirror-induced scattering is the same (in units +of Ωi0). At scale separations much larger than those we are able to simulate currently, +we thus anticipate the nonlinear plateau to be eroded almost instantly compared to the +wave timescales by rapid mirror growth and its associated particle scattering. +Our final piece of evidence that the nonlinear plateau is maintained at subcritical +NP mode amplitudes and eroded at supercritical amplitudes is also the most direct. +In figure 11 we show the ion velocity distribution functions f(v∥, w⊥) measured within +the δB∥ < 0 region from two runs having λ∥ = 2000ρi = 4λ⊥ and either α = 0.4 +(left) or 0.8 (right). The top plots depict in colour the differences between f(v∥, w⊥) +and bi-Maxwellian fits based on the parallel and perpendicular ion temperatures. The +bottom plots show v∥ slices of the distribution functions averaged between v⊥ ≃ 3.6vth,i + +High-β collisionless magnetosonic modes +21 +°2 +°1 +0 +1 +2 +vk/vth,i +2.5 +3.0 +3.5 +w?/vth,i +°2 +°1 +0 +1 +2 +vk/vth,i +°0.5 +0.0 +0.5 +vk/vth,i +0.020 +0.025 +0.030 +0.035 +0.040 +f(vk) +Æ = 0.4 +°0.5 +0.0 +0.5 +vk/vth,i +0.03 +0.04 +0.05 +0.06 +Æ = 0.8 +°0.10 +°0.08 +°0.06 +°0.04 +°0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +f(vk, w?) ° F fit +Max(vk, w?) +Figure 11: Ion velocity distribution functions f(v∥, w⊥) measured within the regions +where δB∥ < 0 of two simulations with λ∥ = 2000ρi0 = 4λ⊥. The left panels, +corresponding to α = 0.4, exhibit a nonlinear plateau around v∥ ∼ 0. The right panels, +corresponding to α = 0.8, show a smooth Maxwellian-like distribution. The bottom +panels are slices in v∥ integrated between 3.6vth,i and 3.7vth,i). The colour bar for the +top panels has been allowed to saturate for the purpose of showing detail. Dotted lines +represent isocontours of total energy, w2 +⊥ + v2 +∥ = const. +and 3.7vth,i (the averaging is performed to reduce sampling noise). In the α = 0.4 +run, the distribution is reduced with respect to the bi-Maxwellian at low pitch angles +where particles are well trapped, and the parallel velocity distribution f(v∥) exhibits +flattening about v∥ ∼ 0 – a nonlinear plateau. No such features are visible in the α = 0.8 +distribution, with betatron heating of the trapped particles evidenced by an increase in +particle phase-space density over the bi-Maxwellian fit at large perpendicular velocities. +2.2.6. Dependence on scale separation +The effective collision frequency predicted by (2.24) suggests that, if the initial NP +mode amplitude and wavenumber obliquity were held constant, then increasing the +wavelength of the mode should have no effect on the collision frequency. This can be recast + +22 +S. Majeski, M. W. Kunz, and J. Squire +Figure 12: Maximum value of the measured mirror-induced effective collision frequency +νeff,max vs. NP mode wavelength at two different wavenumber obliquities and two different +initial amplitudes. The predicted scaling ν/(k∥vth,i) ∝ λ∥ is shown (dashed black line), +normalized to the fiducial case (red circle at λ∥ = 2000ρi0). +as a more illustrative relationship between the thermal crossing time and the collision +frequency, νeff/(k∥vth,i) ∝ λ∥. Figure 12 shows the maximum value of the box-averaged +effective collision frequency normalized to k∥vth,i for a few different NP mode wavelengths, +wavenumber obliquities, and amplitudes. The measured values exhibit good agreement +with the proportional expectation at both wavenumber obliquities. This evidence implies +that, at yet longer wavelengths, the collision frequency will continue to increase compared +to the bounce time. Note that the measured collisionality for α = 0.6 is approximately +a factor of two smaller than for α = 0.8, in qualitative agreement with the prediction +featured in figure 3(a) that the scattering should decrease with decreasing NP mode +amplitude. The fact that the simulated NP mode with α = 0.4 and λ∥ = 1000ρi0 does not +have its nonlinear saturation interrupted by mirrors is also consistent with the prediction +in figure 3(b). +As conjectured in §2.1.6, the linear scaling of νeff/(k∥vth,i) with λ∥ suggests a possible +fluid-like regime at sufficiently long NP-mode wavelengths. To investigate this regime, if +only approximately, we examine the linear decay rate of NP modes in the presence of a +constant pitch-angle scattering rate, shown in figure 13. The details of how we determined +this decay rate are given in appendix B; note that the real part of the frequency is zero +for all scattering rates, i.e., the mode remains non-oscillatory. On the left-hand side of the +plot, the collision frequency is small and the collisionless NP mode is recovered; on the +right-hand side, the collision frequency is large and the mode becomes the MHD entropy +mode. The MHD entropy mode is similar to the kinetic NP mode in that it too has no +real frequency, but in the fully collisional limit it involves only a density perturbation. For +the employed values of k⊥/k∥ = 4 and βi0 = 16, the transition between these two regimes +occurs at ν ≈ 3k∥vth,i. Using an asymptotic expansion at high β and k ≃ k⊥, one can show +that the transitional collisionality scales approximately as ν ∼ (3/4)√βik∥vth,i. With +νeff/Ωi0 ∼ 10−2β−1 +i0 +for α ≳ 0.6 (see figures 3 and 12), we estimate that the transition to +the collisional regime requires a scale separation of at least λ∥/ρi0 ∼ 103β3/2 +i0 . Under this +condition, the mirror-induced scattering will both isotropize the pressure perturbation +and prevent resonant particles from continuously sapping energy from the wave, thereby + +High-β collisionless magnetosonic modes +23 +Figure 13: Linear decay rate of the NP mode obtained from the Landau-fluid CGL-MHD +equations (B 1) (see appendix B for details). The dimensionless (complex) frequency +ζ .= ω/(|k∥|vth,i) is computed numerically as a function of collisionality ν/(|k∥|vth,i) for +k⊥ = 4|k∥|, βi0 = 16, and Te = Ti0. Overlaid are red circles marking the maximum +box-averaged scattering rates measured in our hybrid-kinetic simulations (see figure 12). +reducing the decay rate and morphing the collisionless NP mode into the MHD entropy +mode. Unfortunately, unless the scale separation is extremely large (e.g., λ∥/ρi0 ≳ 105 +for our parameters), the decay rate will not be much slower than in the ν = 0 case. In +the absence of affordable numerical simulations to test this point, we simply conjecture +that at asymptotic wavelengths the reduction in the decay rate would allow these NP +structures to become long lived once again, much like their below-threshold, non-linearly +saturated counterparts. +2.3. Summary of key results on the NP mode +For the reader’s benefit, we summarize here the essential findings of our investigation +of the NP mode in magnetized, high-β, collisionless plasmas: +• +Transit-time (Barnes) damping of NP modes nonlinearly saturates before substantial +collisionless decay occurs when the mode amplitude |δB∥/B0| ≳ β−2 +i0 . +• +The perpendicular pressure balance associated with the polarization of the NP +eigenmode produces large positive βi∆ and only weakly negative βi∆. +• +Above a threshold amplitude of |δB∥/B0| ≈ 0.3, this pressure anisotropy becomes +unstable to the mirror instability; at no point is the plasma firehose unstable. +• +Once the growing mirror fluctuations become nonlinear, they pitch-angle scatter +particles at a rate νeff, which, in accordance with (2.24), is independent of the NP +mode wavelength. +• +At wavelengths sufficiently long so that νeff satisfies √βi +≳ νeff/(k∥vth,i) ≳ +|δB∥/B0|1/2, the induced scattering is only fast enough to erode the nonlinear +plateau, causing the mode to resume its decay close to the linear (collisionless) rate. +• +At yet longer wavelengths for which νeff satisfies νeff/(k∥vth,i) ≫ √βi, transit-time +damping will be interrupted entirely. We predict that in this limit the mode will +behave more like the MHD entropy mode. + +24 +S. Majeski, M. W. Kunz, and J. Squire +3. Fast modes: Wave steepening and viscous damping +3.1. Theory +3.1.1. Model equations and assumptions +Collisionless fast magnetosonic waves are in many ways simpler than their non- +propagating counterparts, particularly so if their wavevectors are nearly perpendicular +to the background magnetic field, viz. k⊥ ≫ k∥. In this limit, collisionless damping is +extremely weak, and magnetic tension plays virtually no role in the mode’s propagation. +In fact, for exactly perpendicular propagation (k∥ = 0), Landau and Barnes damping are +entirely absent at long wavelengths due to the limited cross-field transport of magnetized +particles. In this case, no kinetic information about these modes other than their pressure +anisotropy is needed, and they can be described entirely within double-adiabatic MHD – +a model that results from taking the first three fluid moments of the drift-kinetic system +(see appendix B) and dropping the heat fluxes. Setting B = Bˆy and ∇ = ˆx ∂/∂x, these +equations are +Dn +Dt = −n∂u⊥ +∂x , +(3.1a) +minDu⊥ +Dt += − ∂ +∂x +� +p⊥i + pe + B2 +8π +� +, +(3.1b) +DB +Dt = −B ∂u⊥ +∂x , +(3.1c) +D +Dt +�p⊥i +nB +� += 0, +(3.1d) +D +Dt +�p∥iB2 +n3 +� += 0, +(3.1e) +where D/Dt .= ∂/∂t + u⊥∂/∂x. Although the right-hand side of (3.1b) is independent +of the parallel pressure, and so (3.1e) is not needed to close this set of equations, it is +nevertheless useful for calculating the fast-wave pressure anisotropy. As in §2, we adopt +a simple equation of state for the electrons, pe = nTe, with Te being constant.5 +In what follows, we investigate analytically two features of fast-wave propagation in +a collisionless, magnetized plasma, adopting the simple but illustrative case of k∥ = 0. +First, we demonstrate that such waves nonlinearly steepen quicker in double-adiabatic +MHD than they do in standard (pressure-isotropic) MHD, a direct consequence of the +proportional relationship between T⊥ and B associated with µ conservation, equa- +tion (3.1d). Second, we show how the resulting pressure anisotropy can destabilize +the plasma to both firehose and mirror instabilities. We then estimate the effective +scattering frequency introduced into the plasma by these instabilities and discuss how +the consequent regulation of the pressure anisotropy affects the characteristics of the fast +wave. +Before proceeding, it is useful to linearize (3.1) to obtain the fast-wave dispersion +relation and eigenmode. Perturbing the plasma about a uniform background having +density n0, isotropic ion pressure pi0, and magnetic-field strength B0, we find that +δp⊥,i +pi0 += 2δB +B0 +and +δp∥i +pi0 += δn +n0 += δB +B0 +. +(3.2) +These equations state that the density and pressure anisotropy are positively correlated +5Having the electrons respond double-adiabatically would simply double the pressure anisotropy +associated with the fast wave and send Te/2Ti0 → Te/Ti0 in (3.3). + +High-β collisionless magnetosonic modes +25 +with the magnetic-field strength, with the parallel ion temperature remaining constant. +The dispersion relation of this double-adiabatic (‘da’) fast wave is +ω = k⊥vA +� +1 + βi0 +� +1 + Te +2Ti0 +� +.= k⊥vms,da, +(3.3) +so that the bulk velocity u⊥ = vms,da(δB/B0). For comparison, the dispersion relation +of a fast wave in single-adiabatic (‘sa’) MHD is +ω = k⊥vA +� +1 + βi0 +�γ +2 + Te +2Ti0 +� +.= k⊥vms,sa, +(3.4) +where γ is the adiabatic index of the ions. The proportional relation between the +magnetic-field strength and the density in the double-adiabatic model means that +vms,da > vms,sa. This increase will play a role in allowing double-adiabatic fast waves to +form shocks faster than single-adiabatic fast waves, especially so at high β. +3.1.2. Wave steepening in double- versus single-adiabatic MHD +For waves in which the perturbed quantities determine the wave propagation speed, +steepening may result. Large-amplitude waves in particular generate significant differ- +ences in the propagation speed between the peaks and the troughs, a situation expected +to occur in both double- and single-adiabatic MHD fast waves. In this section, we +perform a series of manipulations to the system (3.1) in order to quantify this effect. +Before proceeding, it is convenient to renormalize quantities using the Alfv´en speed +vA = B0/(4πmin0)1/2 and the wavelength λ as follows: u⊥ = �u⊥vA, B = �BB0, n = �nn0, +x = �xλ, t = �tλ/vA, and p⊥,i = �p⊥imin0v2 +A. We also note that, if the perturbations +satisfy δ�n = δ �B at t = 0, then these two quantities will remain equal for all times (see +equations (3.1a) and (3.1c)); we can then eliminate δ�n in favour of δ �B.6 Meanwhile, if δ �B +is small and its associated perturbations in �p⊥,i and �n are given by (3.2), equation (3.1d) +becomes +∂ +∂�t +� �p⊥,i +�n �B +� +≈ −�u⊥ +βi0 +2 +∂(δ �B)2 +∂�x +∼ O +� +(δ �B)3� +. +(3.5) +Hence, to second order in δ �B, we may treat �p⊥,i = (βi0/2) �B2 as the equation of state if +the initial condition is an eigenmode. +Under these conditions, equations (3.1) may be combined to obtain the following +system: +∂ +∂�t +� +� �u⊥ +δ �B +� +� + +� +� +�u⊥ +1 + βi0 +� +1 + Te/2Ti0 +1 + δ �B +� +1 + δ �B +�u⊥ +� +� ∂ +∂�x +� +� �u⊥ +δ �B +� +� = 0. +(3.6) +Defining W = [�u⊥, δ �B]T, equation (3.6) can be rewritten as ∂�tW +A(W )∂�xW = 0, with +A(W ) being the evolution matrix. By first finding the eigenvalues l(i) and left eigenvectors +L(i) of A(W ), this system can be solved via its characteristic equations, which are given +by L(i) · dW = 0. These characteristic equations are obeyed along space-time trajectories +following d�x/d�t = l(i). However, because our equation of state is only valid up to second +order in the wave amplitude, we need only to retain those terms of first order in the +6This reduction is equivalent to assuming an adiabatic index of γ = 2. In fact, when comparing +the results of this analysis to an MHD treatment with isothermal electrons, the substitution +γ = 2 recovers the double-adiabatic result (see (3.12) and (3.13)). + +26 +S. Majeski, M. W. Kunz, and J. Squire +Figure 14: Approximate solution (3.11) to the fast-wave steepening problem with initial +amplitude α = 0.3 and βi0 = 25. The solution has just begun to form a shock, indicating +a shock-formation time of k⊥vAts ∼ 0.4. +evolution matrix, and hence in its eigenvalues. Therefore, we expand the characteristic +equations to first order and integrate them to find that the combinations +η± = �u⊥ ± �vms,daδ �B +(3.7) +are approximately constant along +�d�x +d�t +�± += η+ + η− +2 +± �vms,da +� +1 + 1 + βi0 +4�v3 +ms,da +(η+ − η−) +� +. +(3.8) +These can be reformulated as two nonlinear advection equations,7 +∂η± +∂�t + +�d�x +d�t +�± ∂η± +∂�x = 0. +(3.9) +Note that, if the initial conditions are those of the fast eigenmode (as previously assumed +in the assertion that δ �B = δ�n for all time), then η− = 0 for all time. We are then left +with +∂η+ +∂�t + +�η+ +2 + �vms,da +� +1 + 1 + βi0 +4�v3 +ms,da +η+ +��∂η+ +∂�x = 0, +(3.10) +the solution of which for �u⊥ is given by the method of characteristics as +�u⊥(�t, �x) = δ�u⊥0 +� +�t, �x − �vms,da�t +� +1 + δ �B0(�xi) + 1 + βi0 +2�v2 +ms,da +δ �B0(�xi) +�� +, +(3.11) +where the subscript ‘0’ denotes an initial value, and xi is the x-position of the source of +a given characteristic. +7This process is analogous to that used in the derivation of approximate Riemann solvers for +numerical solutions of the MHD equations (e.g., Roe 1981; Toro 1999; Miyoshi & Kusano 2005; +Stone et al. 2008). Commonly, the left eigenvector is assumed to be constant when integrating +the characteristic equations. Here, we keep terms up to first order in δB within L to more +accurately resolve the wave steepening. The careful reader will note that these expressions do +not transform directly back to an approximate form of (3.6). This approach focuses on the +characteristics of A, so the leading-order behaviour of (3.6) and the eigenvalues/vectors of A are +approximated accurately; this is in contrast to expanding A itself and truncating past the first +correction in δ �B. + +High-β collisionless magnetosonic modes +27 +The time-dependent solution for an example large-amplitude, double-adiabatic fast +wave is shown in figure 14. This solution is strictly valid only until a shock has formed, +at a time that may be determined by evaluating the eigenvalue l+ at the location x0 +where its derivative achieves its largest negative value: +tda +s += +� +l+(x0) +�−1 ≈ +� +αk⊥vms,da +� +1 + +v2 +A +v2 +ms,da +1 + βi0 +2 +��−1 +, +(3.12) +where α .= δ �B(0) is the initial fast-wave amplitude. This double-adiabatic (‘da’) shock- +formation time is to be compared to the corresponding time in a single-adiabatic MHD +plasma, in which pn−γ = p0n−γ +0 . The general problem of fast-wave steepening in MHD +plasmas has been studied thoroughly under many conditions (Hada & Kennel 1985; +¨Odblom 1998; Sujith 2005). Following an analogous process to that used for the double- +adiabatic fast wave, we find the single-adiabatic (‘sa’) shock-formation time +tsa +s ≈ +� +αk⊥vms,sa +� +1 + +v2 +A +v2ms,sa +1 + γ(γ − 1)βi0/2 +2 +��−1 +. +(3.13) +Simplifying (3.12) and (3.13) at high β, and setting Te = Ti0 and γ = 5/3, yields +k⊥vAtda +s +≈ +√ +6 +4α√βi0 +and +k⊥vAtsa +s ≈ +12 +√ +3 +29α√βi0 +≃ 1.17k⊥vAtda +s . +(3.14) +The single-adiabatic shock-formation time is thus larger than the double-adiabatic shock- +formation time. When Te/Ti0 = 0, their ratio reaches a maximum of ≃1.23; for Te ≫ Ti0, +it approaches unity. This increase is a consequence of the direct correlation between the +magnetic-field strength and the perpendicular (ion) pressure in double-adiabatic MHD, +which amplifies local changes in the mode propagation speed. +3.1.3. Pressure anisotropy and its regulation by kinetic instabilities +By contrast with the NP mode, the fast wave generates a fluctuating pressure +anisotropy as the wave propagates. At sufficiently large β, both firehose and mirror +instabilities may therefore be triggered. With δp⊥,i and δp∥,i given by (3.2), the amplitude +threshold for triggering both firehose and mirror instabilities is +���� +δB +B0 +���� ≳ 2 +βi +(fast-wave amplitude threshold). +(3.15) +At high β, this criterion can be satisfied for even small-amplitude fluctuations, justifying +the use of the linear eigenvector and unperturbed βi in determining the threshold. +To assess whether these micro-instabilities will be able to grow, we compare their +linear growth rates to the linear frequency of the fast wave at high β, ωfast ∼ k⊥vth,i. We +adopt the maximal mirror growth rate from (2.16), and use the maximal oblique firehose +growth rate γf ≈ 0.3ΩiΛ1/2 +f +where Λf .= |∆ + 2/βi| (Yoon et al. 1993; A.F.A. Bott et +al., in preparation), both of which are appropriate for the near-threshold conditions we +anticipate in our fast-wave simulations. Assuming |δB/B0| ≳ 2β−1 +i +, we find that +γm +ωfast +∼ 0.01β−1 +i +λ⊥ +ρi +and +γf +ωfast +∼ 0.1β−1/2 +i +λ⊥ +ρi +, +(3.16) +where λ⊥ = 2π/k⊥ is the wavelength of the fast wave. It is immediately apparent from +(3.16) that, at high β, very large scale separation between the fast-wave wavelength and +the ion-Larmor scale is necessary to allow enough time for mirror fluctuations to grow +and become nonlinear. The scaling with βi is much weaker for the firehose instability, and + +28 +S. Majeski, M. W. Kunz, and J. Squire +so there will exist wavelengths at which mirror regulation of the pressure anisotropy is +effectively non-existent but the firehose regulation is rapid. For this reason our Pegasus++ +simulations, which focus on βi0 = 25, require λ⊥ ≫ 103ρi0 to realize both mirror and +firehose regulation. +The unstable Larmor-scale fluctuations will ultimately grow to amplitudes at which +the particles’ rate of pitch-angle scattering is sufficient to hold the pressure anisotropy +at marginal stability. This rate may be estimated by calculating the pressure anisotropy +driven by a small-amplitude fast wave in a weakly collisional plasma (following Braginskii +1965) and asking what value of effective collisionality νeff would be required to keep +|∆| ∼ 2β−1 +i +. With the former given in the collisional regime by ∆ ∼ −(∇ · u)/νeff ∼ +(k⊥vms/νeff)|δB/B0|, the limiting collisionality is +νeff ∼ k⊥vms +βi +2 +���� +δB +B0 +����. +(3.17) +Note its explicit dependence upon the scale of the fast wave, an indirect consequence +of the pressure anisotropy of the fast wave being continuously driven by the fluctuating +wave. This is very different from the case with the zero-frequency NP mode, in which the +pressure anisotropy – an essential feature of the mode’s perpendicular pressure balance +– actually decays in time through transit-time damping. +3.1.4. Viscous damping and collisional propagation +The estimate of the effective collisionality (3.17) suggests that, depending on the wave +amplitude, one should see a variety of fast-wave behaviour. For example, if |δB/B0| ≫ +2β−1 +i0 , then the implied collisionality can be large enough to push the fast wave into +the collisional Braginskii-MHD regime (ν ≫ ω). If we make the presently unjustified +yet instructive assumption that this collisionality is distributed uniformly in space, the +fast-wave dispersion relation at arbitrary ν can be obtained after including isotropizing +collisional terms −ν∆p/nB and ν∆pB2/n3 on the right-hand sides of (3.1d) and (3.1e) +respectively, then linearizing the resulting system of equations. We find that +ω3 − iνω2 − ωk2 +⊥v2 +ms,da + iνk2 +⊥v2 +ms,sa = 0. +(3.18) +The numerical solution to (3.18) is shown in figure 15. In the collisionless limit ν → 0, +one recovers propagation at the double-adiabatic fast speed; taking ν → ∞ returns +propagation at the single-adiabatic fast speed. Viscous damping occurs at intermediate +values of ν ∼ Re(ω) ∼ k⊥vth,i around the transition between the double- and single- +adiabatic regimes, where the scattering rate is comparable to the wave’s oscillation +frequency. The damping rate is always small compared to the wave frequency. +The dispersion relation (3.18) alongside the amplitude threshold (3.15) and the pre- +dicted effective collision frequency (3.17) imply three regimes for the behaviour of per- +pendicularly propagating fast modes in a high-β plasma. For small amplitudes satisfying +|δB/B0| < 2β−1 +i0 , the mode propagates normally as a collisionless fast mode. It will +steepen and eventually form a shock on the double-adiabatic shock time tda +s . In the +near-threshold regime where |δB/B0| ≳ 2β−1 +i0 , the scattering rate from triggered mirror +and firehose instabilities will not quite reach the value (3.17), though scattering is still +expected to occur and result in some viscous damping. The wave will also steepen to form +a shock, but only a fraction of the wavelength will be kinetically unstable and therefore +the shock will occur on a hybrid of the double- and single-adiabatic shock times. Lastly, +at amplitudes well above the threshold, the scattering rate should be given by (3.17). +The viscous damping will be very weak, the wave will host firehose/mirror scattering + +High-β collisionless magnetosonic modes +29 +Figure 15: Exact solution to the dispersion relation (3.18) for a k∥ = 0 fast wave in a +plasma having collision frequency ν, βi0 = 25, and Te/Ti0 = 1. +sites throughout most of its wavelength, and the shock time should be better represented +by the single-adiabatic model. +We now test these ideas using numerical simulations. +3.2. Numerical results +3.2.1. Method of solution and initial conditions +Due to the large scale separations needed to obtain asymptotic νeff for both firehose +and mirror fluctuations (§3.1.3), we use a combination of Pegasus++ and (much cheaper) +Landau-fluid CGL-MHD simulations. All simulations initialize a k∥ = 0 fast wave in an +otherwise Maxwellian plasma using the collisionless eigenmode (3.2), viz., +B(0, x) = B0 +� +1 + α sin(k⊥x) +�ˆy, +u(0, x) = vms,daα sin(k⊥x)ˆy, +n(0, x) +n0 += p∥i(0, x) +pi0 += 1 + α sin(k⊥x), +p⊥i(0, x) +pi0 += +� +1 + α sin(k⊥x) +�2, +(3.19) +where k⊥ = 2π/λ⊥ and α is a dimensionless number quantifying the mode amplitude. +For the Pegasus++ runs, the mesh is two-dimensional and elongated in the propagation +direction, with size Lx × Ly = λ⊥ × 100ρi0. The size of the domain in the y direction +is large enough to capture all relevant firehose and mirror fluctuations. We set βi0 = 25 +and Te = Ti0; the slightly larger value of βi0, as compared to that used in the simulations +of the NP mode (βi0 = 16), results in a shorter numerical integration time (and thus +computational savings) without changing the physical character of the fast wave. The +spatial resolution and the number of macro-particles per cell are the same as in the +NP simulations (§2.2.1). In the manuscript we only show results from a Pegasus++ run +having λ⊥ = 8000ρi0, corresponding to the largest domain size that we simulated. We +found that this value of λ⊥/ρi0 was the minimum required for the mirrors to have time to +grow and begin scattering particles before the wave oscillates and the sign of the driven +pressure anisotropy reverses. +In the accompanying Landau-fluid simulations, the full system of CGL-MHD equations +is solved using a new Riemann solver implemented in a version of the finite-volume +Athena++ simulation code (Stone et al. 2020) that includes Landau-fluid heat fluxes +(J. Squire et al., in preparation). These equations are given in appendix B; they reduce +to (3.1) in our chosen geometry. For these runs, βi0 is varied between 1 and 100 to study +the variance of the shock time. In order to approximate the effect of the kinetic micro- +instabilities, a ‘limiter’ collisionality νlim is set either to 0 or to αβi0k⊥vms,da, depending + +30 +S. Majeski, M. W. Kunz, and J. Squire +Figure 16: Shock-formation time versus βi0 and α for a double-adiabatic fast wave +computed from CGL-MHD simulations (lines) and predicted analytically using (3.12) +(circles). The simulated waves are estimated to have formed a shock at the time when +the rate of change of the maximum density gradient drops below half of its own peak +value. +on whether the focus is on wave steepening and shock formation (ν = 0) or the effects of +the instability-induced scattering. This anomalous scattering rate is active only within +regions of the domain where the pressure anisotropy would be kinetically unstable, viz., +where βi∆ ⩽ −2 and βi∆ ⩾ 1; elsewhere it is zero. It serves to isotropize the plasma +pressure where mirror or firehose fluctuations would otherwise do so in a kinetic system, +by contributing a term proportional to −νlim∆p to the right-hand sides of the evolution +equations for p⊥ and p∥. +As in §2, ⟨ · ⟩ denotes a spatial average taken over the entire domain, while ⟨ · ⟩k denotes +a spatial average performed along the wavefront (in this case, the y direction). +3.2.2. Wave steepening and shock formation +Our first goal is to test the expression (3.12) for the shock-formation time ts. We +perform a parameter survey by varying βi0 and the wave amplitude α using the CGL- +MHD code with the micro-instability-limiting scattering turned off. At each time step +in the simulation, the local density gradient (using a four-cell average) is calculated +throughout the domain and its maximum value is recorded as a measure of the wave +steepening. As a fast wave steepens, the growth rate of this maximum gradient increases +until eventually the shock forms and the maximum gradient in the domain begins to +plateau. We define the numerically calculated shock-formation time to be the time at +which the rate of change of this maximum gradient drops below half of its own peak +value. The resulting times are compared with (3.12) in figure 14. When testing the +dependence on βi0 (blue, left), the perturbation amplitude is set to α = 0.01; when +testing the dependence on amplitude (red, right), βi0 = 25. +Overall, the agreement between (3.12) and the numerically calculated shock-formation +times is quite good. Small variations occur due to differences in the rates at which the +maximum gradients plateau and to minute fluctuations in the maximum value of the +gradient after the shock is formed (this value does not necessarily reach a perfect steady +state). Perhaps unsurprisingly, at high β where vms,da ≈ vA +� +3βi0/2, the ratio of the +wave-crossing time and the shock-formation time is tcross/ts,da ≈ 4α/3. This means the +number of wavelengths propagated prior to forming a shock is dependent upon the mode +amplitude only. + +High-β collisionless magnetosonic modes +31 +(a) Pressure anisotropy times the ion beta from a Pegasus++ simulation of a collisionless +fast wave, showing that the compression and rarefaction of the magnetic-field lines generates +oppositely signed anisotropies that move with the wavefront. Some sloshing due to firehose +regulation of the negative pressure anisotropy causes an additional reversal of ∆ in the final +time frame. +(b) Zoomed-in regions showing δBy and δBz, with the contribution from the background fast +wave removed. Recall that the mean field is oriented in the y direction. In the left set of +panels, the mirror instability, with its oblique orientation and dominance in δB∥ = δBy, grows +relatively slowly in the co-moving region of fast-wave compression from k⊥vAt = 0.08 to 0.39. +The firehose instability in the right set of panels is predominantly oblique and exhibits rapid +growth and saturation; smaller-amplitude parallel firehoses appear in δBx (not shown). These +firehose fluctuations reside downstream of the mirrors, where the fast-wave δB < 0. +Figure 17 +3.2.3. Generation of pressure anisotropy and triggering of kinetic instabilities +Prior to shock formation, the linearized fluctuations (3.2) suggest that pressure +anisotropy at a level capable of triggering both mirror and firehose instabilities will +exist when the fast-wave amplitude satisfies |δB/B0| ≳ 2/βi. For these supercritical +amplitudes, the wavefront should carry with it rapidly growing firehose fluctuations and +more slowly growing mirror fluctuations, as per (3.16). To test this idea, we performed +a large-scale Pegasus++ simulation, the parameters of which are described in §3.2.1; the +initial wave amplitude α = 0.1 and βi0 = 25. +Figure 17(a) depicts the pressure anisotropy generated by the fast wave as it propagates +through space at three different times (k⊥vAt = 0.0, 0.08, 0.39; note that the aspect +ratio of the plotted domain is far from unity, and that the mean magnetic field is in +the y direction). Initially, the positive and negative pressure anisotropies in the wave are +equal in magnitude. Shortly thereafter, the (unstable) negative anisotropy is reduced +significantly due to the rapid growth of the (primarily oblique) firehose instability. +The positive pressure anisotropy does not show a comparable decrease, and in fact + +3.0 +50 +1.5 +0'0=+VaT +0 +Pio) +50 +0.0 +β;△ +k1At =0.08 +0 +-1.5 +50 +klUAt =0.39 +-3.0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +(pio) + (By/ Bo +Bz/ Bo +-0.10 +-0.05 +0.00 +0.05 +0.10 +-0.2 +-0.1 +0.1 +0.0 +0.2 +100 +kIVAt =0.08 +kIUAt =0.39 +[klUAt =0.08 +k↓UAt =0.39 +75 +(0.d) +50 +9 +25 +0 +2125 2150 2175 +4525 4550 4575 +7225 7250 7275 +1625 1650 1675 +(pio)32 +S. Majeski, M. W. Kunz, and J. Squire +increases somewhat from its initial value. This is likely because the rapid change in the +negative-anisotropy regions, which perturbs the wave and causes some deviation from the +eigenmode, is not matched by a comparable regulation from the positive side because of +the relatively slow mirror growth. Figure 17(b) zooms in on the corresponding magnetic- +field fluctuations that emerge in two separate co-moving regions where the plasma is +mirror unstable (left) or firehose unstable (right). To accentuate these fluctuations, +the large-scale contribution from the fast wave has been removed. At k⊥vAt = 0.08, +oblique firehose fluctuations are strong and nonlinear; parallel firehose fluctuations are +also present, though subdominant, in δBx (not shown). At this time, there is only a hint +of mirror fluctuations emerging above the noise level. In the final frame (k⊥vAt = 0.39) +however, highly oblique mirror modes have grown to large amplitudes in the region +encompassed approximately by x/ρi0 ∈ [4000, 5000]. The scale separation achieved in +this simulation (Lx/ρi0 = 8000) was the minimum at which we could observe mirror +fluctuations with strengths comparable to their firehose counterparts; increasing the scale +separation further would come at considerable computational expense. +3.2.4. Effective collisionality: particle scattering +Following §2.2.3, the effective collisionality was determined for the fast wave shown in +figure 17 by tracking thousands of ion macro-particles and measuring the frequency at +which their µ changes statistically by a factor of κ = 1.2 or more. Figure 18 depicts this +scattering rate as a function of the position along the wave (x/ρi0) and the time (k⊥vAt). +Sites of strong scattering are associated with the firehose modes, which appear more or +less instantly and travel along with the trough of the wave. The trail of the scattering +sites indicates that the trough of the wave moves at ≈6vA, as expected for a fast mode +with βi0 = 25. In this simulation, the rapid regulation of the pressure anisotropy by +the firehose instability causes sloshing. The sloshing temporarily drives a higher positive +pressure anisotropy, and therefore enhanced mirror growth, for a short period beginning +at kvAt ≈ 0.4. The measured scattering rate in the firehose-unstable regions is comparable +to the predicted asymptotic scattering rate for a βi0 = 25 fast wave with α = 0.1 and +Te = Ti0, viz. νeff ≈ 16k⊥vA (see (3.17)). The mirror instability in this case also scatters +particles at an average rate of a few times k⊥vA, but these scattering sites are much +less coherent and do not coincide with the peak in the positive pressure anisotropy. This +delayed growth is a result of the limited achievable scale separation in our simulations, +which only barely allows mirrors to grow to nonlinear levels within a fast-wave crossing +time. +The effects of the induced scattering on the fast wave’s pressure anisotropy are visible +in figure 19, which shows ⟨βi∆⟩y at the same times as in figure 17. The negative anisotropy +is strongly regulated by the firehose instability to well above βi∆ = −2 within a very +short time. This regulation persists, but is not matched on the mirror-unstable side. +Some steepening has also occurred, as expected, but the positive anisotropy has not been +driven down near marginal mirror stability. In order for mirror fluctuations to regulate +the positive pressure anisotropy to marginal stability, they would need to grow faster +with respect to the fast-wave crossing time; equation (3.16) suggests that this could be +achieved by increasing λ/ρi0 even further (beyond λ⊥/ρi0 = 104). Unfortunately, such +large scale separations become prohibitively expensive to simulate using Pegasus++, and +so from this point onward we employ the CGL-MHD code with pressure-anisotropy +limiters. + +High-β collisionless magnetosonic modes +33 +Figure 18: Space-time diagram of the effective collision frequency measured in a +Pegasus++ fast wave. The simulation parameters are βi0 = 25, α = 0.1, and Te/Ti0 = 1; +using these numbers in (3.17) predicts νeff ≈ 16k⊥vA. +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +x (ρi0) +−2 +0 +2 +⟨βi∆⟩y +k⊥vAt =0.0 +k⊥vAt =0.08 +k⊥vAt =0.39 +Figure 19: Wavefront-averaged βi∆ in the fast wave for the same time frames as figure 17. +Pressure-anisotropy regulation from the firehose instability maintains βi∆ ≳ −2, while +the mirror fluctuations cause some distortion of the mode above βi∆ ≈ 1 but are unable +to regulate fully the positive anisotropy to marginally unstable values. An increase in the +rate at which positive pressure anisotropy is generated by the steepened wave and the +asymmetry in the anisotropy’s regulation by micro-instabilities causes an enhancement +of the positive pressure anisotropy in the final time shown. +3.2.5. Viscous damping and collisional steepening +To study fast-wave behaviour at asymptotically large scale separations, we employ +the Landau-fluid CGL-MHD code. These simulations are performed using a larger β +parameter than used in the Pegasus++ run, βi0 = 100 rather than 25, and with α = 0.2. +These parameters have the advantage that a large portion of the fast wave is initially +above the threshold for instability while the wave remains somewhat linear in amplitude. +As discussed in §3.2.1, this code introduces a user-specified constant scattering rate in +(and only in) the kinetically unstable regions of the plasma. We set this scattering rate +according to (3.17) using the initial mode amplitude. In reality, this scattering rate should + +34 +S. Majeski, M. W. Kunz, and J. Squire +(a) +(b) +Figure 20: (a) Propagation of an α = 0.2 fast wave with βi0 = 100 and νlim set by (3.17). +The top panel shows wave steepening in the fluid velocity, with no noticeable viscous +decay on the timescale of shock formation. The bottom panel shows regulation of the +pressure anisotropy to near the mirror and firehose thresholds. A peak appears in βi∆ due +to the rapid generation of positive pressure anisotropy in the steepening wavefront. (b) +The maximum density gradient found within the domain of the same α = 0.2, βi = 100 +fast wave, compared against an equivalent run with νlim = 0. The predicted shock times +are labelled by tda +s +and tsa +s , and the shock times detected by the same method used for +figure 14 are denoted by circular markers. The growth of the maximum gradient continues +for a longer time in the single-adiabatic case than in the double-adiabatic case, indicating +delayed shock formation. +decay alongside the amplitude, and so our treatment will not precisely reproduce the +results that would be obtained from a more rigorous kinetic calculation. +In figure 20, the propagation and nonlinear steepening of the CGL-MHD fast wave are +presented. The top panel in figure 20(a) shows the bulk fluid velocity perpendicular to +the background field at three different times, exhibiting steepening without a significant +change in wave amplitude. This indicates that no significant viscous dissipation occurs on +a timescale comparable to the shock-formation timescale. The bottom panel shows the +pressure anisotropy of the wave at the same times, multiplied by βi. The anisotropy is +substantially reduced below what it would be in the absence of the limiting collisionality, +particularly on the firehose-unstable side, although it is not perfectly regulated to the +instability thresholds. In particular, a peak in the positive pressure anisotropy becomes +prominent starting from k⊥vAt ≈ 0.1. This is a result of wave steepening, as the sharp +gradient at the wavefront generates positive ∆ much faster than the slow decline in the +wake generates negative ∆, as well as faster than our (constant) limiting collisionality is +able to regulate. Figure 20(b) displays the evolution of the maximum absolute value of +the density gradient from this run, alongside that from a comparable run with νlim = 0. +On the abscissa is the simulation time normalized by the double-adiabatic shock time +tda +s +(see (3.12)). We calculated the shock time for each run using the same detection +method as in figure 16; these times, marked by filled circles in the figure, agree reasonably +well with the predicted values of tda +s +and tsa +s +for the collisionless and collisional cases, +respectively. The difference in steepening rate between the two runs can be interpreted +as νlim forcing a more MHD-like, rather than collisionless, evolution in the fast wave. + +High-β collisionless magnetosonic modes +35 +The collisional isotropization at the peaks of the wave (which are also the most rapidly +moving regions) effectively changes the local adiabatic index of the ions, slowing down +the steepening process and yielding better agreement with tsa +s than with tda +s . In this sense +then, all of the essential characteristics of large-amplitude, high-β, collisionless fast waves +approach that of single-adiabatic MHD as a result of induced micro-instabilities. +4. Summary and discussion +This exploration of microphysically unstable magnetosonic modes brings closure to a +systematic investigation of isolated waves in collisionless, high-β plasmas that started +with the discovery of self-interrupting Alfv´en waves (Squire et al. 2016, 2017a) and +continued with the demonstration of self-sustaining sound (Kunz et al. 2020). In sum- +mary, through the action of adiabatic invariance, the consequent production of pressure +anisotropy, and the excitation of rapidly growing, micro-scale kinetic instabilities. . . +• +collisionless linearly polarized Alfv´en waves with amplitudes satisfying (δB⊥/B0)2 ≳ +2/βi0 retard their own propagation and spur their own viscous decay; +• +collisionless IAWs with amplitudes satisfying |δn/n| ≳ 2/βi0 avert their otherwise +potent Landau damping and propagate in a manner akin to sound waves in a weakly +collisional fluid; +• +collisionless NP modes with amplitudes satisfying |δB∥/B0| ≳ 0.4 and wavelengths +λ∥ ≫ 103β3/2 +i0 ρi0 interrupt their transit-time damping and behave similarly to MHD +entropy modes (at smaller wavelengths, these large-amplitude NP modes decay via +transit-time damping, which is sustained against its nonlinear saturation by weak +mirror-induced collisionality); and +• +collisionless fast waves with amplitudes satisfying |δB/B0| ≳ 2/βi0 and wavelengths +λ⊥ ≫ 102βi0ρi0 acquire an effective adiabatic index of 5/3 and therefore propagate +and nonlinearly steepen at single-adiabatic rates. +Notwithstanding the somewhat narrow focus on the behaviour of isolated eigenmodes, +the simple demonstration that micro-scale physics effectively filters out what kinds of +macro-scale fluctuations are allowed in a high-β plasma is of broad relevance to observed +space and astrophysical systems and to theories for electromagnetic turbulence. The +most immediate application to the former is the near-Earth solar wind. For example, +Verscharen et al. (2016) used linear theory to conjecture that plasma instabilities could +be driven by compressive fluctuations in the β ≳ 1 solar wind through the adiabatic +production of pressure anisotropy, leading to ‘collisionless isotropization’ of solar-wind +protons. Our work supports this idea quantitatively from first principles. Verscharen et al. +(2017) then measured the polarization of compressive fluctuations within the solar wind +at 1 au using data from the Wind spacecraft, finding that the eigenmode relationships +detected were best represented by MHD, rather than collisionless, slow modes. Coburn +et al. (2022) approached this same issue from a different angle, measuring the dispersion +relation of compressive modes in the solar wind and determining which scattering rates +best reproduced them. They concluded that the mean free path predicted by their wave +measurements is ∼103 times smaller than that set by Coulomb collisions, finding that the +dispersion relation of the measured fluctuations most closely resembles that of Braginskii- +MHD slow modes. Both of these observational results find a natural explanation in the +context of our paper, at least for those portions of the wind having β ≳ 1 that have been + +36 +S. Majeski, M. W. Kunz, and J. Squire +measured to be constrained by the firehose and mirror instability thresholds (Kasper +et al. 2002; Hellinger et al. 2006; Bale et al. 2009; Chen et al. 2016). +To the extent that nonlinearly interacting fluctuations in strong electromagnetic tur- +bulence retain some characteristics of their linear eigenmodes, the above conclusions +cast doubt on whether some well-established pillars of MHD and gyrokinetic turbulence +theory (Goldreich & Sridhar 1995; Lithwick & Goldreich 2001; Schekochihin et al. 2009; +Schekochihin 2022) are applicable to high-β plasmas. For example, with each fluctuation +generating and responding to pressure anisotropy in an amplitude-, wavelength-, and +polarization-dependent way, it is suspect that inertial-range compressive fluctuations are +simply passively mixed by the Alfv´en-wave cascade and, in turn, exert no back-reaction +on the Alfv´enic fluctuations. The mutual interactions between what are conventionally +considered to be energetically decoupled cascades, and the impact of this coupling +on the constant flux of energy and the locality of interactions in k space, ought be +investigated. Some progress on this front has recently been made by Arzamasskiy et al. +(2022), who showed using hybrid-kinetic simulations that strong Alfv´enic turbulence with +(δB⊥/B0)2 ≳ 2/βi0 self-consistently produces a parallel viscous scale comparable to the +driving scale of the cascade and involves non-local energy transfers in k space associated +with the excitation of ion-Larmor-scale kinetic instabilities. Incorporating compressive +fluctuations into the turbulence forcing would be informative, not only with regards to the +dynamics but also concerning the partition of turbulent energy into ion versus electron +heating (cf. Kawazura et al. 2020). It is additionally unclear how all this additional +physics plays out within a turbulent cascade governed by a scale-by-scale ‘critical balance’ +between the characteristic linear and nonlinear frequencies, an organizing principle for +strong turbulence that appears to hold (albeit in a modified form) even in the presence +of strong pressure anisotropies (Bott et al. 2021; Arzamasskiy et al. 2022). +Acknowledgements +S.M. and M.W.K. were supported in part by NSF CAREER Award No. 1944972. +Support for J.S. was provided by Rutherford Discovery Fellowship RDF-U001804, which +is managed through the Royal Society Te Ap¯arangi. High-performance computing re- +sources were provided by: the Texas Advanced Computer Center at The University of +Texas at Austin under Stampede2 allocation TG-AST160068 and Frontera allocation +AST20010; and the PICSciE-OIT TIGRESS High Performance Computing Center and +Visualization Laboratory at Princeton University. This work used the Extreme Science +and Engineering Discovery Environment (XSEDE), which is supported by National +Science Foundation grant number ACI-1548562. The authors thank Archie Bott, Eliot +Quataert, Alex Schekochihin, and the participants of the 13th Plasma Kinetics Working +Meeting at the Wolfgang Pauli Institute in Vienna for useful discussions. M.W.K. addi- +tionally thanks the Institut de Plan´etologie et d’Astrophysique de Grenoble (IPAG) for +its hospitality and visitor support while this work was being completed. +Appendix A. Hermite–Laguerre solution to linear KMHD +In this appendix, we detail our numerical method for calculating the time-dependent +pressure anisotropy generated by a linear NP mode. The task is to integrate the sys- +tem (2.1) numerically from an appropriate set of initial conditions. Before providing those +conditions, we take the time derivative of (2.1b) and use (2.1c) to obtain the following + +High-β collisionless magnetosonic modes +37 +wave equation for the E × B drift velocity: +� d2 +dt2 + k2v2 +A +� +u⊥ = − ik⊥ +min0 +d +dt +� +δp⊥i + Teδn +� +. +(A 1) +The right-hand side of this equation is calculated by taking the zeroth and second +moments of the linearized Vlasov equation (2.1a). After assuming an isotropic Maxwellian +background, F0 = FM(v), and rewriting the electric and magnetic-mirror forces using +(2.1c) and (2.1d), equation (2.1a) reduces to +� ∂ +∂t + ik∥v∥ +� +δf + +� +ik⊥u⊥ +w2 +⊥ +v2 +th,i ++ ik∥v∥ +Te +Ti0 +δn +n0 +� +FM = 0. +(A 2) +Equations (A 1) and (A 2) are solved numerically as follows. +We express the v∥ dependence of δf in terms of Hermite polynomials Hn and the w2 +⊥ +dependence in terms of Laguerre polymonials Lm: +δf(t, k∥, k⊥, v∥, w⊥) = FM(v) +∞ +� +m,n=0 +gm,nHn +� v∥ +vth,i +� +Lm +� w2 +⊥ +v2 +th,i +� +. +(A 3) +This spectral decomposition allows the required moments to be calculated simply as +δn +n0 += g0,0, +δp⊥i +pi0 += g0,0 − g1,0, +δp∥i +pi0 += g0,0 + 4g0,2, +(A 4) +so that (A 1) becomes +� d2 +dt2 + k2v2 +A +� u⊥ +vth,i += −ik⊥vth,i +2 +d +dt +�� +1 + Te +Ti0 +� +g0,0 − g1,0 +� +. +(A 5) +Because the Hermite and Laguerre polynomials form orthonormal bases with respect to +Gaussian and exponential weights, respectively, equation (A 2) may be easily transformed +to Hermite–Laguerre space to find +dgm,0 +dt ++ ik∥vth,igm,1 + i(δm,0 − δm,1)k⊥u⊥ = 0, +(A 6a) +dgm,1 +dt ++ ik∥vth,i +� +2gm,2 + 1 +2gm,0 +� ++ i Te +2Ti0 +δm,0g0,0 = 0, +(A 6b) +dgm,n +dt ++ ik∥vth,i +� +(n + 1)gm,n+1 + 1 +2gm,n−1 +� += −νn4gm,n, +n ⩾ 2. +(A 6c) +Note that the term k∥v∥δf representing the parallel phase mixing of the perturbed +distribution function couples together different Hermite moments, representing the gen- +eration of fine-scale structure in v∥. Because the magnetic field suppresses phase mixing +across the magnetic field, there is no cascade to higher w⊥ moments and only the first +two Laguerre polynomials (m = 0, 1) are needed. To the right-hand side of (A 6c) we +have appended a fourth-order hyper-collision operator; the restriction of the collision +operator to n ⩾ 2 guarantees that number and momentum are conserved. The hyper- +collisionality is added because only a finite number of Hermite polynomials are usable, +so the series must be truncated somewhere. A hard truncation in which the final v∥ +moment is arbitrarily set to zero will result in numerical instability unless a collisionality +is employed to ensure the velocity-space cascade (associated with parallel phase mixing +of the perturbed distribution function) decays to zero amplitude before the last resolved +moment is reached. +A code was written in Fortran 90 to solve (A 5) and (A 6). Equation (A 6) is solved + +38 +S. Majeski, M. W. Kunz, and J. Squire +and δf updated in time using a semi-implicit Crank–Nicholson method; the moments g0,0 +and g1,0 are then used in (A 5) to update the drift velocity using centered differencing in +time. The discrete time axes on which gm,n and u⊥ are stored are staggered to maintain +appropriate centering for all derivatives. The matrix inversion needed to update gm,n is +performed using the Thomas Tridiagonal Matrix Algorithm (TDMA). +For the initial conditions, we start from isothermal pressure balance, with g1,0 = g0,2 = +0 and g0,0 ̸= 0 (but arbitrary). The reasoning behind this choice is discussed in §2.2.1. +These initial conditions transition rapidly into the NP eigenmode by launching small- +amplitude (relative to the amplitude of the NP mode) fast waves that facilitate the +adjustment. The linear evolution of the NP mode from this initial condition is shown in +figure 1 and discussed in §2.1.3. +Appendix B. Magnetosonic modes with arbitrary scattering +frequency +To obtain the linear dispersion relation of kinetic hydromagnetic modes at arbitrary +ν, we must use a model that accurately captures the effects of adiabatic invariants, heat +fluxes, and collisional isotropization. One such model is given by the Chew et al. (1956) +equations supplemented, by collisional isotropization and closed by so-called Landau-fluid +heat fluxes (Snyder et al. 1997). Assuming isothermal electrons, these equations are: +Dn +Dt = −n∇ · u, +(B 1a) +minDu +Dt = −∇ +� +p⊥i + nTe + B2 +8π +� ++ ∇ · +� +ˆbˆb +� +∆pi + B2 +4π +�� +, +(B 1b) +DB +Dt = (B · ∇)u − B∇ · u, +(B 1c) +nB D +Dt +�p⊥i +nB +� += −∇ · +� +q⊥iˆb +� +− q⊥i∇ · ˆb − 1 +3ν∆pi, +(B 1d) +n3 +B2 +D +Dt +�p∥iB2 +n3 +� += −∇ · +� +q∥iˆb +� ++ 2q⊥i∇ · ˆb + 2 +3ν∆pi, +(B 1e) +where D/Dt .= ∂/∂t+u · ∇ is the convective derivative for the bulk velocity u, ˆb .= B/B +is the unit vector in the direction of the local magnetic field, ∆pi .= p⊥i − p∥i is the +dimensional ion pressure anisotropy, ν is the isotropizing collision frequency, and q∥i and +q⊥i represent the field-parallel flow of parallel and perpendicular ion heat. For linear +perturbations to the ion temperature (δT∥i, δT⊥i) and magnetic-field strength (δB∥) +having parallel wavenumber k∥, the latter may be adopted from equations (48) and (49) +of Snyder et al. (1997): +q∥i,k = − +4nv2 +th∥,i +2√π|k∥|vth∥,i + (3π − 8)ν ik∥δT∥i, +(B 2) +q⊥i,k = − +nv2 +th∥,i +√ +2π|k∥|vth∥,i + 2ν +� +ik∥δT⊥i + ik∥T⊥i∆i +δB∥ +B +� +. +(B 3) +These ‘3+1’ heat fluxes accurately reproduce the linear Landau–Barnes damping of the +kinetic hydromagnetic modes in the collisionless limit (Snyder et al. 1997, §VIII) and +take on a form akin to that obtained by Braginskii (1965) in the collisional limit. Because +Braginskii-MHD does not accurately capture the linear heat fluxes when ν ≲ |k∥|vth,i, +the Landau-fluid CGL equations are used to describe the linear propagation of these + +High-β collisionless magnetosonic modes +39 +modes at arbitrary ν, bridging the gap between the fully collisionless (ν = 0) and the +weakly collisional (ν ≫ k∥vth,i). Note that, in the absence of heat fluxes and collisionality, +equations (B 1d) and (B 1e) guarantee conservation of the adiabatic invariants µ and J +associated with Larmor gyrations and bounce motion. One of the advantages of using the +Landau-fluid CGL equations over a Vlasov approach is the former’s lack of dependence +on the plasma dispersion function Z(ζ), whose dependence on ζ .= ω/|k∥|vth,i can only +be expressed analytically in the asymptotic limits ζ ≫ 1 and ζ ≪ 1. Instead, the ‘3+1’ +heat fluxes yield polynomial dispersion relations for the modes at all frequencies. As a +result, if one wishes to derive an analytic expression for the frequency and damping rate +of the oblique IAW, which has ζ ∼ 1 when Te/Ti0 ∼ 1, they can then do so with ease. +Proceeding with the linear analysis, we assume zero background pressure anisotropy, +neglect all nonlinear terms, and Fourier transform (B 1)–(B 3) in space and time, so +that D/Dt → −iω and ∇ → ik. The result is a straightforward algebraic system, some +solutions of which are shown in figure 21. In total there are 8 modes associated with 8 +unique time derivatives (∇ · B = 0 fixes one of the components of δB⊥). The modes +not displayed in figure 21 are the Alfv´en waves (which would be lines at ζ = ±β−1/2 +i0 +) +and both fast waves (which are shown in figure 15). Considering that there exists one +additional time derivative in CGL-MHD than in collisional MHD due to the splitting of +the thermal pressure into two components, there should be a mode that vanishes in the +collisional limit. Indeed, after bifurcation one branch of the oblique IAW becomes non- +propagating and is damped at a rate approximately equal to ν as ν → ∞. This strong +damping is due to the mode’s polarization, having opposing perpendicular and parallel +pressure perturbations that satisfy |δp⊥| ≫ |δp∥| when k⊥ ≫ k∥. Hence the reason we +have termed this mode the “anisotropy mode” in figure 21: it remains anisotropic even +at arbitrarily large ν, causing it to damp increasingly rapidly. +The NP mode is often associated with the collisionless limit of the MHD slow mag- +netosonic mode (e.g., Verscharen et al. 2017), and is frequently referred to as the +collisionless slow mode. This may be due to the fact that the Braginskii-MHD dispersion +relation predicts a non-propagating slow mode at sufficiently low ν, one which remains +non-propagating as ν → 0. In reality, the slow mode does propagate once again at +sufficiently low collisionality, and the NP mode is better identified as the kinetic extension +of the MHD entropy mode. In the MHD entropy mode, no pressure perturbation is +permitted by the parallel momentum equation, only a density perturbation. However, +at lower scattering rates the pressure separates into its field-parallel and perpendicular +components, and perpendicular pressure balance becomes achievable (see (B 1b)). The +assertion that the NP mode is connected to the MHD entropy mode, rather than the +slow mode, is likely more desirable as it also avoids degeneracy in different branches of +the dispersion relation. Careful inspection of figure 21 shows that there exists a band +in which both the NP and oblique ion-acoustic modes possess zero real frequency. If it +were the case that the MHD slow mode became the NP mode, this branch would have to +cross with the kinetic entropy mode and both would have identical decay rates, making +them degenerate. Therefore, in our argument for the behaviour of above-threshold NP +modes in high-β plasmas, we expect that at very large scale separation, and hence large +ν/|k∥|vth,i, the NP mode will become more akin to the MHD entropy mode. +The oblique ion-acoustic wave (IAW) also deserves special attention, not least because +it possesses a non-propagating band beginning near ν ∼ k∥vth,i. Somewhat paradoxically, +this is the collisionless extension of the MHD slow mode, never mind the fact that at +high β it propagates faster than the Alfv´en speed. Even in the collisionless Landau-fluid +CGL model, this mode evades a simple general expression for its frequency. However, + +40 +S. Majeski, M. W. Kunz, and J. Squire +Figure 21: Linear dispersion relation of the Landau-fluid CGL-MHD equations (B 1). The +dimensionless (complex) frequency ζ .= ω/|k∥|vth,i is computed numerically as a function +of collisionality ν/|k∥|vth,i for k⊥ = 4|k∥|, βi0 = 16, and Te = Ti0. +in the limit of k⊥ ≫ k∥ and β ≫ 1 with Te = Ti0, one can obtain the dispersion +relation numerically; we find that ζ ≈ 1 − 0.43i. This mode therefore has a very +similar dispersion relation to its parallel-propagating variant, especially with regards +to its rapidly damped nature. Asymptotic analysis for k⊥ ≫ k∥ reveals that this mode +develops a non-propagating band when β ≳ 7.1, occurring in the approximate range of +scattering frequencies satisfying ν/k∥vth,i ∈ [2, (3/4)√β]. When β ∼ O(1) and smaller, +the Braginskii slow mode smoothly transitions into the oblique IAW as ν → 0. However, +at high β, an increasingly large gap forms between the two propagating portions of this +mode. This phenomenon is not present in parallel-propagating IAWs at any β. +Appendix C. Oblique IAWs and micro-instabilities +Of the collisionless hydromagnetic modes that do not propagate parallel to the back- +ground magnetic field, we have yet to discuss one in the context of high-β plasmas +and micro-instabilities: the oblique IAW. Given that oblique IAWs share many traits +with their parallel propagating counterparts (§B), generalizing the results of Kunz et al. +(2020) to the oblique case should not require dramatic changes. Even when propagating +across the background magnetic field, at high β these waves are still largely driven by a +perturbation to the parallel pressure. As a result, the magnetic tension plays essentially +no role, and no interruption-like process can occur as in the case of linearly polarized +Alfv´en waves. Furthermore, the oblique IAW generates equivalent positive and negative +pressure anisotropies (there is no pressure balance as in the NP mode). For this reason, +both mirror and firehose instabilities can be triggered by this mode. The only notable +difference between the oblique and parallel IAWs is the existence of a non-propagating +band at certain values of ν in the dispersion relation of the oblique mode. To see how + +High-β collisionless magnetosonic modes +41 +this difference affects propagation in the presence of instability-induced scattering, we +perform an analysis similar to that carried out in §3.1.3. +Our first task is to determine the amplitude limit above which the anisotropic pressure +perturbation in the oblique IAW is unstable to both the mirror and firehose instabilities. +Taking the k⊥ ≫ k∥ and β ≫ 1 limit, the parallel and perpendicular temperature +perturbations in the oblique IAW are +δT∥ +Ti0 +≈ − +� +2 + +� +1 + ik∥vth,i +ω√π +�−1�� +1 + 2ik∥vth,i +ω√π +�−1 δB∥ +B0 +, +(C 1a) +δT⊥ +Ti0 +≈ +� +1 + ik∥vth,i +ω√π +�−1 δB∥ +B0 +. +(C 1b) +Substituting in ω/k∥vth,i ≈ 1 − 0.43i, equations (C 1) yield an ion pressure anisotropy +∆ = (1.88 − 3.03i)(δB∥/B0). This implies the following amplitude threshold for oblique +IAWs to trigger both the firehose and mirror instabilities: +���� +δB∥ +B0 +���� ≳ 1 +βi +(oblique IAW amplitude threshold). +(C 2) +We argue that, above this threshold, the scattering induced by micro-instabilities will be +that required to maintain marginal stability, or ∆ ∼ β−1 +i +. Through the same logic as was +applied to the fast mode, this scattering rate is +ν ∼ Re +� +3ωβi +�δB∥ +B0 +− 2 +3 +δn +n0 +�� +≈ 3.7k∥vth,iβi +���� +δB∥ +B0 +����. +(C 3) +As in the case of the fast wave, the above expression for the limiting collisionality is +only valid in the limit that ν ≫ ω. This constraint is nearly satisfied at the amplitude +threshold, therefore this scattering rate is likely to be a good approximation even for +mode amplitudes of only a few times β−1 +i +. +With the scaling of the induced scattering rate now known, we may return to the +dispersion relation shown in figure 21 to surmise how micro-instabilities might modify +the propagation of oblique IAWs. Recall from appendix B that the oblique IAW becomes +non-propagating for βi ≳ 7.1 when ν/k∥vth,i ∈ [2, (3/4)√β]. The form of the effective +scattering rate (being dependent on δB∥) then suggests that the fate of an oblique IAW +rests on the amplitude of the initial perturbation. For amplitudes within the range β−1 ≲ +|δB∥/B0| ≲ β−1/2, the oblique IAW will become a viscously damped mode which does not +propagate, while above |δB∥/B0| ≳ β−1/2 it will become a Braginskii-like propagating +sound wave. The latter of the two regimes is essentially the result obtained by Kunz +et al. 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G. & Quataert, E. 2016 Collisionless +isotropization of the solar-wind protons by compressive fluctuations and plasma +instabilities. Astrophys. J. 831, 128. +Verscharen, D., Chen, C. H. K. & Wicks, R. T. 2017 On kinetic slow modes, fluid slow +modes, and pressure-balanced structures in the solar wind. Astrophys. J. 840, 106. + +44 +S. Majeski, M. W. Kunz, and J. Squire +Yoon, P. H., Wu, C. S. & de Assis, A. S. 1993 Effect of finite ion gyroradius on the fire-hose +instability in a high beta plasma. Phys. Fluids B 5, 1971. + diff --git a/XtE0T4oBgHgl3EQfWADS/content/tmp_files/load_file.txt b/XtE0T4oBgHgl3EQfWADS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c947b747ea1c5df261b012f8171c6a6f814ad510 --- /dev/null +++ b/XtE0T4oBgHgl3EQfWADS/content/tmp_files/load_file.txt @@ -0,0 +1,2025 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf,len=2024 +page_content='DRAFT 1 Microphysically modified magnetosonic modes in collisionless, high-β plasmas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski 1†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz1,2, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire3 1Department of Astrophysical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Peyton Hall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' NJ 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' USA 2Princeton Plasma Physics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' PO Box 451,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' NJ 08543,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' University of Otago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 730 Cumberland St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' North Dunedin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Dunedin 9016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' New Zealand (compiled on 9 January 2023) With the support of hybrid-kinetic simulations and analytic theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' we describe the nonlinear behaviour of long-wavelength non-propagating (NP) modes and fast magne- tosonic waves in high-β collisionless plasmas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' with particular attention to their excitation of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' and reaction to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' kinetic micro-instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The perpendicularly pressure balanced polarization of NP modes produces an excess of perpendicular pressure over parallel pressure in regions where the plasma β is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For mode amplitudes δB/B0 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, this excess excites the mirror instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Particle scattering off these micro-scale mirrors frustrates the nonlinear saturation of transit-time damping, ensuring that large- amplitude NP modes continue their decay to small amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At asymptotically large wavelengths, we predict that the mirror-induced scattering will be large enough to inter- rupt transit-time damping entirely, isotropizing the pressure perturbations and morphing the collisionless NP mode into the magnetohydrodynamic (MHD) entropy mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In fast waves, a fluctuating pressure anisotropy drives both mirror and firehose instabilities when the wave amplitude satisfies δB/B0 ≳ 2β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The induced particle scattering leads to delayed shock formation and MHD-like wave dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Taken alongside prior work on self-interrupting Alfv´en waves and self-sustaining ion-acoustic waves, our results establish a foundation for new theories of electromagnetic turbulence in low-collisionality, high-β plasmas such as the intracluster medium, radiatively inefficient accretion flows, and the near-Earth solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Context and motivation Nearly half of all the baryonic matter in the Universe resides in a hot and dilute plasma state, in which Coulomb collisions are relatively rare and cosmic magnetic fields greatly influence the trajectories of the constituent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Examples include the warm- hot intergalactic medium, having number densities n ≳ 10−6 cm−3 and temperatures T ∼ 105–107 K, and the intracluster medium of galaxy clusters, with n ≳ 10−3 cm−3 and T ∼ 107–108 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Radiatively inefficient accretion flows such as that onto the supermassive black hole at the Galactic centre, as well as the Solar wind that pervades interplanetary space, provide smaller-scale examples of systems characterized by large collisional mean free paths and small particle gyro-radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A key feature of these systems is that the transport of momentum and heat are anisotropic with respect to the magnetic- field direction, even when the magnetic energy is much less than the thermal pressure, † Email address for correspondence: smajeski@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='02273v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='HE] 5 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= 8πnT/B2 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This spatial anisotropy is a direct result of the velocity-space anisotropy in the particle distribution function, which is allowed by the rarity of particle- particle collisions and shaped by the particles’ primary allegiance to the local magnetic- field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In high-β plasmas, such field-biased deviations from local thermodynamic equilibrium can have important dynamical consequences on both the large ‘fluid’ scales and the small plasma-kinetic ‘micro’ scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is this multi-scale connection between a high-β plasma’s thermodynamics and its fluid dynamics that is the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In particular, by elucidating the non-linear behaviour of long-wavelength magnetosonic modes, and placing our findings in the company of complementary work on Alfv´enic and acoustic fluctuations, we demonstrate that even textbook examples of plasma dynamics, such as basic waves, can be fundamentally different in weakly collisional, high-β plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Pressure anisotropy, micro-instabilities, and collisionless damping Collisionless and weakly collisional plasmas possess particles whose motions are bound by adiabatic invariants that are otherwise broken in highly collisional MHD plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' While there are three adiabatic invariants most commonly considered in plasma physics, two of them – the magnetic moment µ for cross-field gyro-motion and the bounce invariant J for field-parallel bounce motion – are associated with frequencies that are generally large enough for these invariants to be approximately conserved even when some collisions are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For describing collective behaviour, these invariants are often adapted into the form of the double adiabats p⊥/nB and p∥B2/n3, which are conserved in time along the flow of the plasma if the density n and magnetic-field strength B change slowly relative to the periodic (gyro- or bounce) motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, the thermal pressure p is split up into components along and across the magnetic- field direction, p∥ and p⊥ respectively, a result of the invariants each being associated with different components of the particles’ motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In essence, the random thermal motions of a collisionless or weakly collisional plasma are restricted differently depending on whether they are along or across the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Their dynamical importance with respect to the magnetic field can also be defined separately, as β⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= 8πp⊥/B2 and β∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= 8πp∥/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Natural variations in the plasma density and magnetic-field strength that are present, coupled with approximate double-adiabatic invariance, lead to the development of pressure anisotropy ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= p⊥/p∥ − 1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Often, the magnitude of ∆ is small and may have little effect on a plasma’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, in high-β plasmas where the thermal pressure is much larger than the magnetic energy, even small deviations from thermal isotropy may be significant enough to grant the pressure anisotropy a role comparable to that of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Two mechanisms by which the pressure anisotropy plays this elevated role are the modification of magnetic-field-line tension and the triggering of rapidly growing, kinetic micro-instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' An illustration of the former mechanism is a process named ‘Alfv´en wave interruption’ (Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2016, 2017a,b), in which a linearly polarized Alfv´en wave whose amplitude satisfies (δB⊥/B)2 ≳ 2/β adiabatically generates a pressure anisotropy large enough to nullify the restoring magnetic tension and prevent the wave’s propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this paper, we are focused primarily on large-scale compressive fluctuations, for which magnetic tension ends up being of little importance at high β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Our focus is therefore primarily on the connection that pressure anisotropy has with ion-Larmor-scale kinetic instabilities, specifically the firehose and mirror instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The firehose instability is triggered in pressure-anisotropic plasmas satisfying β∥∆ ≲ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This threshold is commonly referred to as the ‘fluid firehose’ threshold, and corre- sponds to an exact balance between the restoring magnetic tension force and the desta- High-β collisionless magnetosonic modes 3 bilizing viscous stress from the negative pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 In this case, when small perpendicular fluctuations in the magnetic field are present, the excess parallel pressure leads to a centrifugal force that acts in the bends of the magnetic-field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' When the pressure anisotropy is sufficiently negative, this force cannot be stably balanced by the magnetic tension and the bends grow very rapidly (Parker 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Vedenov & Sagdeev 1958), increasingly so on smaller lengthscales (down to the ion-Larmor scale, where they are stabilized by finite-Larmor-radius effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kennel & Sagdeev 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Davidson & V¨olk 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Hellinger & Matsumoto 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In a driven system, the unstable pressure anisotropy is regulated through a combination of the particles pitch- angle scattering off of these bends and the compensating positive pressure anisotropy associated with the growing magnetic perturbations (Schekochihin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Rosin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Conversely, the mirror instability is triggered when an excessively positive pressure anisotropy satisfies β⊥∆ ≳ 1 (Barnes 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Hasegawa 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, the enhanced perpendicular pressure is able to push out against local decrements in the magnetic-field strength, causing ion-Larmor-scale ‘magnetic mirrors’ to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These mirrors resonantly confine particles with large pitch angles (v⊥ > v∥) through their conservation of µ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Southwood & Kivelson 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The anisotropic thermal energy of these resonant particles reinforces the outward push against the field lines, further growing the fluctuations (and thus the confining mirror force) until the ends of the mirrors become so kinked that the particles can pitch-angle scatter off of their sharp edges and regulate the pressure anisotropy (Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Riquelme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Rincon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2020) demonstrated that these kinetic instabilities interfere with the collisionless damping of long-wavelength, parallel-propagating ion-acoustic waves (IAWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Namely, IAW amplitudes satisfying |δn/n| ≳ 2/β generate a pressure anisotropy large enough to drive firehose and mirror instabilities, whose associated scattering and trapping impede the maintenance of Landau resonances that enable such waves’ otherwise potent decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The result is self-sustaining wave dynamics that evince a weakly collisional plasma: the ion distribution function is near-Maxwellian, the field-parallel flow of heat resembles its Braginskii form (except in regions where large-amplitude magnetic mirrors strongly suppress particle transport), and the relations between various thermodynamic quantities are more ‘fluid-like’ than kinetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Non-propagating modes, fast waves, and oblique IAWs In this work, a combination of elements from both Alfv´en waves and IAWs is in- vestigated in the study of collisionless magnetosonic modes – namely, non-propagating (NP) modes (in §2), fast waves (in §3), and to a more limited extent oblique IAWs (in appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We investigate fast waves in the limit of perpendicular propagation, in which magnetic tension and collisionless damping play no role, but the associated fluctuations in B and n drive unstable pressure anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The NP modes, on the other hand, are highly oblique, perpendicular-pressure-balanced structures, in which collisionless transit-time damping (or ‘Barnes damping’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Barnes 1966) is responsible for the entirety of the modes’ dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Barnes damping is a form of Landau (1946) damping in which sinusoidal fluctuations in magnetic-field strength caused by an oblique 1Certain conditions can lead to the dominance of a resonant oblique firehose instability having a less stringent threshold of β∥∆ ≲ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 (Hellinger & Matsumoto 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As none of the magnetosonic fluctuations investigated in this paper are subject to self-interruption, the difference between −2 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 is of little consequence, and we generically refer to the ‘firehose threshold’ as being at −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire perturbation (magnetic ‘mirrors’) resonantly confine µ-conserving particles and perform work on their guiding centres, thereby transferring free energy from the electromagnetic perturbations to the particles and damping the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For large values of β, the damping rate of the NP mode is relatively slow, and nonlinear saturation of the damping process can occur before the mode decays by a significant fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, trapped particles in near resonance with the mode are rearranged in phase space, flattening the velocity distribution function of the particles f(v∥) in the vicinity of the phase velocity (in this case, v∥ ∼ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Once (∂f/∂v∥)|0 ∼ 0, there is no more free energy left to be gained by the distribution from rearranging particles, and the damping process stalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This swapping of phase-space positions occurs on the order of a bounce time, ∼Ω−1 b , which is the time it takes for a (just barely) trapped particle to make a full orbit of its confining magnetic mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The larger amplitude a mode, the shorter its bounce time, so the nonlinear saturation ensures that large-amplitude NP modes are longer lived than their small-amplitude counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The principal question here is to what extent the pressure anisotropy associated with these modes affects their character and longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Non-propagating modes: Suppression of nonlinear saturation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Theory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Model equations and assumptions The linear evolution of the NP mode at long wavelengths can be treated analytically in the drift-kinetic approximation, in which all relevant time- and lengthscales are much larger than those associated with the particles’ gyromotion and the velocity distribution function of the particles is gyrotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We adopt this framework, and further simplify the calculation by treating the electrons as a massless, neutralizing, isothermal fluid having constant temperature Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 In this model the velocity of magnetic-field lines, and equivalently the perpendicular fluid flow, is captured by the E × B drift velocity u⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The perpendicular velocity peculiar to this drift, denoted by w⊥, then describes the perpendicular particle motion relative to the field lines and the fluid flow, under the constraint that the magnetic moment µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= miw2 ⊥/2B is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The component of the particle velocity directed along the local magnetic-field direction is denoted by v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In what follows, we solve for the evolution of small perturbations δf(t, r, v∥, w⊥) to a spatially uniform ‘background’ ion velocity distribution function F0(v∥, w⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The parallel (∥) and perpendicular (⊥) coordinate directions are fixed with respect to a uniform background magnetic field, B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Assuming that spatial variations in the plasma are due only to a sinusoidal perturbation having wavenumbers k∥ and k⊥, the relevant equations in their linearized forms are the drift-kinetic Vlasov equation, � ∂ ∂t + ik∥v∥ � � δf + δB∥ B0 w⊥ 2 ∂F0 ∂w⊥ � + e mi δE∥ ∂F0 ∂v∥ − ik∥ δB∥ B0 w2 ⊥ 2 ∂F0 ∂v∥ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a) the force equation for the evolution of the drift velocity, du⊥ dt = − ik⊥ min0 � δp⊥i + Teδn � − ik⊥v2 A δB∥ B0 + ik∥v2 A δB⊥ B0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1b) 2The choice of isothermal electrons is for consistency with the simulations performed using the Pegasus++ hybrid-kinetic particle-in-cell code (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2), though it can be justified physically in some weakly collisional plasmas such as the ICM, where the electrons are collisional enough to remain near-Maxwellian and fast enough to be approximately isothermal along perturbed magnetic-field lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Kunz 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 5 the ideal induction equation governing the parallel and perpendicular components of the perturbed magnetic field δB, d dt δB∥ B0 = −ik⊥u⊥ and d dt δB⊥ B0 = ik∥u⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1c) and a generalized Ohm’s law for the parallel electric field, δE∥ = −ik∥ Te e δn n0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1d) The perturbed number density and perpendicular ion pressure are given by δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= � d3v δf and δp⊥i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= � d3v 1 2miw2 ⊥δf, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2) respectively, with d3v = 2πw⊥dw⊥dv∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The other symbols have their usual meanings: e is the elementary charge, mi is the ion mass, and vA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= B0/(4πmin0)1/2 is the Alfv´en speed given B0 and a uniform background density n0 (the zeroth moment of F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Note that u⊥ is not an explicit moment of the perturbed distribution function, and must be evolved independently using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This combination of the drift-kinetic equation with a fluid equation for the drift velocity and a frozen-in magnetic field is commonly referred to as ‘kinetic MHD’ (Kulsrud 1964, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At this point we take F0 to be a stationary, isotropic, Maxwell–Boltzmann distribution, F0 = FM(v), with � d3v FM(v) = n0 and � d3v miv2FM(v) = 3n0Ti0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= 3pi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This not only simplifies the analysis, but also ensures that the background distribution function itself is not kinetically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a) can then be readily integrated in time to obtain δf(t, w⊥, v∥) = δf(0, w⊥, v∥) e−ik∥v∥t − � t 0 dt′ FM(v) e−ik∥v∥(t−t′) � ik∥v∥ Te Ti0 δn(t′) n0 − w2 ⊥ v2 th,i d dt′ δB∥(t′) B0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) where vth,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= (2Ti0/mi)1/2 is the ion thermal speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) represents the parallel phase mixing of the initial perturbation by the free streaming of particles along the (unperturbed) magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' If δf(0, w⊥, v∥) ∝ FM(v), then any velocity-space moment of this term will decay as exp[−(k∥vth,it/2)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) captures the self-consistent response of the plasma to the induced parallel electric field (∝δn/n0) and the magnetic mirror force (∝δB∥/B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is this eigenmode response that we first calculate and discuss, before moving on to take the second moments of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) and compute the time-dependent pressure anisotropy in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Eigenmode response for the NP mode If we take the fluctuation amplitudes to be proportional to exp(−iωt) with complex frequency ω, the dispersion relation that results after combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1) may be written as D(ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= � ω2 − k2v2 A �� 1 + Ti0 Te + ζZ(ζ) � +k2 ⊥v2 th,i ζZ(ζ) � 1 + Ti0 Te + 1 2ζZ(ζ) � = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4) where k2 = k2 ∥ + k2 ⊥, ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ω/|k∥|vth,i is the dimensionless phase speed, and Z(ζ) is the plasma dispersion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The first term in parentheses captures the combined restoring force of the magnetic pressure and tension, and indicates that we are examining magnetosonic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Indeed, setting the accompanying multiplicative term in square brackets to zero provides the dispersion relation for a Landau-damped IAW in the 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire limit (me/mi)1/2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The final term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4), proportional to k2 ⊥v2 th,i, couples these Alfv´enic and acoustic responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' its presence can be traced back to the final term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) representing the mirror force, and thus introduces collisionless damping of the mode through transit-time damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In order to isolate the NP mode, we focus specifically on highly oblique wavenumbers (k⊥ ≫ k∥) and low frequencies (ζ ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this limit, the plasma dispersion function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4) can be approximated as Z(ζ) ≈ i√π, and we may simplify the dispersion relation further by neglecting terms of order ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The result is an approximate expression for the decay rate of the NP mode: ζ ≃ − i √πβi0 k2 k2 ⊥ , where βi0 = 8πpi0 B2 0 = v2 th,i v2 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5) For ζ ≪ 1 to be satisfied by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5), we require that βi0 ≫ k∥/k⊥, which is easily satisfied by our obliqueness assumption and aligns well with our interest in high-β plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Other useful properties of the NP mode, such as the proportionalities between δn, δp⊥,i, and δB∥, can be found by taking moments of the kinetic equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a): δn n0 = −ζZ(ζ) � 1 + Te Ti0 � 1 + ζZ(ζ) ��−1 δB∥ B0 ≃ − 1 β0 k2 k2 ⊥ δB∥ B0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6a) δp⊥i pi0 = − Te Ti0 δn n0 + 2 ω2 − k2v2 A k2 ⊥v2 th,i δB∥ B0 ≃ � 2 + Te Ti0 �δn n0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6b) where β0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= βi0(1 + Te/Ti0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6b) implies approximate perpendicular pressure balance when k∥ ≪ k⊥, since then δp⊥i + δpe + δB2 8π ≃ − k2 ∥ k2 ⊥ δB2 4π ≪ δB2 4π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6c) Additionally, the parallel ion pressure perturbation is given by δp∥i pi0 = − Te Ti0 δn n0 − 2ζ2� 1 + ζZ(ζ) ��δB∥ B0 + Te Ti0 δn n0 � ≃ − Te Ti0 δn n0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6d) so that δp∥i+δpe ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) highlight some of the essential properties of the NP mode, namely, that it does not oscillate but rather decays slowly at high β, and that its perturbations to the magnetic-field strength and the density are anti-correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The physical mechanism behind the damping rate is primarily transit-time magnetic pumping, in which Landau-resonant particles (technically, their guiding centres) that are trapped between large-scale magnetic mirrors formed by an oblique perturbation in the magnetic field extract energy from the mirror force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' They experience net heating by betatron acceleration because the number of particles heated in regions where |B| increases (lower v∥ particles) is greater than the number of particles cooled where |B| decreases (higher v∥ particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At higher plasma β this difference is smaller, hence the β−1 dependence of the damping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This type of collisionless damping is susceptible to nonlinear saturation, whereby the particles in the well explore the phase space available to them by µ conservation, phase-mixing out the original Maxwellian according to their differing bounce times and flattening the distribution function in the magnetic well to create a plateau around v∥ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This effectively increases the plasma β of the resonant particles, and the damping rate weakens dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Because of the slow nature of the NP mode’s decay rate at high β, nonlinear saturation occurs comparatively rapidly, at a rate comparable to the bounce High-β collisionless magnetosonic modes 7 frequency of a thermal particle, Ωb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= 1 2k∥vth,i ���� δB∥ B0 ���� 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7) For |δB∥/B0| ≳ β−2 i0 , the bounce frequency will be larger than the decay rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5), and thus nonlinear saturation will be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Because of our interest in plasmas with β ≫ 1, even modes that may often be considered ‘linear’ in amplitude will thus decay by only a small amount before experiencing nonlinear saturation, the implication being that these structures should be long lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' That is, unless some process is able to erode the resonant plateau in the perturbed distribution function on a timescale ≲Ω−1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Generation of pressure anisotropy and triggering of the mirror instability The eigenmode (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) implies a dimensionless pressure anisotropy in the ions given by ∆NP ≃ 2 � 1 + Te Ti0 �δn n0 ≃ − 2 βi0 k2 k2 ⊥ δB∥ B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8) This suggests that, for δB∥/B0 ∼ 1, the pressure anisotropy associated with the NP mode is sufficient to excite both the firehose and mirror instabilities, the former occurring in regions where δB∥ > 0, the latter occurring in regions where δB∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' There are two considerations that complicate this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The first complication concerns the additional pressure anisotropy that is generated when the initial perturbation to the distribution function is anisotropically phase mixed by particles streaming freely along, but not across, the field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To see this effect, let us return to the time-dependent solution for the perturbed distribution function, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3), and suppose that, at t = 0, the plasma hosts an isothermal, pressure- balanced perturbation with δf(0, w⊥, v∥) = δn(0) n0 FM(v) = − 2 β0 δB∥(0) B0 FM(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9) This initial condition guarantees that the pressure anisotropy that develops as the particles free stream and the plasma settles into the NP eigenmode is generated self- consistently and not put in by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Calculating the difference of the (1/2)miw2 ⊥ and miv2 ∥ moments of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) with the initial condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9) yields the following expression for the time-dependent pressure anisotropy: ∆NP(t) = 2 �k∥vth,it 2 �2 e−(k∥vth,it/2)2� 1 + Te Ti0 �δn(0) n0 + � t 0 dt′ e−[k∥vth,i(t−t′)/2]2 d dt′ δB∥(t′) B0 + 2 � t 0 dt′ �k∥vth,i(t − t′) 2 �2 e−[k∥vth,i(t−t′)/2]2 d dt′ � Te Ti0 δn(t′) n0 + δB∥(t′) B0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10) All terms involving the combination k∥vth,it/2 describe the damping effect of phase mixing on the moments of the perturbed distribution function due to the production of fine-scale structure along v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As discussed by Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2020, their equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7)), the first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10) captures a transiently produced pressure anisotropy resulting from the anisotropy of particle motion: as the magnetized particles free stream along, but not across, the field, the w2 ⊥ and v2 ∥ moments of δf(0) phase mix differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The integral terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10) capture the pressure anisotropy driven by 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire (a) (b) Figure 1: (a) Solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10) using the method presented in appendix A for the time- dependent root-mean-square pressure anisotropy of a linear NP mode with wavenumber k∥ and dimensionless initial amplitude α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= δB∥(0)/B0 for βi0 = 16 and various Te/Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The small oscillations present after the initial adjustment are due to fast waves generated as the isothermal, pressure-balanced initial condition settles into the NP eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The approximate analytic solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) is shown with the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (b) Maximum pressure anisotropy (divided by α) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Te/Ti0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' its values at Te/Ti0 = 1/2, 1, and 2 are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' adiabatic invariance as the mode is excited and then decays in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is this contribution to ∆NP(t) that includes the pressure anisotropy of the eigenmode, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The integrals in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10) can be computed numerically (see appendix A) and the pressure anisotropy ∆NP(t) determined for a given initial mode amplitude α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ���� δB∥(0) B0 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='11) The result is shown in figure 1(a) at a selection of values of Te/Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The initial rise in ∆NP is due to a combination of the anisotropic phase mixing of the initially perturbed density and the pressure anisotropy adiabatically produced as the system settles into the NP eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' After approximately one thermal-crossing time of the mode’s parallel wavelength, the eigenmode is established and the slow exponential decay of ∆NP seen in the figure reflects the Barnes damping of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (The higher-frequency oscillations seen on top of this slow decay are caused by fast modes excited by the initial conditions and represent rapid oscillations about perpendicular pressure balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=') An approximate analytic solution for ∆NP(t) may be obtained in the limit of βi0 ≫ 1, (k∥/k⊥)2 ≪ 1, and Te/Ti0 ∼ 1 upon substituting the damped eigenmode (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6a) into the time integrals in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The result is that ∆NP(t) ≃ 2τ 2e−τ 2� 1 + Te Ti0 �δn(0) n0 − � erf(τ) − τ √πe−τ 2� 2 βi0 δB∥(t) B0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12a) = − � 2τ 2e−τ 2 + e−2iζτ � erf(τ) − τ √πe−τ 2�� 2 βi0 δB∥(0) B0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12b) where τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= k∥vth,it/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The term in square brackets goes as ∼2τ 2 + τ/√π for early times, suggesting that the plasma would become mirror-unstable at a time tm ∼ (√αk∥vth,i)−1, comparable to the inverse of the bounce frequency (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With the mode then slowly High-β collisionless magnetosonic modes 9 decaying exponentially, the maximum value of the pressure anisotropy may be estimated by setting exp(−2iζτ) ≃ 1 and maximizing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12b) with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The result is a maximum pressure anisotropy ≃2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6αβ−1 i0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8)) occurring at k∥vth,it ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The approximate solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) is traced by the dashed line in figure 1(a), and is a manifestly good description of the full solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The second complication when using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8) to determine the kinetic stability of the NP mode is related to how the mode perturbs the perpendicular and parallel plasma β parameters that feature in the firehose and mirror instability thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) and that δB⊥ = −(k∥/k⊥)δB∥, one obtains β∥i ≃ βi0 � 1 + 2δB∥ B0 + k2 k2 ⊥ δB2 ∥ B2 0 �−1� 1 − k2 k2 ⊥ 1 βi0 Te Ti0 � 1 + Te Ti0 �−1 δB∥ B0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='13a) β⊥i ≃ βi0 � 1 + 2δB∥ B0 + k2 k2 ⊥ δB2 ∥ B2 0 �−1� 1 − k2 k2 ⊥ 1 βi0 � 2 + Te Ti0 �� 1 + Te Ti0 �−1 δB∥ B0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='13b) The final terms in both of these expressions may be dropped in the limit of βi0 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Combining the result with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8) yields β∥i∆NP ≈ β⊥i∆NP ≈ −2δB∥ B0 � 1 + 2δB∥ B0 + k2 k2 ⊥ δB2 ∥ B2 0 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) indicates that is impossible to produce a pressure anisotropy that is sufficiently negative to destabilize the plasma to the firehose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Regions in which ∆NP < 0 also have a reduced plasma β, and so the more negative the anisotropy becomes (for larger δB∥ > 0), the further the firehose threshold (≈ − 2/β∥i) moves away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In contrast, the plasma in regions where δB∥ < 0 that acquire a positive pressure anisotropy have an easier time of reaching the reduced mirror threshold (≈1/β⊥i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Setting the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) to unity and solving for δB∥ = −|δB∥| then provides the following amplitude threshold for the NP mode to trigger the mirror instability: ���� δB∥ B0 ���� ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 (NP mode amplitude threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='15) When this criterion is satisfied, we anticipate regions of kinetically unstable plasma to be localized to where δB∥ < 0 and to host ion-Larmor-scale mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With these predictions borne in mind, we now determine the spatial extent of these mirror-unstable regions and discuss how the mirrors growing within them evolve to regulate the pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Regulation of pressure anisotropy by the mirror instability In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, we showed that the plasma where δB∥ < 0 becomes mirror-unstable at tm ∼ (√αk∥vth,i)−1 if initialized from isothermal pressure balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', when instability is possible), this time is comparable to the timescale over which the NP mode’s pressure anisotropy is set up (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We may then view the mirror instability as growing on top of an otherwise weakly decaying positive pressure anisotropy satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) with δB∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The maximum growth rate of the instability depends on how far the local pressure anisotropy ventures beyond the instability threshold, Λm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ∆ − β−1 ⊥i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the asymptotic limit β⊥iΛm ≪ 1, the maximum mirror growth rate and associated wavenumber are given by (Hellinger 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', in preparation) γm/Ωi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='06β⊥iΛ2 m, k∥,mρi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2β⊥iΛm, k⊥,mρi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6(β⊥iΛm)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16) 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire (a) (b) Figure 2: (a) Perpendicular (k⊥,m) and parallel (k∥,m) wavenumbers of the fastest-growing mirror mode having growth rate γm, all computed from linear Vlasov–Maxwell theory using the instability parameter Λm corresponding to a NP mode with α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= |δB∥/B0| and k⊥/k∥ = 4 in a βi0 = 16 plasma (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' these values are weakly dependent upon βi0 and k⊥/k∥ so long as βi0 ≳ 10 and k ≃ k⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dotted lines trace the asymptotic expressions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16), valid when β⊥iΛm ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (b) The predicted number of mirrors Nm within the δB∥ < 0 region of a NP mode having wavelength λ∥ and amplitude α (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, because of the sensitive dependence of the instability parameter β⊥iΛm on the NP mode amplitude (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14)), with its value ranging from ∼1 to ∼100 for α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9], only very marginally unstable NP modes (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', α ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) satisfy the ordering used to derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For arbitrary NP mode amplitude α, the growth rate and wavenumber of the fastest-growing mirrors can be calculated numerically by solving the linearized Vlasov–Maxwell equations for a bi-Maxwellian plasma (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bott, private communication) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) specifying the pressure anisotropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' their values are shown versus α in figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For this figure we used βi0 = 16 and k⊥/k∥ = 4, although the values shown are fairly insensitive to either parameter as long as βi0 ≳ 10 and (k/k⊥)2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The asymptotic expressions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16) are shown by the dotted lines, and appear to be accurate only for α ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As the NP mode amplitude approaches unity, the maximal mirror instability growth rate and associated wavenumber tend towards γm/Ωi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2Λm, k∥,mρi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6, k⊥,mρi ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In order for the mirror instability to be relevant to the linear evolution of the NP mode, two criteria must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' First, the mirror growth rate must be much larger than the rate at which the NP mode decays (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', γm √πβi ≫ k∥vth,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This condition appears to be trivially satisfied in high-β plasmas for unstable NP modes with parallel wavelengths λ∥ ≳ 103ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The second criterion is that the mirror modes must actually fit inside the length of the region that is mirror unstable, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2π/k∥,m ≲ ℓmirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We estimate ℓmirror by asking where in the NP mode the quantity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) is larger than unity: Λm ≃ 1 βi0 � −1 − 4δB∥ B0 − k2 k2 ⊥ δB2 ∥ B2 0 � ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18) Because the leading-order eigenvector components are all real, we can take δB∥ = −αB0 cos(k∥x + k⊥y) (as used in our simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Courtesy of our assump- High-β collisionless magnetosonic modes 11 tion that k⊥ ≫ k∥, we have that δB⊥ ≪ δB∥, so the field lines are approximately straight everywhere and the paths taken by the trapped particles as they bounce are approximately parallel to B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Then, taking the appropriate root of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18) to ensure that the inverse cosine is defined for mirror-unstable amplitudes, we find that the length of the mirror-unstable portion of the wave satisfies ℓmirror ≈ λ∥ π cos−1 �2 − � 4 − k2/k2 ⊥ α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= fmλ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='19) For α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9 and k∥ ≪ k⊥, fm ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The number of maximally growing mirrors that can fit within ℓmirror is then Nm ≈ fm 2 �k∥,mρi 2π ��λ∥ ρi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='20) In writing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='20), we have included an additional factor of ≈1/2 to account for the fact that the pressure anisotropy is not expected to be uniform within the mirror-unstable region and so the full extent of ℓmirror is unlikely to be filled with mirrors of identical wavelengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' the bespoke factor of ≈1/2 was obtained empirically from examining the hybrid-kinetic simulations of unstable NP modes presented in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A further, and final, adjustment to Nm accounts for the fact that the ion-Larmor radius ρi ∝ √T⊥i/B in the mirror-unstable region is larger than ρi0, primarily because of the decrease in the local magnetic-field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) to express δT⊥i in terms of δB∥, we find that ρi ρi0 ≈ � 1 − α βi0 k2 k2 ⊥ �1/2� 1 − 2α + k2 k2 ⊥ α2 �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='21) With k∥,mρi taken from figure 2(a), we can assemble (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='21) to predict Nm for a given λ∥/ρi0, α, and k∥/k⊥ of the NP mode at βi0 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The result of this procedure is shown in figure 2(b) as the open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Note that the number of mirrors Nm is fairly independent of the NP mode amplitude for α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4, with the consequence that several mirrors can fit within the mirror-unstable region of a NP mode with λ∥ ∼ 103ρi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, at the critical amplitude α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, only one or two mirrors are predicted to fit if λ∥ ∼ 103ρi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, the mirror instability might be ineffective at regulating the pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In summary, we predict that a NP mode with α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 and λ∥ ≳ 103ρi0 should be able to support a robust collection of mirror-unstable fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Effective collisionality induced by the mirror instability We now seek an estimate for the effective collision frequency instigated by these mirror- unstable distortions in the magnetic-field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For this, we follow the arguments of Newman (2020) for the pitch-angle diffusion of charged particles in regions of Larmor- scale magnetic irregularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' First, we conjecture that each encounter of an ion with the edges of a single mirror depletes the plasma’s temperature anisotropy A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= w2 ⊥/2 − v2 ∥ by a fraction χ (here, the overline indicates an average over the ion distribution function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Following Newman (2020), we identify χ with (3/2) sin2 ϑ, where ϑ is the local deflection angle of the perturbed magnetic-field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We estimate sin2 ϑ ∼ |δBm/B|2, and argue that the energy of the mirror modes will be comparable to the free energy available to 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire them in the unstable distribution function, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' |δBm/B|2 ∼ Λm (Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 The result is that χ ∼ Λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='22) In words, larger pressure anisotropies produce larger mirror fluctuations, which in turn are able to decrease by larger amounts the pitch angles of trapped particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To obtain the effective collision frequency νeff, we then multiply χ by the number of Larmor-scale mirrors per unit time encountered by a typical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For a NP mode with amplitude α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4, the criterion for a particle to be able to pass through the NP mode’s enhancement in |B| is v∥/w⊥ ≳ � 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In other words, for a near-Maxwellian distribution of particle velocities, a majority of the particles will be confined to the trough of the NP mode where ion-Larmor-scale mirrors should be present, passing through this mirror-unstable region twice per bounce time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, Nm scattering mirrors are encountered by each trapped particle every transit time ∆t ≈ πΩ−1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The average rate of change of the ion anisotropy is then ∆A ∆t ≈ −χ πNmΩbA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= −νeffA, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='23) where in the last equality we have introduced the effective collision frequency νeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Assembling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='23), we find that νeff ≈ Gβ−1 i0 Ωi0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24a) where G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= k∥,mρi 4π2 � α−2α2 + k2 k2 ⊥ α3 �1/2� −1+4α− k2 k2 ⊥ α2 � cos−1 �2 − � 4 − k2/k2 ⊥ α � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24b) is a function of only the amplitude and wavenumber obliquity of the NP mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24) states that the predicted νeff is independent of the wavelength of the NP mode and increases with increasing α, key features that are tested (and confirmed) in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The predicted dependence of νeff upon α at βi0 = 16 and k⊥/k∥ = 4 is shown in figure 3(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' νeff may be obtained for any βi0 by multiplying the plotted values by 16/βi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The predicted collision frequency drops gradually between α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5, and then falls sharply by more than an order of magnitude to νeff ∼ 10−4β−1 i0 Ωi0 at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In panel (b), we plot the minimum parallel wavelength λ∥ of a NP mode for which νeff∆t ⩾ 1, where ∆t = πΩ−1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Such modes should host mirrors whose scattering frequency is comparable to the transit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Note that, for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 and βi0 = 16, λ∥/ρi0 must be ≳105 for the scattering frequency to be larger than the inverse transit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is worth bearing these numbers in mind when interpreting the simulation results presented in §§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Suppression of nonlinear saturation of the NP mode Once νeff becomes competitive with the bounce frequency, the induced scattering will isotropize the ion distribution function faster than the nonlinear saturation can maintain the plateau in δf(v∥) around v∥ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, the nonlinear saturation is suppressed and the NP mode should resume its decay at a rate comparable to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At some point during this decay, the mode amplitude will pass below its critical threshold for triggering the mirror instability (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='15), and the mirror modes themselves will become 3For plasmas in which the pressure anisotropy is persistently driven (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', by a background shear flow or double-adiabatic compression) rather than supplied as an initial condition, the mirror instability can grow to amplitudes δBm/B ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 before saturating through strong pitch-angle scattering (Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Riquelme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Sironi & Narayan 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 13 (a) (b) Figure 3: (a) Predicted scattering frequency νeff (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24)) caused by the mirror instability for a NP mode with amplitude α, using the values of k∥,mρi in figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (b) Minimum parallel wavelength λ∥ of a NP mode for which νeff∆t ⩾ 1, where ∆t = πΩ−1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Such modes should host mirrors whose scattering frequency is comparable to the transit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The data in both panels correspond to βi0 = 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' to re-scale them for any βi0, multiply νeff by 16/βi0 and λ∥ by βi0/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' short-wavelength decaying NP modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Near the mirror-instability threshold, these short- wavelength NP modes decay very slowly, and so the associated magnetic-field-strength fluctuations will remain nonlinear for some time after the large-scale NP mode is no longer formally mirror unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We therefore conjecture that the NP mode will continue to decay until the mirror fluctuations (and their induced scattering) have had sufficient time to dissipate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Excepting perhaps the case of asymptotically long NP mode wavelengths, then, there should be some delay between when the NP mode passes below threshold and when its nonlinear saturation is re-established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The preceding arguments imply that four distinct regimes exist for collisionless NP modes in high-β plasmas: (i) When the mode amplitude satisfies |δB∥/B0| ≲ β−2 i0 , its behaviour is nearly linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The rate of Barnes damping is faster than the bounce frequency, therefore allowing a substantial fraction of the initial mode amplitude to decay prior to the onset of nonlinear saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (ii) If β−2 i0 ≲ |δB∥/B0| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, the pressure anisotropy associated with the mode is too small to trigger the mirror instability, but the rate at which nonlinear saturation flattens the distribution function is greater than the Barnes damping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At these amplitudes then, nonlinear saturation occurs before the mode can decay appreciably, implying these pressure-balanced structures are thus long-lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (iii) When |δB∥/B0| ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, the pressure anisotropy of the NP mode triggers the mirror instability in regions where δB∥ < 0 and eventually introduces an effective collisionality that, for sufficiently large NP mode wavelengths, suppresses the maintenance of a nonlinear plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As a result, linear decay resumes until the NP mode decays back well below its amplitude threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (iv) Because the induced scattering rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24) does not scale with the wavelength of the NP mode, one might expect a fourth fluid-like regime at very long wavelengths, when νeff ≫ k∥vth,i and the collisionless damping is arrested altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We discuss the realizability of this fourth regime and speculate on its behaviour in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Numerical results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Method of solution and initial conditions To test the theory presented in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 and explore the nonlinear evolution of a mirror- infested NP mode, we employ the hybrid-kinetic particle-in-cell code Pegasus++ (Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Arzamasskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Pegasus++ evolves the ion distribution function f(t, r, v) using a collection of positively charged macro-particles that interact with the self-consistent electromagnetic fields E(t, r) and B(t, r), which are in turn evolved on a discrete mesh using Faraday’s law and a generalized Ohm’s law that includes the inductive electric field, the Hall effect, and a thermoelectric field caused by pressure gradients in the (assumed massless) electron fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The latter ensures quasi- neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For simplicity, we adopt an isothermal equation of state for the electrons with temperature Te = Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Both the interpolation of fields to the macro-particle locations, and the deposition of the macro-particles’ phase-space information on the mesh, are performed using second-order-accurate triangle-shaped stencils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' All simulations of the NP mode are performed on a two-dimensional mesh that is elongated in the direction of a mean magnetic field B0 = B0ˆx and spans one full NP mode wavelength, Lx × Ly = λ∥ × λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The latter ranges from λ∥ = 1000ρi0 to 4000ρi0, with aspect ratios of either λ∥/λ⊥ = 4 or 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' When varying these two dimensions, the transverse dimension is never smaller than 250ρi0, thereby guaranteeing sufficient scale separation between the NP mode and any ion-Larmor-scale instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In all runs, the spatial resolution is ∆x = ∆y ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3ρi0 and the number of macro-particles per cell is either 104 or 5 × 103 (the latter used only in our largest simulations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' these values are similar to those used in previously published Pegasus simulations of collisionless Alfv´en waves (Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2017a) and IAWs (Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2020) in firehose/mirror-susceptible plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At t = 0 we perturb the magnetic field using the vector potential A(x, y) = −αB0 |k| sin(k∥x + k⊥y)ˆz, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='25) where k∥ = 2π/λ∥, k⊥ = 2π/λ⊥, and α is a dimensionless number quantifying the mode amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To excite the NP mode, the associated change in the magnetic pressure, δB2 8π = −αB2 0 8π cos(k∥x + k⊥y) �2k⊥ |k| − α cos(k∥x + k⊥y) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='26) must be exactly balanced by a perturbation to the perpendicular pressure of the plasma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In order to keep the initialization of the latter relatively simple, we choose to begin not from an exact NP eigenmode but rather from an isothermal perturbation to the plasma density δn, in which case the perturbed perpendicular pressure is simply δp⊥ = δn(Ti0 + Te).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Balancing this expression by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='26) and solving for δn leads to the initial ion distribution function f(0, x, y, v) = FM(v) � 1 + α β0 cos(k∥x + k⊥y) �2k⊥ |k| − α cos(k∥x + k⊥y) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='27) In this case, the initial total (magnetic plus thermal) pressure in the simulation domain is constant and equal to (B2 0/8π)(1 + β0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' recall that β0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= βi0(1 + Te/Ti0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Starting from a pressure-isotropic plasma has the advantage that any pressure anisotropy that develops is generated self-consistently and not put in by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is also consistent with the assumptions made to obtain the analytic solution for ∆NP(t), equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In all of our simulations, we set βi0 = 16 so that it is large enough to agree with asymptotic expressions derived in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, but not so large that we cannot capture a full High-β collisionless magnetosonic modes 15 decay time of the linear NP decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We vary α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8], spanning the predicted NP amplitude threshold for triggering the mirror instability (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Special attention is paid to the case with λ∥ = 2000ρi0, λ⊥ = 500ρi0, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' we refer to this as our fiducial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Hereafter, ⟨ · ⟩ denotes a spatial average taken over the entire domain, and ⟨ · ⟩k denotes a spatial average taken over the y-direction while accounting for the changing position of the wavefront (so as to align all of the perturbed and unperturbed regions within the domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The latter is referred to as a ‘wavefront average’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' note that it leaves the x-coordinate (in the direction of B0) unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Overall evolution of the fiducial run We begin our discussion of the Pegasus++ results by using the fiducial run to make contact with some of the predictions laid out in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These predictions include the excitation and subsequent linear collisionless damping of the NP mode, its nonlinear saturation, the simultaneous generation of mirror-unstable pressure anisotropy in the regions of the mode where δB∥ < 0, and the resumption of linear damping following the pitch-angle scattering of trapped ions by the saturated Larmor-scale mirrors at a rate larger than the bounce frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 4 illustrates these evolutionary phases by depicting the amplitude of the NP mode versus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' After a rapid adjustment from the isothermal pressure-balanced initial condition, the NP mode emerges and decays at the linear rate (black line) for approximately one bounce time, Ω−1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Immediately thereafter, the decay stalls (blue line) as nonlinear saturation sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 5 demonstrates that, meanwhile, the NP mode has produced a large, positive pressure anisotropy in the regions where δB∥ < 0 and almost zero pressure anisotropy elsewhere, consistent with the prediction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The mirror-unstable region (with ⟨β⊥i∆⟩k above the dotted line in figure 5) is seen to occupy ∼40% of the NP mode wavelength, consistent with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='19) for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is in this mirror-unstable region that the magnetic field acquires moderate-amplitude, oblique fluctuations in its strength on ion-Larmor scales, which are clearly apparent in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The strongest fluctuations occupy roughly a quarter of the box length and acquire amplitudes comparable to that of the mean field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The associated distortions in the field lines ultimately scatter particles at a rate comparable to the bounce frequency (see figure 7 and the accompanying discussion in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As a result, the NP mode amplitude enters a ‘suppressed saturation’ phase (figure 4, red line), during which the nonlinear plateau is eroded by the mirror-induced collisionality and the Barnes damping resumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 In the remainder of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2, we examine these phases in more detail and their dependence on mode amplitude and scale separation, starting with the mirror-induced scattering and its impact on the NP mode’s pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Effective collisionality: particle scattering and trapping Figure 7 displays the evolution of the mirror-induced effective collisionality νeff in the fiducial run, calculated following the method used in Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2014a, 2020), Melville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2016), and Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Namely, the individual magnetic moments of ∼104 particles are tracked and monitored for (both abrupt and accumulated) changes by at least a factor of κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 (as used by Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2020 to measure firehose/mirror- induced scattering in unstable IAWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The times at which these changes are registered are stored, along with the locations at which they occurred, and a spatially dependent effective collision frequency νeff is calculated from the mean scattering time τ using νeff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= 4We were not able to discern any fluctuations above the noise floor in the out-of-plane component Bz, which would be indicative of the ion-cyclotron instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Gary & Lee 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire 0 2 4 6 8 10 12 14 k||vth,it 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 δB∥(k|| = 2π/Lx)/B0 γ = k2/(k2 ⊥ √πβi0) decay saturation suppressed saturation Ω−1 b Figure 4: Amplitude of the magnetic-field-strength perturbation of the NP mode vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' time from the fiducial run, with the different phases of the predicted evolution labelled and colour-coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dashed line indicates the linear decay rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5) of the NP mode in a pressure-isotropic plasma with βi0 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' See §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 0 5 10 15 ⟨β⊥i∆⟩k mirror k||vth,it = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 λ|| = 2000ρi0 = 4λ⊥ λ|| = 1000ρi0 = 4λ⊥ λ|| = 2000ρi0 = 8λ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 x/λ|| −5 0 5 10 15 ⟨β||i∆⟩k firehose Figure 5: Wavefront-averaged profiles of β⊥i∆ and β∥i∆ at k∥vth,it = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, when the pressure anisotropy is near its maximum value, compared against the theoretical predictions from the linear eigenmode (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14), for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 and different NP mode wavelengths λ∥ and λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The fiducial run corresponds to the solid black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Positive values of βi∆ far exceeding the mirror threshold occur in the regions where δB∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Elsewhere, negative pressure anisotropy is compensated by a decrease in βi to avoid exciting the firehose instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (ln κ)2/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This calculation was also performed using κ ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5], with no significant dependence of νeff on κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the bottom panel of figure 7, the box-averaged effective collisionality (black line) and maximum value of the wavefront-averaged effective collisionality (red line) are shown High-β collisionless magnetosonic modes 17 0 250 500 750 1000 1250 1500 1750 2000 x (ρi0) 0 100 200 300 400 500 y (ρi0) mirror δBx/B0, k||vth,it = 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 Figure 6: The x-component of the magnetic-field perturbation, filtered to remove wavenumbers associated with the α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 NP mode, at k∥vth,it = 25 in the fiducial run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' By this time, the mirror instability is fully nonlinear, causing large-amplitude, small- wavelength deflections in the magnetic-field direction that pitch-angle scatter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' as functions of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Both exhibit rapid growth during the initial phase of the mirror instability and then reach a quasi-steady state, with max(⟨νeff⟩k) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0035Ωi0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5Ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We have found the timescale for the scattering rate to reach this steady state to be largely independent of the wavelength of the NP mode, although it increases somewhat with decreasing α because of the slower linear growth rate of the mirror instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The space- time diagram of the wavefront-averaged collisionality shown in the top panel indicates that the maximum value of νeff is localized to the centre of the mirror-unstable region, with slightly smaller values occurring near this region’s boundaries where the mirror amplitudes are smaller (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A large fraction of the thermal plasma is subject to this collisionality, because the mode amplitude is large enough that most of the plasma particles are confined in the regions where δB∥ < 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', large δn > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For example, when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8, particles whose pitch angles satisfy v∥/w⊥ ⩽ � max(B)/min(B) − 1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 would be mirror-confined in the absence of collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Outside of these regions, where the plasma is stable, the collisionality is very low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' as a result, the box-averaged collisionality is more than a factor of 5 smaller than the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The top panel also shows the path of a single tracked particle as a grey line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The initial evolution demonstrates bouncing within the δB∥ < 0 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Once the mirror fluctuations reach nonlinear amplitudes, the particle is temporarily trapped within a growing mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Eventually, it scatters enough in pitch angle to become de-trapped and traverses the δB∥ > 0 region, breaking its resonance with the NP mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Evolution of pressure anisotropy The top panel of figure 8 shows the evolution of the maximum of the wavefront-averaged ∆ and β⊥i∆ in the fiducial run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The bottom panel depicts the growth of the root-mean- square amplitude of the mirror fluctuations, averaged over the mirror-unstable region where δB∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These fluctuations grow large enough to scatter particles and restore the linear decay of the NP mode, through which the pressure anisotropy decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Indeed, ⟨∆⟩k is similar to the linear prediction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12), denoted here by the blue dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Likewise, ⟨β⊥i∆⟩k is modeled well by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) with the substitution δB∥/B0 = α exp(−iζk∥vth,it) where ζ is the linear eigenvalue (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This expression is traced by the dashed red line 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire Figure 7: Effective collisionality νeff caused by the mirror instability in the fiducial run with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 and λ∥ = 2000ρi0 = 4λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Top: Space-time diagram of ⟨νeff⟩k (colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A illustrative particle trajectory is shown with the grey line, exhibiting resonant bouncing, followed by trapping within a mirror fluctuation, and eventual scattering out of resonance with the NP mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bottom: Box-averaged (black) and maximum wavefront-averaged (red) collision frequencies vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' in figure 8, where we have started the decay at k∥vth,it = 6 and set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='75 in order to account for the delay due to the (temporary) nonlinear saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At larger scale separations, we anticipate that faster pitch-angle scattering induced by the mirrors will be able to regulate the pressure anisotropy more efficiently than its linear decay, at which point the mode will no longer resemble the collisionless linear NP eigenmode (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The growth of mirrors leads to modifications in the shape of the NP mode profile, as shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The evolution of the wavefront-averaged profile of β⊥i∆ in the fiducial run at k∥vth,it = 3, 6, 11, and 27 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The profile in the region where the mirror instability is active has flattened, although the mode seems to remain close to the linear eigenmode, as evidenced by figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The reduction in β⊥i∆ occurs considerably faster than the linear decay of ∆ by itself, which highlights the importance of β⊥i in achieving marginal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This reinforces the idea that the mirror fluctuations do not so much act directly on the anisotropy to achieve β⊥i∆ = 1, but rather they enable the NP mode to decay and reduce both ∆ and β⊥i to achieve marginal stability more rapidly than would otherwise occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10 max(⟨∆⟩k) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='11) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='13) 0 5 10 15 20 25 k∥vth,it 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 ⟨δBrms m ⟩m/B0 0 5 10 15 max(⟨β⊥i∆⟩k) Figure 8: Top: Maximum of the wavefront-averaged ∆ (solid blue line) and β⊥i∆ (solid red line) versus time in the fiducial run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The evolution of ⟨∆⟩k matches well the predicted linear evolution (blue dashed line), suggesting that the rapid reduction of β⊥i∆ is due mostly to the resumed decay of the NP mode and the decrease in β⊥i caused by the growing mirror fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bottom: Root-mean-square amplitude of the mirror fluctuations, averaged over the mirror unstable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The growth of the mirror instability coincides with a drop in ⟨β⊥i∆⟩k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 x/λ|| 0 5 10 15 ⟨β⊥i∆⟩k mirror k||vth,it = 3 k||vth,it = 6 k||vth,it = 11 k||vth,it = 27 Figure 9: Temporal evolution of the wavefront-averaged profile of β⊥i∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Four times are shown: just after the adjustment into the NP eigenmode during the initial decay phase (black line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' an intermediate time during which the NP mode decay is saturated (blue line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' after the mirrors become nonlinear and scatter particles fast enough to suppress the NP mode’s saturation (red line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' and later once β⊥i∆ has been reduced enough that the mirrors are marginally stable (grey line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Suppression of nonlinear saturation and resumption of transit-time damping The effects of nonlinear saturation and mirror-induced collisionality across a variety of NP mode amplitudes can be seen in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For reasons of computational cost, for these runs we used λ∥ = 1000ρi rather than the fiducial 2000ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A Fourier transform is used to select the magnitude of the box-wavelength perturbation to the background field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', the amplitude of the NP mode);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' this quantity is plotted as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire (a) (b) 0 2 4 6 8 k∥vth,it 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9 α−1δB∥(k∥ = 2π/Lx) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 0 10 20 30 40 k∥vth,it 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 Figure 10: Amplitude of the magnetic-field-strength perturbation of the NP mode, normalized to its initial value, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' time for λ∥ = 1000ρi0 = 4λ⊥ and different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (a) Early times, during which the NP mode nonlinearly saturates after approximately one bounce time ∼Ω−1 b (vertical dotted lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dashed line indicates the linear decay rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (b) Late times, showing suppression of nonlinear saturation and resumption of linear damping for amplitudes α ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' panel (a), the initial phase of evolution is featured, at first demonstrating linear decay at a rate similar to the prediction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5) (shown by a black dashed line), approximately independent of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' After roughly one bounce time (marked by dotted lines of matching colour), the decay begins to stall and the mode amplitude tends towards a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This nonlinear saturation occurs at earlier times for larger mode amplitudes, trending with the α−1/2 scaling of the bounce time (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At amplitudes α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4, more than 90% of the original mode amplitude is preserved by the nonlinear saturation, suggesting that large-amplitude collisionless NP modes at high β can be rather long lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 10(b) shows the behaviour of these modes over longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For amplitudes α ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4, nonlinear saturation remains and the linear decay rate is never again realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' By contrast, the larger values of pressure anisotropy in α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 NP modes produce mirror-unstable fluctuations with amplitude large enough to interfere with the maintenance of the nonlinear plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As a result, the linear decay rate is almost re- established at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 and is restored fully at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These modes are then able to decay further and convert magnetic energy into particle energy through a balance between plateau generation and pitch-angle scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With the value of λ∥ used in these runs being twice smaller than that in the fiducial run, it is notable that the time at which near-linear decay is restored by mirror-induced scattering is the same (in units of Ωi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At scale separations much larger than those we are able to simulate currently, we thus anticipate the nonlinear plateau to be eroded almost instantly compared to the wave timescales by rapid mirror growth and its associated particle scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Our final piece of evidence that the nonlinear plateau is maintained at subcritical NP mode amplitudes and eroded at supercritical amplitudes is also the most direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In figure 11 we show the ion velocity distribution functions f(v∥, w⊥) measured within the δB∥ < 0 region from two runs having λ∥ = 2000ρi = 4λ⊥ and either α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 (left) or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The top plots depict in colour the differences between f(v∥, w⊥) and bi-Maxwellian fits based on the parallel and perpendicular ion temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The bottom plots show v∥ slices of the distribution functions averaged between v⊥ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6vth,i High-β collisionless magnetosonic modes 21 °2 °1 0 1 2 vk/vth,i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 w?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='/vth,i °2 °1 0 1 2 vk/vth,i °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 vk/vth,i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='040 f(vk) Æ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 vk/vth,i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='06 Æ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='06 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='04 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10 f(vk, w?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=') ° F fit Max(vk, w?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=') Figure 11: Ion velocity distribution functions f(v∥, w⊥) measured within the regions where δB∥ < 0 of two simulations with λ∥ = 2000ρi0 = 4λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The left panels, corresponding to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4, exhibit a nonlinear plateau around v∥ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The right panels, corresponding to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8, show a smooth Maxwellian-like distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The bottom panels are slices in v∥ integrated between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6vth,i and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7vth,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The colour bar for the top panels has been allowed to saturate for the purpose of showing detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Dotted lines represent isocontours of total energy, w2 ⊥ + v2 ∥ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7vth,i (the averaging is performed to reduce sampling noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 run, the distribution is reduced with respect to the bi-Maxwellian at low pitch angles where particles are well trapped, and the parallel velocity distribution f(v∥) exhibits flattening about v∥ ∼ 0 – a nonlinear plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' No such features are visible in the α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8 distribution, with betatron heating of the trapped particles evidenced by an increase in particle phase-space density over the bi-Maxwellian fit at large perpendicular velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Dependence on scale separation The effective collision frequency predicted by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24) suggests that, if the initial NP mode amplitude and wavenumber obliquity were held constant, then increasing the wavelength of the mode should have no effect on the collision frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This can be recast 22 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire Figure 12: Maximum value of the measured mirror-induced effective collision frequency νeff,max vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' NP mode wavelength at two different wavenumber obliquities and two different initial amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The predicted scaling ν/(k∥vth,i) ∝ λ∥ is shown (dashed black line), normalized to the fiducial case (red circle at λ∥ = 2000ρi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' as a more illustrative relationship between the thermal crossing time and the collision frequency, νeff/(k∥vth,i) ∝ λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 12 shows the maximum value of the box-averaged effective collision frequency normalized to k∥vth,i for a few different NP mode wavelengths, wavenumber obliquities, and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The measured values exhibit good agreement with the proportional expectation at both wavenumber obliquities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This evidence implies that, at yet longer wavelengths, the collision frequency will continue to increase compared to the bounce time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Note that the measured collisionality for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 is approximately a factor of two smaller than for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8, in qualitative agreement with the prediction featured in figure 3(a) that the scattering should decrease with decreasing NP mode amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The fact that the simulated NP mode with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 and λ∥ = 1000ρi0 does not have its nonlinear saturation interrupted by mirrors is also consistent with the prediction in figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As conjectured in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6, the linear scaling of νeff/(k∥vth,i) with λ∥ suggests a possible fluid-like regime at sufficiently long NP-mode wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To investigate this regime, if only approximately, we examine the linear decay rate of NP modes in the presence of a constant pitch-angle scattering rate, shown in figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The details of how we determined this decay rate are given in appendix B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' note that the real part of the frequency is zero for all scattering rates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', the mode remains non-oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' On the left-hand side of the plot, the collision frequency is small and the collisionless NP mode is recovered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' on the right-hand side, the collision frequency is large and the mode becomes the MHD entropy mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The MHD entropy mode is similar to the kinetic NP mode in that it too has no real frequency, but in the fully collisional limit it involves only a density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For the employed values of k⊥/k∥ = 4 and βi0 = 16, the transition between these two regimes occurs at ν ≈ 3k∥vth,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Using an asymptotic expansion at high β and k ≃ k⊥, one can show that the transitional collisionality scales approximately as ν ∼ (3/4)√βik∥vth,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With νeff/Ωi0 ∼ 10−2β−1 i0 for α ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 (see figures 3 and 12), we estimate that the transition to the collisional regime requires a scale separation of at least λ∥/ρi0 ∼ 103β3/2 i0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Under this condition, the mirror-induced scattering will both isotropize the pressure perturbation and prevent resonant particles from continuously sapping energy from the wave, thereby High-β collisionless magnetosonic modes 23 Figure 13: Linear decay rate of the NP mode obtained from the Landau-fluid CGL-MHD equations (B 1) (see appendix B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dimensionless (complex) frequency ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ω/(|k∥|vth,i) is computed numerically as a function of collisionality ν/(|k∥|vth,i) for k⊥ = 4|k∥|, βi0 = 16, and Te = Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Overlaid are red circles marking the maximum box-averaged scattering rates measured in our hybrid-kinetic simulations (see figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' reducing the decay rate and morphing the collisionless NP mode into the MHD entropy mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Unfortunately, unless the scale separation is extremely large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', λ∥/ρi0 ≳ 105 for our parameters), the decay rate will not be much slower than in the ν = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the absence of affordable numerical simulations to test this point, we simply conjecture that at asymptotic wavelengths the reduction in the decay rate would allow these NP structures to become long lived once again, much like their below-threshold, non-linearly saturated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Summary of key results on the NP mode For the reader’s benefit, we summarize here the essential findings of our investigation of the NP mode in magnetized, high-β, collisionless plasmas: Transit-time (Barnes) damping of NP modes nonlinearly saturates before substantial collisionless decay occurs when the mode amplitude |δB∥/B0| ≳ β−2 i0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The perpendicular pressure balance associated with the polarization of the NP eigenmode produces large positive βi∆ and only weakly negative βi∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Above a threshold amplitude of |δB∥/B0| ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, this pressure anisotropy becomes unstable to the mirror instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' at no point is the plasma firehose unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Once the growing mirror fluctuations become nonlinear, they pitch-angle scatter particles at a rate νeff, which, in accordance with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='24), is independent of the NP mode wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At wavelengths sufficiently long so that νeff satisfies √βi ≳ νeff/(k∥vth,i) ≳ |δB∥/B0|1/2, the induced scattering is only fast enough to erode the nonlinear plateau, causing the mode to resume its decay close to the linear (collisionless) rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At yet longer wavelengths for which νeff satisfies νeff/(k∥vth,i) ≫ √βi, transit-time damping will be interrupted entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We predict that in this limit the mode will behave more like the MHD entropy mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 24 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Fast modes: Wave steepening and viscous damping 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Theory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Model equations and assumptions Collisionless fast magnetosonic waves are in many ways simpler than their non- propagating counterparts, particularly so if their wavevectors are nearly perpendicular to the background magnetic field, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' k⊥ ≫ k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this limit, collisionless damping is extremely weak, and magnetic tension plays virtually no role in the mode’s propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In fact, for exactly perpendicular propagation (k∥ = 0), Landau and Barnes damping are entirely absent at long wavelengths due to the limited cross-field transport of magnetized particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this case, no kinetic information about these modes other than their pressure anisotropy is needed, and they can be described entirely within double-adiabatic MHD – a model that results from taking the first three fluid moments of the drift-kinetic system (see appendix B) and dropping the heat fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Setting B = Bˆy and ∇ = ˆx ∂/∂x, these equations are Dn Dt = −n∂u⊥ ∂x , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a) minDu⊥ Dt = − ∂ ∂x � p⊥i + pe + B2 8π � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1b) DB Dt = −B ∂u⊥ ∂x , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1c) D Dt �p⊥i nB � = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1d) D Dt �p∥iB2 n3 � = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1e) where D/Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ∂/∂t + u⊥∂/∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Although the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1b) is independent of the parallel pressure, and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1e) is not needed to close this set of equations, it is nevertheless useful for calculating the fast-wave pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As in §2, we adopt a simple equation of state for the electrons, pe = nTe, with Te being constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 In what follows, we investigate analytically two features of fast-wave propagation in a collisionless, magnetized plasma, adopting the simple but illustrative case of k∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' First, we demonstrate that such waves nonlinearly steepen quicker in double-adiabatic MHD than they do in standard (pressure-isotropic) MHD, a direct consequence of the proportional relationship between T⊥ and B associated with µ conservation, equa- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Second, we show how the resulting pressure anisotropy can destabilize the plasma to both firehose and mirror instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We then estimate the effective scattering frequency introduced into the plasma by these instabilities and discuss how the consequent regulation of the pressure anisotropy affects the characteristics of the fast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Before proceeding, it is useful to linearize (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1) to obtain the fast-wave dispersion relation and eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Perturbing the plasma about a uniform background having density n0, isotropic ion pressure pi0, and magnetic-field strength B0, we find that δp⊥,i pi0 = 2δB B0 and δp∥i pi0 = δn n0 = δB B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2) These equations state that the density and pressure anisotropy are positively correlated 5Having the electrons respond double-adiabatically would simply double the pressure anisotropy associated with the fast wave and send Te/2Ti0 → Te/Ti0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 25 with the magnetic-field strength, with the parallel ion temperature remaining constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dispersion relation of this double-adiabatic (‘da’) fast wave is ω = k⊥vA � 1 + βi0 � 1 + Te 2Ti0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= k⊥vms,da, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3) so that the bulk velocity u⊥ = vms,da(δB/B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For comparison, the dispersion relation of a fast wave in single-adiabatic (‘sa’) MHD is ω = k⊥vA � 1 + βi0 �γ 2 + Te 2Ti0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= k⊥vms,sa, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4) where γ is the adiabatic index of the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The proportional relation between the magnetic-field strength and the density in the double-adiabatic model means that vms,da > vms,sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This increase will play a role in allowing double-adiabatic fast waves to form shocks faster than single-adiabatic fast waves, especially so at high β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Wave steepening in double- versus single-adiabatic MHD For waves in which the perturbed quantities determine the wave propagation speed, steepening may result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Large-amplitude waves in particular generate significant differ- ences in the propagation speed between the peaks and the troughs, a situation expected to occur in both double- and single-adiabatic MHD fast waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this section, we perform a series of manipulations to the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1) in order to quantify this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Before proceeding, it is convenient to renormalize quantities using the Alfv´en speed vA = B0/(4πmin0)1/2 and the wavelength λ as follows: u⊥ = �u⊥vA, B = �BB0, n = �nn0, x = �xλ, t = �tλ/vA, and p⊥,i = �p⊥imin0v2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We also note that, if the perturbations satisfy δ�n = δ �B at t = 0, then these two quantities will remain equal for all times (see equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1c));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' we can then eliminate δ�n in favour of δ �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6 Meanwhile, if δ �B is small and its associated perturbations in �p⊥,i and �n are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2), equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1d) becomes ∂ ∂�t � �p⊥,i �n �B � ≈ −�u⊥ βi0 2 ∂(δ �B)2 ∂�x ∼ O � (δ �B)3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5) Hence, to second order in δ �B, we may treat �p⊥,i = (βi0/2) �B2 as the equation of state if the initial condition is an eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Under these conditions, equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1) may be combined to obtain the following system: ∂ ∂�t � � �u⊥ δ �B � � + � � �u⊥ 1 + βi0 � 1 + Te/2Ti0 1 + δ �B � 1 + δ �B �u⊥ � � ∂ ∂�x � � �u⊥ δ �B � � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) Defining W = [�u⊥, δ �B]T, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) can be rewritten as ∂�tW +A(W )∂�xW = 0, with A(W ) being the evolution matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' By first finding the eigenvalues l(i) and left eigenvectors L(i) of A(W ), this system can be solved via its characteristic equations, which are given by L(i) · dW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These characteristic equations are obeyed along space-time trajectories following d�x/d�t = l(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, because our equation of state is only valid up to second order in the wave amplitude, we need only to retain those terms of first order in the 6This reduction is equivalent to assuming an adiabatic index of γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In fact, when comparing the results of this analysis to an MHD treatment with isothermal electrons, the substitution γ = 2 recovers the double-adiabatic result (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 26 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire Figure 14: Approximate solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='11) to the fast-wave steepening problem with initial amplitude α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3 and βi0 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The solution has just begun to form a shock, indicating a shock-formation time of k⊥vAts ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' evolution matrix, and hence in its eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Therefore, we expand the characteristic equations to first order and integrate them to find that the combinations η± = �u⊥ ± �vms,daδ �B (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7) are approximately constant along �d�x d�t �± = η+ + η− 2 ± �vms,da � 1 + 1 + βi0 4�v3 ms,da (η+ − η−) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='8) These can be reformulated as two nonlinear advection equations,7 ∂η± ∂�t + �d�x d�t �± ∂η± ∂�x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='9) Note that, if the initial conditions are those of the fast eigenmode (as previously assumed in the assertion that δ �B = δ�n for all time), then η− = 0 for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We are then left with ∂η+ ∂�t + �η+ 2 + �vms,da � 1 + 1 + βi0 4�v3 ms,da η+ ��∂η+ ∂�x = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10) the solution of which for �u⊥ is given by the method of characteristics as �u⊥(�t, �x) = δ�u⊥0 � �t, �x − �vms,da�t � 1 + δ �B0(�xi) + 1 + βi0 2�v2 ms,da δ �B0(�xi) �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='11) where the subscript ‘0’ denotes an initial value, and xi is the x-position of the source of a given characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 7This process is analogous to that used in the derivation of approximate Riemann solvers for numerical solutions of the MHD equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Roe 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Toro 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Miyoshi & Kusano 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Commonly, the left eigenvector is assumed to be constant when integrating the characteristic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Here, we keep terms up to first order in δB within L to more accurately resolve the wave steepening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The careful reader will note that these expressions do not transform directly back to an approximate form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This approach focuses on the characteristics of A, so the leading-order behaviour of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='6) and the eigenvalues/vectors of A are approximated accurately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' this is in contrast to expanding A itself and truncating past the first correction in δ �B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 27 The time-dependent solution for an example large-amplitude, double-adiabatic fast wave is shown in figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This solution is strictly valid only until a shock has formed, at a time that may be determined by evaluating the eigenvalue l+ at the location x0 where its derivative achieves its largest negative value: tda s = � l+(x0) �−1 ≈ � αk⊥vms,da � 1 + v2 A v2 ms,da 1 + βi0 2 ��−1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) where α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= δ �B(0) is the initial fast-wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This double-adiabatic (‘da’) shock- formation time is to be compared to the corresponding time in a single-adiabatic MHD plasma, in which pn−γ = p0n−γ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The general problem of fast-wave steepening in MHD plasmas has been studied thoroughly under many conditions (Hada & Kennel 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' ¨Odblom 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Sujith 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Following an analogous process to that used for the double- adiabatic fast wave, we find the single-adiabatic (‘sa’) shock-formation time tsa s ≈ � αk⊥vms,sa � 1 + v2 A v2ms,sa 1 + γ(γ − 1)βi0/2 2 ��−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='13) Simplifying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='13) at high β, and setting Te = Ti0 and γ = 5/3, yields k⊥vAtda s ≈ √ 6 4α√βi0 and k⊥vAtsa s ≈ 12 √ 3 29α√βi0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17k⊥vAtda s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='14) The single-adiabatic shock-formation time is thus larger than the double-adiabatic shock- formation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' When Te/Ti0 = 0, their ratio reaches a maximum of ≃1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' for Te ≫ Ti0, it approaches unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This increase is a consequence of the direct correlation between the magnetic-field strength and the perpendicular (ion) pressure in double-adiabatic MHD, which amplifies local changes in the mode propagation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Pressure anisotropy and its regulation by kinetic instabilities By contrast with the NP mode, the fast wave generates a fluctuating pressure anisotropy as the wave propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At sufficiently large β, both firehose and mirror instabilities may therefore be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With δp⊥,i and δp∥,i given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2), the amplitude threshold for triggering both firehose and mirror instabilities is ���� δB B0 ���� ≳ 2 βi (fast-wave amplitude threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='15) At high β, this criterion can be satisfied for even small-amplitude fluctuations, justifying the use of the linear eigenvector and unperturbed βi in determining the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To assess whether these micro-instabilities will be able to grow, we compare their linear growth rates to the linear frequency of the fast wave at high β, ωfast ∼ k⊥vth,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We adopt the maximal mirror growth rate from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16), and use the maximal oblique firehose growth rate γf ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3ΩiΛ1/2 f where Λf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= |∆ + 2/βi| (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', in preparation), both of which are appropriate for the near-threshold conditions we anticipate in our fast-wave simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Assuming |δB/B0| ≳ 2β−1 i , we find that γm ωfast ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='01β−1 i λ⊥ ρi and γf ωfast ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1β−1/2 i λ⊥ ρi , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16) where λ⊥ = 2π/k⊥ is the wavelength of the fast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is immediately apparent from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16) that, at high β, very large scale separation between the fast-wave wavelength and the ion-Larmor scale is necessary to allow enough time for mirror fluctuations to grow and become nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The scaling with βi is much weaker for the firehose instability, and 28 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire so there will exist wavelengths at which mirror regulation of the pressure anisotropy is effectively non-existent but the firehose regulation is rapid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For this reason our Pegasus++ simulations, which focus on βi0 = 25, require λ⊥ ≫ 103ρi0 to realize both mirror and firehose regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The unstable Larmor-scale fluctuations will ultimately grow to amplitudes at which the particles’ rate of pitch-angle scattering is sufficient to hold the pressure anisotropy at marginal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This rate may be estimated by calculating the pressure anisotropy driven by a small-amplitude fast wave in a weakly collisional plasma (following Braginskii 1965) and asking what value of effective collisionality νeff would be required to keep |∆| ∼ 2β−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With the former given in the collisional regime by ∆ ∼ −(∇ · u)/νeff ∼ (k⊥vms/νeff)|δB/B0|, the limiting collisionality is νeff ∼ k⊥vms βi 2 ���� δB B0 ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17) Note its explicit dependence upon the scale of the fast wave, an indirect consequence of the pressure anisotropy of the fast wave being continuously driven by the fluctuating wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This is very different from the case with the zero-frequency NP mode, in which the pressure anisotropy – an essential feature of the mode’s perpendicular pressure balance – actually decays in time through transit-time damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Viscous damping and collisional propagation The estimate of the effective collisionality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17) suggests that, depending on the wave amplitude, one should see a variety of fast-wave behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For example, if |δB/B0| ≫ 2β−1 i0 , then the implied collisionality can be large enough to push the fast wave into the collisional Braginskii-MHD regime (ν ≫ ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' If we make the presently unjustified yet instructive assumption that this collisionality is distributed uniformly in space, the fast-wave dispersion relation at arbitrary ν can be obtained after including isotropizing collisional terms −ν∆p/nB and ν∆pB2/n3 on the right-hand sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1d) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1e) respectively, then linearizing the resulting system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We find that ω3 − iνω2 − ωk2 ⊥v2 ms,da + iνk2 ⊥v2 ms,sa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18) The numerical solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18) is shown in figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the collisionless limit ν → 0, one recovers propagation at the double-adiabatic fast speed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' taking ν → ∞ returns propagation at the single-adiabatic fast speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Viscous damping occurs at intermediate values of ν ∼ Re(ω) ∼ k⊥vth,i around the transition between the double- and single- adiabatic regimes, where the scattering rate is comparable to the wave’s oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The damping rate is always small compared to the wave frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dispersion relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18) alongside the amplitude threshold (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='15) and the pre- dicted effective collision frequency (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17) imply three regimes for the behaviour of per- pendicularly propagating fast modes in a high-β plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For small amplitudes satisfying |δB/B0| < 2β−1 i0 , the mode propagates normally as a collisionless fast mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It will steepen and eventually form a shock on the double-adiabatic shock time tda s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the near-threshold regime where |δB/B0| ≳ 2β−1 i0 , the scattering rate from triggered mirror and firehose instabilities will not quite reach the value (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17), though scattering is still expected to occur and result in some viscous damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The wave will also steepen to form a shock, but only a fraction of the wavelength will be kinetically unstable and therefore the shock will occur on a hybrid of the double- and single-adiabatic shock times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Lastly, at amplitudes well above the threshold, the scattering rate should be given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The viscous damping will be very weak, the wave will host firehose/mirror scattering High-β collisionless magnetosonic modes 29 Figure 15: Exact solution to the dispersion relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='18) for a k∥ = 0 fast wave in a plasma having collision frequency ν, βi0 = 25, and Te/Ti0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' sites throughout most of its wavelength, and the shock time should be better represented by the single-adiabatic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We now test these ideas using numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Numerical results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Method of solution and initial conditions Due to the large scale separations needed to obtain asymptotic νeff for both firehose and mirror fluctuations (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3), we use a combination of Pegasus++ and (much cheaper) Landau-fluid CGL-MHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' All simulations initialize a k∥ = 0 fast wave in an otherwise Maxwellian plasma using the collisionless eigenmode (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', B(0, x) = B0 � 1 + α sin(k⊥x) �ˆy, u(0, x) = vms,daα sin(k⊥x)ˆy, n(0, x) n0 = p∥i(0, x) pi0 = 1 + α sin(k⊥x), p⊥i(0, x) pi0 = � 1 + α sin(k⊥x) �2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='19) where k⊥ = 2π/λ⊥ and α is a dimensionless number quantifying the mode amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For the Pegasus++ runs, the mesh is two-dimensional and elongated in the propagation direction, with size Lx × Ly = λ⊥ × 100ρi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The size of the domain in the y direction is large enough to capture all relevant firehose and mirror fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We set βi0 = 25 and Te = Ti0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' the slightly larger value of βi0, as compared to that used in the simulations of the NP mode (βi0 = 16), results in a shorter numerical integration time (and thus computational savings) without changing the physical character of the fast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The spatial resolution and the number of macro-particles per cell are the same as in the NP simulations (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the manuscript we only show results from a Pegasus++ run having λ⊥ = 8000ρi0, corresponding to the largest domain size that we simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We found that this value of λ⊥/ρi0 was the minimum required for the mirrors to have time to grow and begin scattering particles before the wave oscillates and the sign of the driven pressure anisotropy reverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the accompanying Landau-fluid simulations, the full system of CGL-MHD equations is solved using a new Riemann solver implemented in a version of the finite-volume Athena++ simulation code (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2020) that includes Landau-fluid heat fluxes (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These equations are given in appendix B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' they reduce to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1) in our chosen geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For these runs, βi0 is varied between 1 and 100 to study the variance of the shock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In order to approximate the effect of the kinetic micro- instabilities, a ‘limiter’ collisionality νlim is set either to 0 or to αβi0k⊥vms,da, depending 30 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire Figure 16: Shock-formation time versus βi0 and α for a double-adiabatic fast wave computed from CGL-MHD simulations (lines) and predicted analytically using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The simulated waves are estimated to have formed a shock at the time when the rate of change of the maximum density gradient drops below half of its own peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' on whether the focus is on wave steepening and shock formation (ν = 0) or the effects of the instability-induced scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This anomalous scattering rate is active only within regions of the domain where the pressure anisotropy would be kinetically unstable, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', where βi∆ ⩽ −2 and βi∆ ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' elsewhere it is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It serves to isotropize the plasma pressure where mirror or firehose fluctuations would otherwise do so in a kinetic system, by contributing a term proportional to −νlim∆p to the right-hand sides of the evolution equations for p⊥ and p∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As in §2, ⟨ · ⟩ denotes a spatial average taken over the entire domain, while ⟨ · ⟩k denotes a spatial average performed along the wavefront (in this case, the y direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Wave steepening and shock formation Our first goal is to test the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) for the shock-formation time ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We perform a parameter survey by varying βi0 and the wave amplitude α using the CGL- MHD code with the micro-instability-limiting scattering turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At each time step in the simulation, the local density gradient (using a four-cell average) is calculated throughout the domain and its maximum value is recorded as a measure of the wave steepening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As a fast wave steepens, the growth rate of this maximum gradient increases until eventually the shock forms and the maximum gradient in the domain begins to plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We define the numerically calculated shock-formation time to be the time at which the rate of change of this maximum gradient drops below half of its own peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The resulting times are compared with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) in figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' When testing the dependence on βi0 (blue, left), the perturbation amplitude is set to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' when testing the dependence on amplitude (red, right), βi0 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Overall, the agreement between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12) and the numerically calculated shock-formation times is quite good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Small variations occur due to differences in the rates at which the maximum gradients plateau and to minute fluctuations in the maximum value of the gradient after the shock is formed (this value does not necessarily reach a perfect steady state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Perhaps unsurprisingly, at high β where vms,da ≈ vA � 3βi0/2, the ratio of the wave-crossing time and the shock-formation time is tcross/ts,da ≈ 4α/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This means the number of wavelengths propagated prior to forming a shock is dependent upon the mode amplitude only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 31 (a) Pressure anisotropy times the ion beta from a Pegasus++ simulation of a collisionless fast wave, showing that the compression and rarefaction of the magnetic-field lines generates oppositely signed anisotropies that move with the wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Some sloshing due to firehose regulation of the negative pressure anisotropy causes an additional reversal of ∆ in the final time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (b) Zoomed-in regions showing δBy and δBz, with the contribution from the background fast wave removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Recall that the mean field is oriented in the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the left set of panels, the mirror instability, with its oblique orientation and dominance in δB∥ = δBy, grows relatively slowly in the co-moving region of fast-wave compression from k⊥vAt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The firehose instability in the right set of panels is predominantly oblique and exhibits rapid growth and saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' smaller-amplitude parallel firehoses appear in δBx (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These firehose fluctuations reside downstream of the mirrors, where the fast-wave δB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Generation of pressure anisotropy and triggering of kinetic instabilities Prior to shock formation, the linearized fluctuations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2) suggest that pressure anisotropy at a level capable of triggering both mirror and firehose instabilities will exist when the fast-wave amplitude satisfies |δB/B0| ≳ 2/βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For these supercritical amplitudes, the wavefront should carry with it rapidly growing firehose fluctuations and more slowly growing mirror fluctuations, as per (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To test this idea, we performed a large-scale Pegasus++ simulation, the parameters of which are described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' the initial wave amplitude α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 and βi0 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 17(a) depicts the pressure anisotropy generated by the fast wave as it propagates through space at three different times (k⊥vAt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' note that the aspect ratio of the plotted domain is far from unity, and that the mean magnetic field is in the y direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Initially, the positive and negative pressure anisotropies in the wave are equal in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Shortly thereafter, the (unstable) negative anisotropy is reduced significantly due to the rapid growth of the (primarily oblique) firehose instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The positive pressure anisotropy does not show a comparable decrease, and in fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content="5 0'0=+VaT 0 Pio) 50 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='△ k1At =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5 50 klUAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 1000 2000 3000 4000 5000 6000 7000 (pio) (By/ Bo Bz/ Bo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 100 kIVAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 kIUAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39 [klUAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 k↓UAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39 75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='d) 50 9 25 0 2125 2150 2175 4525 4550 4575 7225 7250 7275 1625 1650 1675 (pio)32 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire increases somewhat from its initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This is likely because the rapid change in the negative-anisotropy regions, which perturbs the wave and causes some deviation from the eigenmode, is not matched by a comparable regulation from the positive side because of the relatively slow mirror growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 17(b) zooms in on the corresponding magnetic- field fluctuations that emerge in two separate co-moving regions where the plasma is mirror unstable (left) or firehose unstable (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To accentuate these fluctuations, the large-scale contribution from the fast wave has been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At k⊥vAt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08, oblique firehose fluctuations are strong and nonlinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' parallel firehose fluctuations are also present, though subdominant, in δBx (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' At this time, there is only a hint of mirror fluctuations emerging above the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the final frame (k⊥vAt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39) however, highly oblique mirror modes have grown to large amplitudes in the region encompassed approximately by x/ρi0 ∈ [4000, 5000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The scale separation achieved in this simulation (Lx/ρi0 = 8000) was the minimum at which we could observe mirror fluctuations with strengths comparable to their firehose counterparts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' increasing the scale separation further would come at considerable computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Effective collisionality: particle scattering Following §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3, the effective collisionality was determined for the fast wave shown in figure 17 by tracking thousands of ion macro-particles and measuring the frequency at which their µ changes statistically by a factor of κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 18 depicts this scattering rate as a function of the position along the wave (x/ρi0) and the time (k⊥vAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Sites of strong scattering are associated with the firehose modes, which appear more or less instantly and travel along with the trough of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The trail of the scattering sites indicates that the trough of the wave moves at ≈6vA, as expected for a fast mode with βi0 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this simulation, the rapid regulation of the pressure anisotropy by the firehose instability causes sloshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The sloshing temporarily drives a higher positive pressure anisotropy, and therefore enhanced mirror growth, for a short period beginning at kvAt ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The measured scattering rate in the firehose-unstable regions is comparable to the predicted asymptotic scattering rate for a βi0 = 25 fast wave with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 and Te = Ti0, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' νeff ≈ 16k⊥vA (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The mirror instability in this case also scatters particles at an average rate of a few times k⊥vA, but these scattering sites are much less coherent and do not coincide with the peak in the positive pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This delayed growth is a result of the limited achievable scale separation in our simulations, which only barely allows mirrors to grow to nonlinear levels within a fast-wave crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The effects of the induced scattering on the fast wave’s pressure anisotropy are visible in figure 19, which shows ⟨βi∆⟩y at the same times as in figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The negative anisotropy is strongly regulated by the firehose instability to well above βi∆ = −2 within a very short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This regulation persists, but is not matched on the mirror-unstable side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Some steepening has also occurred, as expected, but the positive anisotropy has not been driven down near marginal mirror stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In order for mirror fluctuations to regulate the positive pressure anisotropy to marginal stability, they would need to grow faster with respect to the fast-wave crossing time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='16) suggests that this could be achieved by increasing λ/ρi0 even further (beyond λ⊥/ρi0 = 104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Unfortunately, such large scale separations become prohibitively expensive to simulate using Pegasus++, and so from this point onward we employ the CGL-MHD code with pressure-anisotropy limiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 33 Figure 18: Space-time diagram of the effective collision frequency measured in a Pegasus++ fast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The simulation parameters are βi0 = 25, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, and Te/Ti0 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' using these numbers in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17) predicts νeff ≈ 16k⊥vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 0 1000 2000 3000 4000 5000 6000 7000 8000 x (ρi0) −2 0 2 ⟨βi∆⟩y k⊥vAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='0 k⊥vAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='08 k⊥vAt =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='39 Figure 19: Wavefront-averaged βi∆ in the fast wave for the same time frames as figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Pressure-anisotropy regulation from the firehose instability maintains βi∆ ≳ −2, while the mirror fluctuations cause some distortion of the mode above βi∆ ≈ 1 but are unable to regulate fully the positive anisotropy to marginally unstable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' An increase in the rate at which positive pressure anisotropy is generated by the steepened wave and the asymmetry in the anisotropy’s regulation by micro-instabilities causes an enhancement of the positive pressure anisotropy in the final time shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Viscous damping and collisional steepening To study fast-wave behaviour at asymptotically large scale separations, we employ the Landau-fluid CGL-MHD code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These simulations are performed using a larger β parameter than used in the Pegasus++ run, βi0 = 100 rather than 25, and with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These parameters have the advantage that a large portion of the fast wave is initially above the threshold for instability while the wave remains somewhat linear in amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, this code introduces a user-specified constant scattering rate in (and only in) the kinetically unstable regions of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We set this scattering rate according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17) using the initial mode amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In reality, this scattering rate should 34 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire (a) (b) Figure 20: (a) Propagation of an α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2 fast wave with βi0 = 100 and νlim set by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The top panel shows wave steepening in the fluid velocity, with no noticeable viscous decay on the timescale of shock formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The bottom panel shows regulation of the pressure anisotropy to near the mirror and firehose thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A peak appears in βi∆ due to the rapid generation of positive pressure anisotropy in the steepening wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (b) The maximum density gradient found within the domain of the same α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2, βi = 100 fast wave, compared against an equivalent run with νlim = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The predicted shock times are labelled by tda s and tsa s , and the shock times detected by the same method used for figure 14 are denoted by circular markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The growth of the maximum gradient continues for a longer time in the single-adiabatic case than in the double-adiabatic case, indicating delayed shock formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' decay alongside the amplitude, and so our treatment will not precisely reproduce the results that would be obtained from a more rigorous kinetic calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In figure 20, the propagation and nonlinear steepening of the CGL-MHD fast wave are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The top panel in figure 20(a) shows the bulk fluid velocity perpendicular to the background field at three different times, exhibiting steepening without a significant change in wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This indicates that no significant viscous dissipation occurs on a timescale comparable to the shock-formation timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The bottom panel shows the pressure anisotropy of the wave at the same times, multiplied by βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The anisotropy is substantially reduced below what it would be in the absence of the limiting collisionality, particularly on the firehose-unstable side, although it is not perfectly regulated to the instability thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In particular, a peak in the positive pressure anisotropy becomes prominent starting from k⊥vAt ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This is a result of wave steepening, as the sharp gradient at the wavefront generates positive ∆ much faster than the slow decline in the wake generates negative ∆, as well as faster than our (constant) limiting collisionality is able to regulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Figure 20(b) displays the evolution of the maximum absolute value of the density gradient from this run, alongside that from a comparable run with νlim = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' On the abscissa is the simulation time normalized by the double-adiabatic shock time tda s (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We calculated the shock time for each run using the same detection method as in figure 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' these times, marked by filled circles in the figure, agree reasonably well with the predicted values of tda s and tsa s for the collisionless and collisional cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The difference in steepening rate between the two runs can be interpreted as νlim forcing a more MHD-like, rather than collisionless, evolution in the fast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-β collisionless magnetosonic modes 35 The collisional isotropization at the peaks of the wave (which are also the most rapidly moving regions) effectively changes the local adiabatic index of the ions, slowing down the steepening process and yielding better agreement with tsa s than with tda s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In this sense then, all of the essential characteristics of large-amplitude, high-β, collisionless fast waves approach that of single-adiabatic MHD as a result of induced micro-instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Summary and discussion This exploration of microphysically unstable magnetosonic modes brings closure to a systematic investigation of isolated waves in collisionless, high-β plasmas that started with the discovery of self-interrupting Alfv´en waves (Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2016, 2017a) and continued with the demonstration of self-sustaining sound (Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In sum- mary, through the action of adiabatic invariance, the consequent production of pressure anisotropy, and the excitation of rapidly growing, micro-scale kinetic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' collisionless linearly polarized Alfv´en waves with amplitudes satisfying (δB⊥/B0)2 ≳ 2/βi0 retard their own propagation and spur their own viscous decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' collisionless IAWs with amplitudes satisfying |δn/n| ≳ 2/βi0 avert their otherwise potent Landau damping and propagate in a manner akin to sound waves in a weakly collisional fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' collisionless NP modes with amplitudes satisfying |δB∥/B0| ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='4 and wavelengths λ∥ ≫ 103β3/2 i0 ρi0 interrupt their transit-time damping and behave similarly to MHD entropy modes (at smaller wavelengths, these large-amplitude NP modes decay via transit-time damping, which is sustained against its nonlinear saturation by weak mirror-induced collisionality);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' and collisionless fast waves with amplitudes satisfying |δB/B0| ≳ 2/βi0 and wavelengths λ⊥ ≫ 102βi0ρi0 acquire an effective adiabatic index of 5/3 and therefore propagate and nonlinearly steepen at single-adiabatic rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Notwithstanding the somewhat narrow focus on the behaviour of isolated eigenmodes, the simple demonstration that micro-scale physics effectively filters out what kinds of macro-scale fluctuations are allowed in a high-β plasma is of broad relevance to observed space and astrophysical systems and to theories for electromagnetic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The most immediate application to the former is the near-Earth solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For example, Verscharen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2016) used linear theory to conjecture that plasma instabilities could be driven by compressive fluctuations in the β ≳ 1 solar wind through the adiabatic production of pressure anisotropy, leading to ‘collisionless isotropization’ of solar-wind protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Our work supports this idea quantitatively from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Verscharen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2017) then measured the polarization of compressive fluctuations within the solar wind at 1 au using data from the Wind spacecraft, finding that the eigenmode relationships detected were best represented by MHD, rather than collisionless, slow modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Coburn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2022) approached this same issue from a different angle, measuring the dispersion relation of compressive modes in the solar wind and determining which scattering rates best reproduced them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' They concluded that the mean free path predicted by their wave measurements is ∼103 times smaller than that set by Coulomb collisions, finding that the dispersion relation of the measured fluctuations most closely resembles that of Braginskii- MHD slow modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Both of these observational results find a natural explanation in the context of our paper, at least for those portions of the wind having β ≳ 1 that have been 36 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire measured to be constrained by the firehose and mirror instability thresholds (Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Hellinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To the extent that nonlinearly interacting fluctuations in strong electromagnetic tur- bulence retain some characteristics of their linear eigenmodes, the above conclusions cast doubt on whether some well-established pillars of MHD and gyrokinetic turbulence theory (Goldreich & Sridhar 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Lithwick & Goldreich 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Schekochihin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Schekochihin 2022) are applicable to high-β plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For example, with each fluctuation generating and responding to pressure anisotropy in an amplitude-, wavelength-, and polarization-dependent way, it is suspect that inertial-range compressive fluctuations are simply passively mixed by the Alfv´en-wave cascade and, in turn, exert no back-reaction on the Alfv´enic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The mutual interactions between what are conventionally considered to be energetically decoupled cascades, and the impact of this coupling on the constant flux of energy and the locality of interactions in k space, ought be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Some progress on this front has recently been made by Arzamasskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2022), who showed using hybrid-kinetic simulations that strong Alfv´enic turbulence with (δB⊥/B0)2 ≳ 2/βi0 self-consistently produces a parallel viscous scale comparable to the driving scale of the cascade and involves non-local energy transfers in k space associated with the excitation of ion-Larmor-scale kinetic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Incorporating compressive fluctuations into the turbulence forcing would be informative, not only with regards to the dynamics but also concerning the partition of turbulent energy into ion versus electron heating (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kawazura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' It is additionally unclear how all this additional physics plays out within a turbulent cascade governed by a scale-by-scale ‘critical balance’ between the characteristic linear and nonlinear frequencies, an organizing principle for strong turbulence that appears to hold (albeit in a modified form) even in the presence of strong pressure anisotropies (Bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Arzamasskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Acknowledgements S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' were supported in part by NSF CAREER Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1944972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Support for J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' was provided by Rutherford Discovery Fellowship RDF-U001804, which is managed through the Royal Society Te Ap¯arangi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' High-performance computing re- sources were provided by: the Texas Advanced Computer Center at The University of Texas at Austin under Stampede2 allocation TG-AST160068 and Frontera allocation AST20010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' and the PICSciE-OIT TIGRESS High Performance Computing Center and Visualization Laboratory at Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The authors thank Archie Bott, Eliot Quataert, Alex Schekochihin, and the participants of the 13th Plasma Kinetics Working Meeting at the Wolfgang Pauli Institute in Vienna for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' addi- tionally thanks the Institut de Plan´etologie et d’Astrophysique de Grenoble (IPAG) for its hospitality and visitor support while this work was being completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Hermite–Laguerre solution to linear KMHD In this appendix, we detail our numerical method for calculating the time-dependent pressure anisotropy generated by a linear NP mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The task is to integrate the sys- tem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1) numerically from an appropriate set of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Before providing those conditions, we take the time derivative of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1b) and use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1c) to obtain the following High-β collisionless magnetosonic modes 37 wave equation for the E × B drift velocity: � d2 dt2 + k2v2 A � u⊥ = − ik⊥ min0 d dt � δp⊥i + Teδn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (A 1) The right-hand side of this equation is calculated by taking the zeroth and second moments of the linearized Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' After assuming an isotropic Maxwellian background, F0 = FM(v), and rewriting the electric and magnetic-mirror forces using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1c) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1d), equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1a) reduces to � ∂ ∂t + ik∥v∥ � δf + � ik⊥u⊥ w2 ⊥ v2 th,i + ik∥v∥ Te Ti0 δn n0 � FM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (A 2) Equations (A 1) and (A 2) are solved numerically as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' We express the v∥ dependence of δf in terms of Hermite polynomials Hn and the w2 ⊥ dependence in terms of Laguerre polymonials Lm: δf(t, k∥, k⊥, v∥, w⊥) = FM(v) ∞ � m,n=0 gm,nHn � v∥ vth,i � Lm � w2 ⊥ v2 th,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (A 3) This spectral decomposition allows the required moments to be calculated simply as δn n0 = g0,0, δp⊥i pi0 = g0,0 − g1,0, δp∥i pi0 = g0,0 + 4g0,2, (A 4) so that (A 1) becomes � d2 dt2 + k2v2 A � u⊥ vth,i = −ik⊥vth,i 2 d dt �� 1 + Te Ti0 � g0,0 − g1,0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (A 5) Because the Hermite and Laguerre polynomials form orthonormal bases with respect to Gaussian and exponential weights, respectively, equation (A 2) may be easily transformed to Hermite–Laguerre space to find dgm,0 dt + ik∥vth,igm,1 + i(δm,0 − δm,1)k⊥u⊥ = 0, (A 6a) dgm,1 dt + ik∥vth,i � 2gm,2 + 1 2gm,0 � + i Te 2Ti0 δm,0g0,0 = 0, (A 6b) dgm,n dt + ik∥vth,i � (n + 1)gm,n+1 + 1 2gm,n−1 � = −νn4gm,n, n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (A 6c) Note that the term k∥v∥δf representing the parallel phase mixing of the perturbed distribution function couples together different Hermite moments, representing the gen- eration of fine-scale structure in v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Because the magnetic field suppresses phase mixing across the magnetic field, there is no cascade to higher w⊥ moments and only the first two Laguerre polynomials (m = 0, 1) are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To the right-hand side of (A 6c) we have appended a fourth-order hyper-collision operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' the restriction of the collision operator to n ⩾ 2 guarantees that number and momentum are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The hyper- collisionality is added because only a finite number of Hermite polynomials are usable, so the series must be truncated somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A hard truncation in which the final v∥ moment is arbitrarily set to zero will result in numerical instability unless a collisionality is employed to ensure the velocity-space cascade (associated with parallel phase mixing of the perturbed distribution function) decays to zero amplitude before the last resolved moment is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A code was written in Fortran 90 to solve (A 5) and (A 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Equation (A 6) is solved 38 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire and δf updated in time using a semi-implicit Crank–Nicholson method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' the moments g0,0 and g1,0 are then used in (A 5) to update the drift velocity using centered differencing in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The discrete time axes on which gm,n and u⊥ are stored are staggered to maintain appropriate centering for all derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The matrix inversion needed to update gm,n is performed using the Thomas Tridiagonal Matrix Algorithm (TDMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For the initial conditions, we start from isothermal pressure balance, with g1,0 = g0,2 = 0 and g0,0 ̸= 0 (but arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The reasoning behind this choice is discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' These initial conditions transition rapidly into the NP eigenmode by launching small- amplitude (relative to the amplitude of the NP mode) fast waves that facilitate the adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The linear evolution of the NP mode from this initial condition is shown in figure 1 and discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Magnetosonic modes with arbitrary scattering frequency To obtain the linear dispersion relation of kinetic hydromagnetic modes at arbitrary ν, we must use a model that accurately captures the effects of adiabatic invariants, heat fluxes, and collisional isotropization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' One such model is given by the Chew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (1956) equations supplemented, by collisional isotropization and closed by so-called Landau-fluid heat fluxes (Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Assuming isothermal electrons, these equations are: Dn Dt = −n∇ · u, (B 1a) minDu Dt = −∇ � p⊥i + nTe + B2 8π � + ∇ · � ˆbˆb � ∆pi + B2 4π �� , (B 1b) DB Dt = (B · ∇)u − B∇ · u, (B 1c) nB D Dt �p⊥i nB � = −∇ · � q⊥iˆb � − q⊥i∇ · ˆb − 1 3ν∆pi, (B 1d) n3 B2 D Dt �p∥iB2 n3 � = −∇ · � q∥iˆb � + 2q⊥i∇ · ˆb + 2 3ν∆pi, (B 1e) where D/Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ∂/∂t+u · ∇ is the convective derivative for the bulk velocity u, ˆb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= B/B is the unit vector in the direction of the local magnetic field, ∆pi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= p⊥i − p∥i is the dimensional ion pressure anisotropy, ν is the isotropizing collision frequency, and q∥i and q⊥i represent the field-parallel flow of parallel and perpendicular ion heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For linear perturbations to the ion temperature (δT∥i, δT⊥i) and magnetic-field strength (δB∥) having parallel wavenumber k∥, the latter may be adopted from equations (48) and (49) of Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (1997): q∥i,k = − 4nv2 th∥,i 2√π|k∥|vth∥,i + (3π − 8)ν ik∥δT∥i, (B 2) q⊥i,k = − nv2 th∥,i √ 2π|k∥|vth∥,i + 2ν � ik∥δT⊥i + ik∥T⊥i∆i δB∥ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (B 3) These ‘3+1’ heat fluxes accurately reproduce the linear Landau–Barnes damping of the kinetic hydromagnetic modes in the collisionless limit (Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1997, §VIII) and take on a form akin to that obtained by Braginskii (1965) in the collisional limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Because Braginskii-MHD does not accurately capture the linear heat fluxes when ν ≲ |k∥|vth,i, the Landau-fluid CGL equations are used to describe the linear propagation of these High-β collisionless magnetosonic modes 39 modes at arbitrary ν, bridging the gap between the fully collisionless (ν = 0) and the weakly collisional (ν ≫ k∥vth,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Note that, in the absence of heat fluxes and collisionality, equations (B 1d) and (B 1e) guarantee conservation of the adiabatic invariants µ and J associated with Larmor gyrations and bounce motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' One of the advantages of using the Landau-fluid CGL equations over a Vlasov approach is the former’s lack of dependence on the plasma dispersion function Z(ζ), whose dependence on ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ω/|k∥|vth,i can only be expressed analytically in the asymptotic limits ζ ≫ 1 and ζ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Instead, the ‘3+1’ heat fluxes yield polynomial dispersion relations for the modes at all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As a result, if one wishes to derive an analytic expression for the frequency and damping rate of the oblique IAW, which has ζ ∼ 1 when Te/Ti0 ∼ 1, they can then do so with ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Proceeding with the linear analysis, we assume zero background pressure anisotropy, neglect all nonlinear terms, and Fourier transform (B 1)–(B 3) in space and time, so that D/Dt → −iω and ∇ → ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The result is a straightforward algebraic system, some solutions of which are shown in figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In total there are 8 modes associated with 8 unique time derivatives (∇ · B = 0 fixes one of the components of δB⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The modes not displayed in figure 21 are the Alfv´en waves (which would be lines at ζ = ±β−1/2 i0 ) and both fast waves (which are shown in figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Considering that there exists one additional time derivative in CGL-MHD than in collisional MHD due to the splitting of the thermal pressure into two components, there should be a mode that vanishes in the collisional limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Indeed, after bifurcation one branch of the oblique IAW becomes non- propagating and is damped at a rate approximately equal to ν as ν → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This strong damping is due to the mode’s polarization, having opposing perpendicular and parallel pressure perturbations that satisfy |δp⊥| ≫ |δp∥| when k⊥ ≫ k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Hence the reason we have termed this mode the “anisotropy mode” in figure 21: it remains anisotropic even at arbitrarily large ν, causing it to damp increasingly rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The NP mode is often associated with the collisionless limit of the MHD slow mag- netosonic mode (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Verscharen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 2017), and is frequently referred to as the collisionless slow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This may be due to the fact that the Braginskii-MHD dispersion relation predicts a non-propagating slow mode at sufficiently low ν, one which remains non-propagating as ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In reality, the slow mode does propagate once again at sufficiently low collisionality, and the NP mode is better identified as the kinetic extension of the MHD entropy mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In the MHD entropy mode, no pressure perturbation is permitted by the parallel momentum equation, only a density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, at lower scattering rates the pressure separates into its field-parallel and perpendicular components, and perpendicular pressure balance becomes achievable (see (B 1b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The assertion that the NP mode is connected to the MHD entropy mode, rather than the slow mode, is likely more desirable as it also avoids degeneracy in different branches of the dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Careful inspection of figure 21 shows that there exists a band in which both the NP and oblique ion-acoustic modes possess zero real frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' If it were the case that the MHD slow mode became the NP mode, this branch would have to cross with the kinetic entropy mode and both would have identical decay rates, making them degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Therefore, in our argument for the behaviour of above-threshold NP modes in high-β plasmas, we expect that at very large scale separation, and hence large ν/|k∥|vth,i, the NP mode will become more akin to the MHD entropy mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The oblique ion-acoustic wave (IAW) also deserves special attention, not least because it possesses a non-propagating band beginning near ν ∼ k∥vth,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Somewhat paradoxically, this is the collisionless extension of the MHD slow mode, never mind the fact that at high β it propagates faster than the Alfv´en speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Even in the collisionless Landau-fluid CGL model, this mode evades a simple general expression for its frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, 40 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Majeski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Kunz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Squire Figure 21: Linear dispersion relation of the Landau-fluid CGL-MHD equations (B 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The dimensionless (complex) frequency ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='= ω/|k∥|vth,i is computed numerically as a function of collisionality ν/|k∥|vth,i for k⊥ = 4|k∥|, βi0 = 16, and Te = Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' in the limit of k⊥ ≫ k∥ and β ≫ 1 with Te = Ti0, one can obtain the dispersion relation numerically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' we find that ζ ≈ 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='43i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This mode therefore has a very similar dispersion relation to its parallel-propagating variant, especially with regards to its rapidly damped nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Asymptotic analysis for k⊥ ≫ k∥ reveals that this mode develops a non-propagating band when β ≳ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1, occurring in the approximate range of scattering frequencies satisfying ν/k∥vth,i ∈ [2, (3/4)√β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' When β ∼ O(1) and smaller, the Braginskii slow mode smoothly transitions into the oblique IAW as ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' However, at high β, an increasingly large gap forms between the two propagating portions of this mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This phenomenon is not present in parallel-propagating IAWs at any β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Oblique IAWs and micro-instabilities Of the collisionless hydromagnetic modes that do not propagate parallel to the back- ground magnetic field, we have yet to discuss one in the context of high-β plasmas and micro-instabilities: the oblique IAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Given that oblique IAWs share many traits with their parallel propagating counterparts (§B), generalizing the results of Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2020) to the oblique case should not require dramatic changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Even when propagating across the background magnetic field, at high β these waves are still largely driven by a perturbation to the parallel pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' As a result, the magnetic tension plays essentially no role, and no interruption-like process can occur as in the case of linearly polarized Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Furthermore, the oblique IAW generates equivalent positive and negative pressure anisotropies (there is no pressure balance as in the NP mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For this reason, both mirror and firehose instabilities can be triggered by this mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The only notable difference between the oblique and parallel IAWs is the existence of a non-propagating band at certain values of ν in the dispersion relation of the oblique mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' To see how High-β collisionless magnetosonic modes 41 this difference affects propagation in the presence of instability-induced scattering, we perform an analysis similar to that carried out in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Our first task is to determine the amplitude limit above which the anisotropic pressure perturbation in the oblique IAW is unstable to both the mirror and firehose instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Taking the k⊥ ≫ k∥ and β ≫ 1 limit, the parallel and perpendicular temperature perturbations in the oblique IAW are δT∥ Ti0 ≈ − � 2 + � 1 + ik∥vth,i ω√π �−1�� 1 + 2ik∥vth,i ω√π �−1 δB∥ B0 , (C 1a) δT⊥ Ti0 ≈ � 1 + ik∥vth,i ω√π �−1 δB∥ B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (C 1b) Substituting in ω/k∥vth,i ≈ 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='43i, equations (C 1) yield an ion pressure anisotropy ∆ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='88 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='03i)(δB∥/B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This implies the following amplitude threshold for oblique IAWs to trigger both the firehose and mirror instabilities: ���� δB∥ B0 ���� ≳ 1 βi (oblique IAW amplitude threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (C 2) We argue that, above this threshold, the scattering induced by micro-instabilities will be that required to maintain marginal stability, or ∆ ∼ β−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Through the same logic as was applied to the fast mode, this scattering rate is ν ∼ Re � 3ωβi �δB∥ B0 − 2 3 δn n0 �� ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='7k∥vth,iβi ���� δB∥ B0 ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (C 3) As in the case of the fast wave, the above expression for the limiting collisionality is only valid in the limit that ν ≫ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' This constraint is nearly satisfied at the amplitude threshold, therefore this scattering rate is likely to be a good approximation even for mode amplitudes of only a few times β−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' With the scaling of the induced scattering rate now known, we may return to the dispersion relation shown in figure 21 to surmise how micro-instabilities might modify the propagation of oblique IAWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Recall from appendix B that the oblique IAW becomes non-propagating for βi ≳ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='1 when ν/k∥vth,i ∈ [2, (3/4)√β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The form of the effective scattering rate (being dependent on δB∥) then suggests that the fate of an oblique IAW rests on the amplitude of the initial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' For amplitudes within the range β−1 ≲ |δB∥/B0| ≲ β−1/2, the oblique IAW will become a viscously damped mode which does not propagate, while above |δB∥/B0| ≳ β−1/2 it will become a Braginskii-like propagating sound wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The latter of the two regimes is essentially the result obtained by Kunz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' (2020) for parallel-propagating IAWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' The former limit of moderate amplitude becomes increasingly important at high β where its range of relevance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' In plasmas with β ≲ 10 however (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', the solar wind), this range is either extremely narrow or nonexistent, leading to evolution that closely resembles that of the parallel IAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' REFERENCES Arzamasskiy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Kunz, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Schekochihin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Abel, I.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' ¨Odblom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 1998 On the formation of shocks in warm magnetized plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A 249 (1–2), 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Parker, E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' 447, L45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' Riquelme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} +page_content=', Quataert, E.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE0T4oBgHgl3EQfWADS/content/2301.02273v1.pdf'} diff --git a/Y9FAT4oBgHgl3EQf3h6y/content/tmp_files/2301.08721v1.pdf.txt b/Y9FAT4oBgHgl3EQf3h6y/content/tmp_files/2301.08721v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3050b8a7c729ecd9522f6f30583554280fb3823 --- /dev/null +++ b/Y9FAT4oBgHgl3EQf3h6y/content/tmp_files/2301.08721v1.pdf.txt @@ -0,0 +1,1720 @@ +Batch Prompting: Efficient Inference with Large Language Model APIs +Zhoujun Cheng +Shanghai Jiao Tong University +blankcheng@sjtu.edu.cn +Jungo Kasai +University of Washington +jkasai@cs.washington.edu +Tao Yu +University of Hong Kong +tyu@cs.hku.hk +Abstract +Performing inference on hundreds of thou- +sands of samples with large language mod- +els (LLMs) can be computationally and finan- +cially costly. We propose batch prompting, a +simple alternative prompting approach that en- +ables the LLM to run inference in batches, in- +stead of one sample at a time. Our method +reduces both token and time costs while re- +taining downstream performance. +We theo- +retically demonstrate that under a few-shot in- +context learning setting, the inference costs de- +crease almost inverse linearly with the num- +ber of samples in each batch. We extensively +validate the effectiveness of batch prompting +on ten datasets across commonsense QA, arith- +metic reasoning, and NLI/NLU: batch prompt- +ing significantly (up to 5× with six samples +in batch) reduces the LLM (Codex) inference +token and time costs while achieving better or +comparable performance. Our analysis shows +that the number of samples in each batch and +the complexity of tasks affect its performance. +Further, batch prompting can be applied across +different LLMs and reasoning methods. Our +code will be available at https://github.com/ +HKUNLP/batch-prompting. +1 +Introduction +Large language models (LLMs) have shown their +strong capabilities under zero/few-shot settings +with in-context learning (Brown et al., 2020; Chen +et al., 2021; Chowdhery et al., 2022). Much recent +work has made progress in in-context learning by +eliciting reasoning steps (Wei et al., 2022; Wang +et al., 2022; Khot et al., 2022; Cheng et al., 2022), +selecting representative in-context exemplars (Liu +et al., 2022; Su et al., 2022; Agrawal et al., 2022), +and designing prompt templates (Jiang et al., 2020; +Bach et al., 2022; Arora et al., 2022). +Using LLMs can be costly in terms of token +and time usage, especially when many LLM calls +are needed, such as when benchmarking a large +Standard Prompting +Batch Prompting +# K-shot in-context exemplars +Q: {question} +A: {answer} +Q: {question} +A: {answer} +… +# One sample to inference +Q: Ali had $21. Leila gave him half of her + $100. How much does Ali have now? +----------------------------------------------- +# Response +A: Leila gave 100/2=50 to Ali. Ali now has + $21+$50 = $71. The answer is 71. +# K-shot in-context exemplars in K/b batches +Q[1]: {question} +Q[2]: {question} +A[1]: {answer} +A[2]: {answer} +… +# b samples in a batch to inference +Q[1]: Ali had $21. Leila gave him half of her + $100. How much does Ali have now? +Q[2]: A robe takes 2 bolts of blue fiber and + half that white fiber. How many bolts? +----------------------------------------------- +# Responses to a batch +A[1]: Leila gave 100/2=50 to Ali. Ali now has + $21+$50 = $71. The answer is 71. +A[2]: It takes 2/2=1 bolt of white fiber. The + total amount is 2+1=3. The answer is 3. +b(=2) samples +in one batch +Figure 1: Illustration of batch prompting compared +with standard prompting. Batch prompting groups mul- +tiple samples in one batch (b=2 in the figure) and lets +the LLM generate multiple responses (highlighted in +yellow) for the batch in inference. +dataset or addressing a high volume of customer +inquiries for businesses. For example, the widely- +adopted OpenAI API service1 of LLMs requires +about $400 and 10 hours to perform inference on +10K samples.2 If the rate limits of maximum API +requests per minute are also considered, the costs +will be even higher, preventing users from building +massive LLM applications. +We propose batch prompting, an alternative ap- +proach for prompting LLMs, which allows the +1https://openai.com/api/. +2This assumes each LLM call consumes 2, 000 tokens, +including both the input prompt tokens and generated tokens. +arXiv:2301.08721v1 [cs.CL] 19 Jan 2023 + +model to perform inference on multiple samples +at once, instead of one sample at a time. This re- +duces token and time costs while still retaining +downstream performance, without any change in +APIs. As shown in Figure 1, standard prompting +generates a response (answer) to one sample at a +time, which takes N inference runs of an LLM for +a test set of size N. For our batch prompting, on +the other hand, an LLM generates responses to b +samples in a single inference run and only takes +N/b runs for the same N samples. +We first demonstrate theoretically that under +the few-shot in-context learning setting, most to- +kens consumed during the API call are the few- +shot exemplars, and only a small portion of token +budgets are used for the particular inference sam- +ple(s) (Section 2). Therefore, increasing the num- +ber of samples b in a batch of batch prompting +reduces the token and time costs in an inverse lin- +ear fashion. We extensively validate the effective- +ness of batch prompting on ten diverse downstream +datasets across commonsense QA, arithmetics, and +NLI/NLU using Codex, a strong variant of GPT-3 +finetuned on code data (Section 3). Batch prompt- +ing significantly decreases the tokens and run time +of using LLMs while achieving comparable or even +better performance on all ten datasets. In further +analysis (Section 4), we find the number of sam- +ples in batch and the complexity of tasks affect +its performance. Moreover, we show that batch +prompting works well across different LLMs (e.g., +Codex, ChatGPT, and GPT-3) and reasoning meth- +ods (e.g., end-to-end, Chain-of-Thought, and code +generation), suggesting that batch prompting is an +efficient drop-in substitute for conventional prompt- +ing. +2 +Approach +We first introduce batch prompting, an efficient +alternative to standard prompting. We then com- +pare the token and time costs of batch and stan- +dard prompting, demonstrating the efficiency of +our method. +2.1 +Problem Setup +The conventional paradigm (i.e., standard prompt- +ing in Figure 1) to prompt LLMs for in-context +learning is as follows: K in-context few-shot ex- +emplars with both a context (e.g., question) and +an output (e.g., answer) are selected to build the +input prompt, one test sample with context only is +appended at the end of the prompt, and the LLM is +used to generate the response for the test sample. +In this paper, we focus on a realistic scenario +with N test samples, which is common when +benchmarking on a dataset or handling a large vol- +ume of customer requests. In this case, it takes +N separate calls of the LLM inference under the +standard prompting paradigm. +2.2 +Batch Prompting +Batch prompting enables the LLM to generate re- +sponses for multiple samples in one batch in a sin- +gle inference run, so that it reduces the LLM infer- +ence time from N to N/b, where b is the number +of samples in one batch. Specifically, as shown +in Figure 1, our prompt groups the K in-context +exemplars into K/b batches with b exemplars each +as demonstrations. In every batch, demonstration +contexts are arranged in a specific order at the be- +ginning, with their corresponding outputs placed +in the same order afterwards. Then, b test sam- +ple contexts are grouped together at the end of the +input prompt. In this way, the LLM learns from +the in-context demonstrations and generates cor- +responding responses for the entire batch of test +samples. We add a position identifier “[index]” +within each batch to 1) assist the LLM with iden- +tifying the order correspondence of input contexts +and generated responses and 2) ease the process of +parsing the generated responses. +2.3 +Token Cost +The costs of one LLM call scale linearly with the +number of tokens, including both the input prompt +tokens (few-shot and instruction) and generated +tokens (according to, for example, OpenAI’s pric- +ing). Most tokens are consumed by the prompt +tokens in standard prompting because the num- +ber of prompt tokens is usually far more than the +number of generated tokens so that the LLM can +better learn from in-context exemplar. Thus, the +larger the portion of tokens spent on generated to- +kens, the more economical the total cost is. +We define token efficiency η as the portion of +tokens spent on generated tokens in one LLM call. +For standard prompting and batch prompting (the +instruction tokens are omitted if any for brevity): +ηstandard = +1 +K + 1 +ηbatch = +b +K + b +(1) + +(a) Token-CommonsenseQA +(b) Token-GSM8K +(c) Token-RTE +(d) Time-CommonsenseQA +(e) Time-GSM8K +(f) Time-RTE +Figure 2: Token and time costs per sample on three datasets for illustrations (other datasets show similar trends). +Batch prompting significantly lowers both token and time costs as the number of samples in each batch increases. +When K ≫ 1 and b < K, ηbatch scales almost +inverse linearly with b, and thus increasing b of +batch prompting can greatly reduce token costs. +2.4 +Time Cost +Intuitively, batch prompting reduces the inference +time by decreasing the number of API calls from +N to N/b. If considering the time of Transformer +(Vaswani et al., 2017) decoding, the cost will in- +crease with b in batch prompting since longer re- +sponses will be generated compared with standard +prompting. We give a detailed derivation regarding +this Transformer architecture perspective in Ap- +pendix A. +However, as most end-users are accustomed to +and only have access to LLM API services, this +part of time cost is marginal (observed in main +experiments), relative to the overhead of API call +and request rate limits per minute set by a company, +such as OpenAI. Besides, cases may occur when +network connections are unstable or slow, and the +users seek to finish a task with as few LLM calls +as possible. +Therefore, in practice, reducing the number of +calls from N to N/b with batch prompting can +essentially lower the time costs. Note that when +the API call overhead and rate limits are no longer +the major bottlenecks of time costs in the future, +then the increased decoding time to generate longer +sequences discussed in Appendix A cannot be over- +looked, and the time reduction of batch prompt- +ing will not be as pronounced (e.g., the latest text- +davinci-003). Since LLM infrastructure/services +can change over time, token costs are easier to +measure in experiments than time costs. +3 +Experiments +We extensively evaluate batch prompting across +ten diverse datasets. Our results suggest that batch +prompting can achieve at most 5× token and time +efficiency (with six samples in batches) improve- +ment with similar or even better downstream per- +formance. +3.1 +Datasets +We evaluate batch prompting on ten datasets +across commonsense question answering, arith- +metic reasoning, and natural language under- +standing/inference: +CommonsenseQA (Talmor +et al., 2019), StrategyQA (Geva et al., 2021), +GSM8K (Cobbe et al., 2021), SVAMP (Patel et al., +2021), AQuA (Ling et al., 2017), AddSub (Hos- +seini et al., 2014), MultiArith (Roy and Roth, 2015), +RTE (Bentivogli et al., 2009), MNLI (Williams +et al., 2018), and SST-5 (Socher et al., 2013). For +CommonsenseQA, AQuA, AddSub, MultiArith, +and RTE, we evaluate the whole dev/test sets. For + +1100 +1000 +900 +sample +800 +700 +Method +Tokensper +Standard +600 +Batch Prompting +# +500 +400 +300 +200 +2 +3 +4 +6 +#Samplesinbatch1100 +1000 +900 +sample +800 +700 +Method +#Tokens per +Standard +600 +Batch Prompting +500 +400 +300 +200 +2 +3 +4 +6 +#Samplesinbatch1100 +1000 +900 +sample +800 +700 +Method +#Tokens per +Standard +600 +Batch Prompting +500 +400 +300 +200 +2 +3 +4 +6 +#Samplesinbatch8 +sample +5 +Method +per +Standard +Time(s) +4 +Batch Prompting +3 - +2 - +1 +2 +3 +4 +6 +#Samplesinbatch8. +rsampl +e +7. +Method +per +6 +Standard +Batch Prompting +Time( +5 +4- +3 +2 +3 +4 +6 +#Samplesinbatch9 +(s)persample +4 - +Method +Standard +Batch Prompting +Time( +3 +2 - +X +1 +2 +3 +4 +6 +#SamplesinbatchTask +Dataset +Standard +Batch +Commonsense +CSQA +77.2 +77.4(+0.2) +StrategyQA +73.3 +71.0(−2.3) +Arithmetic +GSM8K +55.7 +58.7(+3.0) +SVAMP +83.7 +81.3(−2.4) +AQuA +46.1 +42.1(−4.0) +AddSub +86.6 +84.8(−1.8) +MultiArith +97.5 +98.7(+1.2) +NLI/NLU +RTE +76.9 +74.7(−2.2) +MNLI +65.3 +65.7(+0.4) +SST-5 +51.3 +49.7(−1.6) +Table 1: +Accuracy of standard and batch prompt- +ing on ten datasets: CommonsenseQA (Talmor et al., +2019), StrategyQA (Geva et al., 2021), SVAMP (Patel +et al., 2021), AQuA (Ling et al., 2017), AddSub (Hos- +seini et al., 2014), MultiArith (Roy and Roth, 2015), +RTE (Bentivogli et al., 2009), MNLI (Williams et al., +2018), and SST-5 (Socher et al., 2013). Batch prompt- +ing shows comparable or even better performance. +the other five datasets, we evaluate the first 300 test +samples considering the costs of LLM APIs. +3.2 +Experimental Setups +We use OpenAI Codex (code-davinci-002) as the +LLM in our experiments (in Section 4.5, different +LLMs are discussed). Codex is currently provided +for free, but the token consumption strategy is the +same as the other LLMs, ensuring that the token +costs in experiments are general. The decoding +temperature is set as 0. For each dataset, we manu- +ally select 12-shot samples from the training set as +in-context exemplars, with Chain-of-Thought (Wei +et al., 2022, CoT) reasoning steps in the answers (in +Section 4.4, other reasoning methods beyond CoT +are discussed). We choose 12 exemplars because +12 is the least common multiple of 2, 3, 4, 6, and +thus it is easy to analyze the effects of grouping +them into batches of 2, 3, 4, 6 samples in our ab- +lation studies. More experimental details and full +results are listed in Appendix B. +3.3 +Results +Figure 2 compares the token and time costs of +standard and batch prompting. As shown, batch +prompting substantially (up to 5× with 6 samples +in each batch) reduces both the token and time costs +of standard prompting with Codex. Further, the de- +crease of costs scales almost inverse linearly with +the number of samples in each batch, verifying our +analysis in Sections 2.3 and 2.4. Note the time costs +include the API call overhead and rate limit blocks, +which exist in the commonly-used OpenAI services. +For LLM services where these are not bottlenecks +Figure 3: Accuracy over varying numbers of batch sam- +ples b on five datasets using batch prompting. The per- +formance decreases with larger b. +of time, e.g. the latest GPT-3 (text-davinci-003), +the decoding time increase from larger b should not +be overlooked as discussed in Section 2.4. As the +LLM infrastructure can change anytime, the token +efficiency improvement is easier to compare than +time; the token reduction in Figure 2 should hold +for any LLM over time. +Table 1 shows that batch prompting (with the +best b, i.e., the number of samples in each batch) +performs comparably or even better than standard +prompting over all ten datasets. We thus recom- +mend that LLM users consider applying batch +prompting to save money and time while main- +taining good performance in realistic applications. +4 +Analysis +In this section, we first analyze factors that may +affect the performance of batch prompting and +explore the tradeoff of balancing the costs and +downstream performance. +We further demon- +strate batch prompting can be applied to different +LLMs (e.g., GPT-3, and ChatGPT) and prompting +methods (e.g., end-to-end, and code generation). +4.1 +Number of Batch Samples +Figure 3 shows how the number of samples in each +batch b affects the benchmark performance in batch +prompting. Firstly, the performance generally de- +creases as b becomes larger. When b = 6, a large +drop is seen across four of these five datasets. In- +terestingly, however, the best performance is not +always achieved when b=2. Setting b = 3 or 4 usu- +ally achieves good performance while saving more +tokens and time than smaller b. The reduction of +time/token costs diminishes when b becomes larger, +indicating that setting b < 6 (given 12 shots in- +context exemplars in experiments) tends to provide + +06 +Dataset +CSQA +GSM8K +SVAMP +80 +AddSub +RTE +Accuracy +70 +60 +50 +40 +2 +4 +6 +#Samples in batchDataset +Random +Similar +Diverse +CSQA +77.4 +77.4 +78.2 +GSM8K +58.7 +57.7 +55.7 +SVAMP +81.3 +81.3 +80.7 +AddSub +84.8 +83.2 +84.1 +RTE +74.7 +70.4 +70.8 +Table 2: Accuracy from various batching methods on +five representative datasets. +Similarity or diversity- +based methods do not achieve performance gains. +a good tradeoff between the costs and downstream +performance. +4.2 +Selection of Batch Samples +Here we examine whether the selection of samples, +i.e. how samples are grouped into batches, will +affect the performance of batch prompting. We +study two widely-adopted sample selection meth- +ods in in-context learning when grouping the test +samples: grouping more similar (Rubin et al., 2021; +Liu et al., 2022) or more diverse (Su et al., 2022; +Agrawal et al., 2022) samples into batches. Specifi- +cally, given N test samples, to group similar ones, +we use k-means clustering and post-process each +cluster into equal size b by moving redundant sam- +ples to their closest groups with size < b. To group +diverse ones, we apply the vote-k method (Su et al., +2022) to iteratively select diverse and representa- +tive groups of samples. +As listed in Table 2, both similarity and diversity- +based selections do not show improvements over +random grouping. We suspect that the reason may +be that both methods assume in-batch samples can +benefit from similar or diverse samples presented +before them, i.e., samples in the front of the batch. +However, these earlier samples are not with ground +truth outputs and thus may lead to error propaga- +tions to the rest of the in-batch samples. Devel- +oping effective strategies for selecting samples for +batch prompting could be a promising area for fu- +ture research to further enhance the performance +of batch prompting. +4.3 +Complexity of Tasks +We further analyze how the complexity of tasks +influences the performance of batch prompting. In +Table 1, the largest drop (from 46.1 to 42.1) oc- +curs on the AQuA dataset, which is an arithmetic +reasoning task in a multi-choice QA format. One +interpretation is that AQuA is more difficult than +other datasets with the lowest absolute accuracy +46.1%, and thus LLMs are more likely to be dis- +Figure 4: Accuracy on WikiTQ of various table input +strategies and b (the number of samples in each batch). +This studies how the input length affects batch prompt- +ing performance. +b = 1 means standard prompting. +Average input tokens per table are 24, 58, and 216 to- +kens. As the number of batch samples increases, batch +prompting suffers in downstream performance. +turbed when input contexts are grouped together. +We study another aspect of the task that may +affect performance: the longer the input contexts, +the more substantially batch prompting hurts per- +formance. We validate our assumption with Wik- +iTQ (Pasupat and Liang, 2015), a challenging Table +QA dataset over Wikitables. Tables contain longer +input tokens for their multiple rows and columns. +We experiment with increasing table input lengths: +a simplified table schema (i.e., column names with- +out column types; avg. 24 tokens/table), a table +schema (avg. 58 tokens/table), and a table schema +with three table rows (avg. 216 tokens/table). We +follow Binder (Cheng et al., 2022) to generate +Binder-SQL programs to solve the questions. +As shown in Figure 4, in standard prompting +(b = 1), inputting table schemas with three rows +dominates QA performance. However, it also sees +the steepest performance drop when b increases us- +ing batch prompting. The shorter the input contexts, +the steadier the performance with batch prompting. +This suggests that long task inputs are more likely +to lead to confusion and performance drops when +batch prompting is applied. +4.4 +Reasoning Methods +In our main experiments (Section 3), we used the +Chain-of-Thought (CoT) for all ten datasets. Here +we examine whether batch prompting is suitable +for other common LLM reasoning methods. We +experiment with two more reasoning methods: end- +to-end (i.e., directly prompt the LLM to output the + +Table Input +Schema(Simple)--24 tokens +Schema +--58tokens +Schema(3 rows)--216 tokens +55 +Accuracy +45 +35 +1 +2 +3 +6 +#Samples in batchDataset +End-to-end +CoT +Program +Standard Batch +Standard Batch +Standard Batch +CSQA +81.5 +80.4 +77.2 +77.4 +- +- +GSM8K +21.3 +17.3 +55.7 +58.7 +72.7 +73.0 +SVAMP +70.7 +68.3 +83.7 +81.3 +86.0 +86.3 +RTE +85.2 +83.4 +76.9 +74.7 +- +- +WikiTQ +- +- +- +- +54.3 +50.7 +Table 3: Accuracy of different reasoning methods with +standard and batch prompting. Batch prompting can be +applied well with different reasoning methods showing +similar or better performance. +Dataset +Codex +GPT-3 +ChatGPT +Standard Batch +Standard Batch +Standard Batch +CSQA +77.2 +77.4 +78.3 +75.8 +66.7 +75.0 +GSM8K +55.7 +58.7 +58.0 +55.0 +71.7 +66.7 +SVAMP +83.7 +81.3 +86.7 +85.8 +- +- +AddSub +86.6 +84.8 +99.2 +98.3 +- +- +RTE +76.9 +74.7 +88.3 +88.3 +81.7 +85.0 +Table 4: Accuracy of different language models with +standard prompting and batch prompting using CoT +prompts. Language models are Codex (code-davinci- +002), GPT-3 (text-davinci-003), and ChatGPT. Batch +prompting can be applied well on different LLMs show- +ing similar or better performance. +answers without intermediate steps) and program- +based, (i.e., prompt the LLM to generate programs +to answer the question). For the program-based +methods, we adopt Binder (Cheng et al., 2022) +on WikiTQ and Program-of-Thought (Chen et al., +2022, PoT) on GSM8K and SVAMP. +As seen in Table 3, both end-to-end and program- +based methods can benefit from the efficiency of +batch prompting while maintaining similar or even +better performance on the task. This indicates batch +prompting is a drop-in replacement that can be +combined with various reasoning methods under +diverse scenarios. +4.5 +Language Models +We experiment with LLMs other than Codex. For +GPT-3, we use the latest OpenAI version, text- +davinci-003. For ChatGPT3, as no official code- +integrated API is provided yet, we manually test 60 +samples for each dataset in the browser for evalua- +tion (ChatGPT prompt is provided in Appendix C). +Table 4 shows performance from these LLMs. +Both GPT-3 and ChatGPT demonstrate capabil- +ities similar to Codex: batch prompting retains +downstream performance across datasets. As dis- +cussed in Section 3, the token efficiency from batch +prompting should hold for different LLMs, though +the decrease in time may vary depending on the +LLM inference implementation. +3https://chat.openai.com/. +5 +Related Work +Improve In-Context Learning. The impressive +capabilities of large language models (Brown et al., +2020; Chen et al., 2021; Chowdhery et al., 2022, +LLM) have sparked a surge of recent research aim- +ing to enhance in-context learning (ICL) perfor- +mance. Several works propose different reason- +ing methods to prompt LLMs (Wei et al., 2022; +Zhou et al., 2022; Khot et al., 2022), showing great +improvements over directly prompting LLMs to +output answers. Other works (Cheng et al., 2022; +Chen et al., 2022; Gao et al., 2022) generate pro- +grams to solve reasoning tasks. Another line of +work (Liu et al., 2022; Su et al., 2022; Agrawal +et al., 2022) focuses on selecting better in-context +exemplars. This work adds a new dimension to +ICL for large-scale real-world applications: batch +prompting to save budget and time while achieving +good or even better performance. +Efficient Language Generation. +Much recent +work proposed methods for efficient language +generation, including machine translation (Ka- +sai et al., 2020, 2021a,b) and language model- +ing (Katharopoulos et al., 2020; Peng et al., 2021, +2022). Many of them introduce alternative archi- +tectures to the standard transformer to achieve such +efficiency gains. While we share the same moti- +vation towards efficient generation, our method is +a simple modification to recent prompting meth- +ods, and thus it is applicable to any off-the-shelf +language model APIs, such as OpenAI GPT-3 and +ChatGPT, without any additional training or cus- +tomized model hosting. +6 +Conclusion +We present batch prompting, a new way to prompt +LLMs that performs inference on samples in a +batched fashion. +With batch prompting, multi- +ple samples can be handled in one API call so +that the costs of tokens and time can be signif- +icantly reduced. +Extensive experiments on ten +datasets across commonsense QA, arithmetics, and +NLI/NLU show that batch prompting can achieve +better or similar performance compared to standard +prompting, with much lower token and time costs. +We hope our batch prompting method opens a new +avenue for research: exploring ways to achieve ef- +ficient inference for large-scale applications using +widely available language model APIs. + +References +Sweta Agrawal, Chunting Zhou, Mike Lewis, Luke +Zettlemoyer, and Marjan Ghazvininejad. 2022. 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Learning to retrieve prompts for in-context +learning. +Richard Socher, Alex Perelygin, Jean Wu, Jason +Chuang, Christopher D Manning, Andrew Y Ng, +and Christopher Potts. 2013. Recursive deep mod- +els for semantic compositionality over a sentiment +treebank. In Proc. of EMNLP. +Hongjin Su, Jungo Kasai, Chen Henry Wu, Weijia Shi, +Tianlu Wang, Jiayi Xin, Rui Zhang, Mari Ostendorf, +Luke Zettlemoyer, Noah A Smith, and Tao Yu. 2022. +Selective annotation makes language models better +few-shot learners. +Alon Talmor, Jonathan Herzig, Nicholas Lourie, and +Jonathan Berant. 2019. CommonsenseQA: A ques- +tion answering challenge targeting commonsense +knowledge. In Proc. of NAACL. +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob +Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz +Kaiser, and Illia Polosukhin. 2017. Attention is all +you need. In Proc. of NeurIPS. +Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, +Ed Chi, and Denny Zhou. 2022. Self-consistency +improves chain of thought reasoning in language +models. +Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten +Bosma, Ed Chi, Quoc Le, and Denny Zhou. 2022. +Chain of thought prompting elicits reasoning in large +language models. In Proc. of NeurIPS. +Adina Williams, Nikita Nangia, and Samuel Bowman. +2018. A broad-coverage challenge corpus for sen- +tence understanding through inference. In Proc. of +NAACL. +Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, +Nathan Scales, Xuezhi Wang, Dale Schuurmans, +Olivier Bousquet, Quoc Le, and Ed Chi. 2022. +Least-to-most prompting enables complex reasoning +in large language models. + +A +Time Cost Analysis Regarding +Transformer Architecture +In batch prompting, assume there are K in-context +exemplars (C tokens per sample on average), b sam- +ples in a batch to be inference. Standard prompting +is a special case where b = 1. Since most current +LLMs (e.g.,GPT-3, Codex, PaLM) are based on the +Transformer decoder-only architecture, we focus +on the time cost of the auto-regressive decoder. +The plain transformer time complexity for decod- +ing one token is O(n2d), i.e., the time for encoding +the embeddings of input tokens, where n is the +length of input tokens and d is the dimension of +embeddings. With the caching of previous tokens, +the time complexity to decode each of the rest to- +kens is O(nd). We omit d since it is a constant. +Thus, the time of one inference to decode C · b +tokens: +Tencode = (CK)2 +Tdecode = (CK + 1) + . . . (CK + Cb) +T = Tencode + Tdecode +(2) +where Tencode is the time for encoding the input +tokens in the decoder, and Tdecode is the time for +decoding the rest tokens. C can be seen as a con- +stant. One inference time T regarding K and b is: +T = C2K2 + Cb · CK + Cb(Cb + 1) +2 += C2(K2 + bK + b2 +2 ) + Cb +2 +(3) +Thus, increasing b in batch prompting will also in- +crease the time cost of one inference. The influence +of b also increases with its value and is relatively +marginal when b is small, especially when b ≪ K, +which is a common practice (b = 1) in few-shot +in-context learning. +We can see a few examples by setting K =12 (as +in experiments), C =100 with varying b in Table 5 +according to equation 3. +b +Time per inference +1 +1565050 +2 +1700100 +3 +1845150 +4 +2000200 +6 +2340300 +12 +3600600 +Table 5: Time(no unit) per inference with K =12, C = +100 and various b. +Though the numbers are not accurate consider- +ing the constant coefficients of Big O time com- +plexity, we can learn the decoding time increase +can not be overlooked as b becomes large. We +do not emphasize this part in Section 2.4 because +the overhead and rate limit blocking time of the +OpenAI API make up the most proportion of time +cost, and thus reducing the N times of API calls to +N/b times almost inverse linearly reduce the time +cost (see Figure 2). +However, if the overhead and rate limits are no +longer the bottlenecks, e.g., rate limits are strict +for Codex (code-davinci-002) but not a big issue to +GPT-3 (text-davinci-003), then the decoding time +increase will be non-negligible. +B +More Experimental Results +We list results for all experiments (Tables 6-9). +For the WikiTQ experiment with Binder, the LLM +generation temperature is 0.4 following its paper. +For the other experiments, the temperature is 0. +For all experiments, top_p = 1, sampling_n = 1, +logprobs =1, and stop_tokens =\n\n. Five Ope- +nAI keys are used as a polling pool on rotation to +request the OpenAI API of Codex (the rate limit er- +rors still occur in the experiments and are counted +into time cost since it is a practical issue). If fewer +OpenAI keys are used, there should be more rate +limit errors because the request interval for one key +will be shorter. +C +Prompts +In the section, we list the prompt templates we +use for each dataset (Tables 10-16). We follow +CoT (Wei et al., 2022) to build the prompts of +CommonsenseQA, StrategyQA, GSM8K, SVAMP, +AQuA, +AddSub, +MutliArith. +We +follow +Binder (Cheng et al., 2022) and Program-of- +Thought (Chen et al., 2022) to build the prompts +of WikiTQ, GSM8K (program), and SVAMP (pro- +gram). For RTE, MNLI, SST-5, we design the +prompts ourselves using Chain-of-Thought. For +prompts with fewer than 12 in-context exemplars, +we manually add to 12 samples using samples from +the training set. We show batch prompting prompts +with b = 4 as examples. For different b, we group +the same 12 samples according to b. When using +ChatGPT in Section 4.5, the prompt format differs +from Codex and GPT-3 because its conversational +capability. See Table 17. + +Task +Dataset +Standard Prompting +Batch Prompting +b=2 +3 +4 +6 +Commonsense +CSQA +77.2 +76.0 +77.4 +77.4 +77.2 +StrategyQA +73.3 +69.0 +67.7 +71.0 +67.7 +Arithmetic +GSM8K +55.7 +55.7 +58.7 +55.0 +49.7 +SVAMP +83.7 +81.3 +80.7 +75.7 +76.0 +AQuA +46.1 +41.3 +42.1 +33.1 +37.4 +AddSub +86.6 +84.8 +80.8 +80.3 +68.1 +MultiArith +97.5 +98.0 +98.7 +96.5 +96.3 +NLI/NLU +RTE +76.9 +70.8 +71.8 +74.7 +67.1 +MNLI +65.3 +65.7 +64.7 +65.3 +64.7 +SST-5 +51.3 +48.0 +45.0 +49.7 +48.7 +Table 6: Batch prompting accuracy with different b (the number of samples in batch) compared with standard +prompting on ten datasets. All use Codex (code-davinci-002) as the LLM and Chain-of-Thought as the reasoning +method. +Task +Dataset +Standard Promting +Batch Prompting +b=2 +3 +4 +6 +Commonsense +CSQA +7.37 +3.77 +2.57 +1.96 +1.40 +StrategyQA +7.62 +3.63 +2.85 +2.42 +1.99 +Arithmetic +GSM8K +8.78 +4.55 +3.91 +3.75 +3.61 +SVAMP +7.25 +3.69 +2.46 +2.50 +1.92 +AQuA +7.02 +3.62 +2.60 +2.45 +1.77 +AddSub +7.79 +4.32 +2.41 +1.58 +1.45 +MultiArith +6.80 +3.56 +2.51 +1.89 +1.38 +NLI/NLU +RTE +6.50 +4.56 +2.73 +2.40 +1.29 +MNLI +7.11 +3.78 +2.54 +2.22 +1.32 +SST-5 +7.42 +3.23 +2.69 +2.22 +1.18 +Table 7: Batch prompting time per sample with different b (the number of samples in batch) compared with +standard prompting on ten datasets. All use Codex (code-davinci-002) as the LLM and Chain-of-Thought as the +reasoning method. +Table Input +Standard Prompting +Batch Prompting +b=2 +3 +4 +6 +Schema(Simple) +45.7 +41.7 +42.0 +40.0 +41.3 +Schema +54.3 +50.7 +48.7 +48.7 +47.3 +Schema(3 table rows) +60.3 +51.3 +46.3 +50.3 +38.0 +Table 8: Accuracy on WikiTQ of various table input strategies and b (number of samples in batch) using +Binder (Cheng et al., 2022) to generate programs with Codex (code-davinci-002). +Dataset +Standard Prompting +Batch Prompting +b=2 +3 +4 +6 +GSM8K +72.7 +66.3 +70.7 +73.0 +51.5 +SVAMP +86.0 +86.3 +83.0 +80.7 +84.3 +Table 9: Accuracy on GSM8K and SVAMP with varying b (number of samples in batch) using Program-of- +Thought (Chen et al., 2022) to generate programs with Codex (code-davinci-002). + +CommonsenseQA Prompt +Q[1]: What do people use to absorb extra ink from a fountain pen? +Answer Choices[1]: (a) shirt pocket (b) calligrapher’s hand (c) inkwell (d) desk drawer (e) blotter +Q[2]: What home entertainment equipment requires cable? +Answer Choices[2]: (a) radio shack (b) substation (c) television (d) cabinet +Q[3]: The fox walked from the city into the forest, what was it looking for? +Answer Choices[3]: (a) pretty flowers (b) hen house (c) natural habitat (d) storybook +Q[4]: Sammy wanted to go to where the people were. Where might he go? +Answer Choices[4]: (a) populated areas (b) race track (c) desert (d) apartment (e) roadblock +A[1]: The answer must be an item that can absorb ink. Of the above choices, only blotters are used to +absorb ink. So the answer is (e). +A[2]: The answer must require cable. Of the above choices, only television requires cable. So the answer +is (c). +A[3]: The answer must be something in the forest. Of the above choices, only natural habitat is in the forest. +So the answer is (b). +A[4]: The answer must be a place with a lot of people. Of the above choices, only populated areas have a +lot of people. So the answer is (a). +Q[1]: Where do you put your grapes just before checking out? +Answer Choices[1]: (a) mouth (b) grocery cart (c)supermarket (d) fruit basket (e) fruit market +Q[2]: Google Maps and other highway and street GPS services have replaced what? +Answer Choices[2]: (a) united states (b) mexico (c) countryside (d) atlas +Q[3]: Before getting a divorce, what did the wife feel who was doing all the work? +Answer Choices[3]: (a) harder (b) anguish (c) bitterness (d) tears (e) sadness +Q[4]: James went to the tennis court that was located in his home what? +Answer Choices[4]: (a) country club (b) park (c) michigan (d) sports (e) town +A[1]: The answer should be the place where grocery items are placed before checking out. Of the above +choices, grocery cart makes the most sense for holding grocery items. So the answer is (b). +A[2]: The answer must be something that used to do what Google Maps and GPS services do, which is to +give directions. Of the above choices, only atlases are used to give directions. So the answer is (d). +A[3]: The answer should be the feeling of someone getting divorced who was doing all the work. Of the +above choices, the closest feeling is bitterness. So the answer is (c). +A[4]: The answer must be a place where tennis courts are located. Of the above choices, only home town +has tennis courts. So the answer is (e). +Q[1]: What does you body do when you exercise? +Answer Choices[1]: (a) need for food (b) thirst (c) work out (d) sweating (e) injury +Q[2]: In order to see a story on the big screen what must you do? +Answer Choices[2]: (a) go to movies (b) visualize (c) reading (d) open book (e) sketching a picture +Q[3]: He followed the train tracks hoping to get home, he had gotten lost in the Yooperland where? +Answer Choices[3]: (a) ghetto (b) michigan (c) new york (d) canada (e) train station +Q[4]: What would you get if you want a painting but cannot afford the original? +Answer Choices[4]: (a) reproduction (b) derivative (c) reproduction (d) simile (e) remake +A[1]: The answer must be something that happens when you exercise. Of the above choices, only sweating +happens when you exercise. So the answer is (d). +A[2]: The answer must be something that you do to see a story on the big screen. Of the above choices, +only going to movies makes sense. So the answer is (a). +A[3]: The answer should be a place that relates to Yooperland. Of the above choices, only michigan is +related to Yooperland. So the answer is (b). +A[4]: The answer must be something that is similar to the original. Of the above choices, only +reproduction is similar to the original. So the answer is (a). +Table 10: CommonsenseQA Prompt. + +StrategyQA Prompt +Q[1]: Do hamsters provide food for any animals? +Q[2]: Could Brooke Shields succeed at University of Pennsylvania? +Q[3]: Hydrogen’s atomic number squared exceeds number of Spice Girls? +Q[4]: Is it common to see frost during some college commencements? +A[1]: Hamsters are prey animals. Prey are food for predators. Thus, hamsters provide food for some +animals. So the answer is yes. +A[2]: Brooke Shields went to Princeton University. Princeton University is about as academically +rigorous as the University of Pennsylvania. Thus, Brooke Shields could also succeed at the University of +Pennsylvania. So the answer is yes. +A[3]: Hydrogen has an atomic number of 1. 1 squared is 1. There are 5 Spice Girls. Thus, Hydrogen’s +atomic number squared is less than 5. So the answer is no. +A[4]: College commencement ceremonies can happen in December, May, and June. December is in the +winter, so there can be frost. Thus, there could be frost at some commencements. So the answer is yes. +Q[1]: Could a llama birth twice during War in Vietnam (1945-46)? +Q[2]: Would a pear sink in water? +Q[3]: Can an Arvanite Greek understand some of the Albanian Declaration of Independence? +Q[4]: Can Burundi’s communicate with citizens of New Brunswick? +A[1]: The War in Vietnam was 6 months. The gestation period for a llama is 11 months, which is more than +6 months. Thus, a llama could not give birth twice during the War in Vietnam. So the answer is no. +A[2]: The density of a pear is about 0.6g/cm3, which is less than water. Objects less dense than water +float. Thus, a pear would float. So the answer is no. +A[3]: The Arvanite Greek’s are a major Tosk speaking group of southern Albania. Thus, they can understand +some of the Albanian Declaration of Independence. So the answer is yes. +A[4]: French is one of the official languages of Burundi. Thus, Burundi’s can communicate with citizens of +New Brunswick. So the answer is yes. +Q[1]: Are quadrupeds represented on Chinese calendar? +Q[2]: Can actress Dafne Keen win the Eurovision Song Contest finals in 2020? +Q[3]: Would a student in eleventh grade be unable to run for president of the United States? +Q[4]: Does the judo rank system reach the triple digits? +A[1]: The Chinese calendar has a number of symbols including monkeys, goats, and tigers. Tigers have four +paws and balance themselves by walking on their toes. Thus, quadrupeds are represented on the Chinese +calendar. So the answer is yes. +A[2]: Contestants must be at least 16 years of age to compete in the finals of Eurovision Song Contest. +Dafne Keen is 15 years old in 2020. Thus, Dafne Keen cannot win the Eurovision Song Contest finals in 2020. +So the answer is no. +A[3]: Students in the eleventh grade are typically 16–17 years of age. To serve as president, one must be at +least 35 years old. Thus, a student in eleventh grade would be unable to run for president of the United States. +So the answer is yes. +A[4]: A triple digit number would be equal to at least 100. The judo dan-rank system was capped at 10th +dan after the death of judo’s founder, Kan¯o Jigor¯o. Thus, the judo rank system does not reach the triple +digits. So the answer is no. +Table 11: StrategyQA Prompt. + +GSM8K, SVAMP, AddSub, MultiArith Prompt +Q[1]: There are 15 trees in the grove. Grove workers will plant trees in the grove today. After they are done, +there will be 21 trees. How many trees did the grove workers plant today? +Q[2]: If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot? +Q[3]: Leah had 32 chocolates and her sister had 42. If they ate 35, how many pieces do they have left +in total? +Q[4]: Jason had 20 lollipops. He gave Denny some lollipops. Now Jason has 12 lollipops. How many lollipops +did Jason give to Denny? +A[1]: There are 15 trees originally. Then there were 21 trees after some more were planted. So there must have +been 21 - 15 = 6. The answer is 6. +A[2]: There are originally 3 cars. 2 more cars arrive. 3 + 2 = 5. The answer is 5. +A[3]: Originally, Leah had 32 chocolates. Her sister had 42. So in total they had 32 + 42 = 74. After eating 35, +they had 74 - 35 = 39. The answer is 39. +A[4]: Jason started with 20 lollipops. Then he had 12 after giving some to Denny. So he gave Denny 20 - 12 += 8. The answer is 8. +Q[1]: Shawn has five toys. For Christmas, he got two toys each from his mom and dad. How many toys does he +have now? +Q[2]: There were nine computers in the server room. Five more computers were installed each day, from monday +to thursday. How many computers are now in the server room? +Q[3]: Michael had 58 golf balls. On tuesday, he lost 23 golf balls. On wednesday, he lost 2 more. How many golf +balls did he have at the end of wednesday? +Q[4]: Olivia has $23. She bought five bagels for $3 each. How much money does she have left? +A[1]: Shawn started with 5 toys. If he got 2 toys each from his mom and dad, then that is 4 more toys. 5 + 4 = 9. +The answer is 9. +A[2]: There were originally 9 computers. For each of 4 days, 5 more computers were added. So 5 * 4 = 20 +computers were added. 9 + 20 is 29. The answer is 29. +A[3]: Michael started with 58 golf balls. After losing 23 on tuesday, he had 58 - 23 = 35. After losing 2 more, he had +35 - 2 = 33 golf balls. The answer is 33. +A[4]: Olivia had 23 dollars. 5 bagels for 3 dollars each will be 5 x 3 = 15 dollars. So she has 23 - 15 dollars left. +23 - 15 is 8. The answer is 8. +Q[1]: A garden produced 237 potatoes, 60 fewer cucumbers and twice as many peppers than the cucumbers. How +many vegetables did the garden produce? +Q[2]: John’s cow weighs 400 pounds. It increased its weight to 1.5 times its starting weight. He is able to sell the cow +for $3 per pound. How much more is it worth after gaining the weight? +Q[3]: John writes 20 pages a day. How long will it take him to write 3 books that are 400 pages each? +Q[4]: James has a rainwater collection barrel. For each inch of rain he collects 15 gallons. On Monday it rained 4 inches +and on Tuesday it rained 3 inches. He can sell water for $1.2 per gallon. How much money did he make from selling +all the water? +A[1]: The garden produced 237 - 60 = 177 cucumbers. The garden produced 177 * 2 = 354 peppers. The garden +produced 237 + 177 + 354 = 768 vegetables. The answer is 768. +A[2]: The cow initially weighs 400 * 1.5 = 600 pounds. So it gained 600 - 400 = 200 pounds. It is worth 200 * 3 = 600 +dollars more. The answer is 600. +A[3]: He wants to write 3 * 400 = 1200 pages. So it will take him 1200 / 20= 60 days The answer is 60. +A[4]: It rained 3 + 4 = 7 inches So he collected 7 * 15 = 105 gallons So he makes 105 * 1.2 = 126 from selling the +water. The answer is 126. +Table 12: GSM8K, SVAMP, AddSub, MultiArith Prompt. + +AQuA Prompt +Q[1]: John found that the average of 15 numbers is 40. If 10 is added to each number then the mean of the +numbers is? Answer Choices[1]: (a) 50 (b) 45 (c) 65 (d) 78 (e) 64 +Q[2]: If a / b = 3/4 and 8a + 5b = 22,then find the value of a. +Answer Choices[2]: (a) 1/2 (b) 3/2 (c) 5/2 (d) 4/2 (e) 7/2 +Q[3]: A person is traveling at 20 km/hr and reached his destiny in 2.5 hr then find the distance? +Answer Choices[3]: (a) 53 km (b) 55 km (c) 52 km (d) 60 km (e) 50 km +Q[4]: How many keystrokes are needed to type the numbers from 1 to 500? +Answer Choices[4]: (a) 1156 (b) 1392 (c) 1480 (d) 1562 (e) 1788 +A[1]: If 10 is added to each number, then the mean of the numbers also increases by 10. So the new mean +would be 50. The answer is (a). +A[2]: If a / b = 3/4, then b = 4a / 3. So 8a + 5(4a / 3) = 22. This simplifies to 8a + 20a / 3 = 22, which means +44a / 3 = 22. So a is equal to 3/2. The answer is (b). +A[3]: The distance that the person traveled would have been 20 km/hr * 2.5 hrs = 50 km. The answer is (e). +A[4]: There are 9 one-digit numbers from 1 to 9. There are 90 two-digit numbers from 10 to 99. There are +401 three-digit numbers from 100 to 500. 9 + 90(2) + 401(3) = 1392. The answer is (b). +Q[1]: A number X equals 80% of the average of 5, 7, 14 and a number Y. If the average of X and Y is 26, the +value of Y is? +Answer Choices[1]: (a) 13 (b) 26 (c) 39 (d)36 (e) None of these +Q[2]: A shopkeeper gave an additional 20 per cent concession on the reduced price after giving 30 per +cent standard concession on an article. If Arun bought that article for 1,120, what was the original price? +Answer Choices[2]: (a) 3,000 (b) 4,000 (c) 2,400 (d) 2,000 (e) None of these +Q[3]: A and B invests Rs.3000 and Rs.7000 respectively in a business. If A doubles his capital after 6 months. +In what ratio should A and B divide that year’s profit? +Answer Choices[3]: (a) 9:6 (b) 9:8 (c) 9:14 (d) 9:9 (e) 9:5 +Q[4]: The angle between two hands at 3.45 is? +Answer Choices[4]: (a) 110 degree (b) 115 degree (c) 112 1/2 degree (d) 117 degree (e) 157 1/2 degree +A[1]: Average of 5, 7, 14 and Y = (5 + 7 + 14 + Y) / 4. Therefore, X = 80% of (5 + 7 + 14 + y) / 4 = (80/100) +x (26 + Y)/4 => X = (26 + Y)/5, i.e., 5X - Y = 26. Also, (X + Y) / 2 = 26. Thus, (26 + Y) / 5 + Y = 52, then Y += 39. The answer is (c). +A[2]: The total discount should be (1 - 0.3) * (1 - 0.2) = 0.56. Thus, the original price should be 1120 / 0.56 += 2000. The answer is (d). +A[3]: The ratio should be (3 * 6 + 6 * 6): (7 * 12) = 54:84. It simplifies to 9:14. The answer is (c). +A[4]: The hour hand is (45/60) * (360/12) = 22.5 degree from 3 o’clock. So the angle between the hour hand and +the minute hand is (9-3) * (360/12) - 22.5 = 157.5. The answer is (e). +Q[1]: Find the sum of first 30 natural numbers. +Answer Choices[1]: (a) 470 (b) 468 (c) 465 (d) 463 (e) 487 +Q[2]: What will come in place of the x in the following Number series? 46080, 3840, ?, 48, 8, 2, 1. +Answer Choices[2]: (a) 1 (b) 384 (c) 5 (d) 7 (e) 9 +Q[3]: A password of a computer used two digits where they are from 0 and 9. What is the probability that the +password solely consists of prime numbers and zero? +Answer Choices[3]: (a) 1/32 (b) 1/16 (c) 1/8 (d) 2/5 (e) 1/4 +Q[4]: If k3 is divisible by 120, what is the least possible value of integer k? +Answer Choices[4]: (a) 12 (b) 30 (c) 60 (d) 90 (e) 120 +A[1]: The sum of first 30 natural numbers is 30 * (30 + 1) / 2 = 465. The answer is (c). +A[2]: The ratio of the numbers is 10:8:6:4:2:1. So the next number should be 384. The answer is (b). +A[3]: 0, 2, 3, 5, 7 are five prime digits(including zero). So there are 5 * 5 = 25 two-digit numbers with +only prime numbers and zero. The probability is 25/100 = 1/4. The answer is (e). +A[4]: 120 can be factored as 2 * 2 * 2 * 3 * 5. So the least k be 2 * 3 * 5 = 30. The answer is (b). +Table 13: AQuA Prompt. + +RTE Prompt +Premise[1]: No Weapons of Mass Destruction Found in Iraq Yet. +Hypothesis[1]: Weapons of Mass Destruction Found in Iraq. +Premise[2]: A place of sorrow, after Pope John Paul II died, became a place of celebration, as Roman +Catholic faithful gathered in downtown Chicago to mark the installation of new Pope Benedict XVI. +Hypothesis[2]: Pope Benedict XVI is the new leader of the Roman Catholic Church. +Premise[3]: Libya’s case against Britain and the US concerns the dispute over their demand +for extradition of Libyans charged with blowing up a Pan Am jet over Lockerbie in 1988. +Hypothesis[3]: One case involved the extradition of Libyan suspects in the Pan Am Lockerbie bombing. +Premise[4]: Argentina sought help from Britain on its privatization program and encouraged British +investment. +Hypothesis[4]: Argentina sought UK expertise on privatization and agriculture. +Answer[1]: No Weapons of Mass Destruction Found, which contradicts the hypothesis. So the +answer is False. +Answer[2]: As Roman Catholic faithful gathered in downtown Chicago to mark the installation of new +Pope Benedict XVI. So the answer is True. +Answer[3]: Libya’s case suspects in the Pan Am Lockerbie bombing. So the answer is True. +Answer[4]: Argentina sought help from Britain on its privatization program, not agriculture, which +contradicts the hypothesis. So the answer is False. +Premise[1]: Startling new research into mobile phones claims they may reduce a man’s sperm count by +up to 30%. +Hypothesis[1]: Male fertility may be affected by use of a mobile phones. +Premise[2]: It rewrites the rules of global trade, established by the General Agreement on Tariffs and +Trade, or GATT, in 1947, and modified in multiple rounds of negotiations since then. +Hypothesis[2]: GATT was formed in 1947. +Premise[3]: The cost of the consumer of the United States fell in June. +Hypothesis[3]: U.S. consumer spending dived in June. +Premise[4]: Israeli Prime Minister Ariel Sharon has said that Mahmoud Abbas is a man that Israel can do +business with. Hypothesis[4]: Palestinian leader, Mahmoud Abbas, may be someone Israel can talk with. +Answer[1]: New research claims mobile phones reduce a man’s sperm count, i.e., affects male fertility. So +the answer is True. +Answer[2]: GATT is rewritten in 1947, not formed in 1947, which contradicts the hypothesis. So the answer +is False. +Answer[3]: The consumer cost fell in June, not the spending, which contradicts the hypothesis. So the +answer is False. +Answer[4]: Mahmoud Abbas is a man that Israel can do business with, i.e., he may be someone Israel +can talk with. So the answer is True. +Premise[1]: In October, however, amid rising tensions between the government and opposition groups, +a car bomb seriously injured an opposition politician and killed his driver, in Beirut. +Hypothesis[1]: A member of the opposition was injured in a car bomb attack in Beirut. +Premise[2]: Ruth’s 1927 single season record of 60 home runs stood unsurpassed until Roger Maris hit 61 in 1961. +Hypothesis[2]: Babe Ruth hit 60 home runs in his lifetime. +Premise[3]: The German technology was employed to build Shanghai’s existing maglev line, the first +in the world to be used commercially. +Hypothesis[3]: Maglev is commercially used. +Premise[4]: Twelve of Jupiter’s moons are relatively small and seem to have been more likely captured +than to have been formed in orbit around Jupiter. +Hypothesis[4]: Jupiter has Twelve moons. +Answer[1]: A car bomb seriously injured an opposition politician in Beirut. So the answer the True. +Answer[2]: Babe Ruth hit 60 home runs in a single season, not his lifetime, which contradicts the hypothesis. +So the answer is False. +Answer[3]: The German technology was employed to build Shanghai’s existing maglev line, i.e., Maglev +is commercially used. So the answer is True. +Answer[4]: Twelve of Jupiter’s moons are relatively small, not Jupiter has Twelve moons, which contradicts +the hypothesis. So the answer is False. +Table 14: RTE Prompt. + +MNLI Prompt +Premise[1]: Conceptually cream skimming has two basic dimensions - product and geography. +Hypothesis[1]: Product and geography are what make cream skimming work. +Premise[2]: One of our number will carry out your instructions minutely. +Hypothesis[2]: A member of my team will execute your orders with immense precision. +Premise[3]: Analyzing Postal Service accounts for depreciation, fuel, and maintenance for +city delivery carriers, we have estimated the average city delivery vehicle cost per route. +Hypotheis[3]: Driving cost estimates can be averaged with sufficient data. +Premise[4]: Consider the United States Postal Service. +Hypothesis[4]: Forget the United States Postal Service. +Answer[1]: The answer is Neutral. +Answer[2]: The answer is True. +Answer[3]: The answer is Neutral. +Answer[4]: The answer is False. +Premise[1]: Take a remarkable statistic that Shesol cites but lets pass relatively unexamined. +Hypothesis[1]: They had data that was very relevant but under used. +Premise[2]: The man on the ground thinks for a moment and yells back, You must work in management. +Hypothesis[2]: There was no one on the ground, man or woman. +Premise[3]: Hello, Ben. +Hypothesis[3]: I ignored Ben. +Premise[4]: How can you prove it? +Hypothesis[4]: Can you tell me how to prove it? +Answer[1]: The answer is True. +Answer[2]: The answer is False. +Answer[3]: The answer is False. +Answer[4]: The answer is True. +Premise[1]: In the midst of this amazing amalgam of cultures is a passion for continuity. +Hypothesis[1]: A passion for continuity is not the most important of these cultures. +Premise[2]: Poirot, I exclaimed, with relief, and seizing him by both hands, I dragged him into the room. +Hypothesis[2]: Poirot was now back and I was sorry that he would take over what I now considered +my own investigation. +Premise[3]: There’s a uh a couple called um oh i’m going to forgot his name now uh Dirkson. +Hypothesis[3]: I can’t remember their name. +Premise[4]: It’s not that the questions they asked weren’t interesting or legitimate (though most did fall +under the category of already asked and answered). +Hypothesis[4]: All of the questions were interesting according to a focus group consulted on the subject. +Answer[1]: The answer is Neutral. +Answer[2]: The answer is False. +Answer[3]: The answer is True. +Answer[4]: The answer is Neutral. +Table 15: MNLI Prompt. + +SST-5 Prompt +Q[1]: a stirring , funny and finally transporting re-imagining of beauty and the beast and 1930s +horror films. +Q[2]: they presume their audience wo n’t sit still for a sociology lesson, however entertainingly +presented, so they trot out the conventional science-fiction elements of bug-eyed monsters and +futuristic women in skimpy clothes. +Q[3]: um , no.. +Q[4]: jonathan parker’s bartleby should have been the be-all-end-all of the modern-office anomie films. +A[1]: The tone is very positive. +A[2]: The tone is negative. +A[3]: The tone is neutral. +A[4]: The tone is positive. +Q[1]: lacks the inspiration of the original and has a bloated plot that stretches the running time +about 10 minutes past a child’s interest and an adult’s patience. +Q[2]: the santa clause 2 proves itself a more streamlined and thought out encounter than the original +could ever have hoped to be. +Q[3]: you might say tykwer has done all that heaven allows, if you wanted to make as anti-kieslowski +a pun as possible. +Q[4]: otto-sallies has a real filmmaker’s eye. +A[1]: The tone is very negative. +A[2]: The tone is positive. +A[3]: The tone is neutral. +A[4]: The tone is positive. +Q[1]: with a confrontational stance, todd solondz takes aim on political correctness and suburban +families. +Q[2]: verall , cletis tout is a winning comedy that excites the imagination and tickles the funny bone. +Q[3]: with its parade of almost perpetually wasted characters ... margarita feels like a hazy high +that takes too long to shake. +Q[4]: an ugly-duckling tale so hideously and clumsily told it feels accidental. +A[1]: The tone is neutral. +A[2]: The tone is very positive. +A[3]: The tone is negative. +A[4]: The tone is very negative. +Table 16: SST-5 Prompt. + +ChatGPT Prompt for GSM8K +1st round input: +You can answer questions. I will give you a few batches of exemplars in format Q[idx]:question, A[idx]:answer. +Say "continue" to get the next input. Finally, a batch of test samples with only contexts Q[idx]:question are input. +Answer the test samples in format A[idx]:answer. Say "okay" if you understand. +1st round output: +okay. +2nd round input: +Q[1]: There are 15 trees in the grove. Grove workers will plant trees in the grove today. After they are done, +there will be 21 trees. How many trees did the grove workers plant today? +Q[2]: If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot? +Q[3]: Leah had 32 chocolates and her sister had 42. If they ate 35, how many pieces do they have left +in total? +Q[4]: Jason had 20 lollipops. He gave Denny some lollipops. Now Jason has 12 lollipops. How many lollipops +did Jason give to Denny? +A[1]: There are 15 trees originally. Then there were 21 trees after some more were planted. So there must have +been 21 - 15 = 6. The answer is 6. +A[2]: There are originally 3 cars. 2 more cars arrive. 3 + 2 = 5. The answer is 5. +A[3]: Originally, Leah had 32 chocolates. Her sister had 42. So in total they had 32 + 42 = 74. After eating 35, +they had 74 - 35 = 39. The answer is 39. +A[4]: Jason started with 20 lollipops. Then he had 12 after giving some to Denny. So he gave Denny 20 - 12 += 8. The answer is 8. +2nd round output: +continue. +3rd round input: +Q[1]: Shawn has five toys. For Christmas, he got two toys each from his mom and dad. How many toys does he +have now? +Q[2]: There were nine computers in the server room. Five more computers were installed each day, from monday +to thursday. How many computers are now in the server room? +Q[3]: Michael had 58 golf balls. On tuesday, he lost 23 golf balls. On wednesday, he lost 2 more. How many golf +balls did he have at the end of wednesday? +Q[4]: Olivia has $23. She bought five bagels for $3 each. How much money does she have left? +A[1]: Shawn started with 5 toys. If he got 2 toys each from his mom and dad, then that is 4 more toys. 5 + 4 = 9. +The answer is 9. +A[2]: There were originally 9 computers. For each of 4 days, 5 more computers were added. So 5 * 4 = 20 +computers were added. 9 + 20 is 29. The answer is 29. +A[3]: Michael started with 58 golf balls. After losing 23 on tuesday, he had 58 - 23 = 35. After losing 2 more, he had +35 - 2 = 33 golf balls. The answer is 33. +A[4]: Olivia had 23 dollars. 5 bagels for 3 dollars each will be 5 x 3 = 15 dollars. So she has 23 - 15 dollars left. +23 - 15 is 8. The answer is 8. +3rd round output: +continue. +4th round input: +Q[1]: A garden produced 237 potatoes, 60 fewer cucumbers and twice as many peppers than the cucumbers. How +many vegetables did the garden produce? +Q[2]: John’s cow weighs 400 pounds. It increased its weight to 1.5 times its starting weight. He is able to sell the cow +for $3 per pound. How much more is it worth after gaining the weight? +Q[3]: John writes 20 pages a day. How long will it take him to write 3 books that are 400 pages each? +Q[4]: James has a rainwater collection barrel. For each inch of rain he collects 15 gallons. On Monday it rained 4 inches +and on Tuesday it rained 3 inches. He can sell water for $1.2 per gallon. How much money did he make from selling +all the water? +A[1]: The garden produced 237 - 60 = 177 cucumbers. The garden produced 177 * 2 = 354 peppers. The garden +produced 237 + 177 + 354 = 768 vegetables. The answer is 768. +A[2]: The cow initially weighs 400 * 1.5 = 600 pounds. So it gained 600 - 400 = 200 pounds. It is worth 200 * 3 = 600 +dollars more. The answer is 600. +A[3]: He wants to write 3 * 400 = 1200 pages. So it will take him 1200 / 20= 60 days The answer is 60. +A[4]: It rained 3 + 4 = 7 inches So he collected 7 * 15 = 105 gallons So he makes 105 * 1.2 = 126 from selling the +water. The answer is 126. +4th round output: +continue. +Test round input: +{four test questions} +Test round output: +{four test answers.} +Table 17: An example ChatGPT prompt we use for batch prompting. Specifically, the task instruction is input in +the first round of the conversation. In the next a few rounds, one batch of in-context exemplars is input in one +round. In the final round, test samples’ contexts are input and ChatGPT outputs the answers. + diff --git a/Y9FAT4oBgHgl3EQf3h6y/content/tmp_files/load_file.txt b/Y9FAT4oBgHgl3EQf3h6y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99cee113a730c881ff6eecf84510e17769144c38 --- /dev/null +++ b/Y9FAT4oBgHgl3EQf3h6y/content/tmp_files/load_file.txt @@ -0,0 +1,1347 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf,len=1346 +page_content='Batch Prompting: Efficient Inference with Large Language Model APIs Zhoujun Cheng Shanghai Jiao Tong University blankcheng@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='cn Jungo Kasai University of Washington jkasai@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='edu Tao Yu University of Hong Kong tyu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='hk Abstract Performing inference on hundreds of thou- sands of samples with large language mod- els (LLMs) can be computationally and finan- cially costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We propose batch prompting, a simple alternative prompting approach that en- ables the LLM to run inference in batches, in- stead of one sample at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Our method reduces both token and time costs while re- taining downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We theo- retically demonstrate that under a few-shot in- context learning setting, the inference costs de- crease almost inverse linearly with the num- ber of samples in each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We extensively validate the effectiveness of batch prompting on ten datasets across commonsense QA, arith- metic reasoning, and NLI/NLU: batch prompt- ing significantly (up to 5× with six samples in batch) reduces the LLM (Codex) inference token and time costs while achieving better or comparable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Our analysis shows that the number of samples in each batch and the complexity of tasks affect its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Further, batch prompting can be applied across different LLMs and reasoning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Our code will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='com/ HKUNLP/batch-prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 1 Introduction Large language models (LLMs) have shown their strong capabilities under zero/few-shot settings with in-context learning (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Much recent work has made progress in in-context learning by eliciting reasoning steps (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022), selecting representative in-context exemplars (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022), and designing prompt templates (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Bach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Using LLMs can be costly in terms of token and time usage, especially when many LLM calls are needed, such as when benchmarking a large Standard Prompting Batch Prompting # K-shot in-context exemplars Q: {question} A: {answer} Q: {question} A: {answer} … # One sample to inference Q: Ali had $21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Leila gave him half of her $100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much does Ali have now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' ----------------------------------------------- # Response A: Leila gave 100/2=50 to Ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Ali now has $21+$50 = $71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' # K-shot in-context exemplars in K/b batches Q[1]: {question} Q[2]: {question} A[1]: {answer} A[2]: {answer} … # b samples in a batch to inference Q[1]: Ali had $21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Leila gave him half of her $100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much does Ali have now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: A robe takes 2 bolts of blue fiber and half that white fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many bolts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' ----------------------------------------------- # Responses to a batch A[1]: Leila gave 100/2=50 to Ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Ali now has $21+$50 = $71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: It takes 2/2=1 bolt of white fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The total amount is 2+1=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' b(=2) samples in one batch Figure 1: Illustration of batch prompting compared with standard prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Batch prompting groups mul- tiple samples in one batch (b=2 in the figure) and lets the LLM generate multiple responses (highlighted in yellow) for the batch in inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' dataset or addressing a high volume of customer inquiries for businesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For example, the widely- adopted OpenAI API service1 of LLMs requires about $400 and 10 hours to perform inference on 10K samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 If the rate limits of maximum API requests per minute are also considered, the costs will be even higher, preventing users from building massive LLM applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We propose batch prompting, an alternative ap- proach for prompting LLMs, which allows the 1https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='com/api/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2This assumes each LLM call consumes 2, 000 tokens, including both the input prompt tokens and generated tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='08721v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='CL] 19 Jan 2023 model to perform inference on multiple samples at once, instead of one sample at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' This re- duces token and time costs while still retaining downstream performance, without any change in APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As shown in Figure 1, standard prompting generates a response (answer) to one sample at a time, which takes N inference runs of an LLM for a test set of size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For our batch prompting, on the other hand, an LLM generates responses to b samples in a single inference run and only takes N/b runs for the same N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We first demonstrate theoretically that under the few-shot in-context learning setting, most to- kens consumed during the API call are the few- shot exemplars, and only a small portion of token budgets are used for the particular inference sam- ple(s) (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Therefore, increasing the num- ber of samples b in a batch of batch prompting reduces the token and time costs in an inverse lin- ear fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We extensively validate the effective- ness of batch prompting on ten diverse downstream datasets across commonsense QA, arithmetics, and NLI/NLU using Codex, a strong variant of GPT-3 finetuned on code data (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Batch prompt- ing significantly decreases the tokens and run time of using LLMs while achieving comparable or even better performance on all ten datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In further analysis (Section 4), we find the number of sam- ples in batch and the complexity of tasks affect its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Moreover, we show that batch prompting works well across different LLMs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', Codex, ChatGPT, and GPT-3) and reasoning meth- ods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', end-to-end, Chain-of-Thought, and code generation), suggesting that batch prompting is an efficient drop-in substitute for conventional prompt- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2 Approach We first introduce batch prompting, an efficient alternative to standard prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We then com- pare the token and time costs of batch and stan- dard prompting, demonstrating the efficiency of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 Problem Setup The conventional paradigm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', standard prompt- ing in Figure 1) to prompt LLMs for in-context learning is as follows: K in-context few-shot ex- emplars with both a context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', question) and an output (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', answer) are selected to build the input prompt, one test sample with context only is appended at the end of the prompt, and the LLM is used to generate the response for the test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In this paper, we focus on a realistic scenario with N test samples, which is common when benchmarking on a dataset or handling a large vol- ume of customer requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In this case, it takes N separate calls of the LLM inference under the standard prompting paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 Batch Prompting Batch prompting enables the LLM to generate re- sponses for multiple samples in one batch in a sin- gle inference run, so that it reduces the LLM infer- ence time from N to N/b, where b is the number of samples in one batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Specifically, as shown in Figure 1, our prompt groups the K in-context exemplars into K/b batches with b exemplars each as demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In every batch, demonstration contexts are arranged in a specific order at the be- ginning, with their corresponding outputs placed in the same order afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Then, b test sam- ple contexts are grouped together at the end of the input prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In this way, the LLM learns from the in-context demonstrations and generates cor- responding responses for the entire batch of test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We add a position identifier “[index]” within each batch to 1) assist the LLM with iden- tifying the order correspondence of input contexts and generated responses and 2) ease the process of parsing the generated responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 Token Cost The costs of one LLM call scale linearly with the number of tokens, including both the input prompt tokens (few-shot and instruction) and generated tokens (according to, for example, OpenAI’s pric- ing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Most tokens are consumed by the prompt tokens in standard prompting because the num- ber of prompt tokens is usually far more than the number of generated tokens so that the LLM can better learn from in-context exemplar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, the larger the portion of tokens spent on generated to- kens, the more economical the total cost is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We define token efficiency η as the portion of tokens spent on generated tokens in one LLM call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For standard prompting and batch prompting (the instruction tokens are omitted if any for brevity): ηstandard = 1 K + 1 ηbatch = b K + b (1) (a) Token-CommonsenseQA (b) Token-GSM8K (c) Token-RTE (d) Time-CommonsenseQA (e) Time-GSM8K (f) Time-RTE Figure 2: Token and time costs per sample on three datasets for illustrations (other datasets show similar trends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Batch prompting significantly lowers both token and time costs as the number of samples in each batch increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' When K ≫ 1 and b < K, ηbatch scales almost inverse linearly with b, and thus increasing b of batch prompting can greatly reduce token costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 Time Cost Intuitively, batch prompting reduces the inference time by decreasing the number of API calls from N to N/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If considering the time of Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2017) decoding, the cost will in- crease with b in batch prompting since longer re- sponses will be generated compared with standard prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We give a detailed derivation regarding this Transformer architecture perspective in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' However, as most end-users are accustomed to and only have access to LLM API services, this part of time cost is marginal (observed in main experiments), relative to the overhead of API call and request rate limits per minute set by a company, such as OpenAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Besides, cases may occur when network connections are unstable or slow, and the users seek to finish a task with as few LLM calls as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Therefore, in practice, reducing the number of calls from N to N/b with batch prompting can essentially lower the time costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Note that when the API call overhead and rate limits are no longer the major bottlenecks of time costs in the future, then the increased decoding time to generate longer sequences discussed in Appendix A cannot be over- looked, and the time reduction of batch prompt- ing will not be as pronounced (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', the latest text- davinci-003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Since LLM infrastructure/services can change over time, token costs are easier to measure in experiments than time costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3 Experiments We extensively evaluate batch prompting across ten diverse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Our results suggest that batch prompting can achieve at most 5× token and time efficiency (with six samples in batches) improve- ment with similar or even better downstream per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 Datasets We evaluate batch prompting on ten datasets across commonsense question answering, arith- metic reasoning, and natural language under- standing/inference: CommonsenseQA (Talmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2019), StrategyQA (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021), GSM8K (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021), SVAMP (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021), AQuA (Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2017), AddSub (Hos- seini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2014), MultiArith (Roy and Roth, 2015), RTE (Bentivogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2009), MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2018), and SST-5 (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For CommonsenseQA, AQuA, AddSub, MultiArith, and RTE, we evaluate the whole dev/test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='Tokensper ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='#Samplesinbatch8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' rsampl e 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Method per 6 Standard Batch Prompting Time( 5 4- 3 2 3 4 6 #Samplesinbatch9 (s)persample 4 - Method Standard Batch Prompting Time( 3 2 - X 1 2 3 4 6 #SamplesinbatchTask Dataset Standard Batch Commonsense CSQA 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2) StrategyQA 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0(−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3) Arithmetic GSM8K 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7(+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0) SVAMP 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3(−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4) AQuA 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1(−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0) AddSub 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8) MultiArith 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2) NLI/NLU RTE 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7(−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2) MNLI 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4) SST-5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='6) Table 1: Accuracy of standard and batch prompt- ing on ten datasets: CommonsenseQA (Talmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2019), StrategyQA (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021), SVAMP (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021), AQuA (Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2017), AddSub (Hos- seini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2014), MultiArith (Roy and Roth, 2015), RTE (Bentivogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2009), MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2018), and SST-5 (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Batch prompt- ing shows comparable or even better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' the other five datasets, we evaluate the first 300 test samples considering the costs of LLM APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 Experimental Setups We use OpenAI Codex (code-davinci-002) as the LLM in our experiments (in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5, different LLMs are discussed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Codex is currently provided for free, but the token consumption strategy is the same as the other LLMs, ensuring that the token costs in experiments are general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The decoding temperature is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For each dataset, we manu- ally select 12-shot samples from the training set as in-context exemplars, with Chain-of-Thought (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022, CoT) reasoning steps in the answers (in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4, other reasoning methods beyond CoT are discussed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We choose 12 exemplars because 12 is the least common multiple of 2, 3, 4, 6, and thus it is easy to analyze the effects of grouping them into batches of 2, 3, 4, 6 samples in our ab- lation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' More experimental details and full results are listed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 Results Figure 2 compares the token and time costs of standard and batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As shown, batch prompting substantially (up to 5× with 6 samples in each batch) reduces both the token and time costs of standard prompting with Codex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Further, the de- crease of costs scales almost inverse linearly with the number of samples in each batch, verifying our analysis in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Note the time costs include the API call overhead and rate limit blocks, which exist in the commonly-used OpenAI services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For LLM services where these are not bottlenecks Figure 3: Accuracy over varying numbers of batch sam- ples b on five datasets using batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The per- formance decreases with larger b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' the latest GPT-3 (text-davinci-003), the decoding time increase from larger b should not be overlooked as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As the LLM infrastructure can change anytime, the token efficiency improvement is easier to compare than time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' the token reduction in Figure 2 should hold for any LLM over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 1 shows that batch prompting (with the best b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', the number of samples in each batch) performs comparably or even better than standard prompting over all ten datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We thus recom- mend that LLM users consider applying batch prompting to save money and time while main- taining good performance in realistic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4 Analysis In this section, we first analyze factors that may affect the performance of batch prompting and explore the tradeoff of balancing the costs and downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We further demon- strate batch prompting can be applied to different LLMs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', GPT-3, and ChatGPT) and prompting methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', end-to-end, and code generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 Number of Batch Samples Figure 3 shows how the number of samples in each batch b affects the benchmark performance in batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Firstly, the performance generally de- creases as b becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' When b = 6, a large drop is seen across four of these five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In- terestingly, however, the best performance is not always achieved when b=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Setting b = 3 or 4 usu- ally achieves good performance while saving more tokens and time than smaller b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The reduction of time/token costs diminishes when b becomes larger, indicating that setting b < 6 (given 12 shots in- context exemplars in experiments) tends to provide 06 Dataset CSQA GSM8K SVAMP 80 AddSub RTE Accuracy 70 60 50 40 2 4 6 #Samples in batchDataset Random Similar Diverse CSQA 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 GSM8K 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 SVAMP 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 AddSub 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 RTE 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 Table 2: Accuracy from various batching methods on five representative datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Similarity or diversity- based methods do not achieve performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' a good tradeoff between the costs and downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 Selection of Batch Samples Here we examine whether the selection of samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' how samples are grouped into batches, will affect the performance of batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We study two widely-adopted sample selection meth- ods in in-context learning when grouping the test samples: grouping more similar (Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) or more diverse (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) samples into batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Specifi- cally, given N test samples, to group similar ones, we use k-means clustering and post-process each cluster into equal size b by moving redundant sam- ples to their closest groups with size < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' To group diverse ones, we apply the vote-k method (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) to iteratively select diverse and representa- tive groups of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As listed in Table 2, both similarity and diversity- based selections do not show improvements over random grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We suspect that the reason may be that both methods assume in-batch samples can benefit from similar or diverse samples presented before them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', samples in the front of the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' However, these earlier samples are not with ground truth outputs and thus may lead to error propaga- tions to the rest of the in-batch samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Devel- oping effective strategies for selecting samples for batch prompting could be a promising area for fu- ture research to further enhance the performance of batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 Complexity of Tasks We further analyze how the complexity of tasks influences the performance of batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Table 1, the largest drop (from 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 to 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1) oc- curs on the AQuA dataset, which is an arithmetic reasoning task in a multi-choice QA format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' One interpretation is that AQuA is more difficult than other datasets with the lowest absolute accuracy 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1%, and thus LLMs are more likely to be dis- Figure 4: Accuracy on WikiTQ of various table input strategies and b (the number of samples in each batch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' This studies how the input length affects batch prompt- ing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' b = 1 means standard prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Average input tokens per table are 24, 58, and 216 to- kens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As the number of batch samples increases, batch prompting suffers in downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' turbed when input contexts are grouped together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We study another aspect of the task that may affect performance: the longer the input contexts, the more substantially batch prompting hurts per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We validate our assumption with Wik- iTQ (Pasupat and Liang, 2015), a challenging Table QA dataset over Wikitables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Tables contain longer input tokens for their multiple rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We experiment with increasing table input lengths: a simplified table schema (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', column names with- out column types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 24 tokens/table), a table schema (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 58 tokens/table), and a table schema with three table rows (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 216 tokens/table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We follow Binder (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) to generate Binder-SQL programs to solve the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As shown in Figure 4, in standard prompting (b = 1), inputting table schemas with three rows dominates QA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' However, it also sees the steepest performance drop when b increases us- ing batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The shorter the input contexts, the steadier the performance with batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' This suggests that long task inputs are more likely to lead to confusion and performance drops when batch prompting is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 Reasoning Methods In our main experiments (Section 3), we used the Chain-of-Thought (CoT) for all ten datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Here we examine whether batch prompting is suitable for other common LLM reasoning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We experiment with two more reasoning methods: end- to-end (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', directly prompt the LLM to output the Table Input Schema(Simple)--24 tokens Schema --58tokens Schema(3 rows)--216 tokens 55 Accuracy 45 35 1 2 3 6 #Samples in batchDataset End-to-end CoT Program Standard Batch Standard Batch Standard Batch CSQA 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 GSM8K 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 SVAMP 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 RTE 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 WikiTQ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 Table 3: Accuracy of different reasoning methods with standard and batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Batch prompting can be applied well with different reasoning methods showing similar or better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Dataset Codex GPT-3 ChatGPT Standard Batch Standard Batch Standard Batch CSQA 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 GSM8K 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 SVAMP 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 AddSub 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 RTE 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 Table 4: Accuracy of different language models with standard prompting and batch prompting using CoT prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Language models are Codex (code-davinci- 002), GPT-3 (text-davinci-003), and ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Batch prompting can be applied well on different LLMs show- ing similar or better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' answers without intermediate steps) and program- based, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', prompt the LLM to generate programs to answer the question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For the program-based methods, we adopt Binder (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) on WikiTQ and Program-of-Thought (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022, PoT) on GSM8K and SVAMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As seen in Table 3, both end-to-end and program- based methods can benefit from the efficiency of batch prompting while maintaining similar or even better performance on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' This indicates batch prompting is a drop-in replacement that can be combined with various reasoning methods under diverse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 Language Models We experiment with LLMs other than Codex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For GPT-3, we use the latest OpenAI version, text- davinci-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For ChatGPT3, as no official code- integrated API is provided yet, we manually test 60 samples for each dataset in the browser for evalua- tion (ChatGPT prompt is provided in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 4 shows performance from these LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Both GPT-3 and ChatGPT demonstrate capabil- ities similar to Codex: batch prompting retains downstream performance across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' As dis- cussed in Section 3, the token efficiency from batch prompting should hold for different LLMs, though the decrease in time may vary depending on the LLM inference implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3https://chat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 5 Related Work Improve In-Context Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The impressive capabilities of large language models (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022, LLM) have sparked a surge of recent research aim- ing to enhance in-context learning (ICL) perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Several works propose different reason- ing methods to prompt LLMs (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022), showing great improvements over directly prompting LLMs to output answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Other works (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) generate pro- grams to solve reasoning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Another line of work (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) focuses on selecting better in-context exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' This work adds a new dimension to ICL for large-scale real-world applications: batch prompting to save budget and time while achieving good or even better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Efficient Language Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Much recent work proposed methods for efficient language generation, including machine translation (Ka- sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2020, 2021a,b) and language model- ing (Katharopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Many of them introduce alternative archi- tectures to the standard transformer to achieve such efficiency gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' While we share the same moti- vation towards efficient generation, our method is a simple modification to recent prompting meth- ods, and thus it is applicable to any off-the-shelf language model APIs, such as OpenAI GPT-3 and ChatGPT, without any additional training or cus- tomized model hosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 6 Conclusion We present batch prompting, a new way to prompt LLMs that performs inference on samples in a batched fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' With batch prompting, multi- ple samples can be handled in one API call so that the costs of tokens and time can be signif- icantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Extensive experiments on ten datasets across commonsense QA, arithmetics, and NLI/NLU show that batch prompting can achieve better or similar performance compared to standard prompting, with much lower token and time costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We hope our batch prompting method opens a new avenue for research: exploring ways to achieve ef- ficient inference for large-scale applications using widely available language model APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' References Sweta Agrawal, Chunting Zhou, Mike Lewis, Luke Zettlemoyer, and Marjan Ghazvininejad.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Luisa Bentivogli, Peter Clark, Ido Dagan, and Danilo Giampiccolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The fifth pascal recognizing tex- tual entailment challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of TAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert- Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Mark Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jerry Tworek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Heewoo Jun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Qiming Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Henrique Ponde de Oliveira Pinto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jared Ka- plan,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, Dan Roth, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Did aristotle use a laptop?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' a question answering benchmark with implicit reasoning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' TACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Mohammad Javad Hosseini, Hannaneh Hajishirzi, Oren Etzioni, and Nate Kushman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Learning to solve arithmetic word problems with verb catego- rization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Zhengbao Jiang, Frank F Xu, Jun Araki, and Graham Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How can we know what language models know?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' TACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jungo Kasai, James Cross, Marjan Ghazvininejad, and Jiatao Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Non-autoregressive machine trans- lation with disentangled context transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jungo Kasai, Nikolaos Pappas, Hao Peng, James Cross, and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Deep encoder, shallow decoder: Reevaluating non-autoregressive machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jungo Kasai, Hao Peng, Yizhe Zhang, Dani Yogatama, Gabriel Ilharco, Nikolaos Pappas, Yi Mao, Weizhu Chen, and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Finetuning pretrained transformers into RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pap- pas, and François Fleuret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Transformers are rnns: Fast autoregressive transformers with linear at- tention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ICML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Tushar Khot, Harsh Trivedi, Matthew Finlayson, Yao Fu, Kyle Richardson, Peter Clark, and Ashish Sab- harwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Decomposed prompting: A modular approach for solving complex tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Wang Ling, Dani Yogatama, Chris Dyer, and Phil Blun- som.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Program induction by rationale genera- tion: Learning to solve and explain algebraic word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jiachang Liu, Dinghan Shen, Yizhe Zhang, Bill Dolan, Lawrence Carin, and Weizhu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' What makes good in-context examples for gpt-3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Panupong Pasupat and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Composi- tional semantic parsing on semi-structured tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Arkil Patel, Satwik Bhattamishra, and Navin Goyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Are NLP models really able to solve simple math word problems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hao Peng, Jungo Kasai, Nikolaos Pappas, Dani Yogatama, Zhaofeng Wu, Lingpeng Kong, Roy Schwartz, and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' ABC: At- tention with bounded-memory control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hao Peng, Nikolaos Pappas, Dani Yogatama, Roy Schwartz, Noah Smith, and Lingpeng Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Random feature attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Subhro Roy and Dan Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Solving general arith- metic word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Ohad Rubin, Jonathan Herzig, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Learning to retrieve prompts for in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D Manning, Andrew Y Ng, and Christopher Potts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Recursive deep mod- els for semantic compositionality over a sentiment treebank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hongjin Su, Jungo Kasai, Chen Henry Wu, Weijia Shi, Tianlu Wang, Jiayi Xin, Rui Zhang, Mari Ostendorf, Luke Zettlemoyer, Noah A Smith, and Tao Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Selective annotation makes language models better few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Alon Talmor, Jonathan Herzig, Nicholas Lourie, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' CommonsenseQA: A ques- tion answering challenge targeting commonsense knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, and Denny Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Self-consistency improves chain of thought reasoning in language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Chain of thought prompting elicits reasoning in large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Adina Williams, Nikita Nangia, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A broad-coverage challenge corpus for sen- tence understanding through inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' of NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, Nathan Scales, Xuezhi Wang, Dale Schuurmans, Olivier Bousquet, Quoc Le, and Ed Chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Least-to-most prompting enables complex reasoning in large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A Time Cost Analysis Regarding Transformer Architecture In batch prompting, assume there are K in-context exemplars (C tokens per sample on average), b sam- ples in a batch to be inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Standard prompting is a special case where b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Since most current LLMs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=',GPT-3, Codex, PaLM) are based on the Transformer decoder-only architecture, we focus on the time cost of the auto-regressive decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The plain transformer time complexity for decod- ing one token is O(n2d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', the time for encoding the embeddings of input tokens, where n is the length of input tokens and d is the dimension of embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' With the caching of previous tokens, the time complexity to decode each of the rest to- kens is O(nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We omit d since it is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, the time of one inference to decode C · b tokens: Tencode = (CK)2 Tdecode = (CK + 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' (CK + Cb) T = Tencode + Tdecode (2) where Tencode is the time for encoding the input tokens in the decoder, and Tdecode is the time for decoding the rest tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' C can be seen as a con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' One inference time T regarding K and b is: T = C2K2 + Cb · CK + Cb(Cb + 1) 2 = C2(K2 + bK + b2 2 ) + Cb 2 (3) Thus, increasing b in batch prompting will also in- crease the time cost of one inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The influence of b also increases with its value and is relatively marginal when b is small, especially when b ≪ K, which is a common practice (b = 1) in few-shot in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We can see a few examples by setting K =12 (as in experiments), C =100 with varying b in Table 5 according to equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' b Time per inference 1 1565050 2 1700100 3 1845150 4 2000200 6 2340300 12 3600600 Table 5: Time(no unit) per inference with K =12, C = 100 and various b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Though the numbers are not accurate consider- ing the constant coefficients of Big O time com- plexity, we can learn the decoding time increase can not be overlooked as b becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We do not emphasize this part in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 because the overhead and rate limit blocking time of the OpenAI API make up the most proportion of time cost, and thus reducing the N times of API calls to N/b times almost inverse linearly reduce the time cost (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' However, if the overhead and rate limits are no longer the bottlenecks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', rate limits are strict for Codex (code-davinci-002) but not a big issue to GPT-3 (text-davinci-003), then the decoding time increase will be non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' B More Experimental Results We list results for all experiments (Tables 6-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For the WikiTQ experiment with Binder, the LLM generation temperature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 following its paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For the other experiments, the temperature is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For all experiments, top_p = 1, sampling_n = 1, logprobs =1, and stop_tokens =\\n\\n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Five Ope- nAI keys are used as a polling pool on rotation to request the OpenAI API of Codex (the rate limit er- rors still occur in the experiments and are counted into time cost since it is a practical issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If fewer OpenAI keys are used, there should be more rate limit errors because the request interval for one key will be shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' C Prompts In the section, we list the prompt templates we use for each dataset (Tables 10-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We follow CoT (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) to build the prompts of CommonsenseQA, StrategyQA, GSM8K, SVAMP, AQuA, AddSub, MutliArith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We follow Binder (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) and Program-of- Thought (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) to build the prompts of WikiTQ, GSM8K (program), and SVAMP (pro- gram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For RTE, MNLI, SST-5, we design the prompts ourselves using Chain-of-Thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For prompts with fewer than 12 in-context exemplars, we manually add to 12 samples using samples from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' We show batch prompting prompts with b = 4 as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For different b, we group the same 12 samples according to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' When using ChatGPT in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5, the prompt format differs from Codex and GPT-3 because its conversational capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' See Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Task Dataset Standard Prompting Batch Prompting b=2 3 4 6 Commonsense CSQA 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 StrategyQA 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 Arithmetic GSM8K 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 SVAMP 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 AQuA 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='4 AddSub 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 MultiArith 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 NLI/NLU RTE 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='1 MNLI 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 SST-5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 Table 6: Batch prompting accuracy with different b (the number of samples in batch) compared with standard prompting on ten datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' All use Codex (code-davinci-002) as the LLM and Chain-of-Thought as the reasoning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Task Dataset Standard Promting Batch Prompting b=2 3 4 6 Commonsense CSQA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='40 StrategyQA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='99 Arithmetic GSM8K 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='61 SVAMP 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='92 AQuA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='77 AddSub 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='79 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='45 MultiArith 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='38 NLI/NLU RTE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='29 MNLI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='32 SST-5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='18 Table 7: Batch prompting time per sample with different b (the number of samples in batch) compared with standard prompting on ten datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' All use Codex (code-davinci-002) as the LLM and Chain-of-Thought as the reasoning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table Input Standard Prompting Batch Prompting b=2 3 4 6 Schema(Simple) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 Schema 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 Schema(3 table rows) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 Table 8: Accuracy on WikiTQ of various table input strategies and b (number of samples in batch) using Binder (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) to generate programs with Codex (code-davinci-002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Dataset Standard Prompting Batch Prompting b=2 3 4 6 GSM8K 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 SVAMP 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3 Table 9: Accuracy on GSM8K and SVAMP with varying b (number of samples in batch) using Program-of- Thought (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 2022) to generate programs with Codex (code-davinci-002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' CommonsenseQA Prompt Q[1]: What do people use to absorb extra ink from a fountain pen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[1]: (a) shirt pocket (b) calligrapher’s hand (c) inkwell (d) desk drawer (e) blotter Q[2]: What home entertainment equipment requires cable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[2]: (a) radio shack (b) substation (c) television (d) cabinet Q[3]: The fox walked from the city into the forest, what was it looking for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[3]: (a) pretty flowers (b) hen house (c) natural habitat (d) storybook Q[4]: Sammy wanted to go to where the people were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Where might he go?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[4]: (a) populated areas (b) race track (c) desert (d) apartment (e) roadblock A[1]: The answer must be an item that can absorb ink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only blotters are used to absorb ink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The answer must require cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only television requires cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The answer must be something in the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only natural habitat is in the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The answer must be a place with a lot of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only populated areas have a lot of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: Where do you put your grapes just before checking out?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[1]: (a) mouth (b) grocery cart (c)supermarket (d) fruit basket (e) fruit market Q[2]: Google Maps and other highway and street GPS services have replaced what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[2]: (a) united states (b) mexico (c) countryside (d) atlas Q[3]: Before getting a divorce, what did the wife feel who was doing all the work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[3]: (a) harder (b) anguish (c) bitterness (d) tears (e) sadness Q[4]: James went to the tennis court that was located in his home what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[4]: (a) country club (b) park (c) michigan (d) sports (e) town A[1]: The answer should be the place where grocery items are placed before checking out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, grocery cart makes the most sense for holding grocery items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The answer must be something that used to do what Google Maps and GPS services do, which is to give directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only atlases are used to give directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The answer should be the feeling of someone getting divorced who was doing all the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, the closest feeling is bitterness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The answer must be a place where tennis courts are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only home town has tennis courts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: What does you body do when you exercise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[1]: (a) need for food (b) thirst (c) work out (d) sweating (e) injury Q[2]: In order to see a story on the big screen what must you do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[2]: (a) go to movies (b) visualize (c) reading (d) open book (e) sketching a picture Q[3]: He followed the train tracks hoping to get home, he had gotten lost in the Yooperland where?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[3]: (a) ghetto (b) michigan (c) new york (d) canada (e) train station Q[4]: What would you get if you want a painting but cannot afford the original?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[4]: (a) reproduction (b) derivative (c) reproduction (d) simile (e) remake A[1]: The answer must be something that happens when you exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only sweating happens when you exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The answer must be something that you do to see a story on the big screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only going to movies makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The answer should be a place that relates to Yooperland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only michigan is related to Yooperland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The answer must be something that is similar to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Of the above choices, only reproduction is similar to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 10: CommonsenseQA Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' StrategyQA Prompt Q[1]: Do hamsters provide food for any animals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: Could Brooke Shields succeed at University of Pennsylvania?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Hydrogen’s atomic number squared exceeds number of Spice Girls?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Is it common to see frost during some college commencements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: Hamsters are prey animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Prey are food for predators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, hamsters provide food for some animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: Brooke Shields went to Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Princeton University is about as academically rigorous as the University of Pennsylvania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, Brooke Shields could also succeed at the University of Pennsylvania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: Hydrogen has an atomic number of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 1 squared is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' There are 5 Spice Girls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, Hydrogen’s atomic number squared is less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: College commencement ceremonies can happen in December, May, and June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' December is in the winter, so there can be frost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, there could be frost at some commencements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: Could a llama birth twice during War in Vietnam (1945-46)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: Would a pear sink in water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Can an Arvanite Greek understand some of the Albanian Declaration of Independence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Can Burundi’s communicate with citizens of New Brunswick?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The War in Vietnam was 6 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The gestation period for a llama is 11 months, which is more than 6 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, a llama could not give birth twice during the War in Vietnam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The density of a pear is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='6g/cm3, which is less than water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Objects less dense than water float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, a pear would float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The Arvanite Greek’s are a major Tosk speaking group of southern Albania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, they can understand some of the Albanian Declaration of Independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: French is one of the official languages of Burundi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, Burundi’s can communicate with citizens of New Brunswick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: Are quadrupeds represented on Chinese calendar?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: Can actress Dafne Keen win the Eurovision Song Contest finals in 2020?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Would a student in eleventh grade be unable to run for president of the United States?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Does the judo rank system reach the triple digits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The Chinese calendar has a number of symbols including monkeys, goats, and tigers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Tigers have four paws and balance themselves by walking on their toes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, quadrupeds are represented on the Chinese calendar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: Contestants must be at least 16 years of age to compete in the finals of Eurovision Song Contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Dafne Keen is 15 years old in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, Dafne Keen cannot win the Eurovision Song Contest finals in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: Students in the eleventh grade are typically 16–17 years of age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' To serve as president, one must be at least 35 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, a student in eleventh grade would be unable to run for president of the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: A triple digit number would be equal to at least 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The judo dan-rank system was capped at 10th dan after the death of judo’s founder, Kan¯o Jigor¯o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, the judo rank system does not reach the triple digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 11: StrategyQA Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' GSM8K, SVAMP, AddSub, MultiArith Prompt Q[1]: There are 15 trees in the grove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Grove workers will plant trees in the grove today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After they are done, there will be 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Leah had 32 chocolates and her sister had 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If they ate 35, how many pieces do they have left in total?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Jason had 20 lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' He gave Denny some lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Now Jason has 12 lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many lollipops did Jason give to Denny?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: There are 15 trees originally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Then there were 21 trees after some more were planted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So there must have been 21 - 15 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: There are originally 3 cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2 more cars arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3 + 2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: Originally, Leah had 32 chocolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Her sister had 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So in total they had 32 + 42 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After eating 35, they had 74 - 35 = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: Jason started with 20 lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Then he had 12 after giving some to Denny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So he gave Denny 20 - 12 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: Shawn has five toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For Christmas, he got two toys each from his mom and dad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many toys does he have now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: There were nine computers in the server room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Five more computers were installed each day, from monday to thursday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many computers are now in the server room?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Michael had 58 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' On tuesday, he lost 23 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' On wednesday, he lost 2 more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many golf balls did he have at the end of wednesday?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Olivia has $23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' She bought five bagels for $3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much money does she have left?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: Shawn started with 5 toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If he got 2 toys each from his mom and dad, then that is 4 more toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 5 + 4 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: There were originally 9 computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For each of 4 days, 5 more computers were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So 5 * 4 = 20 computers were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 9 + 20 is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: Michael started with 58 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After losing 23 on tuesday, he had 58 - 23 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After losing 2 more, he had 35 - 2 = 33 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: Olivia had 23 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 5 bagels for 3 dollars each will be 5 x 3 = 15 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So she has 23 - 15 dollars left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 23 - 15 is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: A garden produced 237 potatoes, 60 fewer cucumbers and twice as many peppers than the cucumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many vegetables did the garden produce?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: John’s cow weighs 400 pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' It increased its weight to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 times its starting weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' He is able to sell the cow for $3 per pound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much more is it worth after gaining the weight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: John writes 20 pages a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How long will it take him to write 3 books that are 400 pages each?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: James has a rainwater collection barrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For each inch of rain he collects 15 gallons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' On Monday it rained 4 inches and on Tuesday it rained 3 inches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' He can sell water for $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 per gallon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much money did he make from selling all the water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The garden produced 237 - 60 = 177 cucumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The garden produced 177 * 2 = 354 peppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The garden produced 237 + 177 + 354 = 768 vegetables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The cow initially weighs 400 * 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 = 600 pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So it gained 600 - 400 = 200 pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' It is worth 200 * 3 = 600 dollars more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: He wants to write 3 * 400 = 1200 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So it will take him 1200 / 20= 60 days The answer is 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: It rained 3 + 4 = 7 inches So he collected 7 * 15 = 105 gallons So he makes 105 * 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 = 126 from selling the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 12: GSM8K, SVAMP, AddSub, MultiArith Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' AQuA Prompt Q[1]: John found that the average of 15 numbers is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If 10 is added to each number then the mean of the numbers is?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[1]: (a) 50 (b) 45 (c) 65 (d) 78 (e) 64 Q[2]: If a / b = 3/4 and 8a + 5b = 22,then find the value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[2]: (a) 1/2 (b) 3/2 (c) 5/2 (d) 4/2 (e) 7/2 Q[3]: A person is traveling at 20 km/hr and reached his destiny in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 hr then find the distance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[3]: (a) 53 km (b) 55 km (c) 52 km (d) 60 km (e) 50 km Q[4]: How many keystrokes are needed to type the numbers from 1 to 500?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[4]: (a) 1156 (b) 1392 (c) 1480 (d) 1562 (e) 1788 A[1]: If 10 is added to each number, then the mean of the numbers also increases by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the new mean would be 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: If a / b = 3/4, then b = 4a / 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So 8a + 5(4a / 3) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' This simplifies to 8a + 20a / 3 = 22, which means 44a / 3 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So a is equal to 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The distance that the person traveled would have been 20 km/hr * 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 hrs = 50 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: There are 9 one-digit numbers from 1 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' There are 90 two-digit numbers from 10 to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' There are 401 three-digit numbers from 100 to 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 9 + 90(2) + 401(3) = 1392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: A number X equals 80% of the average of 5, 7, 14 and a number Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If the average of X and Y is 26, the value of Y is?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[1]: (a) 13 (b) 26 (c) 39 (d)36 (e) None of these Q[2]: A shopkeeper gave an additional 20 per cent concession on the reduced price after giving 30 per cent standard concession on an article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If Arun bought that article for 1,120, what was the original price?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[2]: (a) 3,000 (b) 4,000 (c) 2,400 (d) 2,000 (e) None of these Q[3]: A and B invests Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3000 and Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='7000 respectively in a business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If A doubles his capital after 6 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In what ratio should A and B divide that year’s profit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[3]: (a) 9:6 (b) 9:8 (c) 9:14 (d) 9:9 (e) 9:5 Q[4]: The angle between two hands at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='45 is?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[4]: (a) 110 degree (b) 115 degree (c) 112 1/2 degree (d) 117 degree (e) 157 1/2 degree A[1]: Average of 5, 7, 14 and Y = (5 + 7 + 14 + Y) / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Therefore, X = 80% of (5 + 7 + 14 + y) / 4 = (80/100) x (26 + Y)/4 => X = (26 + Y)/5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 5X - Y = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Also, (X + Y) / 2 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, (26 + Y) / 5 + Y = 52, then Y = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The total discount should be (1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='3) * (1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Thus, the original price should be 1120 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='56 = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The ratio should be (3 * 6 + 6 * 6): (7 * 12) = 54:84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' It simplifies to 9:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The hour hand is (45/60) * (360/12) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 degree from 3 o’clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the angle between the hour hand and the minute hand is (9-3) * (360/12) - 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: Find the sum of first 30 natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[1]: (a) 470 (b) 468 (c) 465 (d) 463 (e) 487 Q[2]: What will come in place of the x in the following Number series?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 46080, 3840, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', 48, 8, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[2]: (a) 1 (b) 384 (c) 5 (d) 7 (e) 9 Q[3]: A password of a computer used two digits where they are from 0 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' What is the probability that the password solely consists of prime numbers and zero?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[3]: (a) 1/32 (b) 1/16 (c) 1/8 (d) 2/5 (e) 1/4 Q[4]: If k3 is divisible by 120, what is the least possible value of integer k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer Choices[4]: (a) 12 (b) 30 (c) 60 (d) 90 (e) 120 A[1]: The sum of first 30 natural numbers is 30 * (30 + 1) / 2 = 465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The ratio of the numbers is 10:8:6:4:2:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the next number should be 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: 0, 2, 3, 5, 7 are five prime digits(including zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So there are 5 * 5 = 25 two-digit numbers with only prime numbers and zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The probability is 25/100 = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: 120 can be factored as 2 * 2 * 2 * 3 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the least k be 2 * 3 * 5 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 13: AQuA Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' RTE Prompt Premise[1]: No Weapons of Mass Destruction Found in Iraq Yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[1]: Weapons of Mass Destruction Found in Iraq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[2]: A place of sorrow, after Pope John Paul II died, became a place of celebration, as Roman Catholic faithful gathered in downtown Chicago to mark the installation of new Pope Benedict XVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[2]: Pope Benedict XVI is the new leader of the Roman Catholic Church.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[3]: Libya’s case against Britain and the US concerns the dispute over their demand for extradition of Libyans charged with blowing up a Pan Am jet over Lockerbie in 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[3]: One case involved the extradition of Libyan suspects in the Pan Am Lockerbie bombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[4]: Argentina sought help from Britain on its privatization program and encouraged British investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[4]: Argentina sought UK expertise on privatization and agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[1]: No Weapons of Mass Destruction Found, which contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[2]: As Roman Catholic faithful gathered in downtown Chicago to mark the installation of new Pope Benedict XVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[3]: Libya’s case suspects in the Pan Am Lockerbie bombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[4]: Argentina sought help from Britain on its privatization program, not agriculture, which contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[1]: Startling new research into mobile phones claims they may reduce a man’s sperm count by up to 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[1]: Male fertility may be affected by use of a mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[2]: It rewrites the rules of global trade, established by the General Agreement on Tariffs and Trade, or GATT, in 1947, and modified in multiple rounds of negotiations since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[2]: GATT was formed in 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[3]: The cost of the consumer of the United States fell in June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[3]: U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' consumer spending dived in June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[4]: Israeli Prime Minister Ariel Sharon has said that Mahmoud Abbas is a man that Israel can do business with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[4]: Palestinian leader, Mahmoud Abbas, may be someone Israel can talk with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[1]: New research claims mobile phones reduce a man’s sperm count, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', affects male fertility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[2]: GATT is rewritten in 1947, not formed in 1947, which contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[3]: The consumer cost fell in June, not the spending, which contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[4]: Mahmoud Abbas is a man that Israel can do business with, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', he may be someone Israel can talk with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[1]: In October, however, amid rising tensions between the government and opposition groups, a car bomb seriously injured an opposition politician and killed his driver, in Beirut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[1]: A member of the opposition was injured in a car bomb attack in Beirut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[2]: Ruth’s 1927 single season record of 60 home runs stood unsurpassed until Roger Maris hit 61 in 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[2]: Babe Ruth hit 60 home runs in his lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[3]: The German technology was employed to build Shanghai’s existing maglev line, the first in the world to be used commercially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[3]: Maglev is commercially used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[4]: Twelve of Jupiter’s moons are relatively small and seem to have been more likely captured than to have been formed in orbit around Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[4]: Jupiter has Twelve moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[1]: A car bomb seriously injured an opposition politician in Beirut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer the True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[2]: Babe Ruth hit 60 home runs in a single season, not his lifetime, which contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[3]: The German technology was employed to build Shanghai’s existing maglev line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=', Maglev is commercially used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[4]: Twelve of Jupiter’s moons are relatively small, not Jupiter has Twelve moons, which contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So the answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 14: RTE Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' MNLI Prompt Premise[1]: Conceptually cream skimming has two basic dimensions - product and geography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[1]: Product and geography are what make cream skimming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[2]: One of our number will carry out your instructions minutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[2]: A member of my team will execute your orders with immense precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[3]: Analyzing Postal Service accounts for depreciation, fuel, and maintenance for city delivery carriers, we have estimated the average city delivery vehicle cost per route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypotheis[3]: Driving cost estimates can be averaged with sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[4]: Consider the United States Postal Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[4]: Forget the United States Postal Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[1]: The answer is Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[2]: The answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[3]: The answer is Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[4]: The answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[1]: Take a remarkable statistic that Shesol cites but lets pass relatively unexamined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[1]: They had data that was very relevant but under used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[2]: The man on the ground thinks for a moment and yells back, You must work in management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[2]: There was no one on the ground, man or woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[3]: Hello, Ben.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[3]: I ignored Ben.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[4]: How can you prove it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[4]: Can you tell me how to prove it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[1]: The answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[2]: The answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[3]: The answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[4]: The answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[1]: In the midst of this amazing amalgam of cultures is a passion for continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[1]: A passion for continuity is not the most important of these cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[2]: Poirot, I exclaimed, with relief, and seizing him by both hands, I dragged him into the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[2]: Poirot was now back and I was sorry that he would take over what I now considered my own investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[3]: There’s a uh a couple called um oh i’m going to forgot his name now uh Dirkson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[3]: I can’t remember their name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Premise[4]: It’s not that the questions they asked weren’t interesting or legitimate (though most did fall under the category of already asked and answered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Hypothesis[4]: All of the questions were interesting according to a focus group consulted on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[1]: The answer is Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[2]: The answer is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[3]: The answer is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer[4]: The answer is Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 15: MNLI Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' SST-5 Prompt Q[1]: a stirring , funny and finally transporting re-imagining of beauty and the beast and 1930s horror films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: they presume their audience wo n’t sit still for a sociology lesson, however entertainingly presented, so they trot out the conventional science-fiction elements of bug-eyed monsters and futuristic women in skimpy clothes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: um , no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='. Q[4]: jonathan parker’s bartleby should have been the be-all-end-all of the modern-office anomie films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The tone is very positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The tone is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The tone is neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The tone is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: lacks the inspiration of the original and has a bloated plot that stretches the running time about 10 minutes past a child’s interest and an adult’s patience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: the santa clause 2 proves itself a more streamlined and thought out encounter than the original could ever have hoped to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: you might say tykwer has done all that heaven allows, if you wanted to make as anti-kieslowski a pun as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: otto-sallies has a real filmmaker’s eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The tone is very negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The tone is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The tone is neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The tone is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[1]: with a confrontational stance, todd solondz takes aim on political correctness and suburban families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: verall , cletis tout is a winning comedy that excites the imagination and tickles the funny bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: with its parade of almost perpetually wasted characters .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' margarita feels like a hazy high that takes too long to shake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: an ugly-duckling tale so hideously and clumsily told it feels accidental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The tone is neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The tone is very positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: The tone is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: The tone is very negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Table 16: SST-5 Prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' ChatGPT Prompt for GSM8K 1st round input: You can answer questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' I will give you a few batches of exemplars in format Q[idx]:question, A[idx]:answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Say "continue" to get the next input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Finally, a batch of test samples with only contexts Q[idx]:question are input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Answer the test samples in format A[idx]:answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Say "okay" if you understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 1st round output: okay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2nd round input: Q[1]: There are 15 trees in the grove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Grove workers will plant trees in the grove today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After they are done, there will be 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Leah had 32 chocolates and her sister had 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If they ate 35, how many pieces do they have left in total?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Jason had 20 lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' He gave Denny some lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Now Jason has 12 lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many lollipops did Jason give to Denny?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: There are 15 trees originally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Then there were 21 trees after some more were planted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So there must have been 21 - 15 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: There are originally 3 cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2 more cars arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3 + 2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: Originally, Leah had 32 chocolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Her sister had 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So in total they had 32 + 42 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After eating 35, they had 74 - 35 = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: Jason started with 20 lollipops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Then he had 12 after giving some to Denny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So he gave Denny 20 - 12 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 2nd round output: continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3rd round input: Q[1]: Shawn has five toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For Christmas, he got two toys each from his mom and dad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many toys does he have now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: There were nine computers in the server room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Five more computers were installed each day, from monday to thursday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many computers are now in the server room?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: Michael had 58 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' On tuesday, he lost 23 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' On wednesday, he lost 2 more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many golf balls did he have at the end of wednesday?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: Olivia has $23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' She bought five bagels for $3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much money does she have left?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: Shawn started with 5 toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' If he got 2 toys each from his mom and dad, then that is 4 more toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 5 + 4 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: There were originally 9 computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For each of 4 days, 5 more computers were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So 5 * 4 = 20 computers were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 9 + 20 is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: Michael started with 58 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After losing 23 on tuesday, he had 58 - 23 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' After losing 2 more, he had 35 - 2 = 33 golf balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: Olivia had 23 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 5 bagels for 3 dollars each will be 5 x 3 = 15 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So she has 23 - 15 dollars left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 23 - 15 is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 3rd round output: continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4th round input: Q[1]: A garden produced 237 potatoes, 60 fewer cucumbers and twice as many peppers than the cucumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How many vegetables did the garden produce?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[2]: John’s cow weighs 400 pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' It increased its weight to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 times its starting weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' He is able to sell the cow for $3 per pound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much more is it worth after gaining the weight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[3]: John writes 20 pages a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How long will it take him to write 3 books that are 400 pages each?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Q[4]: James has a rainwater collection barrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' For each inch of rain he collects 15 gallons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' On Monday it rained 4 inches and on Tuesday it rained 3 inches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' He can sell water for $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 per gallon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' How much money did he make from selling all the water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[1]: The garden produced 237 - 60 = 177 cucumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The garden produced 177 * 2 = 354 peppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The garden produced 237 + 177 + 354 = 768 vegetables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[2]: The cow initially weighs 400 * 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='5 = 600 pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So it gained 600 - 400 = 200 pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' It is worth 200 * 3 = 600 dollars more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[3]: He wants to write 3 * 400 = 1200 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' So it will take him 1200 / 20= 60 days The answer is 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' A[4]: It rained 3 + 4 = 7 inches So he collected 7 * 15 = 105 gallons So he makes 105 * 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='2 = 126 from selling the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' The answer is 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' 4th round output: continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Test round input: {four test questions} Test round output: {four test answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content='} Table 17: An example ChatGPT prompt we use for batch prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' Specifically, the task instruction is input in the first round of the conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FAT4oBgHgl3EQf3h6y/content/2301.08721v1.pdf'} +page_content=' In the next a few rounds, one batch of in-context exemplars is input in one round.' 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a/ZtAyT4oBgHgl3EQfifij/content/tmp_files/2301.00397v1.pdf.txt b/ZtAyT4oBgHgl3EQfifij/content/tmp_files/2301.00397v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..543aed33734b50ffc7dfb0de025bcc9b4ccb9b0c --- /dev/null +++ b/ZtAyT4oBgHgl3EQfifij/content/tmp_files/2301.00397v1.pdf.txt @@ -0,0 +1,1157 @@ +Inflected Forms Are Redundant in Question Generation Models +Xingwu Sun +University of Macau +sunxingwu01@gmail.com +Hongyin Tang +Meituan Inc. +tanghongyin@meituan.com +Chengzhong Xu +University of Macau +czxu@um.edu.mo +Abstract +Neural models with an encoder-decoder frame- +work provide a feasible solution to Question +Generation (QG). However, after analyzing +the model vocabulary we find that current mod- +els (both RNN-based and pre-training based) +have more than 23% inflected forms. +As +a result, the encoder will generate separate +embeddings for the inflected forms, leading +to a waste of training data and parameters. +Even worse, in decoding these models are +vulnerable to irrelevant noise and they suffer +from high computational costs. In this paper, +we propose an approach to enhance the per- +formance of QG by fusing word transforma- +tion. Firstly, we identify the inflected forms +of words from the input of encoder, and re- +place them with the root words, letting the +encoder pay more attention to the repetitive +root words. +Secondly, we propose to adapt +QG as a combination of the following actions +in the encode-decoder framework: generating +a question word, copying a word from the +source sequence or generating a word transfor- +mation type. Such extension can greatly de- +crease the size of predicted words in the de- +coder as well as noise. We apply our approach +to a typical RNN-based model and UNILM to +get the improved versions. We conduct exten- +sive experiments on SQuAD and MS MARCO +datasets. The experimental results show that +the improved versions can significantly outper- +form the corresponding baselines in terms of +BLEU, ROUGE-L and METEOR as well as +time cost. +1 +Introduction +Question Generation (QG) aims to generate natu- +ral language questions from given text. It can aid +several applications: (1) QG can help create educa- +tional materials by generating questions for read- +ing comprehension materials (Heilman and Smith, +2010; Du et al., 2017). (2) It can be used to auto- +matically curate question answering datasets (Duan +Figure 1: Two root words and their corresponding in- +flected forms in the vocabulary of QG models. +et al., 2017). (3) It can also aid dialogue systems +by actively asking meaningful questions. Typically, +QG includes two sub-tasks: (1) determine the tar- +gets that should be asked (e.g., sentences, phrases +or words), and (2) produce the surface-form of the +question. In this work, we focus on the sub-task of +surface-form generation of questions by assuming +that the targets are given. +Recent neural QG models, e.g., RNN-based +sequence-to-sequence models and pre-training +based models, have made great progress in generat- +ing proper questions. In particular, UNILM (Dong +et al., 2019), which applies a pre-training model for +QG, is the state-of-the-art model with the best per- +formance. However, those models all suffer high +computational costs in decoding. We carefully ex- +amine existing neural QG models which have an +encoder-decoder architecture in general and find +that existing models remain two issues. Firstly, in +the encoder, the root word and its inflected forms, +e.g., “commit” and its inflected forms “committed” +and “committing” as shown in Figure 1, might all +occur in the vocabulary of the encoder. Accord- +ing to our statistics, in the widely used QG dataset +SQuAD (Rajpurkar et al., 2016), the top-10000 fre- +quent words contain 3718 inflected forms. More- +over, even though UNILM applies the WordPiece +segmentation to reduce the vocabulary size, its vo- +cabulary still contains about 23.18% (6722/28996) +inflected forms. As a result, the encoder will gener- +ate individual embeddings for these words, which +arXiv:2301.00397v1 [cs.CL] 1 Jan 2023 + +Root Word +Inflected Forms +commit +committed +committing +commits +big +bigger +biggestseems a little redundant for QG. Even worse, they +occupy the space in the vocabulary of encoder. As +a result, the encoder will generate separate embed- +dings for the inflected words, leading to a waste of +training data and parameters. Secondly, in the de- +coder, the same large set of words are employed to +generate a question regardless of the input. There- +fore, these models are vulnerable to irrelevant noise. +Besides, while decoding each word of a question, +probability distribution of words in the entire static +and fairly large vocabulary have to be calculated, +which slows the decoding process down. +To decrease the computational cost and improve +the performance of QG, we propose an approach +named as word transformation approach. The ap- +proach is inspired by the feature of the inflected +language and our daily reading experience. The +former refers to the fact that words with different +inflected forms may divert the focus to QG. The +latter refers to our observations that most words +in the generated question can be copied or simply +transformed from the words in the original source +sequence, except for the question word. Therefore, +in our approach, we treat QG as a combination of +word transformation, word copying and question +word generation. In detail, in the question word +generation, a question word is generated from a +limited question word vocabulary. In the transfor- +mation type generation, an inflected form type is +generated from a small word transformation type +vocabulary. For example, type “##ed” may be gen- +erated and will be used for transforming a verb to +its past tense in the final step. In the word copy- +ing, words are copied from the source sequence. +Besides, we identify the inflected forms of words +from the input of encoder, and replace them with +their root forms, letting the encoder pay more atten- +tion to the repetitive root words and take in more +different words for training. We have applied our +approach to a typical RNN-based model and a pre- +trained transformer-based model UNILM and get +the corresponding improved versions. +The main contributions of this paper can be sum- +marized as follows. +• We find that inflected words take up unnec- +essary spaces in vocabulary and define a se- +ries of word transformation types to facilitate +transforming a word into its inflected forms. +• We only keep the root words in vocabulary of +the encoder to make the best of the training +data and the limited vocabulary space, which +is also applicable in many other NLP models. +• We propose to simplify the decoding process +of QG by question word generation, transfor- +mation type generation and word copying, to +avoid generating each word from a large vo- +cabulary and enhance the performance. +• For evaluating the effectiveness and efficiency +of our proposed approach, we conduct exten- +sive experiments on two large scale datasets, +i.e., SQuAD and MS MARCO datasets and +compare the baselines with the corresponding +improved versions. The experimental results +show that the improved versions can signifi- +cantly outperform the baselines. +2 +Related Work +The existing QG approaches can be broadly classi- +fied into two kinds: rule-based and neural network- +based. The rule-based approaches rely on hand- +crafted rules, while the neural network-based ap- +proaches are data-driven and trainable in an end-to- +end fashion. +Early QG approaches are mostly rule-based +ones (Heilman and Smith, 2009, 2010; Chali and +Hasan, 2015), which leverage rules or templates +to generate questions. The rule-based approaches +employ manually crafted rules for declarative-to- +interrogative sentence transformation, typically +based on syntactic (Mitkov and Ha, 2003; Ali +et al., 2010; Heilman, 2011) or semantic informa- +tion (Chen, 2009), while the template-based ap- +proaches generate questions using manually cre- +ated templates which are predefined with place- +holders to be filled with words from the source +word sequence (Cai et al., 2006; Lindberg et al., +2013; Song and Zhao, 2016). The major limita- +tions of these rule-based approaches include: (1) +they rely heavily on rules and templates, which are +created manually and therefore expensive, (2) these +rules or templates lack diversity, (3) the targets they +can deal with are limited. +To tackle the limitations of rule-based ap- +proaches, the neural network-based approaches +with an encoder-decoder framework are applied +to the task of QG. These approaches do not rely +on hand-crafted rules, and they are instead data +driven and trainable in an end-to-end fashion. The +release of large-scale machine reading comprehen- +sion datasets, e.g. SQuAD, MS MARCO (Nguyen +et al., 2016), Hotpot QA (Yang et al., 2018) and +DROP (Dua et al., 2019), further drives the devel- + +Part-of-speech +Transformation Types +Transformation Type Description +verb +##ing +converting the verb to its present participle +##vs +converting the verb to its singular present +##ed +converting the verb to its past tense +##edp +converting the verb to its past participle +noun +##ns +converting the noun to its plural word +adjective +##jer +converting the adjective to its comparative form +##jest +converting the adjective to its superlative form +adverb +##ver +converting the adverb to its comparative form +##vest +converting the adverb to its superlative form +Table 1: Word transformation types and the corresponding descriptions. +opment of neural QG models. In general, these +datasets contain large-scale manually annotated +triples, i.e., question, answer and the context, and +they can be used as training data of QG models. +As for the RNN-based models, both Du et al. +(2017) and Yuan et al. (2017) apply a sequence-to- +sequence model with an attention mechanism to +generate questions for the text in SQuAD dataset. +Zhou et al. (2017) enrich the RNN-based model +with rich features (i.e., answer position and lexi- +cal features) to generate answer focused questions, +and incorporate a copy mechanism that allows the +model to copy words from the context. +Duan +et al. (2017) propose to combine templates and the +sequence-to-sequence model, in which they mine +question patterns from a question answering com- +munity and employ a sequence-to-sequence model +to generate question patterns for a given text. Tang +et al. (2017) model question answering and ques- +tion generation as dual tasks, which helps generate +better questions and get better question answering +models at the same time. +Further, based on pointer generator network (See +et al., 2017), Sun et al. (2018) propose an answer fo- +cused and position-aware model to effectively lever- +age answer encoding and position features. Zhao +et al. (2018) mainly focus on incorporating para- +graph level context by using gated self-attention +and maxout pointer networks. Nema et al. (2019) +give a refine network to mimic human process of +generating questions. Besides, there has also been +some work on generating questions from knowl- +edge bases (Serban et al., 2016; Indurthi et al., +2017; Liu et al., 2019). +More recently, pre-training models are applied +to QG. For example, Dong et al. (2019) propose +UNILM which makes a great progress in QG. Cur- +rently, UNILM is the QG model with the best per- +formance. +Differing from the previous work, our approach +does not provide a QG model, and instead provides +two extensions to the encoder-decoder framework +for performance improvement. Our approach can +be applied to most existing models, including RNN- +based sequence-to-sequence models plus UNILM. +3 +Our Approach +To begin with, we formally give the QG task def- +inition as follows. Given the text (x1, x2, ..., xTx) +of length Tx as well as lexical features, i.e., named +entity (NE) and part-of-speech (POS). The answer +positions in this text range from l to r. The goal is +to generate a question, which is required to be as +close as the reference question (y1, y2, ..., yTy). +In the following subsections, we describe the +details of the proposed approach to deal with the +issues discussed in the previous sections. Firstly, +we define a series of word transformation types, +which is the basis of our approach. Secondly, we +choose the pointer generator network and UNILM +as baselines, and describe how to apply our ap- +proach by elaborating the working process of the +encoder-decoder in the improved versions, respec- +tively. +3.1 +Word Transformation Types +As the basis of our approach, we define the transfor- +mation types in terms of the part-of-speech of the +word, as listed in Table 1. As for a verb, we define +four types of transformation, i.e., “##ing”, “##vs”, +“##ed” and “##edp”. As for a noun, we define one +transformation type “##ns”, which means convert- +ing the noun to its plural word. For an adjective +or an adverb, we define the transformation to their +comparative form as “##jer” and “##ver”, respec- +tively. Similarly, we define their transformation to +the superlative form as “##jest” and “##vest”. Note + +that we use the Pattern module1 in Python to con- +duct these transformations. As for irregular words, +we manually collect a lookup table, for instance, +“went” can be converted into go and “##ed”. +Recall the two issues mentioned in Section 1. +As for the first issue in the encoder, by a series of +transformation types, we only keep the root words +and the transformation types in the encoder vocab- +ulary. As a result, it can substantially save space +for other words and make the best of training data. +As for the second issue, we directly copy or trans- +form the word in the source sequence to generate +questions except for the question word. A transfor- +mation type vocabulary is introduced as a necessity +in the transformation type generation, which will +be elaborated in the next subsections. +Before model encoding, we first get the root +form of each word in the original input sequence. +For example, in Figure 2, the word “succeeded” +is converted into its root form “succeed” and the +corresponding transformation type is “##ed”. After +transformation, the input sequence is converted into +a sequence (x′ +1, x′ +2, ..., x′ +T ′x) of length T ′ +x and the +answer positions range from l′ to r′. Similarly, The +reference question is converted into (y′ +1, y′ +2, ..., y′ +T ′y) +of length T ′ +y. +3.2 +Reforming Pointer Generator Network +3.2.1 +The Encoder +The baseline which is reformed is the attention- +based pointer generator network (See et al., 2017) +enhanced with various rich features proposed +by Zhou et al. (2017). +These features include +named entity (NE), part-of-speech (POS) and an- +swer position in the embedding layer of the en- +coder. +As shown in Figure 2, the encoder is a +bidirectional GRU, which takes as input the +joint embedding of word, answer position and +lexical features (NE, POS) in the form of +(w1, w2, ..., wT ′x) with wi +∈ +Rdw+da+dn+dp, +where T ′ +x is the input length, wi is the embedding +of x′ +i by concatenating all the feature embeddings +and dw, da, dn, dp are the dimensionalities of word +embedding, answer position embedding, NE em- +bedding and POS embedding, respectively. It pro- +duces a sequence of dh-dimensional hidden states +(h1, h2, ..., hT ′x), each of which is the sum of for- +1https://github.com/clips/pattern +ward and backward GRU representations: +hi = ←−h i + −→h i, +←−h i = GRU(wi, ←−h i+1), +−→h i = GRU(wi, −→h i−1) +(1) +where ←−h i, −→h i are all dh-dimensional vectors. +3.2.2 +The Decoder +The decoder is modified to support three actions: +word copying, transformation type generation and +question word generation. (1) In the word copy- +ing, the decoder generates words from the source +sequence. (2) In the transformation type genera- +tion, the decoder generates from a limited transfor- +mation type vocabulary, which only includes nine +types in Table 1. (3) In the question word genera- +tion, the decoder generates question words from a +restricted question word vocabulary. +Word Copying The decoder is a unidirectional +GRU conditionally taking all the encoded hidden +states as input. At decoding step t, the decoder +reads an input word embedding wt, previous at- +tentional context vector ct−1 and its previous hid- +den state st−1 to update its current hidden state +st ∈ Rdh: +st = GRU([wt; ct−1], st−1) +(2) +The context vector ct is generated through an +attention mechanism (Bahdanau et al., 2014). At +time step t, the context vector ct is calculated as +follows: +ct = +T ′ +x +� +i=1 +αtihi +αti = Softmax(eti) +eti = vT tanh(W T +h hi + W T +s st + b) +(3) +where α is the attention distribution and Wh, Ws, b +and v are all trainable parameters. +The attention distribution can be viewed as a +semantic matching between hidden states of the +encoder and the hidden state of the decoder. It indi- +cates how the decoder cares about different hidden +states of the encoder during decoding. Therefore, +the attention distribution can be regarded as the +probability distribution of the word copying. +Pcopy = αt +(4) + +Figure 2: The improved version based on pointer generator network. “C” represents the context vector. +Transformation Type Generation Before encod- +ing, we have already converted each word to its +root form, i.e., the root word. In the decoding, be- +sides coping words, we should endow the model +the ability to transform some words to their proper +forms, e.g., the “do” in Figure 2 is generated from +question word vocabulary and we need to trans- +form it to “did”. To carry out this transformation, +we introduce the transformation type generation +and generate the transformation type for the root +word, e.g., “##ed” is generated to transform “do” +to “did”. +We get the transformation type distribution by +the following function. +Ptrans = Softmax (g1(st, ct)) +(5) +where g1(·) is a two-layer feedforward neural net- +work with a maxout internal activation. Ptrans ∈ +R|Vtrans| denotes the probability distribution of +transformation types. +Question Word Generation We find that the gen- +eration of question words is mainly determined by +the answer and its surrounding words. For example, +in Figure 2, the answer and its context “in 1970” +already involve the essential information to gen- +erate the question word “when”, which suggests +that the answer and its surrounding words can ben- +efit the question word generation. We also find +that the total number of question words is limited. +Therefore, we introduce a specific vocabulary of +question words to directly and explicitly model the +question words generation. Note that the question +word vocabulary contains top-1000 frequent words +in reference questions of the training set, which +means that it also contains other question-related +words. +As depicted in Figure 2, in the question word +generation, the model generates question words +based on a restricted vocabulary of question words. +This action produces a question word distribution +based on an answer embedding vanswer, the de- +coder state st and the context vector ct: +Pquest = Softmax (g2(vanswer, st, ct)) +(6) +where g2(·) is a two-layer feedforward neural net- +work, Pquest is a |Vquest|-dimensional probability +distribution, and |Vquest| is the size of vocabulary +of question words. We employ the average pooling +function to calculate the answer embedding. +vanswer = +�r′ +t=l′ ht +r′ − l′ + 1 +(7) +Three-action Combination To control the bal- +ance among different actions, we introduce a three- +dimensional switch probability, acting as a three- +way soft switch: +pquest, pcopy, ptrans = Softmax(f(ct,st,wt)) +(8) +where f(·) is a one layer feedforward network. We +compute the final probability distribution through a +weighted summation of the three action probability +distributions: +P(w) = pcopyPcopy(w) + ptransPtrans(w) ++ pquestPquest(w) +(9) +3.3 +Reforming UNILM +UNILM is a pre-trained language model which has +the same structure as BERT(Devlin et al., 2019) +but can be used for natural language generation +tasks. The model is firstly pre-trained on a large +scale corpus which enables it to learn flexible lan- +guage representations. Then, it is fine-tuned on the +dataset of the downstream tasks using the sequence- +to-sequence language model objective. In UNILM, +there is no explicit separation between the encoder +and decoder. In contrast, the source and target se- +quence are fed to the model together. The sequence- +to-sequence generation is implemented by a special +self-attention mask M. + +final + distribution +question word +attention +transformation +distribution +distribution +type distribution +Average Pooling +in +1970 +president nasser +die +##ed +and +is +##ed +succeed +##ed +by ... +when +do +##ed +nasser +die +个 +Transform Infected Words To Their Root Forms +Decoder +in +1970 +president nasser +died +and +was +succeeded +by +Answer +EncoderFigure 3: Model architecture of the improved version +based on UNILM. Note that there should be L trans- +former layers. For simplicity, we only keep two of them +in this figure. +Following +Dong +et +al. +(2019), +we +con- +catenate the answer segment (x′ +l′, ..., x′ +r′) and +the text segment (x′ +1, x′ +2, ..., x′ +T ′x) forming the +new source segment X += +(x∗ +1, ..., x∗ +T ∗ +x ) += +(x′ +1, x′ +2, ..., x′ +T ′x, [EOS], x′ +l′, ..., x′ +r′). Next, we con- +catenate the new source segment and the tar- +get segment Y = (y′ +1, y′ +2, ..., y′ +T ′y) as input, i.e., +[SOS]X[EOS]Y [EOS]. Then we feed the input to +the model. In training, we randomly replace tokens +with [MASK] in Y at a rate of 0.8. +For each input token, its embedding is the sum +of the token embedding, position embedding and +segment embedding. As shown in Figure 3, we get +token embeddings w = (w1, w2, ..., wT ∗ +x +T ′y+3), +which is then feed into L-layer transformers to get +the deep representation Hl = hl +i, i ∈ [1, T ∗ +x + T ′ +y + +3]. +hl +i = Transformer(hl−1 +i +), l = 1, 2, 3, ..., L +(10) +where H0 is initialized with w. The output of the +self-attention head of the l-th transformer is calcu- +lated by: +Ql = Hl−1W Q, Kl = Hl−1W K, Vl = Hl−1W V +(11) +Al = Softmax(QlKT +l +√dk ++ M)Vl +(12) +where M ∈ R(T ∗ +x +T ′ +y+3)×(T ∗ +x +T ′ +y+3). Mij = 0 +means it is allowed to attend, while Mij = −∞ +indicates it is not allowed to attend. We can find +that the special self-attention mask M allows the +token to be generated to attend to all of the source +tokens and the preceding generated tokens. This +mechanism allows the model to generate text in a +sequence-to-sequence fashion. +To adopt the word transformation approach, we +directly compute the distributions associated with +the three actions based on the hidden state of the +current step. The details are shown as follows. +Word Copying The attention distribution over the +source tokens is obtained by an extra attention mod- +ule which computes the attention scores using dot +product: +Pcopy = Softmax(g3(ht)) +(13) +g3(ht) = [g3(ht)i]T ′ +x +i=1 = [ht · hi]T ′ +x +i=1 +(14) +where ht is the hidden state of t-th token output by +the last layer of transformer. The output of g3(ht) +is the dot product between ht and hidden states of +all of the input tokens. +Transformation Type Generation The transfor- +mation type distribution is obtained by computing +the dot product between the current hidden state +and the embeddings of the transformation types. +Ptrans = Softmax(g4(ht)) +(15) +g4(ht) = [g4(ht)i]|Vtrans| +i=1 += +� +ht · eT +i +�|Vtrans| +i=1 +(16) +where eT is the embedding of the transformation +types. g4(ht) is the dot product between ht and +the embeddings of all of the transformation types. +Ptrans denotes the probability distribution of trans- +formation types. +Question Word Generation Similar to the cal- +culation of the distribution of the transformation +types, the distribution of the question words is +obtained by the dot product between the current +hidden state and the embeddings of the question +words. +Pquest = Softmax(g5(ht)) +(17) +g5(ht) = [g5(ht)i]|Vquest| +i=1 += +� +ht · eQ +i +�|Vquest| +i=1 +(18) +where eQ is the embedding of the question words. +g5(ht) is the dot product between ht and the em- +beddings of all of the question words eQ. +Three-action +Combination +The +three- +dimensional +switch +probability +is +obtained +by the current hidden state ht. +pquest, pcopy, ptrans = Softmax(f1(ht)) +(19) +where f1(·) is a one layer feedforward network. +The final probability distribution is computed as +the same as Equation 9. + +final distribution +attention distribution +question word +distribution +Transformer +Block +transformation type +distribution +Transformer +Block +Embedding +Layer +Transform Infected Words To Their Root Forms +[SOS] +x* +[EOS] +yi +[MASK] +[EOS]Model +BLEU1 +BLEU2 +BLEU3 +BLEU4 +ROUGE-L +METEOR +Pointer +40.49 +26.11 +18.94 +14.34 +42.15 +18.71 +Pointer + WT +45.08 +29.56 +21.54 +16.41 +45.40 +20.61 +UNILM +49.82 +34.48 +26.03 +20.39 +49.02 +23.52 +UNILM + WT +51.56 +35.78 +27.06 +21.24 +50.46 +23.93 +(a) +Model +BLEU1 +BLEU2 +BLEU3 +BLEU4 +ROUGE-L +METEOR +Pointer +44.45 +31.85 +23.32 +17.90 +46.07 +20.02 +Pointer + WT +56.14 +39.36 +29.04 +22.10 +59.29 +26.40 +UNILM +60.57 +43.14 +32.38 +25.01 +60.58 +29.31 +UNILM + WT +62.65 +45.33 +34.31 +26.55 +62.85 +29.72 +(b) +Table 2: The main experimental results of baselines and improved versions on SQuAD (a) and MARCO (b). “WT” +means the proposed word transformation approach. +4 +Experiments +4.1 +Experiment Settings +Dataset We conduct the experiments on SQuAD +and MARCO datasets. Since the test sets of these +datasets are not publicly available, we follow Zhou +et al. (2017) to randomly split the development set +into two parts and use them as the development set +and test set for the QG task. In SQuAD, there are +86, 635, 8, 965 and 8, 964 question-answer pairs +in our training set, development set and test set, +respectively. We directly use the extracted features2 +shared by Zhou et al. (2017). In MARCO, there are +74, 097, 4, 539 and 4, 539 question-answer pairs +in our training set, development set and test set, +respectively. We use Stanford CoreNLP3 to extract +lexical features. +Implementation Details We set the cutoff length +of the input sequence as 128 words. The encoder +vocabulary contains the most frequent 30, 000 +words in each training set. The decoder vocabulary +contains two sub-vocabularies. One is the ques- +tion word vocabulary which contains most frequent +1000 words in the reference questions of the train- +ing set. The other one is the transformation type +vocabulary, including 9 transformation types as de- +scribed in Section 3. For the RNN-based model, +we use the pre-trained Glove word vectors4 with +300 dimensions to initialize the word embeddings +that will be further fine-tuned in the training stage. +The representations of answer position feature and +lexical features at the embedding layer of the en- +coder are randomly initialized to 32 dimensional +vectors that are trainable during training stage. The +2https://res.qyzhou.me/redistribute. +zip +3https://nlp.stanford.edu/software/ +4http://nlp.stanford.edu/data +size of hidden states of both the encoder and de- +coder is 512. We use dropout only in the encoder +with a dropout rate 0.20. The size of answer em- +bedding is 512. We use the optimization algorithm +Adam (Kingma and Ba, 2014) with the learning +rate 0.002 and we set the batch size as 32. As +for the UNILM, we adopt the base version of the +UNILM and use the recommended parameters as +detailed by Dong et al. (2019). After training, we +select the best model on the development set for +testing. +Evaluation Metrics We evaluate our approach us- +ing n-gram similarity metrics, i.e., BLEU (Pap- +ineni et al., 2002), ROUGE-L (Lin, 2004) and ME- +TEOR (Lavie and Denkowski, 2009). +Competitors In the experiments, we have the fol- +lowing four competitors for comparisons, where +Pointer and UNILM are baselines and the other +two are corresponding improved versions. +• Pointer generator network (Pointer) It is +a typical RNN-based sequence-to-sequence +model with the copy mechanism (See et al., +2017). To make a fair comparison, the lexical +features are added to the embedding layer as +same as Zhou et al. (2017). +• Pointer generator network plus Word +Transformation approach (Pointer + WT) +We enhance the Pointer model with the pro- +posed word transformation approach. +• UNILM (Dong et al., 2019) It is a pre-trained +language model which can be applied to natu- +ral language generation tasks. +• UNILM + WT We combine the UNILM +model with the proposed word transformation +approach. +Pointer and Pointer + WT are implemented with + +Tensorflow, while UNILM and UNILM + WT are +implemented by PyTorch. +4.2 +Performance Evaluation +Table 2 shows the main results, and we have the +following observations: +• The Pointer + WT model performs better than +the original Pointer model which indicates +that generating questions by word transforma- +tion is effective. +• UNILM + WT model can still significantly +outperform the powerful UNILM, which in- +dicates the effectiveness of applying our ap- +proach to the pre-trained language models for +QG tasks. +• Our approach makes greater improvement +over MARCO than SQuAD. That might be +because MARCO is extracted from the search +engine, which induces a bias over word copy- +ing and word transformation generation from +source text. +Besides generation quality, we also compare +baselines with the corresponding improved ver- +sions on efficiency of decoding. We record the +time of generating a word of a question given the +test text with a beam size of 12 and calculate the +average value. Note that we regard the decoding +of root word and its corresponding transformation +type as one word in efficiency comparison. The +comparison is conducted on a GPU environment +with a single Tesla V100. For the RNN-based mod- +els, the Pointer + WT model is more efficient which +can save more than 39% (from 0.0081s to 0.0049s) +decoding time compared to original model. For +the pre-trained models, the UNILM + WT model +can save 13% decoding time (from 0.0144s to +0.0125s). We also find that UNILM behaves better +than Pointer on generation quality, but it sacrifices +efficiency. Specifically, it is nearly one time slower +than the RNN-based models. From the compar- +isons on both effectiveness and efficiency, we can +conclude that this word transformation approach +can enhance the quality of QG and speed up the +decoding at the same time. +4.3 +Human Evaluation +We further conduct human evaluation to analyze +the generation quality of the Pointer model and +the Pointer + WT model. We randomly sample +200 cases from the SQuAD dataset and ask three +Text: In the March Battle of Fort Bull, French +forces destroyed the fort and large quantities +of supplies , including 45,000 pounds of gun- +powder . +Answer: 45,000 +Reference question: How much gun powder +was destroyed in attack ? +Pointer: How much of gunpowder ’s were +killed in the March Battle of Fort Bull ? +Pointer + WT: How much gunpowder was +destroyed in the March Battle of Fort Bull ? +Table 3: A case where the Pointer model generates un- +related words, i.e., “killed” and “’s”, while the model +incorporating word transformation approach can gener- +ate the question more correctly. +annotators specialized in language to compare the +generation quality. The annotators are shown two +questions, one generated by the Pointer model and +the other one by the Pointer + WT model. They are +asked which one is better by three factors, i.e., flu- +ency, completeness, and answerability. These three +factors are equally important in this evaluation. The +final label is determined by majority voting. By +the metric value, we get that the Pointer + WT +model outperforms the Pointer model. Specifically, +in 73/200 cases, the Pointer + WT model is better +compared to the Pointer model. In 91/200 cases, +the two models behave nearly the same. In 36/200 +cases, the Pointer + WT model behaves worse than +the baseline. +Table 3 gives a case to demonstrate the effec- +tiveness of our approach. As shown in Table 3, +the Pointer model generates unrelated words, i.e., +“killed” and “’s”. It might be because the Pointer +model generates from a large and noisy vocabulary. +However, the Pointer + WT model can generate the +question more correctly, which can be definitely +owed to our approach. +5 +Conclusion +In this paper, we discover two major issues in the +existing neural QG models. To tackle the two is- +sues, we propose this enhancing approach for QG +and apply the approach to two typical sequence- +to-sequence models, i.e., the pointer generator net- +work and UNILM. We further conduct extensive +experiments using SQuAD and MARCO datasets. +The experimental results show that improved ver- +sions of models can significantly enhance the qual- +ity of QG and speed up the decoding. + +References +Husam Ali, Yllias Chali, and Sadid A Hasan. 2010. +Automation of question generation from sentences. +In Proceedings of QG2010: The Third Workshop on +Question Generation, pages 58–67. +Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Ben- +gio. 2014. +Neural machine translation by jointly +learning to align and translate. +arXiv preprint +arXiv:1409.0473. +Zhiqiang Cai, Vasile Rus, Hyun-Jeong Joyce Kim, +Suresh C Susarla, Pavan Karnam, and Arthur C +Graesser. 2006. +Nlgml: A markup language for +question generation. 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In +National CCF Conference on Natural Language +Processing and Chinese Computing, pages 662–671. +Springer. + diff --git a/ZtAyT4oBgHgl3EQfifij/content/tmp_files/load_file.txt b/ZtAyT4oBgHgl3EQfifij/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b18882b573523a8c291239004d17fdc63e157f33 --- /dev/null +++ b/ZtAyT4oBgHgl3EQfifij/content/tmp_files/load_file.txt @@ -0,0 +1,594 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf,len=593 +page_content='Inflected Forms Are Redundant in Question Generation Models Xingwu Sun University of Macau sunxingwu01@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='com Hongyin Tang Meituan Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' tanghongyin@meituan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='com Chengzhong Xu University of Macau czxu@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='mo Abstract Neural models with an encoder-decoder frame- work provide a feasible solution to Question Generation (QG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' However, after analyzing the model vocabulary we find that current mod- els (both RNN-based and pre-training based) have more than 23% inflected forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As a result, the encoder will generate separate embeddings for the inflected forms, leading to a waste of training data and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Even worse, in decoding these models are vulnerable to irrelevant noise and they suffer from high computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In this paper, we propose an approach to enhance the per- formance of QG by fusing word transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Firstly, we identify the inflected forms of words from the input of encoder, and re- place them with the root words, letting the encoder pay more attention to the repetitive root words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Secondly, we propose to adapt QG as a combination of the following actions in the encode-decoder framework: generating a question word, copying a word from the source sequence or generating a word transfor- mation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Such extension can greatly de- crease the size of predicted words in the de- coder as well as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We apply our approach to a typical RNN-based model and UNILM to get the improved versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We conduct exten- sive experiments on SQuAD and MS MARCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The experimental results show that the improved versions can significantly outper- form the corresponding baselines in terms of BLEU, ROUGE-L and METEOR as well as time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 1 Introduction Question Generation (QG) aims to generate natu- ral language questions from given text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' It can aid several applications: (1) QG can help create educa- tional materials by generating questions for read- ing comprehension materials (Heilman and Smith, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2) It can be used to auto- matically curate question answering datasets (Duan Figure 1: Two root words and their corresponding in- flected forms in the vocabulary of QG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (3) It can also aid dialogue systems by actively asking meaningful questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Typically, QG includes two sub-tasks: (1) determine the tar- gets that should be asked (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', sentences, phrases or words), and (2) produce the surface-form of the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In this work, we focus on the sub-task of surface-form generation of questions by assuming that the targets are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Recent neural QG models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', RNN-based sequence-to-sequence models and pre-training based models, have made great progress in generat- ing proper questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In particular, UNILM (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2019), which applies a pre-training model for QG, is the state-of-the-art model with the best per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' However, those models all suffer high computational costs in decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We carefully ex- amine existing neural QG models which have an encoder-decoder architecture in general and find that existing models remain two issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Firstly, in the encoder, the root word and its inflected forms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', “commit” and its inflected forms “committed” and “committing” as shown in Figure 1, might all occur in the vocabulary of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Accord- ing to our statistics, in the widely used QG dataset SQuAD (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2016), the top-10000 fre- quent words contain 3718 inflected forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' More- over, even though UNILM applies the WordPiece segmentation to reduce the vocabulary size, its vo- cabulary still contains about 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='18% (6722/28996) inflected forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As a result, the encoder will gener- ate individual embeddings for these words, which arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='00397v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='CL] 1 Jan 2023 Root Word Inflected Forms commit committed committing commits big bigger biggestseems a little redundant for QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Even worse, they occupy the space in the vocabulary of encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As a result, the encoder will generate separate embed- dings for the inflected words, leading to a waste of training data and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Secondly, in the de- coder, the same large set of words are employed to generate a question regardless of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' There- fore, these models are vulnerable to irrelevant noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Besides, while decoding each word of a question, probability distribution of words in the entire static and fairly large vocabulary have to be calculated, which slows the decoding process down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' To decrease the computational cost and improve the performance of QG, we propose an approach named as word transformation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The ap- proach is inspired by the feature of the inflected language and our daily reading experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The former refers to the fact that words with different inflected forms may divert the focus to QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The latter refers to our observations that most words in the generated question can be copied or simply transformed from the words in the original source sequence, except for the question word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Therefore, in our approach, we treat QG as a combination of word transformation, word copying and question word generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In detail, in the question word generation, a question word is generated from a limited question word vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In the transfor- mation type generation, an inflected form type is generated from a small word transformation type vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For example, type “##ed” may be gen- erated and will be used for transforming a verb to its past tense in the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In the word copy- ing, words are copied from the source sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Besides, we identify the inflected forms of words from the input of encoder, and replace them with their root forms, letting the encoder pay more atten- tion to the repetitive root words and take in more different words for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We have applied our approach to a typical RNN-based model and a pre- trained transformer-based model UNILM and get the corresponding improved versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The main contributions of this paper can be sum- marized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We find that inflected words take up unnec- essary spaces in vocabulary and define a se- ries of word transformation types to facilitate transforming a word into its inflected forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We only keep the root words in vocabulary of the encoder to make the best of the training data and the limited vocabulary space, which is also applicable in many other NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We propose to simplify the decoding process of QG by question word generation, transfor- mation type generation and word copying, to avoid generating each word from a large vo- cabulary and enhance the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For evaluating the effectiveness and efficiency of our proposed approach, we conduct exten- sive experiments on two large scale datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', SQuAD and MS MARCO datasets and compare the baselines with the corresponding improved versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The experimental results show that the improved versions can signifi- cantly outperform the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2 Related Work The existing QG approaches can be broadly classi- fied into two kinds: rule-based and neural network- based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The rule-based approaches rely on hand- crafted rules, while the neural network-based ap- proaches are data-driven and trainable in an end-to- end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Early QG approaches are mostly rule-based ones (Heilman and Smith, 2009, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Chali and Hasan, 2015), which leverage rules or templates to generate questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The rule-based approaches employ manually crafted rules for declarative-to- interrogative sentence transformation, typically based on syntactic (Mitkov and Ha, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Heilman, 2011) or semantic informa- tion (Chen, 2009), while the template-based ap- proaches generate questions using manually cre- ated templates which are predefined with place- holders to be filled with words from the source word sequence (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Lindberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Song and Zhao, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The major limita- tions of these rule-based approaches include: (1) they rely heavily on rules and templates, which are created manually and therefore expensive, (2) these rules or templates lack diversity, (3) the targets they can deal with are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' To tackle the limitations of rule-based ap- proaches, the neural network-based approaches with an encoder-decoder framework are applied to the task of QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' These approaches do not rely on hand-crafted rules, and they are instead data driven and trainable in an end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The release of large-scale machine reading comprehen- sion datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' SQuAD, MS MARCO (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2016), Hotpot QA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2018) and DROP (Dua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2019),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' further drives the devel- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='Part-of-speech ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='Transformation Types ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='Transformation Type Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='verb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##ing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the verb to its present participle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##vs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the verb to its singular present ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the verb to its past tense ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##edp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the verb to its past participle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='noun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the noun to its plural word ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='adjective ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##jer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the adjective to its comparative form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##jest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the adjective to its superlative form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='adverb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##ver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the adverb to its comparative form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='##vest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='converting the adverb to its superlative form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='Table 1: Word transformation types and the corresponding descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' opment of neural QG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In general, these datasets contain large-scale manually annotated triples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', question, answer and the context, and they can be used as training data of QG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for the RNN-based models, both Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017) and Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017) apply a sequence-to- sequence model with an attention mechanism to generate questions for the text in SQuAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017) enrich the RNN-based model with rich features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', answer position and lexi- cal features) to generate answer focused questions, and incorporate a copy mechanism that allows the model to copy words from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017) propose to combine templates and the sequence-to-sequence model, in which they mine question patterns from a question answering com- munity and employ a sequence-to-sequence model to generate question patterns for a given text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017) model question answering and ques- tion generation as dual tasks, which helps generate better questions and get better question answering models at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Further, based on pointer generator network (See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2017), Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2018) propose an answer fo- cused and position-aware model to effectively lever- age answer encoding and position features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2018) mainly focus on incorporating para- graph level context by using gated self-attention and maxout pointer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Nema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2019) give a refine network to mimic human process of generating questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Besides, there has also been some work on generating questions from knowl- edge bases (Serban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Indurthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' More recently, pre-training models are applied to QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For example, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2019) propose UNILM which makes a great progress in QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Cur- rently, UNILM is the QG model with the best per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Differing from the previous work, our approach does not provide a QG model, and instead provides two extensions to the encoder-decoder framework for performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Our approach can be applied to most existing models, including RNN- based sequence-to-sequence models plus UNILM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 3 Our Approach To begin with, we formally give the QG task def- inition as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Given the text (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', xTx) of length Tx as well as lexical features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', named entity (NE) and part-of-speech (POS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The answer positions in this text range from l to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The goal is to generate a question, which is required to be as close as the reference question (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', yTy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In the following subsections, we describe the details of the proposed approach to deal with the issues discussed in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Firstly, we define a series of word transformation types, which is the basis of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Secondly, we choose the pointer generator network and UNILM as baselines, and describe how to apply our ap- proach by elaborating the working process of the encoder-decoder in the improved versions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='1 Word Transformation Types As the basis of our approach, we define the transfor- mation types in terms of the part-of-speech of the word, as listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for a verb, we define four types of transformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', “##ing”, “##vs”, “##ed” and “##edp”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for a noun, we define one transformation type “##ns”, which means convert- ing the noun to its plural word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For an adjective or an adverb, we define the transformation to their comparative form as “##jer” and “##ver”, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Similarly, we define their transformation to the superlative form as “##jest” and “##vest”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Note that we use the Pattern module1 in Python to con- duct these transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for irregular words, we manually collect a lookup table, for instance, “went” can be converted into go and “##ed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Recall the two issues mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for the first issue in the encoder, by a series of transformation types, we only keep the root words and the transformation types in the encoder vocab- ulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As a result, it can substantially save space for other words and make the best of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for the second issue, we directly copy or trans- form the word in the source sequence to generate questions except for the question word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' A transfor- mation type vocabulary is introduced as a necessity in the transformation type generation, which will be elaborated in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Before model encoding, we first get the root form of each word in the original input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For example, in Figure 2, the word “succeeded” is converted into its root form “succeed” and the corresponding transformation type is “##ed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' After transformation, the input sequence is converted into a sequence (x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', x′ T ′x) of length T ′ x and the answer positions range from l′ to r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Similarly, The reference question is converted into (y′ 1, y′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', y′ T ′y) of length T ′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='2 Reforming Pointer Generator Network 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='1 The Encoder The baseline which is reformed is the attention- based pointer generator network (See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2017) enhanced with various rich features proposed by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' These features include named entity (NE), part-of-speech (POS) and an- swer position in the embedding layer of the en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As shown in Figure 2, the encoder is a bidirectional GRU, which takes as input the joint embedding of word, answer position and lexical features (NE, POS) in the form of (w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', wT ′x) with wi ∈ Rdw+da+dn+dp, where T ′ x is the input length, wi is the embedding of x′ i by concatenating all the feature embeddings and dw, da, dn, dp are the dimensionalities of word embedding, answer position embedding, NE em- bedding and POS embedding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' It pro- duces a sequence of dh-dimensional hidden states (h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', hT ′x), each of which is the sum of for- 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='com/clips/pattern ward and backward GRU representations: hi = ←−h i + −→h i, ←−h i = GRU(wi, ←−h i+1), −→h i = GRU(wi, −→h i−1) (1) where ←−h i, −→h i are all dh-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='2 The Decoder The decoder is modified to support three actions: word copying, transformation type generation and question word generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (1) In the word copy- ing, the decoder generates words from the source sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2) In the transformation type genera- tion, the decoder generates from a limited transfor- mation type vocabulary, which only includes nine types in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (3) In the question word genera- tion, the decoder generates question words from a restricted question word vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Word Copying The decoder is a unidirectional GRU conditionally taking all the encoded hidden states as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' At decoding step t, the decoder reads an input word embedding wt, previous at- tentional context vector ct−1 and its previous hid- den state st−1 to update its current hidden state st ∈ Rdh: st = GRU([wt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' ct−1], st−1) (2) The context vector ct is generated through an attention mechanism (Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' At time step t, the context vector ct is calculated as follows: ct = T ′ x � i=1 αtihi αti = Softmax(eti) eti = vT tanh(W T h hi + W T s st + b) (3) where α is the attention distribution and Wh, Ws, b and v are all trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The attention distribution can be viewed as a semantic matching between hidden states of the encoder and the hidden state of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' It indi- cates how the decoder cares about different hidden states of the encoder during decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Therefore, the attention distribution can be regarded as the probability distribution of the word copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pcopy = αt (4) Figure 2: The improved version based on pointer generator network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' “C” represents the context vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Transformation Type Generation Before encod- ing, we have already converted each word to its root form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', the root word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In the decoding, be- sides coping words, we should endow the model the ability to transform some words to their proper forms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', the “do” in Figure 2 is generated from question word vocabulary and we need to trans- form it to “did”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' To carry out this transformation, we introduce the transformation type generation and generate the transformation type for the root word, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', “##ed” is generated to transform “do” to “did”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We get the transformation type distribution by the following function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Ptrans = Softmax (g1(st, ct)) (5) where g1(·) is a two-layer feedforward neural net- work with a maxout internal activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Ptrans ∈ R|Vtrans| denotes the probability distribution of transformation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Question Word Generation We find that the gen- eration of question words is mainly determined by the answer and its surrounding words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For example, in Figure 2, the answer and its context “in 1970” already involve the essential information to gen- erate the question word “when”, which suggests that the answer and its surrounding words can ben- efit the question word generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We also find that the total number of question words is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Therefore, we introduce a specific vocabulary of question words to directly and explicitly model the question words generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Note that the question word vocabulary contains top-1000 frequent words in reference questions of the training set, which means that it also contains other question-related words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As depicted in Figure 2, in the question word generation, the model generates question words based on a restricted vocabulary of question words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' This action produces a question word distribution based on an answer embedding vanswer, the de- coder state st and the context vector ct: Pquest = Softmax (g2(vanswer, st, ct)) (6) where g2(·) is a two-layer feedforward neural net- work, Pquest is a |Vquest|-dimensional probability distribution, and |Vquest| is the size of vocabulary of question words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We employ the average pooling function to calculate the answer embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' vanswer = �r′ t=l′ ht r′ − l′ + 1 (7) Three-action Combination To control the bal- ance among different actions, we introduce a three- dimensional switch probability, acting as a three- way soft switch: pquest, pcopy, ptrans = Softmax(f(ct,st,wt)) (8) where f(·) is a one layer feedforward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We compute the final probability distribution through a weighted summation of the three action probability distributions: P(w) = pcopyPcopy(w) + ptransPtrans(w) + pquestPquest(w) (9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='3 Reforming UNILM UNILM is a pre-trained language model which has the same structure as BERT(Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2019) but can be used for natural language generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The model is firstly pre-trained on a large scale corpus which enables it to learn flexible lan- guage representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Then, it is fine-tuned on the dataset of the downstream tasks using the sequence- to-sequence language model objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In UNILM, there is no explicit separation between the encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In contrast, the source and target se- quence are fed to the model together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The sequence- to-sequence generation is implemented by a special self-attention mask M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' final distribution question word attention transformation distribution distribution type distribution Average Pooling in 1970 president nasser die ##ed and is ##ed succeed ##ed by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' when do ##ed nasser die 个 Transform Infected Words To Their Root Forms Decoder in 1970 president nasser died and was succeeded by Answer EncoderFigure 3: Model architecture of the improved version based on UNILM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Note that there should be L trans- former layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For simplicity, we only keep two of them in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Following Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2019), we con- catenate the answer segment (x′ l′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', x′ r′) and the text segment (x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', x′ T ′x) forming the new source segment X = (x∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', x∗ T ∗ x ) = (x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', x′ T ′x, [EOS], x′ l′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', x′ r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Next, we con- catenate the new source segment and the tar- get segment Y = (y′ 1, y′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', y′ T ′y) as input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', [SOS]X[EOS]Y [EOS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Then we feed the input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In training, we randomly replace tokens with [MASK] in Y at a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For each input token, its embedding is the sum of the token embedding, position embedding and segment embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As shown in Figure 3, we get token embeddings w = (w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', wT ∗ x +T ′y+3), which is then feed into L-layer transformers to get the deep representation Hl = hl i, i ∈ [1, T ∗ x + T ′ y + 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' hl i = Transformer(hl−1 i ), l = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', L (10) where H0 is initialized with w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The output of the self-attention head of the l-th transformer is calcu- lated by: Ql = Hl−1W Q, Kl = Hl−1W K, Vl = Hl−1W V (11) Al = Softmax(QlKT l √dk + M)Vl (12) where M ∈ R(T ∗ x +T ′ y+3)×(T ∗ x +T ′ y+3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Mij = 0 means it is allowed to attend, while Mij = −∞ indicates it is not allowed to attend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We can find that the special self-attention mask M allows the token to be generated to attend to all of the source tokens and the preceding generated tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' This mechanism allows the model to generate text in a sequence-to-sequence fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' To adopt the word transformation approach, we directly compute the distributions associated with the three actions based on the hidden state of the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The details are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Word Copying The attention distribution over the source tokens is obtained by an extra attention mod- ule which computes the attention scores using dot product: Pcopy = Softmax(g3(ht)) (13) g3(ht) = [g3(ht)i]T ′ x i=1 = [ht · hi]T ′ x i=1 (14) where ht is the hidden state of t-th token output by the last layer of transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The output of g3(ht) is the dot product between ht and hidden states of all of the input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Transformation Type Generation The transfor- mation type distribution is obtained by computing the dot product between the current hidden state and the embeddings of the transformation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Ptrans = Softmax(g4(ht)) (15) g4(ht) = [g4(ht)i]|Vtrans| i=1 = � ht · eT i �|Vtrans| i=1 (16) where eT is the embedding of the transformation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' g4(ht) is the dot product between ht and the embeddings of all of the transformation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Ptrans denotes the probability distribution of trans- formation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Question Word Generation Similar to the cal- culation of the distribution of the transformation types, the distribution of the question words is obtained by the dot product between the current hidden state and the embeddings of the question words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pquest = Softmax(g5(ht)) (17) g5(ht) = [g5(ht)i]|Vquest| i=1 = � ht · eQ i �|Vquest| i=1 (18) where eQ is the embedding of the question words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' g5(ht) is the dot product between ht and the em- beddings of all of the question words eQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Three-action Combination The three- dimensional switch probability is obtained by the current hidden state ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' pquest, pcopy, ptrans = Softmax(f1(ht)) (19) where f1(·) is a one layer feedforward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The final probability distribution is computed as the same as Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' final distribution attention distribution question word distribution Transformer Block transformation type distribution Transformer Block Embedding Layer Transform Infected Words To Their Root Forms [SOS] x* [EOS] yi [MASK] [EOS]Model BLEU1 BLEU2 BLEU3 BLEU4 ROUGE-L METEOR Pointer 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='49 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='11 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='94 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='34 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='15 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='71 Pointer + WT 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='08 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='56 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='54 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='41 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='40 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='61 UNILM 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='82 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='48 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='03 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='39 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='02 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='52 UNILM + WT 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='56 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='78 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='06 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='24 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='46 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='93 (a) Model BLEU1 BLEU2 BLEU3 BLEU4 ROUGE-L METEOR Pointer 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='45 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='85 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='32 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='90 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='07 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='02 Pointer + WT 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='14 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='36 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='04 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='10 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='29 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='40 UNILM 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='57 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='14 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='38 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='01 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='58 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='31 UNILM + WT 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='65 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='33 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='31 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='55 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='85 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='72 (b) Table 2: The main experimental results of baselines and improved versions on SQuAD (a) and MARCO (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' “WT” means the proposed word transformation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='1 Experiment Settings Dataset We conduct the experiments on SQuAD and MARCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Since the test sets of these datasets are not publicly available, we follow Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017) to randomly split the development set into two parts and use them as the development set and test set for the QG task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In SQuAD, there are 86, 635, 8, 965 and 8, 964 question-answer pairs in our training set, development set and test set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We directly use the extracted features2 shared by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In MARCO, there are 74, 097, 4, 539 and 4, 539 question-answer pairs in our training set, development set and test set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We use Stanford CoreNLP3 to extract lexical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Implementation Details We set the cutoff length of the input sequence as 128 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The encoder vocabulary contains the most frequent 30, 000 words in each training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The decoder vocabulary contains two sub-vocabularies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' One is the ques- tion word vocabulary which contains most frequent 1000 words in the reference questions of the train- ing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The other one is the transformation type vocabulary, including 9 transformation types as de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For the RNN-based model, we use the pre-trained Glove word vectors4 with 300 dimensions to initialize the word embeddings that will be further fine-tuned in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The representations of answer position feature and lexical features at the embedding layer of the en- coder are randomly initialized to 32 dimensional vectors that are trainable during training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The 2https://res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='qyzhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='me/redistribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' zip 3https://nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='edu/software/ 4http://nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='edu/data size of hidden states of both the encoder and de- coder is 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We use dropout only in the encoder with a dropout rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The size of answer em- bedding is 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We use the optimization algorithm Adam (Kingma and Ba, 2014) with the learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='002 and we set the batch size as 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As for the UNILM, we adopt the base version of the UNILM and use the recommended parameters as detailed by Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' After training, we select the best model on the development set for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Evaluation Metrics We evaluate our approach us- ing n-gram similarity metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', BLEU (Pap- ineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2002), ROUGE-L (Lin, 2004) and ME- TEOR (Lavie and Denkowski, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Competitors In the experiments, we have the fol- lowing four competitors for comparisons, where Pointer and UNILM are baselines and the other two are corresponding improved versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pointer generator network (Pointer) It is a typical RNN-based sequence-to-sequence model with the copy mechanism (See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' To make a fair comparison, the lexical features are added to the embedding layer as same as Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pointer generator network plus Word Transformation approach (Pointer + WT) We enhance the Pointer model with the pro- posed word transformation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' UNILM (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', 2019) It is a pre-trained language model which can be applied to natu- ral language generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' UNILM + WT We combine the UNILM model with the proposed word transformation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pointer and Pointer + WT are implemented with Tensorflow, while UNILM and UNILM + WT are implemented by PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='2 Performance Evaluation Table 2 shows the main results, and we have the following observations: The Pointer + WT model performs better than the original Pointer model which indicates that generating questions by word transforma- tion is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' UNILM + WT model can still significantly outperform the powerful UNILM, which in- dicates the effectiveness of applying our ap- proach to the pre-trained language models for QG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Our approach makes greater improvement over MARCO than SQuAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' That might be because MARCO is extracted from the search engine, which induces a bias over word copy- ing and word transformation generation from source text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Besides generation quality, we also compare baselines with the corresponding improved ver- sions on efficiency of decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We record the time of generating a word of a question given the test text with a beam size of 12 and calculate the average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Note that we regard the decoding of root word and its corresponding transformation type as one word in efficiency comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The comparison is conducted on a GPU environment with a single Tesla V100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For the RNN-based mod- els, the Pointer + WT model is more efficient which can save more than 39% (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='0081s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='0049s) decoding time compared to original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' For the pre-trained models, the UNILM + WT model can save 13% decoding time (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='0144s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='0125s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We also find that UNILM behaves better than Pointer on generation quality, but it sacrifices efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Specifically, it is nearly one time slower than the RNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' From the compar- isons on both effectiveness and efficiency, we can conclude that this word transformation approach can enhance the quality of QG and speed up the decoding at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='3 Human Evaluation We further conduct human evaluation to analyze the generation quality of the Pointer model and the Pointer + WT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We randomly sample 200 cases from the SQuAD dataset and ask three Text: In the March Battle of Fort Bull, French forces destroyed the fort and large quantities of supplies , including 45,000 pounds of gun- powder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Answer: 45,000 Reference question: How much gun powder was destroyed in attack ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pointer: How much of gunpowder ’s were killed in the March Battle of Fort Bull ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Pointer + WT: How much gunpowder was destroyed in the March Battle of Fort Bull ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Table 3: A case where the Pointer model generates un- related words, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', “killed” and “’s”, while the model incorporating word transformation approach can gener- ate the question more correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' annotators specialized in language to compare the generation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The annotators are shown two questions, one generated by the Pointer model and the other one by the Pointer + WT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' They are asked which one is better by three factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', flu- ency, completeness, and answerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' These three factors are equally important in this evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The final label is determined by majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' By the metric value, we get that the Pointer + WT model outperforms the Pointer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Specifically, in 73/200 cases, the Pointer + WT model is better compared to the Pointer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In 91/200 cases, the two models behave nearly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In 36/200 cases, the Pointer + WT model behaves worse than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Table 3 gives a case to demonstrate the effec- tiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' As shown in Table 3, the Pointer model generates unrelated words, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', “killed” and “’s”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' It might be because the Pointer model generates from a large and noisy vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' However, the Pointer + WT model can generate the question more correctly, which can be definitely owed to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 5 Conclusion In this paper, we discover two major issues in the existing neural QG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' To tackle the two is- sues, we propose this enhancing approach for QG and apply the approach to two typical sequence- to-sequence models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=', the pointer generator net- work and UNILM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' We further conduct extensive experiments using SQuAD and MARCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' The experimental results show that improved ver- sions of models can significantly enhance the qual- ity of QG and speed up the decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' References Husam Ali, Yllias Chali, and Sadid A Hasan.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Domain-specific question generation from a knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' CoRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Xingwu Sun, Jing Liu, Yajuan Lyu, Wei He, Yanjun Ma, and Shi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Answer-focused and position-aware neural question generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In Pro- ceedings of the 2018 Conference on Empirical Meth- ods in Natural Language Processing, pages 3930– 3939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Duyu Tang, Nan Duan, Tao Qin, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Question answering and question generation as dual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='02027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Ben- gio, William W Cohen, Ruslan Salakhutdinov, and Christopher D Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Hotpotqa: A dataset for diverse, explainable multi-hop question answer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='09600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Xingdi Yuan, Tong Wang, Caglar Gulcehre, Alessan- dro Sordoni, Philip Bachman, Sandeep Subrama- nian, Saizheng Zhang, and Adam Trischler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Machine comprehension by text-to-text neural ques- tion generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' arXiv preprint arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content='02012.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Neural ques- tion generation from text: A preliminary study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' In National CCF Conference on Natural Language Processing and Chinese Computing, pages 662–671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfifij/content/2301.00397v1.pdf'} diff --git a/aNAzT4oBgHgl3EQf2f7B/vector_store/index.faiss b/aNAzT4oBgHgl3EQf2f7B/vector_store/index.faiss new file mode 100644 index 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University of Central Florida, 4304 +Scorpius St., Orlando, Florida 32816, USA. + +†Equal contribution. + +Correspondence and requests for materials should be addressed to D.C. +(email: Debashis.Chanda@ucf.edu). + + +In nature, vibrant colors as those of many butterflies, birds, octopuses, or fishes, arise +from microscopically textured surfaces. These vivid colors result from the coherent +interaction between light and the structural arrangement of colorless materials found +in their skin. In contrast, all manmade colors are pigment based and rely on the +molecular absorption of their constituents, with each color requiring a different +molecule. While traditional pigment-based colorants offer a viable commercial +platform for large-volume and angle-insensitiveness, they are limited by their +resolution, instability in atmosphere, color fading, and severe environmental toxicity. +These widely used pigment-based paints are destroying the environment, aquatic life, +and adversely affecting global warming by working as heat traps. However, till date +all attempts to industrial production of polarization and angle independent full-range +structural colors have failed due to the angle-dependent colors and fabrication +challenges. Here, we present a subwavelength plasmonic cavity that generates color +by the hybridization of a metallic self-assembly with an ultrathin optical cavity. This +configuration offers both polarization and angle insensitiveness, while simultaneously +providing a full-color gamut of vibrant structural color paints. In this work, we +presented this unique structural color generation mechanism and demonstrated color +generation of these structures across the entire visible spectrum by tuning just the +structural parameters. Further, akin to traditional mixing of different pigments to +produce new colors, we demonstrated in a unique way lateral as well as vertical +“mixing of structures” to expand the available color palette. The self-assembly +facilitates the growth of the structure on large areas and non-conventional substrates +in both diffuse and specular coloration modes. Growing the stack on a sacrificial layer + +we produce a self-standing nanostructured color platform that, when mixed with a +binder, can be transferred to any surface to impart full coloration with a single sub- +micron layer of pigment. This structural color platform offers a highly integrable +ultra-lightweight solution that bridges the gap from proof-of-concept to real-world +industrial applications of non-toxic, fade resistant, and environmentally friendly +colorants. + +Introduction +Color presents one of the richest sources of sensorial information in our daily lives. +Throughout history, the fascination with colors has driven human efforts to produce newer +and better colorants. From the Paleolithic cave paintings to the development of the first +synthetic dyes in the mid nineteenth century the quest for purer, fade-resistant, and +environmentally-friendly colorants has remained very active. In the last decades, adding to +purely decorative applications in textile, cosmetics, or food industries, colorant research +has found relevance, among others, in display technologies, optical storage, sensing and +therapeutics, or functional coatings1,2. +Color engineering can be achieved by controlling the colorant’s absorptive or +reflective response to white light. All commercial colorants/pigments are based on +absorption mechanisms. These colorants absorb photons of energies overlapping with their +molecular electronic transitions. Contrarily, photons with energies not matching these +discrete transitions will be reflected and registered as color by an observer. Although +chemical colorants can be produced in large amounts, most of them are composed of toxic +materials difficult to remove in the recycling process and are responsible for the pollution + +of our lifeline on earth—water3. Being chemically unstable, many colorants fade with time, +a process accelerated with higher temperatures or light exposure. Furthermore, as volumes +of several microns are needed to obtain enough color saturation, they suffer from low +resolution. In contrast, instead of controlling the absorption of light, structural colorants +control the way the light is reflected or scattered. Structural color is the result of optical +phenomena produced by micron- and nano-scale structures4. Remarkably, when in bulk, +the material constituents show completely different hues or are even colorless. Colors +generated by engineered structures such as photonic crystals5–9 or metasurfaces10–13, have +received increasing attention in recent years for their striking advantages over chemical +colorants. Characterized by their intense brilliance and saturation, they exhibit larger +stability to chemical reagents, harsh environmental conditions, and high illuminating +intensities14,15. Additionally, they can offer dynamic tunability and resolutions beating the +diffraction limit, both essential for display applications16–18. Due to the geometrical nature +of their response, however, structural colors usually present directional effects, i.e. their +color varies with the positioning of the observer and the angle and polarization of the +incident light. More importantly, many proposed architectures rely on the use of costly and +low-throughput nanofabrication techniques not compatible with mass-production. Overall, +these constrains prohibit the commercial viability of all previously reported structural color +technologies. It is therefore not surprising that, to date, no angle-independent structural +paints are available in the marketplace19. +Here, we present a subwavelength plasmonic cavity that overcomes these +challenges while offering a tailorable platform for rendering angle and polarization +independent vivid structural colors by coupling incident light with gap-plasmons. The + +structures are fabricated through a large-area, highly versatile, and reproducible technique +where aluminum nanoislands are self-assembled in an electron beam evaporator on top of +a transparent oxide-coated aluminum mirror. The optical response of these artificially +engineered nanostructures can be spectrally tuned across the entire visible spectrum to form +a full color gamut by controlling the gap-plasmon dispersion via the geometrical +parameters. In the proposed architecture the subwavelength optical cavity ensures a large +degree of angle insensitivity while the stochastic nature of the self-assembled layer results +in polarization independence and near 100% absorption at selected spectral bands. The +evaporation process, relying only on widespread industrial techniques, is compatible with +many substrates, and takes on their scattering properties to render diffuse and specular +coloration modes when utilizing micro-corrugated or flat surfaces, respectively. E-beam +evaporators are widely employed in industries such as electronics, semiconductors, optics, +and even aerospace, to name a few. Moreover, Lexus Blue, the only industrially produced +simple Fabry-Perot resonance based structural color is actually fabricated with ebeam +evaporators20 We present mechanisms for expanding the available color space through +lateral and vertical mixing of structures, similar to traditional pigment mixing schemes. +Finally, to demonstrate the commercial capabilities of our platform for inorganic metallic +structural coloration, we formed bi-directional structures on a water-soluble sacrificial +layer that resulted in omni-directional color flakes. These structural color flakes were then +mixed with a commercial binder to develop self-standing structural color paints hundreds +of times lighter than commercially available paints21. Conventional chemical coloration +relies on volumetric absorption of light to produce a color. In contrast to the several microns +required for commercial paints, our ultrathin paint can impart full coloration with a + +thickness of only 150 nm. Consequently, this huge lateral area (few 10s of µm) to thickness +(100 – 150 nm) ratio makes it the lightest paint in the world with a surface density of only +0.4 g/m2. For comparison, while a Boeing 747 requires 500 kg of paint22, our ultralight +paint would require about 1.3 kg, an astonishing potential about 400-fold reduction in +weight, SI Appendix I. Our approach presents the first environmental-friendly, large-scale, +multi-color, and self-standing platform for imparting nanostructured coloration to any +surface, thus bridging the gap from proof of concept to industrial production. +Results +Self-Assembled Plasmonic Surface. Nature presents a rich variety of both chemical and +structural coloration. For example, the pink tint of Formosa azaleas, Figure 1a, is due to +the absorption of cyaniding molecules, a type of anthocyanin pigment23. In contrast, the +bright metallic blue displayed by the Peruvian Morpho didius, Figure 1b, is primarily the +result of the way the blue components are scattered by the lameallae nanostructures found +in this butterfly’s wings24. Oftentimes, however, structural color in animals results from the +combination of the diffraction and scattering of the outer skin layers, and the molecular +absorption of the complementary color by intrinsic pigments of the skin25. This critical +observation inspired us to produce an absorptive structural pigment where the selective +absorption of specific frequencies is the result of the tailored structural resonant response +of metallic nanostructures coupled to a subwavelength optical cavity. Specifically, the +proposed architecture consists of a highly-packed monolayer of self-assembled aluminum +nanoislands on a thin aluminum oxide film that spaces them from the aluminum back- +mirror, Figure 1c. In this configuration the aluminum nanoislands resonantly absorb + +specific wavelengths, while the back mirror strongly back-reflects the non-resonant ones, +rendering vivid colors based on colorless materials. +Contrary to other artificial structural schemes that rely on the use of low- +throughput, multi-step, top-down techniques such as electron beam lithography or focused +ion beam, incompatible with mass-production, the proposed architecture is the result of a +naturally occurring nucleation process in an electron beam evaporator. In the self-assembly +growth, small clusters of aluminium nanoparticles are formed due to the larger affinity of +the aluminium atoms to their own kind over the oxide substrate. With a low enough rate, +the evaporation of nanometric films results in a nanoparticles’ monolayer that exhibit +optical plasmonic resonances. Crucially, this pressure- and temperature-controlled process +ensures high reproducibility over broad areas in a single step, lowering the cost of +production and enabling large-scale fabrication. The dynamics of the self-assembly process +Figure 1 | Structural Absorption for Color Generation. a, Many chemical substances produce color by +selectively absorbing frequencies matching their molecular electronic transitions. Pink color in Formosa +azaleas is due to the absorption of cyaniding molecules. b, An example of structural coloration is found in +the Peruvian Morpho didius. Lamellae nanostructures found in its wings scatter the blue components of +incident light generating its characteristic metallic blue. c, A subwavelength plasmonic cavity formed by a +self-assembly of metallic nanoislands on top of an oxide-coated mirror, generates color by selectively +absorbing certain wavelengths and strongly back-reflecting other. + +are presented in detail in SI Appendix A, while the technical parameters can be found in +Methods. +Optical Response of the Near-Field Coupled Gap Plasmons. The color produced in the +nanostructure is the result of the hybridization of the absorptive response of the aluminum +self-assembled monolayer, and the subwavelength cavity formed by this top layer, the +aluminum back-mirror, and the dielectric spacer sandwiched in between. Geometrical +changes in any of the layers will then result in a change in the perceived color. When +ambient light reaches the monolayer, the electric field of the light at select wavelengths can +drive the free electrons of the aluminum to oscillate resonantly within the nanoparticles’ +geometry. This collective oscillation, termed localized surface plasmon resonance, is +further affected by the coupling between closely-packed neighboring particles and the +presence of the back-mirror interface at a subwavelength distance from the particles’ layer. +This complex hybridization mechanism results in a gap-plasmon mode that leads to strong +optical absorption and tight confinement of the light at the metal/dielectric boundary of the +metallic particles at resonant frequencies26. The spectral position of the absorption band, +and thus the perceived color, depends distinctly on the gap-plasmon dispersion which is +hence controlled by three parameters: (1) the size and spatial distribution of the +nanoislands, (2) the refractive index of their environment, and (3) the thickness of the +spacing layer. +The size of the nanoislands can be simply controlled by tuning the amount of aluminum +evaporated. To investigate the range of colors available with this cumulative process we +use a shutter that controls the partial exposure of the sample during the evaporation. In this +manner, by rotating the sample, we can produce a polar gradient of thicknesses from 0.5 + +nm to 16 nm, in thickness increments of 0.5 nm corresponding to wedges of approximately +11°, Figure 2b. As the thickness mass is increased neighboring nanoislands coalesce to +form larger particles, Figure 2a-top. This increase in the nanoislands’ size red-shifts the +absorption band and results in different hues and saturations that produce a color palette +that covers from the white of the back mirror, at very low thicknesses, to the yellow, +magenta, and blue. It should be noted that, being a subtractive color scheme, a red-shift of +the absorption band results in a blue-shift in perceived color, as the intensity of blue +components in the reflected light augment at the expense of the yellow and red ones. If the +Figure 2 | Color Space based on Tunable Gap Plasmon Dispersion. a, In the Volmer-Weber growth mode, +the size of the nanoislands can be controlled by tuning the amount of aluminum evaporated –top-. If the +process is carried on for long enough semicontinuos films are formed that disable the plasmonic resonances +and thus the color –bottom-. b, Color polar gradient for thicknesses from 0.5 to 16 nm, for fixed 10 nm spacer. +c, CIELAB coordinates for the points in the color wheel compared to ISO DIS 15339-2 cold-set newsprint +and coated premium paper standards (inner and outer hexagon). d, Tuning of the spacer and capping layer +thicknesses expands the available color space. e, f, Show the red-shift of the absorption resonance as the +spacer and capping layer thicknesses are increased. + +a +b +c +100 +Nanoisland Growth +50 +. +0 +0.5nm +8 nm +-50 +Continuous Film Growth +-100 +-50 +0 +50 +100 +tm +a* +100 +L +d +. +0 +8.5 nm +16 nm +2 +6 +10 +14 +, (nm) +tm (nm) +tte +e +f +tts. +tt. +t ts +30 +100 +100 +25 +(wu) +75 +75 +20 +ts: 10 +te: 0 +78nm +ts +50 +50 +15 +6nm +25 +25 +10 +30 +10 +4 nm +0 +0 +0 +2.5 +5 +7.5 +10 +450 +550 +650 +750 +450 +550 +650 +750 +t。 (nm) +Wavelength (nm) +Wavelength (nm)process is carried on for long enough, adjacent nuclei can coalesce to form semi-continuous +films and, eventually, continuous films, Figure 2a-bottom. The thickness at which the +transition from isolated islands to continuous film occurs is the percolation threshold. At +thicknesses above the percolation threshold the free electrons of the metal can find paths +to move through the self-assembly, eliminating the geometrical confinement necessary for + +Figure 3 | Morphology Effect on Optical Response. a, The inhomogeneous broadening of the optical +resonance can be accounted for by introducing size and spatial variability. We simulate the broadening by +averaging the reflection curves of 50 particles with radii within 4 standard deviation of the mean value +obtained from SEM analysis. b, Statistical radii distribution -top- and reflection curves -bottom- for +experimental (solid), FDTD mean value (dashed), and weight averaged (dotted). c, Reflection curves for +FDTD simulations corresponding to 7x7 hemispherical particles with equal size in periodic and disorder +arrangement, and random size and disordered arrangement, equivalent to the 5 nm self-assembly. d, Electric +profiles in three different spectral positions as labeled in panel c. + +Periodic +Disordered +Random +wu +425 +II +425nm +wu +705 +Ithe resonant plasmonic absorption and thus disabling the color production. This can be +observed in the gradient wheel sample at higher thicknesses where the blue fades to white, +Figure 2b. Further details on the growth dynamics can be found in SI Appendix A. +Together with the spectral shift, the increase in the thickness mass results in a +broadening of the optical resonances. We attribute this phenomenon to the inhomogeneous +broadening of the nanoparticles’ resonances arisen from the doubly random nature of both +the morphology and spatial distribution, as can be seen in Figure 3. On the one hand, as +thickness mass increases, larger variability of island size can be observed, SI Appendix +Figure 1a. To further assess this effect, we build a semi-analytical model that defines the +total reflection of the monolayer by weight averaging the reflection of periodic islands of +50 hemispherical radii within 4 standard deviation of the mean value as obtained from the +SEM analysis for the 8 nm thickness mass, Figure 3a and SI Appendix B. This larger +morphological variability translates into a reduction in the reflection contrast and saturation +of the colors produced, with a distinct broadening of the resonance, Figure 3b. On the other +hand, the effect of the spatial distribution can be explained by the well-known dependency +of relative position of interacting plasmonic resonators27. To evaluate this latter effect, we +run a set of simulations for 7x7 hemispherical nanoparticles, for equivalent thickness mass +of 4 nm on top of a 10 nm oxide spacer, in periodic square array, disordered array, and +disordered randomized sizes, Figure 3c. The reflection curves show additionally a spectral +shift that we associate with the different energies of the new available modes resulting from +the laterally-hybridization of nanoparticles, modes otherwise forbidden in the symmetric +arrangement, SI Appendix C. These assumptions are indeed confirmed from the comparison +of the electric profiles in-resonance where we observe the strongly confined fields + +characteristic of the gap plasmon modes, for both ordered and disordered arrangements, as +seen in Figure 3d. Interestingly, we observe clearly that while for the ordered structure the +dipolar resonance is only excited at in-resonance wavelength, both disordered and random +disordered configurations show excitations even well outside the in-resonance spectral +position. The inhomogeneous broadening observed is also in good agreement with the +expected behavior predicted by the classical formula for dipole-dipole interaction energy +given by28: +������������ = ������������������������������������������������ +|������������1||������������2| +������������������������2|������������12|3 +(1) +where ������������������������ is the Coulomb constant, ������������������������ the orientation factor, ������������������������ the refractive index of the +environment, |������������1| and |������������2| the modulus of the dipole moments for two interacting +particles, and |������������12| the modulus of the distance between them. In this near-field +approximation, considering two neighbor particles interacting, if their sizes, and also +shapes, show a large variability, it is expected that the dipole modes corresponding to a +given illuminating wavelength will be indeed weakly excited, and consequently lower +absorption will result with a poorer reflection contrast. Furthermore, the spatial disorder +broadening can be understood by averaging the distance between particles, where some of +them will be constructively interfering, while others will be out of phase and thus +destructively interfering. Finally, it should be noted that unlike transmissive colors, +emitting out of a source, all subtractive colors lack purity (paper print vs. LED displays). +In this case the high-density packing of the self-assembly plays a critical role in bringing +the hybridized modes to the visible range, while ensuring vivid coloration in a single +nanometric layer, it exacerbates the resonance broadening resulting from the dynamic +depolarization of non-spherical particles, see SI Appendix D. Although, all factors + +considered, spectrally purer colors could be achieved with pre-treatment steps prior to the +self-assembly growth, this would be achieved at the expense of the fabrication simplicity +offered in our approach. To better understand the coupled mechanism, we also develop a +theoretical model, and compared it with FDTD simulations, results of this can be found in +SI Appendix D. +To assess the quality of the color gamut generated by our self-assembled plasmonic +structure, we calculated the L*a*b* coordinates from the reflection spectra of each +thickness in the gradient sample (see Methods), and plot them as black dots in the CIELAB +color space, Figure 2c. To compare with two color quality standards used in the printing +industry, we overlay the standards for the cold-set newsprint and coated premium paper +technologies as defined in ISO DIS 15339-2 (inner and outer hexagon respectively). For a +substantial portion of the color space, we find that the self-assembled plasmonic color +exceeds the newsprint standard, and even matches the quality of some colors as produced +in coated premium paper. However, although the color space of the plasmonic structure +can be expanded in some regions by careful selection of the other geometrical parameters, +due to its subtractive nature, the production of green is prohibited for a single particle layer. +To address this limitation, in sections below, we introduce two different color mixing +schemes. +Due to the strong field confinement in the metal-dielectric interface plasmonic +resonances are extremely sensitive to changes in the environment28,29. The addition of a +capping layer on top of the self-assembly presents an opportunity to further tune the color +response by shifting the resonant spectral position of the nanoislands’ layer. For samples +corresponding to 4, 6, and 8 nm mass thicknesses, and fixed 10 nm-thick oxide spacer, we + +monitor the color change as we grow capping layers of alumina in 2.5 nm increments, +Figure 2d. Reflection curves for the 6 nm samples can be seen in Figure 2f. Clearly, the +presence of the capping layer red-shifts the plasmonic resonance producing colors with +higher blue components. This behavior is captured by the classical formula for the dipole- +dipole energy interaction presented in SI Appendix D. As the thickness of the capping layer +increases, more energy is contained within the higher dielectric media and the particle- +particle interaction weakens. This causes lower hybridization energies and results in higher +resonant wavelengths. The effect of this top layer is of particular importance from the +applications point of view. Although aluminum, due to its native oxide layer, is very +chemically stable in atmosphere, we found the structures to be fragile to harsh +contaminants and physical contact. To address this, we capped samples with a commercial +polyurethane clear coat (DuraClear Varnish, Americana). Interestingly, these samples still +maintained vivid colors while offering protection to physical contact and larger chemical +resistance to spills as can be observed in SI Appendix E. +The final element that controls the optical response of the structure is the spacer +defined by the thickness of the transparent aluminum oxide spacer layer. As shown in +Figure 2d for samples corresponding to 4, 6, and 8 nm mass thickness and varying spacers +from 10 to 30 nm, changes in the spacer thickness result in pronounced color changes in +the structure. For the 6 nm nanoparticles’ layer the reflection curves are shown in Figure 2e. +We observe that as the spacer is increased the resonance is shifted to longer wavelengths +and the overall reflection levels increase producing less saturated colors. We explain the +behavior of the multilayer stack using interference theory of a non-symmetric +subwavelength cavity, where the bottom mirror and the top nanostructured self-assembly + +form the two limiting interfaces, and the ultra-thin dielectric (alumina) spacer sandwiched +between them which controls the vertical coupling between two metallic layers. This +configuration is essential to achieve the almost-100% levels of absorption in the +nanostructured plasmonic self-assembled layer, which occur only when field-enhancement +occurs at the nanoparticles layer for wavelengths that fulfill the phase matching condition30. +In contrast to conventional Fabry-Perot resonators, where the phase is simply accumulated +through the propagation in the dielectric and the resonant condition can only be fulfilled +for cavity lengths proportional to the wavelength of light, the dispersive nature of the gap- +plasmon mode excited on the self-assembled Al nanoislands introduces an interface with +non-trivial phase shifts and high losses that can produce absorption resonances even for +deeply subwavelength thicknesses well below the resonant wavelength. As the thickness +of the spacer is further increased the mismatch between phases results in weaker absorption +response and renders less saturated colors. The different nature of this near-field coupled +gap-plasmon mode compared to a far-field Fabry-Perot mode, is further verified for large +enough spacer thicknesses. When the dielectric spacing layer takes values large enough +(multiples of ������������/4������������������������), far field effects become dominant and the phase accumulated through +propagation can fulfill the resonant condition, as in typical Fabry-Perot resonators, +resulting in a sharp dip in reflection, SI Appendix Figure 10. Although this resonance offers +colors with higher saturation, the pure geometrical nature of the mode makes it highly +angle-dependent, thus limiting greatly its practical applications, offering further proof of +the fundamental advantage of the near-field coupled gap-plasmon engineered inside this +novel self-assembled ultra-thin structure which is exploited here. + +A Versatile Platform for Structural Coloration. Growing the structure with +conventional evaporation techniques at low temperatures permits the use of a wide variety +of substrates. To prove the versatility of the proposed plasmonic self-assembled structure +we produced multicolor butterflies by growing several stacks on wing-shaped polyethylene +terephthalate (PET) templates, Figure 4a. The polarization and angle-insensitiveness of +these color structures readily shows their superiority over many other reported structural +approaches. On the one hand, the polarization independency arises from the isotropic +character of the disordered self-assembled layer, where nanoislands show no predominant +direction or orientation of growth. In Figure 4b we show how, as expected, when +photographed with unpolarized, and two orthogonal linearly polarized states, the butterfly +assembly shows no appreciable color difference. This particular feature of the multilayer +structure is highly important for integration in devices that rely on the use of polarized light +such as liquid crystal displays. On the other hand, the subwavelength character of the cavity +makes the structure pretty color insensitive to the angle of incidence. Photographies of the +blue artistic butterfly at three different combinations of zenith and azimuth angles show +clearly that the color is retained regardless of the angle of incidence, Figure 4c. Indeed, +upon further study, SI Appendix F, we observe the structures retaining their color for angles +as large as 60°. +The adaptability to different substrates of this unique large-area, self-assembling +based fabrication method paves the path towards the integration of the stack in elastic +platforms without the loss of color quality. We grow three samples with 5, 8 and 12 nm +nanoparticles’ layers, and fixed 10 nm aluminum oxide spacer, on top of aluminum coated +polyethylene terephthalate (PET) strips. These three configurations, corresponding to the + +three primaries in the CYM color mode, can be seen in Figure 4d. Although vivid and +brilliant, the specular coloration observed in flat substrates is inconvenient in many +applications. For such cases, corrugated substrates can be used to produce diffuse +coloration mode. In the diffuse coloration mode, careful texturing of the substrate can +control the degree of dispersion of light reflected. We produce diffuse coloration by +growing the nanostack on sandblasted PET strips, Figure 4e. In contrast to flat substrates, +that result in specular coloration mode, the use of microtextured substrates result in +surfaces that homogenously diffuse the light without inconvenient light streaks of specular +reflection, while retaining angle and polarization insensitiveness. +Figure 4 | Dual Color Mode, Polarization Independence, and Angle-Insensitiveness. a, Butterfly garden +with an assorted collection of different butterfly wings and colors. b, An artistic butterfly model coated with +structural blue retains its color when photographed with unpolarized –left-, and two orthogonal linearly +polarized states –center and right-. c, The butterfly color is also angle-insensitive, as shown for three different +combinations of azimuth and zenith angles. d-e, The versatility of the self-assembly fabrication process +permits the use of a wide array of substrates. Flat and sandblasted PET strips are used as flexible substrates +to form the three primaries in both d, specular, and e, diffuse coloration mode. + +Polarization Independence +AngleIndependence +SpecularColoration +DiffusiveColorationExpanding the Color Gamut with Mixing of Structures. Changes in the geometrical +parameters can be introduced to tailor the color response of the structure. Often, however, +the production of a larger color palette is difficult, due to the limitation of primary colors. +Guided by the principle of conventional color mixing where multiple pigments are mixed +to produce secondary colors, we demonstrated in a unique way production of new colors +by the “mixing of structures” without needing new materials. Growing side-by-side patches +covered with 5 and 10 nm mass thickness nanoislands, we controlled the final color +appearance by careful selection of the ratio of the area covered by each one of the particles’ +configurations. We define the control parameter ������������ as the ratio of the area covered by the +10 nm equivalent nanoislands to the total area covered by both configurations, Figure 5a. +Using a lithographic mask we define subpixels of 100 µm length and variable 0 to 100 µm +width, in steps of 10 µm, to be covered by 10 nm nanoislands. The rest of the area is then +covered by the 5 nm mass thickness nanoislands producing samples with ������������ values ranging +from 0 to 100. In this manner we fabricated three samples for cavity length values of 10, +15, and 20 nm, Figure 5b. The pixel geometry is purposely selected to be in a chess-board +arrangement with pixels smaller than 100 µm to reduce chromatic aliasing and produce +smooth colored surfaces to the naked eye. Microscopy insets for selected samples can be +seen in Figure 5c. This side-by-side mixing mechanism can be explained by a simple +additive rule. For given reflection curves, ������������������������ and ������������������������, corresponding to the two color bases +A and B with mixing ratio ������������, the total reflection is given by: +������������������������������������������������ = (1 − ������������) ∙ ������������������������ + ������������ ∙ ������������������������ +Reflection curves for the mixtures with fixed 10 nm-oxide layer can be seen in Figure 5d, +where we observe the transition from one basis to the other. This is even clearer in + +Figure 5e, where we have plotted as black dots the L*a*b* coordinates corresponding to +the ������������ values from 0 to 100. For context, the colorspace defined by the thickness wheel +analyzed in Figure 2c is overlaid as dotted white lines. Conveniently enough, any color +contained in the segment defined by the two coordinates corresponding to the bases can be +generated by careful selection of in-plane mixing ratios, SI Appendix G. +Figure 5 | Mixing of Structures to Expand the Color Gamut. a, Controlling the ratio of area covered by +two configurations the reflection curve can be defined by a simple additive rule. b, Camera pictures of +samples with mixing ratios from 0 to 100%, for spacer thicknesses of 10, 15, and 20 nm. c, Microscope +images for the samples highlighted in b. d, As the ratio is increased the reflection curves transition from pure +basis A to pure basis B. e, CIELAB space for the samples corresponding to spacer thickness of 10 nm in +panels b and g. The white dotted line overlay represents the space defined by the color wheel in Figure 3c. f, +New colors can be generated by multilayer structures. g, Green shades inaccessible with a single layer can +be generated by stacking two self-assemblies with different interspacing thicknesses. h, Tuning of the +interspace layer between self-assemblies controls the optical response of the cavity. + +a +b +In-plane +FillingFactorα (%) +Colour Mixing +20 +m +Spacer t, (nm) ++15 +'m +II +III +IV +Area () +α= +Total Area (+) +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +c +d +e +100 um +α +100 +90 +wug +a +45 +50 +：10nm +II +25 +0 +III +0 +tm +IV +450 +550 +650 +750 +Wavelength(nm) +-45 +-45 +0 +45 +90 +a* +f +6 +h +Out-of-plane +Interspacingt,(nm） +100 +Colour Mixing +.6 +75 +50 +A +B +D +R +25 +F +0 +450 +550 +650 +750 +4 +Wavelength(nm) +10.5 +12.5 +15.0 +17.5 +20.0 +22.5In-plane mixing does expand the color palette by offering a route to generate any +color contained in the region defined by the basis employed. However, it does not permit +to generate colors outside of its boundaries. Generating green shades would therefore +require a green basis. Yet, due to its subtractive nature, the production of green is prohibited +for the plasmonic self-assembly. However, this limitation can be broken by growing +multilayers of plasmonic nanoparticles, Figure 5f. In this multilayer configuration two +extra geometrical parameters are introduced to control the color appearance: the thickness +mass of the extra layer and the interspace between the nanoislands films. We produce a +wide variety of green shades by growing, on top of a base structure consisting of an +aluminum mirror, 10 nm oxide layer, and 10 nm equivalent nanoislands, three top layers +corresponding to 4, 5, and 6 nm self-assembled layers with oxide interspaces ranging from +10.5 to 22.5 nm, Figure 5g. The reflection curves for the 5 nm equivalent top layer can be +seen in Figure 5h, while the L*a*b* coordinates for these curves are plotted as dots in +Figure 5e, where we observe how the out-of-plane mixing scheme does indeed expand the +color palette to areas otherwise inaccessible with a single plasmonic layer. Although the +levels of reflection in the bilayer structures are low, due to the double absorption of the +two-fold plasmonic layer, careful study of all geometrical parameters can help mitigate +partially this effect. Indeed, the interspace and top-layer geometrical parameters add to the +bottom self-assembly and the spacer layer thickness to offer extra degrees of freedom to +expand the available color gamut introducing new colors with different saturation and +luminance, SI Appendix H. Interestingly, from the industrial point of view, these new layers +do not increase excessively the complexity of the manufacturing, as they can be grown in +the same chamber as a single process step, with little extra time or cost. The addition of an + +extra plasmonic film can also be incorporated to the analytical model by adding on top of +the single stack two extra layers: the dielectric interspace and an extra effective medium +describing the upper plasmonic layer. +Structural Color Paint. The fabrication process used to grow the self-assembled structure +permits an easy integration in many industrial processes that already use compatible +systems. Despite this, the platform is fundamentally limited by the need to grow in-situ +structures. This restricts critically the applicability in contexts where non-vacuum- +compatible substrates are required or instances where large areas need to be covered, as the +limiting factor would always be the particular specification of the evaporation equipment +used. To present a realistic alternative to commercial chemical colorants the multilayer +structure should ideally be available in a stand-alone platform that can be transferred, after +fabrication, to any substrate. On top of a sacrificial layer we evaporate sequentially a +double-sided mirror-symmetric stack, where each side comprises an alumina protecting +capping layer, a plasmonic self-assembly and an alumina spacer, while the mirror is shared +between them. Removal of the sacrificial layer at the end of the fabrication results in +tunable self-standing doubly-colored flakes, Figure 6. We chose to grow the structures +symmetrically to ensure homogeneous colorization, however, flakes can be grown in +asymmetric configurations, each face showing a different color, to render new mixtures +similar to the in-plane mixing color scheme explored before. Once the structures are flaked +off of the substrate, the flakes can be stored dry in powder form, Figure 6b, or kept in an +organic solvent (here we used acetone), Figure 6c. After lifting off, flakes present irregular +shapes and sizes, with lateral dimensions 20-150 µm. To increase homogeneity and + +improve efficiency when coating, a final ultrasonication and filtering step is performed to +break flakes to lateral sizes of 10s of µm, SI Appendix I. +To demonstrate the commercial potential of this platform for inorganic metallic +pigmentation, we formulated a paint by mixing the structural color flakes with a drying oil +(Linseed oil, Gamblin), Figure 6a. The mixture presents the simplest form of a paint, where +the flakes are the pigment and the oil is the binder that permits transferring to the target +substrate. This mixture can be used then to coat surfaces in applications otherwise +incompatible with vacuum systems. In Figure 6b we show such an example, where we +painted an artistic multicolor butterfly on a black canvas. Although different target surfaces +would require more careful selection of the binder, and possibly the use of other chemical +Figure 6 | Structural Color Paint. a Sequential growing of a bi-directional stack on a sacrificial layer results +on color flakes. b and c, Color flakes can be stored dry or dispersed in a solution. d, A paint can be produced +by mixing the flakes with a drying oil. This simple formulation, where the flakes are the pigments and the oil +the binder, can be adapted to impart the nanostructured coloration to any surface. e, Photography of a +multicolor artistic butterfly on a black canvas painted with a set of linseed oil-based plasmonic paints +demonstrating the commercial feasibility of the platform. Insets correspond to a microscope image –top- and +SEM micrographs –bottom-. Scale bar for the butterfly is 1 inch, whereas for the insets, from top-left to +bottom-right, scale bars corresponds to 1 mm, 100 μm, 75 μm, and 100 nm, respectively. + +StructuralColorFlakes +Structural ColorPaint +NanostructuredColorationadditives in the paint formulation, the structural color paint demonstrated here can be easily +adapted. Indeed, provided that non-corrosive chemicals are used, the flakes self-assemble +structure is a universal platform independent of the particular paint components employed. +Finally, from a commercial point of view, two additional features make this structural color +paint a very promising candidate for industrial production. First, in contrast to chemical +coloration schemes that use toxic and contaminant components, the fabricated flakes avoid +detrimental environmental impacts by employing only nontoxic materials such as +aluminum and its oxide, and a biodegradable water-soluble polymer as a sacrificial layer +(see Methods). And second, the structural color paint offers 100% reflection with only a +single ultra-thin layer of thickness (100 – 150 nm) pigment of extremely low surface +density. 22 +Conclusion +While structural coloration presents a promising opportunity to substitute chemical +colorants with purer and non-toxic options, commercial production of such structural color +poses a challenge due to the anisotropic optical response that results in undesired effects +such as dichroism or iridescence coupled with tedious fabrication processes. In this work +we have demonstrated a color nanostructure that overcomes these challenges offering a +real-world opportunity for industrial production. + +Hybridizing the plasmonic response of a metallic self-assembly with an ultrathin +optical cavity we have demonstrated a large CYM palette that can be produced by simply +changing the geometrical parameters of the structure. Furthermore, we studied mechanisms +for expanding the available color space through multilayers and in-plane addition for cost- +effective color mixing to expand the color gamut further. While the isotropic character of + +the nanoislands’ layer ensures the polarization independence, the angle insensitiveness, +otherwise impossible in conventional Fabry-Perot resonators, is achieved by exploiting the +non-trivial phase discontinuities in the ultrathin cavity, thus avoiding path length effects +for steep angles close to 70°. + +The subwavelength plasmonic cavity is fabricated with a low-temperature process +in an ebeam evaporator. The versatility of the process permits the use of many different +substrates, including flexible platforms required in wearable electronics and roll-to-roll +manufactures, and takes on the scattering properties of the target surface to produce both +diffuse and specular coloration mode. Additionally, being a self-assembly process, the +color consistency is ensured for large areas. We observed that color purity of our structure +is dependent on the size and shape distribution of the nanoislands. We note that although +the use of pre-seeding techniques, higher-temperatures, or alternative materials would +improve control on the assembly morphology it would impose a scale and cost toll on the +manufacturing process. +To demonstrate the commercial capabilities of our platform for inorganic metallic +pigments, we have fabricated self-standing bi-directional color flakes by evaporating on +top of a water-soluble sacrificial layer a double-sided stack. This ultralight pigment, that +offers full coloration with a single layer of flakes, can be then mixed with a binder matrix +to formulate a structural color paint that can be used to coat, after fabrication, any surface. +This approach presents an ultralight, multi-color, large-scale, low-cost, and environmental- +friendly platform for imparting nanostructured coloration to any surface. With an +unbeatable surface density of 0.4 g/m2, hundreds of times lighter than commercially + +available paints, thus paving the way towards industrial production and real-world +application. + +__________________________________________________________ +Methods +Self-Assembled Structural Color Fabrication. The optically thick back-mirror is +produced by evaporating 100 nm of aluminum on a Thermionics e-beam evaporator. +Pressure at the beginning of the evaporation was ~1x10-6 Torr, and evaporation rate was +kept at 0.1 nm/s. A spacer layer of aluminum oxide was then grown by atomic layer +deposition (Savannah 200, Cambridge Nanotech) by pulsing trimetylaluminum and water +at 100 °C. The aluminum nanoparticles were then produced on an UHV AJA electron beam +evaporator. In agreement with previous studies17,31,32, we find that three parameters play a +critical role in the geometry of the self-assembled monolayer: the temperature of the +substrate, the pressure in the chamber, and the rate of growth. We chose these parameters +compromising between the desired higher saturation of colors and the lower requirements +for fabrication that ensures the versatility of the proposed architecture and the viability of +a transition to industrial-scale production. Thus, we chose to keep the temperature of the +substrates at 100 °C, resulting in high color saturation while being below the melting point +of many polymers, essential for flexible substrate applications and lift-off during flake +preparation. For reproducibility and color vividness, nanoparticles’ growth was carried on +at pressures below 5x10-8 Torr, while growth rates were kept constant about 0.1 A/s, both +readily available in conventional UHV evaporators. It should be note that although an ALD +system was used for convenience for the hard substrates, the entire fabrication process +could be carried on in a single UHV ebeam evaporator. To prove this promising ease of + +integration, critical for industrial chain systems, the oxide layer in the pigment flakes +production was grown at room temperature on the Thermionics ebeam evaporator system. +Besides a slight change in color shade, attributed to the change in refractive index of the +oxide layer, no fundamental quality change was observed between the layers grown by the +two systems. +Finite Difference Time Domain Modeling. The reflection spectra and electric field +distribution of the simulated samples were calculated using the experimental geometrical +parameters extracted from the SEM analysis of the samples, with a commercial FDTD +software package (Lumerical FDTD, Lumerical Solutions Inc.). The relative permittivities +of aluminum and aluminum oxide are taken from literature33. To build the weight-averaged +semi-analytical model of SI Appendix Figure 2 we simulated a single particle with periodic +conditions. The values of the unit cell size were chosen to make the area covered by the +aluminum island equivalent to those obtained from the SEM analysis for the different +samples (55%, 60% and 70% for the 4, 8 and 12 nm respectively). 50 simulations with +radii values within 4 standard deviations of the mean value were then performed and the +weight-averaged reflectance calculated as: +������������� = � ������������������������ × ������������������������ +������������ + +where ������������������������ and ������������������������ represent the Gaussian weight and simulated reflection for the particle ������������. +For the analysis of the effect of the disorder as shown in SI Appendix Figure 3, we +simulated an array of 49 particles placed in periodic arrangement, disordered arrangement, +and disordered arrangement of different radii, introducing the disorder parameter following +the method presented by Zhang et al.34. + +Color Gamut Evaluations. To find the L*a*b* coordinates of the fabricated samples we +first obtained the XYZ tristimulus values integrating over the visible spectrum according +to: +������������ = 1 +������������ � ������������̅(������������)������������(������������)������������(������������)������������������������ +������������ = 1 +������������ � ������������̅(������������)������������(������������)������������(������������)������������������������ +������������ = 1 +������������ � ������������̅(������������)������������(������������)������������(������������)������������������������ +������������ = � �������������(������������)������������(������������)������������������������ +where ������������(������������) is the measured reflectance; ������������̅(������������), �������������(������������), and ������������̅(������������) are the color matching +functions; ������������(������������) is the reference illuminant. From the tristimulus values CIEXYZ the +CIELAB coordinates can be calculated from: +������������∗ = 116 ������������������������ − 16 +������������∗ = 500 ������������������������� − ������������������������� +������������∗ = 200 ������������������������� − ������������������������� +where, being ������������������������ = +������������ +������������������������, ������������������������ = +������������ +������������������������, or ������������������������ = +������������ +������������������������; and ������������������������ = 0.9642, ������������������������ = 1.0000, and ������������������������ = +0.8251 the D50 white point coordinates: +������������������������ = +⎩ +⎨ +⎧ +������������������������� +3 +if ������������ > � 6 +29� +3 + +841 ������������������������ +108 + 4 +29 +otherwise + + +As the reference illuminant we chose D50, used in the graphic arts industry for color +proofing as per ISO 3664:2009. Finally, to keep consistency, all colormaps presented in + +the figures in the main manuscript and the SI Appendix correspond to a horizontal slice +with ������������∗ = 75. +Measurements and Images. Reflection measurements were taken at normal incidence +with unpolarized light using a 4x, 0.07 numerical aperture objective and a fiber coupled +spectrometer (HR 2000+, Ocean Optics). An aluminum mirror was used as a reference. +Angular measurements were taken with an integrating sphere (RTC-060-SF, Labsphere) +connected to the spectrometer. To ensure consistency on illumination the samples were +photographed with flash-light at fixed intensity. Photographies for the butterfly models +were taken under sun illumination with a linear polarizer attached to the objective. + +Acknowledgments +This work at University of Central Florida was supported by National Science Foundation +Grant #ECCS-1920840. P.C.A. acknowledges the support from the UCF Preeminent +Postdoctoral Fellowship Program (P3). + + +References +1. +Mustroph, H. Dyes, General Survey. in Ullmann’s Encyclopedia of Industrial Chemistry 1– +38 (American Cancer Society, 2014). +2. +Gregory, P. High-Technology Applications of Organic Colorants. High-Technology +Applications of Organic Colorants (Springer US, 1991). +3. +Brillas, E. & Martínez-Huitle, C. A. Decontamination of wastewaters containing synthetic +organic dyes by electrochemical methods. An updated review. Applied Catalysis B: +Environmental 166–167, 603–643 (2015). +4. +Dushkina, N. & Lakhtakia, A. Structural Colors. in Engineered Biomimicry 267–303 (Elsevier, +2013). +5. +Arsenault, A. C., Puzzo, D. P., Manners, I. & Ozin, G. A. Photonic-crystal full-colour displays. +Nature Photonics 1, 468–472 (2007). +6. +Kim, H. et al. Structural colour printing using a magnetically tunable and lithographically +fixable photonic crystal. 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ACS Nano 11, 4445– +4452 (2017). +13. +Zhu, X., Yan, W., Levy, U., Mortensen, N. A. & Kristensen, A. Resonant laser printing of +structural colors on high-index dielectric metasurfaces. Science Advances 3, e1602487 +(2017). +14. +Roberts, A. S., Pors, A., Albrektsen, O. & Bozhevolnyi, S. I. Subwavelength plasmonic color +printing protected for ambient use. Nano Letters 14, 783–787 (2014). +15. +Cheng, F., Gao, J., Luk, S. T. & Yang, X. Structural color printing based on plasmonic +metasurfaces of perfect light absorption. Scientific Reports 5, 1–10 (2015). +16. +Kumar, K. et al. Printing colour at the optical diffraction limit. Nature Nanotechnology 7, +557–561 (2012). +17. +Franklin, D. et al. Self-assembled plasmonics for angle-independent structural color +displays with actively addressed black states. Proceedings of the National Academy of +Sciences 117, 13350–13358 (2020). +18. +Hu, Q., Lin, K. Te, Lin, H., Zhang, Y. & Jia, B. Graphene Metapixels for Dynamically +Switchable Structural Color. ACS Nano 15, (2021). +19. +Daqiqeh Rezaei, S. et al. Nanophotonic Structural Colors. ACS Photonics 8, 18–33 (2021). +20. +Toyota Motor Corporation. The All-New Lexus LC Structural Blue Edition. +https://www.lexus.eu/discover-lexus/lexus-news/lc-structural- +blue?lexReferrer=https%3A%2F%2Fwww.google.com%2F#hero. +21. +Stoye, D. & Freitag, W. Paints, Coatings and Solvents: Second, Completely Revised Edition. +Paints, Coatings and Solvents: Second, Completely Revised Edition (Wiley Blackwell, 2007). +doi:10.1002/9783527611867. + +22. +The Boeing Company. Painting versus Polishing of Airplane Exterior Surfaces. +https://www.boeing.com/commercial/aeromagazine/aero_05/textonly/fo01txt.html. +23. +Asen, S. & Budin, P. S. Cyanidin 3-arabinoside-5-glucoside, an anthocyanin with a new +glycosidic pattern, from flowers of “Red Wing” azaleas. Phytochemistry 5, 1257–1261 +(1966). +24. +Kinoshita, S., Yoshioka, S. & Kawagoe, K. Mechanisms of structural colour in the Morpho +butterfly: cooperation of regularity and irregularity in an iridescent scale. Proceedings of +the Royal Society of London. Series B: Biological Sciences 269, 1417–1421 (2002). +25. +Kinoshita, S. & Yoshioka, S. Structural Colors in Nature: The Role of Regularity and +Irregularity in the Structure. ChemPhysChem 6, 1442–1459 (2005). +26. +Maier, S. Alexander. Plasmonics : fundamentals and applications. (Springer, 2007). +27. +Kreibig, U. & Vollmer, M. Optical properties of metal clusters. (Springer, 1995). +28. +Schmidl, G. et al. Formation and characterization of silver nanoparticles embedded in +optical transparent materials for plasmonic sensor surfaces. Materials Science and +Engineering: B 193, 207–216 (2015). +29. +Mayer, K. M. & Hafner, J. H. Localized Surface Plasmon Resonance Sensors. Chemical +Reviews 111, 3828–3857 (2011). +30. +Kats, M. A. & Capasso, F. Optical absorbers based on strong interference in ultra-thin films. +Laser & Photonics Reviews 10, 735–749 (2016). +31. +Andersson, T. & Granqvist, C. G. Morphology and size distributions of islands in +discontinuous films. Journal of Applied Physics 48, 1673 (2008). +32. +Lončarić, M. et al. Optical and structural characterization of silver islands films on glass +substrates. Vacuum 84, 188–192 (2009). +33. +Palik, E. D. Handbook of optical constants of solids. vol. 3 (Academic press, 1998). +34. +Mao, P. et al. Manipulating disordered plasmonic systems by external cavity with transition +from broadband absorption to reconfigurable reflection. Nature Communications 2020 +11:1 11, 1–7 (2020). + + + + +Figure 1 | Structural Absorption for Color Generation. a, Many chemical substances produce color by +selectively absorbing frequencies matching their molecular electronic transitions. Pink color in Formosa +azaleas is due to the absorption of cyaniding molecules. b, An example of structural coloration is found in +the Peruvian Morpho didius. Lamellae nanostructures found in its wings scatter the blue components of +incident light generating its characteristic metallic blue. c, A subwavelength plasmonic cavity formed by a +self-assembly of metallic nanoislands on top of an oxide-coated mirror, generates color by selectively +absorbing certain wavelengths and strongly back-reflecting other. + + +Figure 2 | Color Space and Quality of the Plasmonic Cavity. a, In the Volmer-Weber growth mode, the +size of the nanoislands can be controlled by tuning the amount of aluminum evaporated –top-. If the process +is carried on for long enough semicontinuos films are formed that disable the plasmonic resonances and thus +the color –bottom-. b, Color polar gradient for thicknesses from 0.5 to 16 nm, for fixed 10 nm spacer. c, +CIELAB coordinates for the points in the color wheel compared to ISO DIS 15339-2 cold-set newsprint and +coated premium paper standards (inner and outer hexagon). d, Tuning of the spacer and capping layer +thicknesses expands the available color space. e, f, Show the red-shift of the absorption resonance as the +spacer and capping layer thicknesses are increased. + +a +b +c +100 +Nanoisland Growth +50 +. +0 +0.5nm +8 nm +-50 +Continuous Film Growth +-100 +-50 +0 +50 +100 +tm +a* +100 +L +d +. +0 +8.5 nm +16 nm +2 +6 +10 +14 +, (nm) +tm (nm) +tte +e +f +tts. +tt. +t ts +30 +100 +100 +25 +(wu) +75 +75 +20 +ts: 10 +te: 0 +78nm +ts +50 +50 +15 +6nm +25 +25 +10 +30 +10 +4 nm +0 +0 +0 +2.5 +5 +7.5 +10 +450 +550 +650 +750 +450 +550 +650 +750 +t。 (nm) +Wavelength (nm) +Wavelength (nm) + + +Figure 7 | Morphology Effect on Optical Response. a, The inhomogeneous broadening of the optical +resonance can be accounted for by introducing size and spatial variability. We simulate the broadening by +averaging the reflection curves of 50 particles with radii within 4 standard deviation of the mean value +obtained from SEM analysis. b, Statistical radii distribution -top- and reflection curves -bottom- for +experimental (solid), FDTD mean value (dashed), and weight averaged (dotted). c, Reflection curves for +FDTD simulations corresponding to 7x7 hemispherical particles with equal size in periodic and disorder +arrangement, and random size and disordered arrangement, equivalent to the 5 nm self-assembly. d, Electric +profiles in three different spectral positions as labeled in panel c. + +Periodic +Disordered +Random +wu +425 +II +425nm +wu +705 +I + + + +Figure 4 | Dual Color Mode, Polarization Independence, and Angle-Insensitiveness. a, Butterfly garden +with an assorted collection of different butterfly wings and colors. b, An artistic butterfly model coated with +structural blue retains its color when photographed with unpolarized –left-, and two orthogonal linearly +polarized states –center and right-. c, The butterfly color is also angle-insensitive, as shown for three different +combinations of azimuth and zenith angles. d-e, The versatility of the self-assembly fabrication process +permits the use of a wide array of substrates. Flat and sandblasted PET strips are used as flexible substrates +to form the three primaries in both d, specular, and e, diffuse coloration mode. + +Polarization Independence +AngleIndependence +SpecularColoration +DiffusiveColoration + + +Figure 5 | Mixing Schemes for Expanding the Color Palette. a, Controlling the ratio of area covered by +two configurations the reflection curve can be defined by a simple additive rule. b, Camera pictures of +samples with mixing ratios from 0 to 100%, for spacer thicknesses of 10, 15, and 20 nm. c, Microscope +images for the samples highlighted in b. d, As the ratio is increased the reflection curves transition from pure +basis A to pure basis B. e, CIELAB space for the samples corresponding to spacer thickness of 10 nm in +panels b and g. The white dotted line overlay represents the space defined by the color wheel in Figure 3c. f, +New colors can be generated by multilayer structures. g, Green shades inaccessible with a single layer can +be generated by stacking two self-assemblies with different interspacing thicknesses. h, Tuning of the +interspace layer between self-assemblies controls the optical response of the cavity. + +a +b +In-plane +FillingFactorα (%) +Colour Mixing +20 +m +Spacer t, (nm) ++15 +'m +II +III +IV +Area () +α= +Total Area (+) +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +c +d +e +100 um +α +100 +90 +wug +a +45 +50 +：10nm +II +25 +0 +III +0 +tm +IV +450 +550 +650 +750 +Wavelength(nm) +-45 +-45 +0 +45 +90 +a* +f +6 +h +Out-of-plane +Interspacingt,(nm） +100 +Colour Mixing +.6 +75 +50 +A +B +D +R +25 +F +0 +450 +550 +650 +750 +4 +Wavelength(nm) +10.5 +12.5 +15.0 +17.5 +20.0 +22.5 +Figure 6 | Structural Color Paint. a Sequential growing of a bi-directional stack on a sacrificial layer results +on color flakes. b and c, Color flakes can be stored dry or dispersed in a solution. d, A paint can be produced +by mixing the flakes with a drying oil. This simple formulation, where the flakes are the pigments and the oil +the binder, can be adapted to impart the nanostructured coloration to any surface. e, Photography of a +multicolor artistic butterfly on a black canvas painted with a set of linseed oil-based plasmonic paints +demonstrating the commercial feasibility of the platform. Insets correspond to a microscope image –top- and +SEM micrographs –bottom-. Scale bar for the butterfly is 1 inch, whereas for the insets, from top-left to +bottom-right, scale bars corresponds to 1 mm, 100 μm, 75 μm, and 100 nm, respectively. + +StructuralColorFlakes +Structural ColorPaint +NanostructuredColoration \ No newline at end of file diff --git a/bNE0T4oBgHgl3EQf4gLC/content/tmp_files/load_file.txt b/bNE0T4oBgHgl3EQf4gLC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75a2ee547f0d2cdf01dc315764865cafb16b345e --- /dev/null +++ b/bNE0T4oBgHgl3EQf4gLC/content/tmp_files/load_file.txt @@ -0,0 +1,632 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf,len=631 +page_content='Ultralight Plasmonic Structural Color Paint Pablo Cencillo-Abad1,†, Daniel Franklin1, 2, †, Pamela Mastranzo-Ortega1, Debashis Chanda1, 2, 3 1 NanoScience Technology Center, University of Central Florida, 12424 Research Parkway Suite 400, Orlando, Florida 32826, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 2 Department of Physics, University of Central Florida, 4111 Libra Drive, Physical Sciences Bldg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 430, Orlando, Florida 32816, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 3CREOL, The College of Optics and Photonics, University of Central Florida, 4304 Scorpius St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=', Orlando, Florida 32816, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' †Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Correspondence and requests for materials should be addressed to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' (email: Debashis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='Chanda@ucf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In nature, vibrant colors as those of many butterflies, birds, octopuses, or fishes, arise from microscopically textured surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' These vivid colors result from the coherent interaction between light and the structural arrangement of colorless materials found in their skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In contrast, all manmade colors are pigment based and rely on the molecular absorption of their constituents, with each color requiring a different molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' While traditional pigment-based colorants offer a viable commercial platform for large-volume and angle-insensitiveness, they are limited by their resolution, instability in atmosphere, color fading, and severe environmental toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' These widely used pigment-based paints are destroying the environment, aquatic life, and adversely affecting global warming by working as heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' However, till date all attempts to industrial production of polarization and angle independent full-range structural colors have failed due to the angle-dependent colors and fabrication challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Here, we present a subwavelength plasmonic cavity that generates color by the hybridization of a metallic self-assembly with an ultrathin optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This configuration offers both polarization and angle insensitiveness, while simultaneously providing a full-color gamut of vibrant structural color paints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this work, we presented this unique structural color generation mechanism and demonstrated color generation of these structures across the entire visible spectrum by tuning just the structural parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Further, akin to traditional mixing of different pigments to produce new colors, we demonstrated in a unique way lateral as well as vertical “mixing of structures” to expand the available color palette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The self-assembly facilitates the growth of the structure on large areas and non-conventional substrates in both diffuse and specular coloration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Growing the stack on a sacrificial layer we produce a self-standing nanostructured color platform that, when mixed with a binder, can be transferred to any surface to impart full coloration with a single sub- micron layer of pigment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This structural color platform offers a highly integrable ultra-lightweight solution that bridges the gap from proof-of-concept to real-world industrial applications of non-toxic, fade resistant, and environmentally friendly colorants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Introduction Color presents one of the richest sources of sensorial information in our daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Throughout history, the fascination with colors has driven human efforts to produce newer and better colorants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' From the Paleolithic cave paintings to the development of the first synthetic dyes in the mid nineteenth century the quest for purer, fade-resistant, and environmentally-friendly colorants has remained very active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In the last decades, adding to purely decorative applications in textile, cosmetics, or food industries, colorant research has found relevance, among others, in display technologies, optical storage, sensing and therapeutics, or functional coatings1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Color engineering can be achieved by controlling the colorant’s absorptive or reflective response to white light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' All commercial colorants/pigments are based on absorption mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' These colorants absorb photons of energies overlapping with their molecular electronic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Contrarily, photons with energies not matching these discrete transitions will be reflected and registered as color by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although chemical colorants can be produced in large amounts, most of them are composed of toxic materials difficult to remove in the recycling process and are responsible for the pollution of our lifeline on earth—water3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Being chemically unstable, many colorants fade with time, a process accelerated with higher temperatures or light exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Furthermore, as volumes of several microns are needed to obtain enough color saturation, they suffer from low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In contrast, instead of controlling the absorption of light, structural colorants control the way the light is reflected or scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Structural color is the result of optical phenomena produced by micron- and nano-scale structures4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Remarkably, when in bulk, the material constituents show completely different hues or are even colorless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Colors generated by engineered structures such as photonic crystals5–9 or metasurfaces10–13, have received increasing attention in recent years for their striking advantages over chemical colorants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Characterized by their intense brilliance and saturation, they exhibit larger stability to chemical reagents, harsh environmental conditions, and high illuminating intensities14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Additionally, they can offer dynamic tunability and resolutions beating the diffraction limit, both essential for display applications16–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Due to the geometrical nature of their response, however, structural colors usually present directional effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' their color varies with the positioning of the observer and the angle and polarization of the incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' More importantly, many proposed architectures rely on the use of costly and low-throughput nanofabrication techniques not compatible with mass-production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Overall, these constrains prohibit the commercial viability of all previously reported structural color technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' It is therefore not surprising that, to date, no angle-independent structural paints are available in the marketplace19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Here, we present a subwavelength plasmonic cavity that overcomes these challenges while offering a tailorable platform for rendering angle and polarization independent vivid structural colors by coupling incident light with gap-plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The structures are fabricated through a large-area, highly versatile, and reproducible technique where aluminum nanoislands are self-assembled in an electron beam evaporator on top of a transparent oxide-coated aluminum mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The optical response of these artificially engineered nanostructures can be spectrally tuned across the entire visible spectrum to form a full color gamut by controlling the gap-plasmon dispersion via the geometrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In the proposed architecture the subwavelength optical cavity ensures a large degree of angle insensitivity while the stochastic nature of the self-assembled layer results in polarization independence and near 100% absorption at selected spectral bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The evaporation process, relying only on widespread industrial techniques, is compatible with many substrates, and takes on their scattering properties to render diffuse and specular coloration modes when utilizing micro-corrugated or flat surfaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' E-beam evaporators are widely employed in industries such as electronics, semiconductors, optics, and even aerospace, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Moreover, Lexus Blue, the only industrially produced simple Fabry-Perot resonance based structural color is actually fabricated with ebeam evaporators20 We present mechanisms for expanding the available color space through lateral and vertical mixing of structures, similar to traditional pigment mixing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Finally, to demonstrate the commercial capabilities of our platform for inorganic metallic structural coloration, we formed bi-directional structures on a water-soluble sacrificial layer that resulted in omni-directional color flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' These structural color flakes were then mixed with a commercial binder to develop self-standing structural color paints hundreds of times lighter than commercially available paints21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Conventional chemical coloration relies on volumetric absorption of light to produce a color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In contrast to the several microns required for commercial paints, our ultrathin paint can impart full coloration with a thickness of only 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Consequently, this huge lateral area (few 10s of µm) to thickness (100 – 150 nm) ratio makes it the lightest paint in the world with a surface density of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='4 g/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For comparison, while a Boeing 747 requires 500 kg of paint22, our ultralight paint would require about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='3 kg, an astonishing potential about 400-fold reduction in weight, SI Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Our approach presents the first environmental-friendly, large-scale, multi-color, and self-standing platform for imparting nanostructured coloration to any surface, thus bridging the gap from proof of concept to industrial production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Results Self-Assembled Plasmonic Surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Nature presents a rich variety of both chemical and structural coloration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For example, the pink tint of Formosa azaleas, Figure 1a, is due to the absorption of cyaniding molecules, a type of anthocyanin pigment23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In contrast, the bright metallic blue displayed by the Peruvian Morpho didius, Figure 1b, is primarily the result of the way the blue components are scattered by the lameallae nanostructures found in this butterfly’s wings24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Oftentimes, however, structural color in animals results from the combination of the diffraction and scattering of the outer skin layers, and the molecular absorption of the complementary color by intrinsic pigments of the skin25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This critical observation inspired us to produce an absorptive structural pigment where the selective absorption of specific frequencies is the result of the tailored structural resonant response of metallic nanostructures coupled to a subwavelength optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Specifically, the proposed architecture consists of a highly-packed monolayer of self-assembled aluminum nanoislands on a thin aluminum oxide film that spaces them from the aluminum back- mirror, Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this configuration the aluminum nanoislands resonantly absorb specific wavelengths, while the back mirror strongly back-reflects the non-resonant ones, rendering vivid colors based on colorless materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Contrary to other artificial structural schemes that rely on the use of low- throughput, multi-step, top-down techniques such as electron beam lithography or focused ion beam, incompatible with mass-production, the proposed architecture is the result of a naturally occurring nucleation process in an electron beam evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In the self-assembly growth, small clusters of aluminium nanoparticles are formed due to the larger affinity of the aluminium atoms to their own kind over the oxide substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' With a low enough rate, the evaporation of nanometric films results in a nanoparticles’ monolayer that exhibit optical plasmonic resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Crucially, this pressure- and temperature-controlled process ensures high reproducibility over broad areas in a single step, lowering the cost of production and enabling large-scale fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The dynamics of the self-assembly process Figure 1 | Structural Absorption for Color Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, Many chemical substances produce color by selectively absorbing frequencies matching their molecular electronic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Pink color in Formosa azaleas is due to the absorption of cyaniding molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, An example of structural coloration is found in the Peruvian Morpho didius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Lamellae nanostructures found in its wings scatter the blue components of incident light generating its characteristic metallic blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, A subwavelength plasmonic cavity formed by a self-assembly of metallic nanoislands on top of an oxide-coated mirror, generates color by selectively absorbing certain wavelengths and strongly back-reflecting other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' are presented in detail in SI Appendix A, while the technical parameters can be found in Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Optical Response of the Near-Field Coupled Gap Plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The color produced in the nanostructure is the result of the hybridization of the absorptive response of the aluminum self-assembled monolayer, and the subwavelength cavity formed by this top layer, the aluminum back-mirror, and the dielectric spacer sandwiched in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Geometrical changes in any of the layers will then result in a change in the perceived color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' When ambient light reaches the monolayer, the electric field of the light at select wavelengths can drive the free electrons of the aluminum to oscillate resonantly within the nanoparticles’ geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This collective oscillation, termed localized surface plasmon resonance, is further affected by the coupling between closely-packed neighboring particles and the presence of the back-mirror interface at a subwavelength distance from the particles’ layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This complex hybridization mechanism results in a gap-plasmon mode that leads to strong optical absorption and tight confinement of the light at the metal/dielectric boundary of the metallic particles at resonant frequencies26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The spectral position of the absorption band, and thus the perceived color, depends distinctly on the gap-plasmon dispersion which is hence controlled by three parameters: (1) the size and spatial distribution of the nanoislands, (2) the refractive index of their environment, and (3) the thickness of the spacing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The size of the nanoislands can be simply controlled by tuning the amount of aluminum evaporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To investigate the range of colors available with this cumulative process we use a shutter that controls the partial exposure of the sample during the evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this manner, by rotating the sample, we can produce a polar gradient of thicknesses from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 nm to 16 nm, in thickness increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 nm corresponding to wedges of approximately 11°, Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' As the thickness mass is increased neighboring nanoislands coalesce to form larger particles, Figure 2a-top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This increase in the nanoislands’ size red-shifts the absorption band and results in different hues and saturations that produce a color palette that covers from the white of the back mirror, at very low thicknesses, to the yellow, magenta, and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' It should be noted that, being a subtractive color scheme, a red-shift of the absorption band results in a blue-shift in perceived color, as the intensity of blue components in the reflected light augment at the expense of the yellow and red ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' If the Figure 2 | Color Space based on Tunable Gap Plasmon Dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, In the Volmer-Weber growth mode, the size of the nanoislands can be controlled by tuning the amount of aluminum evaporated –top-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' If the process is carried on for long enough semicontinuos films are formed that disable the plasmonic resonances and thus the color –bottom-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, Color polar gradient for thicknesses from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 to 16 nm, for fixed 10 nm spacer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, CIELAB coordinates for the points in the color wheel compared to ISO DIS 15339-2 cold-set newsprint and coated premium paper standards (inner and outer hexagon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, Tuning of the spacer and capping layer thicknesses expands the available color space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' e, f, Show the red-shift of the absorption resonance as the spacer and capping layer thicknesses are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a b c 100 Nanoisland Growth 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5nm 8 nm 50 Continuous Film Growth 100 50 0 50 100 tm a* 100 L d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 nm 16 nm 2 6 10 14 , (nm) tm (nm) tte e f tts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' t ts 30 100 100 25 (wu) 75 75 20 ts: 10 te: 0 78nm ts 50 50 15 6nm 25 25 10 30 10 4 nm 0 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 10 450 550 650 750 450 550 650 750 t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' (nm) Wavelength (nm) Wavelength (nm)process is carried on for long enough, adjacent nuclei can coalesce to form semi-continuous films and, eventually, continuous films, Figure 2a-bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The thickness at which the transition from isolated islands to continuous film occurs is the percolation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' At thicknesses above the percolation threshold the free electrons of the metal can find paths to move through the self-assembly, eliminating the geometrical confinement necessary for Figure 3 | Morphology Effect on Optical Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, The inhomogeneous broadening of the optical resonance can be accounted for by introducing size and spatial variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We simulate the broadening by averaging the reflection curves of 50 particles with radii within 4 standard deviation of the mean value obtained from SEM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, Statistical radii distribution -top- and reflection curves -bottom- for experimental (solid), FDTD mean value (dashed), and weight averaged (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, Reflection curves for FDTD simulations corresponding to 7x7 hemispherical particles with equal size in periodic and disorder arrangement, and random size and disordered arrangement, equivalent to the 5 nm self-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, Electric profiles in three different spectral positions as labeled in panel c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Periodic Disordered Random wu 425 II 425nm wu 705 Ithe resonant plasmonic absorption and thus disabling the color production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This can be observed in the gradient wheel sample at higher thicknesses where the blue fades to white, Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Further details on the growth dynamics can be found in SI Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Together with the spectral shift, the increase in the thickness mass results in a broadening of the optical resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We attribute this phenomenon to the inhomogeneous broadening of the nanoparticles’ resonances arisen from the doubly random nature of both the morphology and spatial distribution, as can be seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' On the one hand, as thickness mass increases, larger variability of island size can be observed, SI Appendix Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To further assess this effect, we build a semi-analytical model that defines the total reflection of the monolayer by weight averaging the reflection of periodic islands of 50 hemispherical radii within 4 standard deviation of the mean value as obtained from the SEM analysis for the 8 nm thickness mass, Figure 3a and SI Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This larger morphological variability translates into a reduction in the reflection contrast and saturation of the colors produced, with a distinct broadening of the resonance, Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' On the other hand, the effect of the spatial distribution can be explained by the well-known dependency of relative position of interacting plasmonic resonators27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To evaluate this latter effect, we run a set of simulations for 7x7 hemispherical nanoparticles, for equivalent thickness mass of 4 nm on top of a 10 nm oxide spacer, in periodic square array, disordered array, and disordered randomized sizes, Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The reflection curves show additionally a spectral shift that we associate with the different energies of the new available modes resulting from the laterally-hybridization of nanoparticles, modes otherwise forbidden in the symmetric arrangement, SI Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' These assumptions are indeed confirmed from the comparison of the electric profiles in-resonance where we observe the strongly confined fields characteristic of the gap plasmon modes, for both ordered and disordered arrangements, as seen in Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Interestingly, we observe clearly that while for the ordered structure the dipolar resonance is only excited at in-resonance wavelength, both disordered and random disordered configurations show excitations even well outside the in-resonance spectral position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The inhomogeneous broadening observed is also in good agreement with the expected behavior predicted by the classical formula for dipole-dipole interaction energy given by28: ������������ = ������������������������������������������������ |������������1||������������2| ������������������������2|������������12|3 (1) where ������������������������ is the Coulomb constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' ������������������������ the orientation factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' ������������������������ the refractive index of the environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' |������������1| and |������������2| the modulus of the dipole moments for two interacting particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' and |������������12| the modulus of the distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this near-field approximation, considering two neighbor particles interacting, if their sizes, and also shapes, show a large variability, it is expected that the dipole modes corresponding to a given illuminating wavelength will be indeed weakly excited, and consequently lower absorption will result with a poorer reflection contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Furthermore, the spatial disorder broadening can be understood by averaging the distance between particles, where some of them will be constructively interfering, while others will be out of phase and thus destructively interfering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Finally, it should be noted that unlike transmissive colors, emitting out of a source, all subtractive colors lack purity (paper print vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' LED displays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this case the high-density packing of the self-assembly plays a critical role in bringing the hybridized modes to the visible range, while ensuring vivid coloration in a single nanometric layer, it exacerbates the resonance broadening resulting from the dynamic depolarization of non-spherical particles, see SI Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although, all factors considered, spectrally purer colors could be achieved with pre-treatment steps prior to the self-assembly growth, this would be achieved at the expense of the fabrication simplicity offered in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To better understand the coupled mechanism, we also develop a theoretical model, and compared it with FDTD simulations, results of this can be found in SI Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To assess the quality of the color gamut generated by our self-assembled plasmonic structure, we calculated the L*a*b* coordinates from the reflection spectra of each thickness in the gradient sample (see Methods), and plot them as black dots in the CIELAB color space, Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To compare with two color quality standards used in the printing industry, we overlay the standards for the cold-set newsprint and coated premium paper technologies as defined in ISO DIS 15339-2 (inner and outer hexagon respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For a substantial portion of the color space, we find that the self-assembled plasmonic color exceeds the newsprint standard, and even matches the quality of some colors as produced in coated premium paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' However, although the color space of the plasmonic structure can be expanded in some regions by careful selection of the other geometrical parameters, due to its subtractive nature, the production of green is prohibited for a single particle layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To address this limitation, in sections below, we introduce two different color mixing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Due to the strong field confinement in the metal-dielectric interface plasmonic resonances are extremely sensitive to changes in the environment28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The addition of a capping layer on top of the self-assembly presents an opportunity to further tune the color response by shifting the resonant spectral position of the nanoislands’ layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For samples corresponding to 4, 6, and 8 nm mass thicknesses, and fixed 10 nm-thick oxide spacer, we monitor the color change as we grow capping layers of alumina in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 nm increments, Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Reflection curves for the 6 nm samples can be seen in Figure 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Clearly, the presence of the capping layer red-shifts the plasmonic resonance producing colors with higher blue components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This behavior is captured by the classical formula for the dipole- dipole energy interaction presented in SI Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' As the thickness of the capping layer increases, more energy is contained within the higher dielectric media and the particle- particle interaction weakens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This causes lower hybridization energies and results in higher resonant wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The effect of this top layer is of particular importance from the applications point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although aluminum, due to its native oxide layer, is very chemically stable in atmosphere, we found the structures to be fragile to harsh contaminants and physical contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To address this, we capped samples with a commercial polyurethane clear coat (DuraClear Varnish, Americana).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Interestingly, these samples still maintained vivid colors while offering protection to physical contact and larger chemical resistance to spills as can be observed in SI Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The final element that controls the optical response of the structure is the spacer defined by the thickness of the transparent aluminum oxide spacer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' As shown in Figure 2d for samples corresponding to 4, 6, and 8 nm mass thickness and varying spacers from 10 to 30 nm, changes in the spacer thickness result in pronounced color changes in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For the 6 nm nanoparticles’ layer the reflection curves are shown in Figure 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We observe that as the spacer is increased the resonance is shifted to longer wavelengths and the overall reflection levels increase producing less saturated colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We explain the behavior of the multilayer stack using interference theory of a non-symmetric subwavelength cavity, where the bottom mirror and the top nanostructured self-assembly form the two limiting interfaces, and the ultra-thin dielectric (alumina) spacer sandwiched between them which controls the vertical coupling between two metallic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This configuration is essential to achieve the almost-100% levels of absorption in the nanostructured plasmonic self-assembled layer, which occur only when field-enhancement occurs at the nanoparticles layer for wavelengths that fulfill the phase matching condition30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In contrast to conventional Fabry-Perot resonators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' where the phase is simply accumulated through the propagation in the dielectric and the resonant condition can only be fulfilled for cavity lengths proportional to the wavelength of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' the dispersive nature of the gap- plasmon mode excited on the self-assembled Al nanoislands introduces an interface with non-trivial phase shifts and high losses that can produce absorption resonances even for deeply subwavelength thicknesses well below the resonant wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' As the thickness of the spacer is further increased the mismatch between phases results in weaker absorption response and renders less saturated colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The different nature of this near-field coupled gap-plasmon mode compared to a far-field Fabry-Perot mode, is further verified for large enough spacer thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' When the dielectric spacing layer takes values large enough (multiples of ������������/4������������������������), far field effects become dominant and the phase accumulated through propagation can fulfill the resonant condition, as in typical Fabry-Perot resonators, resulting in a sharp dip in reflection, SI Appendix Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although this resonance offers colors with higher saturation, the pure geometrical nature of the mode makes it highly angle-dependent, thus limiting greatly its practical applications, offering further proof of the fundamental advantage of the near-field coupled gap-plasmon engineered inside this novel self-assembled ultra-thin structure which is exploited here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' A Versatile Platform for Structural Coloration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Growing the structure with conventional evaporation techniques at low temperatures permits the use of a wide variety of substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To prove the versatility of the proposed plasmonic self-assembled structure we produced multicolor butterflies by growing several stacks on wing-shaped polyethylene terephthalate (PET) templates, Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The polarization and angle-insensitiveness of these color structures readily shows their superiority over many other reported structural approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' On the one hand, the polarization independency arises from the isotropic character of the disordered self-assembled layer, where nanoislands show no predominant direction or orientation of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In Figure 4b we show how, as expected, when photographed with unpolarized, and two orthogonal linearly polarized states, the butterfly assembly shows no appreciable color difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This particular feature of the multilayer structure is highly important for integration in devices that rely on the use of polarized light such as liquid crystal displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' On the other hand, the subwavelength character of the cavity makes the structure pretty color insensitive to the angle of incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Photographies of the blue artistic butterfly at three different combinations of zenith and azimuth angles show clearly that the color is retained regardless of the angle of incidence, Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Indeed, upon further study, SI Appendix F, we observe the structures retaining their color for angles as large as 60°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The adaptability to different substrates of this unique large-area, self-assembling based fabrication method paves the path towards the integration of the stack in elastic platforms without the loss of color quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We grow three samples with 5, 8 and 12 nm nanoparticles’ layers, and fixed 10 nm aluminum oxide spacer, on top of aluminum coated polyethylene terephthalate (PET) strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' These three configurations, corresponding to the three primaries in the CYM color mode, can be seen in Figure 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although vivid and brilliant, the specular coloration observed in flat substrates is inconvenient in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For such cases, corrugated substrates can be used to produce diffuse coloration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In the diffuse coloration mode, careful texturing of the substrate can control the degree of dispersion of light reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We produce diffuse coloration by growing the nanostack on sandblasted PET strips, Figure 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In contrast to flat substrates, that result in specular coloration mode, the use of microtextured substrates result in surfaces that homogenously diffuse the light without inconvenient light streaks of specular reflection, while retaining angle and polarization insensitiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Figure 4 | Dual Color Mode, Polarization Independence, and Angle-Insensitiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, Butterfly garden with an assorted collection of different butterfly wings and colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, An artistic butterfly model coated with structural blue retains its color when photographed with unpolarized –left-, and two orthogonal linearly polarized states –center and right-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, The butterfly color is also angle-insensitive, as shown for three different combinations of azimuth and zenith angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d-e, The versatility of the self-assembly fabrication process permits the use of a wide array of substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Flat and sandblasted PET strips are used as flexible substrates to form the three primaries in both d, specular, and e, diffuse coloration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Polarization Independence AngleIndependence SpecularColoration DiffusiveColorationExpanding the Color Gamut with Mixing of Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Changes in the geometrical parameters can be introduced to tailor the color response of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Often, however, the production of a larger color palette is difficult, due to the limitation of primary colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Guided by the principle of conventional color mixing where multiple pigments are mixed to produce secondary colors, we demonstrated in a unique way production of new colors by the “mixing of structures” without needing new materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Growing side-by-side patches covered with 5 and 10 nm mass thickness nanoislands, we controlled the final color appearance by careful selection of the ratio of the area covered by each one of the particles’ configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We define the control parameter ������������ as the ratio of the area covered by the 10 nm equivalent nanoislands to the total area covered by both configurations, Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Using a lithographic mask we define subpixels of 100 µm length and variable 0 to 100 µm width, in steps of 10 µm, to be covered by 10 nm nanoislands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The rest of the area is then covered by the 5 nm mass thickness nanoislands producing samples with ������������ values ranging from 0 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this manner we fabricated three samples for cavity length values of 10, 15, and 20 nm, Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The pixel geometry is purposely selected to be in a chess-board arrangement with pixels smaller than 100 µm to reduce chromatic aliasing and produce smooth colored surfaces to the naked eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Microscopy insets for selected samples can be seen in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This side-by-side mixing mechanism can be explained by a simple additive rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For given reflection curves, ������������������������ and ������������������������, corresponding to the two color bases A and B with mixing ratio ������������, the total reflection is given by: ������������������������������������������������ = (1 − ������������) ∙ ������������������������ + ������������ ∙ ������������������������ Reflection curves for the mixtures with fixed 10 nm-oxide layer can be seen in Figure 5d, where we observe the transition from one basis to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This is even clearer in Figure 5e, where we have plotted as black dots the L*a*b* coordinates corresponding to the ������������ values from 0 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For context, the colorspace defined by the thickness wheel analyzed in Figure 2c is overlaid as dotted white lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Conveniently enough, any color contained in the segment defined by the two coordinates corresponding to the bases can be generated by careful selection of in-plane mixing ratios, SI Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Figure 5 | Mixing of Structures to Expand the Color Gamut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, Controlling the ratio of area covered by two configurations the reflection curve can be defined by a simple additive rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, Camera pictures of samples with mixing ratios from 0 to 100%, for spacer thicknesses of 10, 15, and 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, Microscope images for the samples highlighted in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, As the ratio is increased the reflection curves transition from pure basis A to pure basis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' e, CIELAB space for the samples corresponding to spacer thickness of 10 nm in panels b and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The white dotted line overlay represents the space defined by the color wheel in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' f, New colors can be generated by multilayer structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' g, Green shades inaccessible with a single layer can be generated by stacking two self-assemblies with different interspacing thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' h, Tuning of the interspace layer between self-assemblies controls the optical response of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=" a b In-plane FillingFactorα (%) Colour Mixing 20 m Spacer t, (nm) +15 'm II III IV Area () α= Total Area (+) 0 10 20 30 40 50 60 70 80 90 100 c d e 100 um α 100 90 wug a 45 50 ：10nm II 25 0 III 0 tm IV 450 550 650 750 Wavelength(nm) 45 45 0 45 90 a* f 6 h Out-of-plane Interspacingt,(nm） 100 Colour Mixing ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='6 75 50 A B D R 25 F 0 450 550 650 750 4 Wavelength(nm) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5In-plane mixing does expand the color palette by offering a route to generate any color contained in the region defined by the basis employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' However, it does not permit to generate colors outside of its boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Generating green shades would therefore require a green basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Yet, due to its subtractive nature, the production of green is prohibited for the plasmonic self-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' However, this limitation can be broken by growing multilayers of plasmonic nanoparticles, Figure 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this multilayer configuration two extra geometrical parameters are introduced to control the color appearance: the thickness mass of the extra layer and the interspace between the nanoislands films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We produce a wide variety of green shades by growing, on top of a base structure consisting of an aluminum mirror, 10 nm oxide layer, and 10 nm equivalent nanoislands, three top layers corresponding to 4, 5, and 6 nm self-assembled layers with oxide interspaces ranging from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 nm, Figure 5g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The reflection curves for the 5 nm equivalent top layer can be seen in Figure 5h, while the L*a*b* coordinates for these curves are plotted as dots in Figure 5e, where we observe how the out-of-plane mixing scheme does indeed expand the color palette to areas otherwise inaccessible with a single plasmonic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although the levels of reflection in the bilayer structures are low, due to the double absorption of the two-fold plasmonic layer, careful study of all geometrical parameters can help mitigate partially this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Indeed, the interspace and top-layer geometrical parameters add to the bottom self-assembly and the spacer layer thickness to offer extra degrees of freedom to expand the available color gamut introducing new colors with different saturation and luminance, SI Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Interestingly, from the industrial point of view, these new layers do not increase excessively the complexity of the manufacturing, as they can be grown in the same chamber as a single process step, with little extra time or cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The addition of an extra plasmonic film can also be incorporated to the analytical model by adding on top of the single stack two extra layers: the dielectric interspace and an extra effective medium describing the upper plasmonic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Structural Color Paint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The fabrication process used to grow the self-assembled structure permits an easy integration in many industrial processes that already use compatible systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Despite this, the platform is fundamentally limited by the need to grow in-situ structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This restricts critically the applicability in contexts where non-vacuum- compatible substrates are required or instances where large areas need to be covered, as the limiting factor would always be the particular specification of the evaporation equipment used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To present a realistic alternative to commercial chemical colorants the multilayer structure should ideally be available in a stand-alone platform that can be transferred, after fabrication, to any substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' On top of a sacrificial layer we evaporate sequentially a double-sided mirror-symmetric stack, where each side comprises an alumina protecting capping layer, a plasmonic self-assembly and an alumina spacer, while the mirror is shared between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Removal of the sacrificial layer at the end of the fabrication results in tunable self-standing doubly-colored flakes, Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We chose to grow the structures symmetrically to ensure homogeneous colorization, however, flakes can be grown in asymmetric configurations, each face showing a different color, to render new mixtures similar to the in-plane mixing color scheme explored before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Once the structures are flaked off of the substrate, the flakes can be stored dry in powder form, Figure 6b, or kept in an organic solvent (here we used acetone), Figure 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' After lifting off, flakes present irregular shapes and sizes, with lateral dimensions 20-150 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To increase homogeneity and improve efficiency when coating, a final ultrasonication and filtering step is performed to break flakes to lateral sizes of 10s of µm, SI Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To demonstrate the commercial potential of this platform for inorganic metallic pigmentation, we formulated a paint by mixing the structural color flakes with a drying oil (Linseed oil, Gamblin), Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The mixture presents the simplest form of a paint, where the flakes are the pigment and the oil is the binder that permits transferring to the target substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This mixture can be used then to coat surfaces in applications otherwise incompatible with vacuum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In Figure 6b we show such an example, where we painted an artistic multicolor butterfly on a black canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Although different target surfaces would require more careful selection of the binder, and possibly the use of other chemical Figure 6 | Structural Color Paint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a Sequential growing of a bi-directional stack on a sacrificial layer results on color flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b and c, Color flakes can be stored dry or dispersed in a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, A paint can be produced by mixing the flakes with a drying oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This simple formulation, where the flakes are the pigments and the oil the binder, can be adapted to impart the nanostructured coloration to any surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' e, Photography of a multicolor artistic butterfly on a black canvas painted with a set of linseed oil-based plasmonic paints demonstrating the commercial feasibility of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Insets correspond to a microscope image –top- and SEM micrographs –bottom-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Scale bar for the butterfly is 1 inch, whereas for the insets, from top-left to bottom-right, scale bars corresponds to 1 mm, 100 μm, 75 μm, and 100 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' StructuralColorFlakes Structural ColorPaint NanostructuredColorationadditives in the paint formulation, the structural color paint demonstrated here can be easily adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Indeed, provided that non-corrosive chemicals are used, the flakes self-assemble structure is a universal platform independent of the particular paint components employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Finally, from a commercial point of view, two additional features make this structural color paint a very promising candidate for industrial production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' First, in contrast to chemical coloration schemes that use toxic and contaminant components, the fabricated flakes avoid detrimental environmental impacts by employing only nontoxic materials such as aluminum and its oxide, and a biodegradable water-soluble polymer as a sacrificial layer (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' And second, the structural color paint offers 100% reflection with only a single ultra-thin layer of thickness (100 – 150 nm) pigment of extremely low surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 22 Conclusion While structural coloration presents a promising opportunity to substitute chemical colorants with purer and non-toxic options, commercial production of such structural color poses a challenge due to the anisotropic optical response that results in undesired effects such as dichroism or iridescence coupled with tedious fabrication processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In this work we have demonstrated a color nanostructure that overcomes these challenges offering a real-world opportunity for industrial production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Hybridizing the plasmonic response of a metallic self-assembly with an ultrathin optical cavity we have demonstrated a large CYM palette that can be produced by simply changing the geometrical parameters of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Furthermore, we studied mechanisms for expanding the available color space through multilayers and in-plane addition for cost- effective color mixing to expand the color gamut further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' While the isotropic character of the nanoislands’ layer ensures the polarization independence, the angle insensitiveness, otherwise impossible in conventional Fabry-Perot resonators, is achieved by exploiting the non-trivial phase discontinuities in the ultrathin cavity, thus avoiding path length effects for steep angles close to 70°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The subwavelength plasmonic cavity is fabricated with a low-temperature process in an ebeam evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The versatility of the process permits the use of many different substrates, including flexible platforms required in wearable electronics and roll-to-roll manufactures, and takes on the scattering properties of the target surface to produce both diffuse and specular coloration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Additionally, being a self-assembly process, the color consistency is ensured for large areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We observed that color purity of our structure is dependent on the size and shape distribution of the nanoislands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We note that although the use of pre-seeding techniques, higher-temperatures, or alternative materials would improve control on the assembly morphology it would impose a scale and cost toll on the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To demonstrate the commercial capabilities of our platform for inorganic metallic pigments, we have fabricated self-standing bi-directional color flakes by evaporating on top of a water-soluble sacrificial layer a double-sided stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This ultralight pigment, that offers full coloration with a single layer of flakes, can be then mixed with a binder matrix to formulate a structural color paint that can be used to coat, after fabrication, any surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This approach presents an ultralight, multi-color, large-scale, low-cost, and environmental- friendly platform for imparting nanostructured coloration to any surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' With an unbeatable surface density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='4 g/m2, hundreds of times lighter than commercially available paints, thus paving the way towards industrial production and real-world application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' __________________________________________________________ Methods Self-Assembled Structural Color Fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The optically thick back-mirror is produced by evaporating 100 nm of aluminum on a Thermionics e-beam evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Pressure at the beginning of the evaporation was ~1x10-6 Torr, and evaporation rate was kept at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='1 nm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' A spacer layer of aluminum oxide was then grown by atomic layer deposition (Savannah 200, Cambridge Nanotech) by pulsing trimetylaluminum and water at 100 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The aluminum nanoparticles were then produced on an UHV AJA electron beam evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' In agreement with previous studies17,31,32, we find that three parameters play a critical role in the geometry of the self-assembled monolayer: the temperature of the substrate, the pressure in the chamber, and the rate of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We chose these parameters compromising between the desired higher saturation of colors and the lower requirements for fabrication that ensures the versatility of the proposed architecture and the viability of a transition to industrial-scale production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Thus, we chose to keep the temperature of the substrates at 100 °C, resulting in high color saturation while being below the melting point of many polymers, essential for flexible substrate applications and lift-off during flake preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For reproducibility and color vividness, nanoparticles’ growth was carried on at pressures below 5x10-8 Torr, while growth rates were kept constant about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='1 A/s, both readily available in conventional UHV evaporators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' It should be note that although an ALD system was used for convenience for the hard substrates, the entire fabrication process could be carried on in a single UHV ebeam evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To prove this promising ease of integration, critical for industrial chain systems, the oxide layer in the pigment flakes production was grown at room temperature on the Thermionics ebeam evaporator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Besides a slight change in color shade, attributed to the change in refractive index of the oxide layer, no fundamental quality change was observed between the layers grown by the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Finite Difference Time Domain Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The reflection spectra and electric field distribution of the simulated samples were calculated using the experimental geometrical parameters extracted from the SEM analysis of the samples, with a commercial FDTD software package (Lumerical FDTD, Lumerical Solutions Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The relative permittivities of aluminum and aluminum oxide are taken from literature33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To build the weight-averaged semi-analytical model of SI Appendix Figure 2 we simulated a single particle with periodic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The values of the unit cell size were chosen to make the area covered by the aluminum island equivalent to those obtained from the SEM analysis for the different samples (55%, 60% and 70% for the 4, 8 and 12 nm respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 50 simulations with radii values within 4 standard deviations of the mean value were then performed and the weight-averaged reflectance calculated as: ������������� = � ������������������������ × ������������������������ ������������ where ������������������������ and ������������������������ represent the Gaussian weight and simulated reflection for the particle ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' For the analysis of the effect of the disorder as shown in SI Appendix Figure 3, we simulated an array of 49 particles placed in periodic arrangement, disordered arrangement, and disordered arrangement of different radii, introducing the disorder parameter following the method presented by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Color Gamut Evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To find the L*a*b* coordinates of the fabricated samples we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='first obtained the XYZ tristimulus values integrating over the visible spectrum according ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='to: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ � ������������̅(������������)������������(������������)������������(������������)������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ � ������������̅(������������)������������(������������)������������(������������)������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ � ������������̅(������������)������������(������������)������������(������������)������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='������������ = � �������������(������������)������������(������������)������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='where ������������(������������) is the measured reflectance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' ������������̅(������������), �������������(������������), and ������������̅(������������) are the color matching functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' ������������(������������) is the reference illuminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' From the tristimulus values CIEXYZ the CIELAB coordinates can be calculated from: ������������∗ = 116 ������������������������ − 16 ������������∗ = 500 ������������������������� − ������������������������� ������������∗ = 200 ������������������������� − ������������������������� where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' being ������������������������ = ������������ ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' ������������������������ = ������������ ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' or ������������������������ = ������������ ������������������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' and ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='9642, ������������������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='0000, and ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='8251 the D50 white point coordinates: ������������������������ = ⎩ ⎨ ⎧ ������������������������� 3 if ������������ > � 6 29� 3 841 ������������������������ 108 + 4 29 otherwise As the reference illuminant we chose D50, used in the graphic arts industry for color proofing as per ISO 3664:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Finally, to keep consistency, all colormaps presented in the figures in the main manuscript and the SI Appendix correspond to a horizontal slice with ������������∗ = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Measurements and Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Reflection measurements were taken at normal incidence with unpolarized light using a 4x, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='07 numerical aperture objective and a fiber coupled spectrometer (HR 2000+, Ocean Optics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' An aluminum mirror was used as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Angular measurements were taken with an integrating sphere (RTC-060-SF, Labsphere) connected to the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' To ensure consistency on illumination the samples were photographed with flash-light at fixed intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Photographies for the butterfly models were taken under sun illumination with a linear polarizer attached to the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Acknowledgments This work at University of Central Florida was supported by National Science Foundation Grant #ECCS-1920840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' acknowledges the support from the UCF Preeminent Postdoctoral Fellowship Program (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Mustroph, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Dyes, General Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' in Ullmann’s Encyclopedia of Industrial Chemistry 1– 38 (American Cancer Society, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Gregory, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' High-Technology Applications of Organic Colorants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' High-Technology Applications of Organic Colorants (Springer US, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Lončarić, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Optical and structural characterization of silver islands films on glass substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Vacuum 84, 188–192 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Palik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Handbook of optical constants of solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 3 (Academic press, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Mao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Manipulating disordered plasmonic systems by external cavity with transition from broadband absorption to reconfigurable reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Nature Communications 2020 11:1 11, 1–7 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Figure 1 | Structural Absorption for Color Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, Many chemical substances produce color by selectively absorbing frequencies matching their molecular electronic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Pink color in Formosa azaleas is due to the absorption of cyaniding molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, An example of structural coloration is found in the Peruvian Morpho didius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Lamellae nanostructures found in its wings scatter the blue components of incident light generating its characteristic metallic blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, A subwavelength plasmonic cavity formed by a self-assembly of metallic nanoislands on top of an oxide-coated mirror, generates color by selectively absorbing certain wavelengths and strongly back-reflecting other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Figure 2 | Color Space and Quality of the Plasmonic Cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, In the Volmer-Weber growth mode, the size of the nanoislands can be controlled by tuning the amount of aluminum evaporated –top-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' If the process is carried on for long enough semicontinuos films are formed that disable the plasmonic resonances and thus the color –bottom-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, Color polar gradient for thicknesses from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 to 16 nm, for fixed 10 nm spacer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, CIELAB coordinates for the points in the color wheel compared to ISO DIS 15339-2 cold-set newsprint and coated premium paper standards (inner and outer hexagon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, Tuning of the spacer and capping layer thicknesses expands the available color space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' e, f, Show the red-shift of the absorption resonance as the spacer and capping layer thicknesses are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a b c 100 Nanoisland Growth 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5nm 8 nm 50 Continuous Film Growth 100 50 0 50 100 tm a* 100 L d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 nm 16 nm 2 6 10 14 , (nm) tm (nm) tte e f tts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' t ts 30 100 100 25 (wu) 75 75 20 ts: 10 te: 0 78nm ts 50 50 15 6nm 25 25 10 30 10 4 nm 0 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 10 450 550 650 750 450 550 650 750 t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' (nm) Wavelength (nm) Wavelength (nm) Figure 7 | Morphology Effect on Optical Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, The inhomogeneous broadening of the optical resonance can be accounted for by introducing size and spatial variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' We simulate the broadening by averaging the reflection curves of 50 particles with radii within 4 standard deviation of the mean value obtained from SEM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, Statistical radii distribution -top- and reflection curves -bottom- for experimental (solid), FDTD mean value (dashed), and weight averaged (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, Reflection curves for FDTD simulations corresponding to 7x7 hemispherical particles with equal size in periodic and disorder arrangement, and random size and disordered arrangement, equivalent to the 5 nm self-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, Electric profiles in three different spectral positions as labeled in panel c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Periodic Disordered Random wu 425 II 425nm wu 705 I Figure 4 | Dual Color Mode, Polarization Independence, and Angle-Insensitiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, Butterfly garden with an assorted collection of different butterfly wings and colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, An artistic butterfly model coated with structural blue retains its color when photographed with unpolarized –left-, and two orthogonal linearly polarized states –center and right-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, The butterfly color is also angle-insensitive, as shown for three different combinations of azimuth and zenith angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d-e, The versatility of the self-assembly fabrication process permits the use of a wide array of substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Flat and sandblasted PET strips are used as flexible substrates to form the three primaries in both d, specular, and e, diffuse coloration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Polarization Independence AngleIndependence SpecularColoration DiffusiveColoration Figure 5 | Mixing Schemes for Expanding the Color Palette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a, Controlling the ratio of area covered by two configurations the reflection curve can be defined by a simple additive rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b, Camera pictures of samples with mixing ratios from 0 to 100%, for spacer thicknesses of 10, 15, and 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' c, Microscope images for the samples highlighted in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, As the ratio is increased the reflection curves transition from pure basis A to pure basis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' e, CIELAB space for the samples corresponding to spacer thickness of 10 nm in panels b and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' The white dotted line overlay represents the space defined by the color wheel in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' f, New colors can be generated by multilayer structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' g, Green shades inaccessible with a single layer can be generated by stacking two self-assemblies with different interspacing thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' h, Tuning of the interspace layer between self-assemblies controls the optical response of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=" a b In-plane FillingFactorα (%) Colour Mixing 20 m Spacer t, (nm) +15 'm II III IV Area () α= Total Area (+) 0 10 20 30 40 50 60 70 80 90 100 c d e 100 um α 100 90 wug a 45 50 ：10nm II 25 0 III 0 tm IV 450 550 650 750 Wavelength(nm) 45 45 0 45 90 a* f 6 h Out-of-plane Interspacingt,(nm） 100 Colour Mixing ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='6 75 50 A B D R 25 F 0 450 550 650 750 4 Wavelength(nm) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content='5 Figure 6 | Structural Color Paint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' a Sequential growing of a bi-directional stack on a sacrificial layer results on color flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' b and c, Color flakes can be stored dry or dispersed in a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' d, A paint can be produced by mixing the flakes with a drying oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' This simple formulation, where the flakes are the pigments and the oil the binder, can be adapted to impart the nanostructured coloration to any surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' e, Photography of a multicolor artistic butterfly on a black canvas painted with a set of linseed oil-based plasmonic paints demonstrating the commercial feasibility of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Insets correspond to a microscope image –top- and SEM micrographs –bottom-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' Scale bar for the butterfly is 1 inch, whereas for the insets, from top-left to bottom-right, scale bars corresponds to 1 mm, 100 μm, 75 μm, and 100 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} +page_content=' StructuralColorFlakes Structural ColorPaint NanostructuredColoration' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQf4gLC/content/2301.02740v1.pdf'} diff --git a/bdAzT4oBgHgl3EQfnf0f/content/tmp_files/2301.01580v1.pdf.txt b/bdAzT4oBgHgl3EQfnf0f/content/tmp_files/2301.01580v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96404e8f8b56114c94e13d84c4e1cfa9f91617ac --- /dev/null +++ b/bdAzT4oBgHgl3EQfnf0f/content/tmp_files/2301.01580v1.pdf.txt @@ -0,0 +1,801 @@ +First on-sky results of ERIS at VLT +Kateryna Kravchenkoa, Yigit Dallilara, Olivier Absili, Alex Agudo Berbela, Andrea Baruffoloh, +Markus J. Bonsef, Alexander Buronc, Yixian Caoa, Angela Cortesc, Felix Dannertf, Richard +Daviesa, Robert J. De Rosad, Matthias Deysenrotha, David S. Doelmang, Frank Eisenhauera, +Simone Espositob, Helmut Feuchtgrubera, Natascha F¨orster Schreibera, Xiaofeng Gaoe, Hans +Gemperleina, Reinhard Genzela, Stefan Gillessena, Christian Ginskig, Adrian M. Glauserf, +Andreas Glindemannc, Paolo Granib, Pierre Haguenauerc, Johannes Hartwiga, Jean Hayozf, +Marianne Heidac, Matthew Kenworthyg, Johann Kolbc, Harald Kuntschnerc, Dieter Lutza, +Daizhong Liua, Mike MacIntoshe, Micha¨el Marssetd, Gilles Orban de Xivryi, Hakan ¨Ozdemira, +Alfio Puglisib, Sascha P. Quanzf, Christian Raua, Armando Riccardib, Daniel Schuppea, Frans +Snikg, Eckhard Sturma, Linda Tacconia, William D. Taylore, and Erich Wiezorreka +aMax Planck Institute for extraterrestrial Physics, Gießenbachstraße 1, 85748 Garching, +Germany +b INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy +cEuropean Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany +dEuropean Southern Observatory, Alonso de C´ordova 3107, Vitacura, Casilla, 19001 Santiago +de Chile, Chile +eUK Astronomy Technology Centre, STFC, Blackford Hill, Edinburgh, EH9 3HJ, UK +fInstitute for Particle Physics and Astrophysics, ETH Z¨urich, Wolfgang-Pauli-Straße 27, +CH-8093 Z¨urich, Switzerland +gLeiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands +hINAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio, 5, 35141 Padova PD, +Italy +iSTAR Institute, Universit´e de Li`ege, All´ee du Six Aoˆut 19c, 4000 Li`ege, Belgium +ABSTRACT +ERIS (Enhanced Resolution Imager and Spectrograph) is a new adaptive optics instrument installed at the +Cassegrain focus of the VLT-UT4 telescope at the Paranal Observatory in Chile. ERIS consists of two near- +infrared instruments: SPIFFIER, an integral field unit (IFU) spectrograph covering J to K bands, and NIX, +an imager covering J to M bands. ERIS has an adaptive optics system able to work with both LGS and NGS. +The Assembly Integration Verification (AIV) phase of ERIS at the Paranal Observatory was carried out starting +in December 2021, followed by several commissioning runs in 2022. This contribution will describe the first +preliminary results of the on-sky performance of ERIS during its commissioning and the future perspectives +based on the preliminary scientific results. +Keywords: ERIS, SPIFFIER, NIX, VLT, integral field spectroscopy, near infrared, imager, adaptive optics, +instrumentation +1. INTRODUCTION +ERIS is a Cassegrain instrument at the VLT-UT4 of the Paranal Observatory in Chile that will operate at +1-5 µm.1 It will take over the fundamental adaptive optics (AO) capabilities at the VLT previously provided by +NACO and SINFONI and, thus, ensure that the VLT remains at the forefront of AO imaging and spectroscopy +into the next decade. +The main scientific drivers of ERIS include resolved studies of high-redshift galaxies, +kkravchenko@mpe.mpg.de +arXiv:2301.01580v1 [astro-ph.IM] 4 Jan 2023 + +Figure 1. Left panel: The overview of ERIS and its main subsystems: the SPIFFIER spectrograph, the NIX imager, the +calibration unit, and the central structure with the LGS and NGS WFS. Right panel: ERIS mounted to the Cassegrain +focus of VLT-UT4. +astrometry in the Galactic Centre, and characterisation of exoplanets. +The ERIS project is being led by a +Consortium of Max-Planck Institute for Extraterrestrial Physics (MPE, leading institute), Istituto Nazionale di +Astrofisica (INAF Arcetri, Abruzzo and Padova), UK Astronomy Technology Centre (UK-ATC), Institute for +Particle Physics and Astrophysics (ETH-Zurich), Netherlands Research School for Astronomy (NOVA Leiden), +and European Southern Observatory (ESO). Fig. 1 displays the overview of ERIS and its main subsystems. +ERIS has two science cameras called SPIFFIER and NIX. SPIFFIER2 is an integral field unit (IFU) spec- +trograph covering the JHK bands, and is an upgraded version of SPIFFI (SPectrometer for Infrared Faint Field +Imaging), which was part of SINFONI.3,4 SPIFFIER provides simultaneous spectroscopy of 32x64 spatial pixels +with a spectral resolution of either ∼5000 or ∼10000 at three image scales: 25, 100, and 250 mas/px, leading +to fields of view (FoV) on the sky of 0.8”x0.8”, 3.2”x3.2” and 8”x8”. NIX5 is an imager operating in the JHK +and LM bands and providing a wide range of modes: standard diffraction-limited imaging in JHK (13 and 27 +mas/px image scales leading to 26”x26” and 55”x55” FoV, respectively) and LM (13 mas/px image scale leading +to 26”x26” FoV) bands, long slit spectroscopy from 3 to 4 µm and high contrast imaging (HCI) modes from +focal/pupil plane coronagraphy to sparse aperture masking interferometry. During science operations, users will +select either ERIS/NIX or ERIS/SPIFFIER for their observations. +The AO module of ERIS provides corrected wavefronts in the J-M bands to NIX and SPIFFIER and has the +following adaptive modes: +• Natural Guide Star (NGS) with an on- or off-axis reference star; +• Laser Guide Star (LGS) with an on-axis LGS and off-axis NGS for tip tilt sensing and truth sensing; +• Seeing enhancer mode where only the on-axis LGS wavefront sensor (WFS) is used for the high-order +correction (in cases when no tip-tilt star is available). +Since ERIS is mounted on UT4 it makes use of the Adaptive Optics Facility (AOF6): one of the four lasers of +4LGSF7 is used to generate an artificial sodium LGS, and the wavefront correction is done by the deformable +secondary mirror (DSM) using the Real Time Computer (RTC) platform called SPARTA.8 +The calibration data for the SPIFFIER and NIX science observations are provided by the Calibration Unit +(CU9), which consists of various internal sources for JHK bands: a Quartz-Tungsten Halogen (QTH) lamp for +flatfielding, four pencil-ray lamps (Ne, Xe, Kr, Ar) for SPIFFIER wavelength calibration and a Laser Driven + +Cable Wrap +Instrument Shutter +NIX +Calibration +Unit +NGS AO +Camera +LGS AO +Camera +SPIFFIER +Height: 2.3m +Cabinet for +Diameter: 3.4m +Electronics +Mass: ≤2.5tCollimator M2 +Grating +wheel +Image +slicer +Detector +Light entering the +instrument +Camera +Pre- +optics +wheel +Filter wheel +Collimator M1 +Collimator M3 +Stiffening structure +Figure 2. An inside view of the SPIFFIER cryostat with indications for the locations of various opto-mechanical elements. +Some housing covers and parts of the stiffening structure are not shown for better illustration. +Light Source (LDLS) for focusing purposes, position adjustments on the detector and distortion correction (only +SPIFFIER). Long-wavelength (LM band) calibrations with NIX are performed exclusively on-sky. +The AIV phase of ERIS at the Paranal Observatory was carried out between December 2021 and Febru- +ary 2022 followed by its first light and subsequent commissioning, which is still ongoing. The following sections +describe technical overview and preliminary performance results of ERIS/SPIFFIER (Sect. 2) and ERIS/NIX +(Sect. 3) at selected observing modes. The description of the AO sub-system design and preliminary commis- +sioning results are reported in 10. +2. SPIFFIER +2.1 Instrument overview +SPIFFIER is the integral field spectrometer and is an upgraded/refurbished version of SPIFFI featuring a new +HAWAII 2RG (2x2k) detector and four new gratings (J, H, K, high-resolution(J,H,K)) providing better spectral +resolution thanks to more symmetric and narrower line spread functions. Addition of the high-resolution grating +leads to twice higher spectral resolution compared to the nominal gratings. SPIFFIER provides simultaneous +spectra of 32x64 spatial pixels (spaxels) at three image scales: 250, 100, and 25 mas/px, leading to field of views +on the sky of 8”x8”, 3”x3”, and 0.8”x0.8”, respectively. Figure 2 shows an image of the open SPIFFIER cryostat +with indications for the locations of the various elements. All components are cooled in a bath cryostat to the +temperature of liquid nitrogen (∼77 K). The liquid nitrogen reservoir sits below the instrument base plate. +The light enters from the top. Below the entrance focal plane baffle, a triplet lens unit collimates the light onto +a cold stop for the suppression of the thermal background. Just in front of the cold stop is the motorized filter +wheel housing the band-pass filters. After the cold stop, the motorized optics wheel provides the interchangeable +lens systems for the three different image scales: 25, 100 and 250 mas/px. The light of the pre-optics is focused +on the image slicer: a stack of 32 small plane mirrors – the so-called small slicer – slices the image and redirects +the light towards the 32 mirrors of the big slicer, which rearranges the slitlets to a long pseudo-slit, which appears +as a brick-wall pattern on the detector. All parts are of Zerodur and are optically contacted (without using any +glue). Each one of the 32 slitlets is imaged onto 64 pixels of the detector. Figure 3 shows the image slicer and the + +Figure 3. Top panels: SPIFFIER image slicer (adapted from 11). The small image slicer B (shown on the top right +panel) cuts the image into stripes and reflects them onto the big image slicer A to create a pseudo long slit to be fed into +spectrometer. Both image slicer components are mounted to a baseplate C. Bottom panel: The layout of the slitlets on a +raw detector frame. +Figure 4. SPIFFIER K-band flat at 25 mas pixel scale. A clump of cold pixels is marked with circle. +positions of the slitlets on a raw SPIFFIER frame. The slitlets run horizontally across the imaging field-of-view +and are numbered from top to bottom on the small slicer. +After the image has been sliced and re-arranged into a pseudo-slit, three diamond turned mirrors (M1, M2 +and M3 in Fig. 2) collimate the light onto the gratings. The first mirror is spherical, and the other two have an +oblate elliptical shape. All mirrors are made from aluminum and are gold-coated for higher reflectivity. In total, +four gratings are implemented on the grating drive. They are based on Zerodur blanks ruled into a gold layer on +the reflecting surface. Three of the gratings cover the J (1.1-1.4 µm), H (1.45-1.85 µm), and K (1.95-2.45 µm) +spectral bands at a resolution of ∼ 5000 superior to the SINFONI gratings by a factor of ∼1.3 in K to ∼2.5 in J. + +Fromthetelescope +A +RC +B +Pseudo-slit32 +3 +2 +17 +18 +19 +20 +21 +22 +23 +24 +15 +16 +31 +30 +29 +28 +27 +26 +25Figure 5. Spectral resolution as a function of wavelength for the low- (dashed lines) and high-resolution (solid lines) JHK +gratings of SPIFFIER. Colors correspond to the three pixel scales (25, 100, and 250 mas). Errorbars represent standard +deviations from the averages over 32 slitlets. +The fourth grating is the high-resolution grating and replaces the previous R∼1500 H+K grating of SINFONI. +This grating doubles the spectral resolution in a given band but reduces the wavelength range by a factor of two. +For each band, users can select either the short, middle or long wavelength regime. A five-lens camera system +then focuses the spectra on the detector. All lenses have a multi-layer anti-reflection coating optimized for the +wavelength range from 1.05-2.45 µm. +A detailed description of the SPIFFI instrument design can be found in 3, with part of the upgrades to +SPIFFIER described in 12. +2.2 Detector +The old Hawaii 2RG detector of SPIFFI was replaced by a new Hawaii 2RG detector because of better persistence +and cosmetics. The new detector was delivered from Teledyne Imaging Sensors. The SPIFFIER detector operates +using the up-the-ramp readout scheme with a frame time of 1.6 seconds. The conversion gain is near 2 e-/ADU +and the read noise is 12 e- rms at the shortest exposures. The minimum noise of ∼7 e- rms is reached around +80 s exposures. The dark current amounts to 0.19 e-/s. +The SPIFFIER detector has randomly distributed bad pixels. These can be interpolated over during data +reduction. The only defect worth noting is a clump of cold pixels (not sensitive to light) about 10 pixels in +diameter, marked with a dark blue circle in Fig. 4. This spot falls into slitlet 16 in all configurations, i.e. in the +middle of the reconstructed image. +2.3 Performance +2.3.1 Spectral resolution +For a given SPIFFIER grating, the spectral resolution can be calculated using wavelength calibration data +provided by the Ne, Xe, Kr and Ar penray lamps of ERIS CU. The procedure is to fit a Gaussian to individual +spectral lines in a single lamp exposure and divide the wavelength of a spectral line by the FWHM of its Gaussian +fit. Using pipeline-processed wavelength calibration maps, the wavelengths and widths of various spectral lines +were extracted for all gratings and pixel scales. The calculated spectral resolution values are illustrated in Fig. 5. +SPIFFIER provides spectral resolution of about 5000 and 10000 for the low- and high-resolution JHK grating +configurations, respectively. The resolution increases for smaller pixel scales and longer wavelengths. + +16000 +25 mas +100 mas +14000 +250 mas +Iresolution +12000 +10000 +pectral +8000 +6000 +S +4000 +2000 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +Wavelength[um]Figure 6. Improvement of instrumental line profile shapes from asymmetric seen in SPIFFI (left, center) to symmetric +with the new gratings of SPIFFIER (right) for the same spectral resolution configuration. +2.3.2 Instrumental line profiles +The instrumental line profiles of SPIFFI were characterized by asymmetric shapes deviating from a Gaussian +shape (left panel of Fig. 6). The SPIFFI gratings were made of NiP coated aluminum with lightweighting. It was +found that the interplay of the lightweighting structure with the stress induced by the NiP coating at cryogenic +temperatures caused deformations of the grating surface and led to degraded instrumental line profiles.12 +The new gratings of SPIFFIER are based on Zerodur blanks without any lightweighting and, therefore, +substantially improve the shapes of the spectral lines. Since line profiles are undersampled on the detector (the +widths are less than two pixels), an approach similar to 13 was used to obtain hypersampled spectral line profiles +for the detailed line-shape analysis. A series of penray lamp calibration exposures with CU was taken for each +band and pixel scale. For a series of exposures of a particular band and pixel scale, the grating wheel was turned +by a few encoder positions between each exposure, which corresponds to a shift of the central wavelength on the +detector by approximately 0.1 pixels. These slightly shifted exposures are referred to as ”babysteps”. In total, +21 babysteps exposures per band and pixel scale were obtained. The combination of the babysteps exposures +allows to create hypersampled line profiles for a given instrument configuration. Fig. 6 illustrates an example of +the resulting oversampled SPIFFIER line profile compared to those from SPIFFI. The instrumental line profiles +of SPIFFIER are symmetric in all bands thanks to the improved design of its diffraction gratings. +3. NIX +3.1 Instrument overview +The NIX imager provides diffraction-limited imaging capabilities in J-M bands (from 1 to 5 µm); focal plane +coronagraphy with Annular Groove Phase Mask (AGPM) in L-M bands;14 pupil plane coronagraphy with grating +vector Apodizing Phase Plate (gvAPP) in K-M bands;15 sparse aperture masking (SAM) in J-M bands; and +long-slit spectroscopy (LSS) in L-band (from 3 to 4 µm). The primary elements of NIX are indicated in Figs. 7 +and 8. Light enters NIX via the NIX selector mirror. This selector mirror is part of the ERIS system and, when +deployed, directs the light from UT4 into the NIX imager instead of the SPIFFIER spectrograph. Inside NIX, +the light passes through the aperture window (Calcium Fluoride) into the NIX cryostat, indicated by the blue +region in Fig. 7 (and also marked in Fig. 8). Nearly all components and mechanisms inside the cryostat are +cooled to 75K to limit thermal background radiation; the detector is the only item cooled further to ∼35K by a +closed cycle cooler. +After the cryostat window the light passes through the aperture wheel located at the telescope focal plane. +It houses various field masks (including a blank position) that are used depending on the observing mode; these +are interchangeable by means of a deployment mechanism driven by a stepper motor providing high positional + +SINF0NI J 250mas +SINFONl post-upgrade J 25Omas +ERIS J 250mas +1.2 +1.2 +1.2 +FWHM = 4.6pixel +FWHM = 4.2pixel +FWHM = 1.6pixel +1.0 +1.0 +1.0 ++ +丰+ +++ +++# +++, +0.8 +0.8 +0.8 +++++ ++ +++ ++# ++ ++ ++ ++++ ++ ++ +0.6 ++ +0.6 +0.6 +++ ++ ++ ++ +++ ++ ++ ++ ++++ +++ +丰 +0.4 +0.4 ++++ ++ +++ +0.4 +x ++++++++ +++ +++ ++ +丰 +++ ++ ++车 ++ +丰 +++ +0.2 +0.2 +0.2 ++ ++ +丰 +0.0 +0.0 +0.0 +-15 +-10 +-5 +0 +5 +10 +15 +-15 +-10 +-5 +0 +5 +10 +15 +-15 +-10 +-5 +0 +5 +10 +15 +pixel +pixel +pixelFigure 7. A sketch of the light path through the NIX cryostat. +Figure 8. A three-dimensional view of NIX instrument. +repeatability. The design and performance of the other wheels is very similar. The next mechanism is the camera +wheel, which contains three different camera barrels. The camera designs are optimized to use the minimum +number of elements to maximize the throughput, while being as axially compact as possible. The camera lenses +are fabricated from Barium Fluoride, IRG2 and Zinc Selenide. +Two cameras are optimised for the shorter +wavelengths (J, H and K), providing spatial scales of 13 mas/px or 27 mas/px. The third camera barrel is for +the longer wavelengths (L and M) delivering 13 mas/px. +From the camera wheel the light passes through the filter and pupil wheels, which are identical mechanisms +housed within a single unit. Both wheels can house up to 18 elements that can be combined in various ways for +the different operating modes of NIX. The filter wheel houses all the optical filters. The pupil wheel contains +a variety of elements that include pupil masks, a grism and additional filters. A set of fold mirrors (the image +selector) then brings the light to a focus on the detector. The image selector has four positions: one for each of + +Camera wheel +Aperture +Pupil Filter Wheel +wheel +Image selector +Detector +box +ERIS +interface. +mount +Electronics +feedthroughs +Closed cycle +Vacuum +cooler in anti- +pumping port +vibration +mount +Nitrogen +purge valve +Emergency +dwndLight from +telescope +Aperture +Camera wheel +Pupil +Detector +Filter +focus +wheel +wheel +wheel +JHK 13 mas/pix +NIX selector +JHK 27mas/pix +Chosen +Chosen +Chosen pupil +Image +mirror +aperture +filter +mask +selector +LM13mas/pix +CryostatFigure 9. Left: Bad pixel map derived with the pipeline. Right: Confidence map in the stacked final product for a dither +pattern with five offset positions. +the three cameras and one to allow pupil imaging. The light is then detected by a Teledyne Hawaii-2RG 5 µm +cutoff detector which is read out using the standard ESO NGC controller. The detector focus stage is used to +adjust internal focus of the NIX detector. +3.1.1 Detector +The NIX detector allows two readout configurations for the user; a slow up-the-ramp configuration with a frame +time of 1.6 seconds and a fast uncorrelated configuration with fastest full frame readout speed of 30 Hz. The +former is configured with an effective gain of 5.2 e-/ADU. The read noise is 19 e- rms at the shortest exposures, +dipping around 9 -e rms near 100 second exposures and the dark current dominates from here onwards with 0.1 +e-/s. The latter is configured with an effective gain of 2.6 e-/ADU and the read noise per read is about 50 e-/s. +The detector configuration comes with several caveats such as odd-even channel effect, for which a fix using +the reference pixels is included with the pipeline. The clumps of bad pixels have the most notable impact in the +design of observations. The users can determine a specific position angle on-sky to perform their observations +to position poorly affected regions to the uninteresting regions and/or a design a dither pattern minimizing the +overlap of bad pixels. Note that aside from these regions, the cosmetics are more typical, < 1%. For example, +high contrast imaging modes are demonstrated to operate well on the upper half of the detector. +Such an example is given in Figure 9 with a choice of five point dither pattern on a 14” box using the +13mas-JHK camera. This certainly reduces the effective exposure time in the regions where there is overlap with +the clump of bad pixels. However, the NIX pipeline delivers stacked variance and confidence maps for users to +properly assess the performance of each pixel in the final product. +3.2 Performance +3.2.1 Astrometric Calibrations - Omega Centauri +After a preliminary coordinate calibration with the telescope offsets observing an isolated star, we opted for +Omega Centauri crowded field observations for astrometric calibrations of the instrument from there onward. +Such an observation provides many benefits in a single attempt taking advantage of GAIA sources sampled in +the field. We can accurately determine the position angle, pixel scale of each camera and the precise pointing +of each frame. Hence, this analysis also serves as input for the coordinate calibration of stages of the ERIS AO +system, similarly a feedback for SPIFFIER as its FOV is small in comparison to perform such accurate analysis. + +Figure 10. Combined image around our original pointing near the center of Omega Centauri (27”x27” mas FOV); the +left image with the 13mas/px and the right image with the 27mas/px camera. Red dot marks the tip-tilt star (2MASS +J13264631-4728402, I=∼11 mag) and available GAIA sources are overlaid with cyan. +Figure 11. Stacked auto-jitter images of NIX Galactic Center observations in L′ band. The markers denote the SiO masers, +the trajectory of which are traced to current date and used for astrometric calibration. +We observed the same field of Omega Centauri as depicted in Figure 10 in two separate occasions. Our first +visit in April 2022 was with sub-optimal AO performance as we were in the earlier stages of the commissioning. +However, the data were useful for the development and testing of our astrometric calibration routines and ERIS +observing blocks. In return, this provided an early feedback for the NIX pipeline and ERIS AO system, and +served as a reference for our second visit in July 2022. In the July observations, we derived a position angle +of -0.4±0.05 degrees and pixel scale of 13.09±0.008 mas/px for the 13mas-JHK camera out of 25 frames, and ++1.5±0.02 degrees and 27.92±0.009 mas/px for the 27mas-JHK camera out of nine frames. The values inside +the parentheses indicate the 1-σ deviation among the frames. +The fields are also selected on purpose in order to match the pointing of recent HST observations. Hence, this + +0 +00000 +0 +0 +0VLT/ERIS vAPP model +Br-a filter (4.05 m) +astrometric +calibration spot +astrometric +calibration spot +1” +VLT/ERIS vAPP +Br-a filter (4.05 µm) +dark hole +dark hole +central leakage term +(2% of stellar flux) +Figure 12. Left panel: Theoretical PSF obtained with the gvAPP mask in Br-α filter of NIX, showing the central faint +reference PSF and the two main PSFs with opposite D-shaped high contrast region. Right panel: First on-sky PSFs of +the standard star ι Cap with the gvAPP in the Br-α filter. +will also serve as accurate distortion characterization for both cameras, for which the analysis is still in progress. +3.2.2 Astrometric Calibration - Galactic Center +For the astrometric calibration of the 13mas-LM camera, we observed the Galactic Center relying on the accu- +rately known trajectories of SiO masers.16 Ten diffraction-limited images with sufficient quality were used for +this purpose. The stacked image is given in Figure 11 with the eight SiO masers used in astrometric calibration +depicted with circles. Our analysis resulted in position angle of -0.4 degrees and pixel scale of 13.03 mas/px. +Poor weather conditions hampered further observations of the Galactic Center and of Omega Cen as additional +reference field. +3.2.3 High-contrast imaging with gvAPP +High contrast imaging is used to image faint circumstellar material or planetary companions, where the diffraction +halo of the central star is the dominant noise source in the regions of interest. NIX provides focal and pupil +plane coronagraphy as well as sparse aperture masking modes. This section is focused solely on the pupil plane +coronagraphy for which the first preliminary on-sky data were obtained. +The pupil plane coronagraph is a grating vector Apodizing Phase Plate (gvAPP15). The gvAPP is a single +optic that goes in the pupil plane of the telescope. It adds a phase pattern across the wavefront of the whole +telescope pupil and therefore modifies the PSF of any point source in the field of view as illustrated in Fig. 12. +The central PSF (referred to as the ”leakage” PSF) is the PSF of the telescope pupil. It contains about 2% +of the transmitted flux and acts as a photometric and astrometric reference. Two additional PSFs appear on +either side of the leakage PSF, with the remaining large fraction of the stellar flux split evenly between the two. +Each of these two PSFs has a dark D-shaped region of suppression on one side of the star. The two D-shaped +regions of both images are on opposite sides of the star, and so provide nearly 360 degrees of suppression with +an approximate inner working angle of three λ/D when combined. +We observed the bright (2.2 mag in L) standard IR star ι Cap in July 2022 using gvAPP in the Br-α filter. In +data pre-processing, the significant near-infrared background noise is calibrated by subtracting pairs of nodding +positions of the target on the detector. The right panel of Fig. 12 illustrates the observed PSF, which is almost +identical to the theoretical PSF calculated for the same instrument configuration (left panel of Fig. 12): both +show the central faint reference PSF and the two main PSFs with opposite D-shaped high contrast region. + +Figure 13. The ADI-processed image of ι Cap in the Br-α filter. White circle marks the location of the detected point +source. +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Separation [arcsec] +10 +3 +Planet-to-star flux ratio +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 + Magnitude +5 + Contrast Limits - Filter: Br-a +3 +4 +5 +6 +7 +8 +9 +Separation [ /D] +# PCA components +5 +10 +15 +20 +25 +30 +40 +50 +60 +70 +80 +90 +100 +Best +Figure 14. Contrast curve corresponding to a 5σ detection for a 735 s exposure with the gvAPP in the Br-α filter of +NIX. The blue-dotted curve indicates the deepest achievable contrast under choice of the optimal number of principal +components. Observing conditions were good with an average seeing of 0.6 arcsec and average precipitable water vapor +(PWV) of 1.2 mm. + +Amag = 8.3 +companion +candidate +classicalADl0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Separation [arcsec] +10 +5 +10 +4 +Planet-to-star flux ratio +8 +9 +10 +11 +12 +13 +14 + Magnitude +5 + Contrast Limits - Filter: Br-g +2 +4 +6 +8 +10 +12 +14 +16 +18 +Separation [ /D] +# PCA components +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +Best +Figure 15. Contrast curve corresponding to a 5σ detection for a 2700 s exposure with the gvAPP in the Br-γ filter of +NIX. The blue-dotted curve indicates the deepest achievable contrast under choice of the optimal number of principal +components. Observing conditions were fair with an average seeing of 0.9 arcsec and average PWV of 6.9 mm. +As next step, the data were post-processed by applying classical angular differential imaging (ADI), which +takes advantage of the rotation of the sky field during the observation (since the telescope tracks the pupil rather +than the field) to remove the PSF of the star (which is static on the detector), while a companion astrophysical +source (e.g. exoplanet) moves on the detector from one exposure to the next. Fig. 13 illustrates the resulting +PSF-subtracted image of ι Cap with the first detection of a point source with separation of 1.6” and a contrast +of roughly 8.3 mag relative to ι Cap in the Br-α filter. With the age of ι Cap (about 0.5 Gyr) the detected point +source is either a low mass M-dwarf companion or a background star. +Fig. 14 illustrates an L-band contrast curve of the gvAPP in the Br-α filter using the ι Cap dataset. The +results are based on fake planet injections at different positions and brightnesses around the host star. We use +the unsaturated regions of the APP PSF as the fake planet signal. The data were reduced using ADI-based +Principle Component Analysis (PCA)17 and detection limits are determined by using the package described in +18 (assuming Gaussian noise). The resulting curve identifies which planets can confidently be rejected after 735 s +of integration time. By conducting a longer observation (1 hour), one can expect the contrast limits to deepen +by about one magnitude. +To demonstrate the contrast performance at shorter wavelengths, we observed the K=3.45 mag star γ Gru +with the gvAPP in the Br-γ filter in July 2022 for a total integration time of 45 min. The data were processed +in the same manner as described for the Br-α filter, and the resulting contrast curve is shown in Fig. 15. The +contrast curves for other NIX filters will be measured during forthcoming commissioning runs. + +Figure 16. Left panel: Stacked SPIFFIER reconstructed image of the central arcsecond of the Galactic center in the +Br-γ line (K band). The spatial scale is 25mas/px. The location of the star S2 is marked. The integration time was +40 min on source. Right panel: Radial velocity (RV) variations of the star S2 during last 20 years observed with different +instruments. The RV of S2 measured with SPIFFIER in April and July 2022 is added and marked with arrow. +4. SCIENCE OUTLOOK +4.1 Galactic Center +SINFONI/SPIFFI was built specifically for observations and studies of the Galactic Center. This allowed major +scientific breakthroughs in the field, for example, the discovery of young B stars (the so-called ”S-stars”) in close +orbits around the SgrA∗ black hole.19 By monitoring the orbits of these stars it was possible to derive the mass +and the distance to the central black hole.20 SPIFFI also led to the discovery of infrared flares of SgrA∗ and +characterization of their spectra.19 The Galactic center with its supermassive black hole is an excellent target +to test general relativity in the strong-field limit, which was done by the peri-center passage of the star S2.21 +Finally, long-term monitoring of the Galacic center with SPIFFI led to the discovery of the gas cloud G2 falling +towards the black hole.22 +ERIS is an important next step in the Galactic center research. First of all, it will allow to continue simul- +taneous monitoring of a large number of stars orbiting around the SgrA∗ black hole (Fig. 16) and, therefore, +continue to improve our knowledge of the distance and the mass of the black hole. In addition, higher Strehl +ratios provided by the AOF as well as the increased throughput and instrumental line profiles of SPIFFIER, will +lead to discovery of new fainter stars, which were not seen by SPIFFI. Finally, exotic events like gas clouds and +flares will continue being observed by SPIFFIER to improve our understanding of the evolution of the Galactic +Center and accretion processes onto its black hole. +4.2 High-redshift galaxies +One of the main science drivers of ERIS is to map the distribution of star formation, physical conditions of the +interstellar medium (ISM), and the motions within galaxies at redshift z ∼ 1–3. At this epoch, galaxies were +forming stars most rapidly so it is a key epoch to study the early assembly of the bulk of their stellar mass. +The key capabilities of ERIS + AOF are the sensitivity, spectral resolution, and spatial resolution. The +performance of AOF is far superior to that of the previous AO at UT4 with much higher Strehl ratios achievable. +Along with improvements from the various SPIFFIER elements in terms of throughput, this leads to a sensitivity +about 4-5 times higher than SINFONI + AO for compact structures on scales of ∼ 0.1”, or ∼ 1 kpc at z ∼ 1–3. +The better AO performance also allows to push the AO star to fainter magnitude substantially increasing the +sky coverage. In addition, the improved spectral resolution of SPIFFIER (especially the R∼10000 capability) + +ER1S-2022-06-13 +06:00:00 +S2 +40x40 LGS AO +4 x 600 sec4000 +3000 +2000 +[km/s +VLSR +1000 +1000 +2000 +2000 +2005 +2010 +2015 +2020 +t [yr]Figure 17. Top panel: H-α and [NII] line intensity maps from SINFONI.23 The integration time was 10 hours on source. +Bottom panel: SPIFFIER reconstructed images in the H-α line; the spatial scale is 100mas/px. The integration time was +1 hour on source. +is crucial to better constrain the kinematics, disentangle non-circular motions that are present within otherwise +globally regular disk rotation kinematics, and measure disk velocity dispersion roughly twice lower than before. +These are key enabling capabilities to study the inner workings of distant galaxies on the physically relevant +Toomre scale (the characteristic fragmentation scale of globally unstable gas-rich disks as observed at high +redshift), which is about 1 kpc at z ∼ 1–3. Science goals at the forefront of galaxy evolution research can now be +addressed with ERIS systematically and for large samples, including gas transport within galaxies, the nature +and fate of giant star-forming complexes, the origin of the elevated gas turbulence, the mass and energetics +of feedback from star formation and active galactic nuclei (AGN), and the spatial distribution of gas-phase +metallicity across galaxies. +The target observed during commissioning, zC406690, has been observed with SINFONI + AO with 10 hours +on-source.23 It is a typical z = 2.2 star-forming galaxy of 4 × 1010M⊙ and the star formation rate (SFR) of +about 250 M⊙/yr. A large fraction of this star formation takes place in kpc-scale ”clumps” along a ring-like +distribution (top panel of Fig. 17). These clumps drive powerful gas outflows, and this galaxy is one of the most +striking example of star formation feedback. Ultimately, with longer integration, the higher spectral and spatial +resolution will provide tighter constraints on the individual clump outflows and their impact on the surrouding +ISM as well as on the clumps longevity. The purpose of the observations was to assess the performance of ERIS + +AOF in realistic cases as will apply to faint distant galaxies, via a direct comparison with existing SINFONI+AO +data. The bottom panel of Fig. 17 illustrates reconstructed SPIFFIER images in the H-α line at the 100 mas/px +spatial scale: the morphology of the galaxy clumps and the diffuse emission along the ring are already recognized +after about one hour of integration on source. +This data set fully matches the expected angular resolution +gain, and the sensitivity improvements for compact sources and extended emission of ERIS+AOF compared to +SINFONI+AO. + +ZC406690 +Hα +[IIN] +z=2.1950 +NGS-AO +(10.0h) +PSF +PSFH-alpha line intensity +H-alpha line S/N +0.79 +1.6 +2.4 +3.2 +4 +4.7 +5.5 +6.3 +7.1ACKNOWLEDGMENTS +We would like to thank all the ESO staff in Garching and Paranal who have supported the installation and +commissioning of ERIS, and to all consortium members who helped to design and build the instrument. +REFERENCES +[1] Davies, R., Esposito, S., Schmid, H. 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K., et al., “New Observations of the Gas Cloud G2 in the Galactic +Center,” ApJ 763, 78 (Feb. 2013). +[23] F¨orster Schreiber, N. M., Renzini, A., Mancini, C., et al., “The SINS/zC-SINF Survey of z ∼ 2 Galaxy Kine- +matics: SINFONI Adaptive Optics-assisted Data and Kiloparsec-scale Emission-line Properties,” ApJS 238, +21 (Oct. 2018). + diff --git a/bdAzT4oBgHgl3EQfnf0f/content/tmp_files/load_file.txt b/bdAzT4oBgHgl3EQfnf0f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..deb6276d93b4c53d0685a3caada6bc9c3eecb063 --- /dev/null +++ b/bdAzT4oBgHgl3EQfnf0f/content/tmp_files/load_file.txt @@ -0,0 +1,620 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf,len=619 +page_content='First on-sky results of ERIS at VLT Kateryna Kravchenkoa, Yigit Dallilara, Olivier Absili, Alex Agudo Berbela, Andrea Baruffoloh, Markus J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Bonsef, Alexander Buronc, Yixian Caoa, Angela Cortesc, Felix Dannertf, Richard Daviesa, Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' De Rosad, Matthias Deysenrotha, David S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Doelmang, Frank Eisenhauera, Simone Espositob, Helmut Feuchtgrubera, Natascha F¨orster Schreibera, Xiaofeng Gaoe, Hans Gemperleina, Reinhard Genzela, Stefan Gillessena, Christian Ginskig, Adrian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Glauserf, Andreas Glindemannc, Paolo Granib, Pierre Haguenauerc, Johannes Hartwiga, Jean Hayozf, Marianne Heidac, Matthew Kenworthyg, Johann Kolbc, Harald Kuntschnerc, Dieter Lutza, Daizhong Liua, Mike MacIntoshe, Micha¨el Marssetd, Gilles Orban de Xivryi, Hakan ¨Ozdemira, Alfio Puglisib, Sascha P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Quanzf, Christian Raua, Armando Riccardib, Daniel Schuppea, Frans Snikg, Eckhard Sturma, Linda Tacconia, William D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Taylore, and Erich Wiezorreka aMax Planck Institute for extraterrestrial Physics, Gießenbachstraße 1, 85748 Garching, Germany b INAF-Osservatorio Astrofisico di Arcetri, Largo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Fermi 5, 50125 Firenze, Italy cEuropean Southern Observatory, Karl-Schwarzschild-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2, 85748 Garching, Germany dEuropean Southern Observatory, Alonso de C´ordova 3107, Vitacura, Casilla, 19001 Santiago de Chile, Chile eUK Astronomy Technology Centre, STFC, Blackford Hill, Edinburgh, EH9 3HJ, UK fInstitute for Particle Physics and Astrophysics, ETH Z¨urich, Wolfgang-Pauli-Straße 27, CH-8093 Z¨urich, Switzerland gLeiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Box 9513, 2300 RA Leiden, The Netherlands hINAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio, 5, 35141 Padova PD, Italy iSTAR Institute, Universit´e de Li`ege, All´ee du Six Aoˆut 19c, 4000 Li`ege, Belgium ABSTRACT ERIS (Enhanced Resolution Imager and Spectrograph) is a new adaptive optics instrument installed at the Cassegrain focus of the VLT-UT4 telescope at the Paranal Observatory in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' ERIS consists of two near- infrared instruments: SPIFFIER, an integral field unit (IFU) spectrograph covering J to K bands, and NIX, an imager covering J to M bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' ERIS has an adaptive optics system able to work with both LGS and NGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The Assembly Integration Verification (AIV) phase of ERIS at the Paranal Observatory was carried out starting in December 2021, followed by several commissioning runs in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This contribution will describe the first preliminary results of the on-sky performance of ERIS during its commissioning and the future perspectives based on the preliminary scientific results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Keywords: ERIS, SPIFFIER, NIX, VLT, integral field spectroscopy, near infrared, imager, adaptive optics, instrumentation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' INTRODUCTION ERIS is a Cassegrain instrument at the VLT-UT4 of the Paranal Observatory in Chile that will operate at 1-5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 It will take over the fundamental adaptive optics (AO) capabilities at the VLT previously provided by NACO and SINFONI and, thus, ensure that the VLT remains at the forefront of AO imaging and spectroscopy into the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The main scientific drivers of ERIS include resolved studies of high-redshift galaxies, kkravchenko@mpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='01580v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='IM] 4 Jan 2023 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Left panel: The overview of ERIS and its main subsystems: the SPIFFIER spectrograph, the NIX imager, the calibration unit, and the central structure with the LGS and NGS WFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Right panel: ERIS mounted to the Cassegrain focus of VLT-UT4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' astrometry in the Galactic Centre, and characterisation of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The ERIS project is being led by a Consortium of Max-Planck Institute for Extraterrestrial Physics (MPE, leading institute), Istituto Nazionale di Astrofisica (INAF Arcetri, Abruzzo and Padova), UK Astronomy Technology Centre (UK-ATC), Institute for Particle Physics and Astrophysics (ETH-Zurich), Netherlands Research School for Astronomy (NOVA Leiden), and European Southern Observatory (ESO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 1 displays the overview of ERIS and its main subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' ERIS has two science cameras called SPIFFIER and NIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' SPIFFIER2 is an integral field unit (IFU) spec- trograph covering the JHK bands, and is an upgraded version of SPIFFI (SPectrometer for Infrared Faint Field Imaging), which was part of SINFONI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3,4 SPIFFIER provides simultaneous spectroscopy of 32x64 spatial pixels with a spectral resolution of either ∼5000 or ∼10000 at three image scales: 25, 100, and 250 mas/px, leading to fields of view (FoV) on the sky of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8”x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8”, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2”x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2” and 8”x8”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' NIX5 is an imager operating in the JHK and LM bands and providing a wide range of modes: standard diffraction-limited imaging in JHK (13 and 27 mas/px image scales leading to 26”x26” and 55”x55” FoV, respectively) and LM (13 mas/px image scale leading to 26”x26” FoV) bands, long slit spectroscopy from 3 to 4 µm and high contrast imaging (HCI) modes from focal/pupil plane coronagraphy to sparse aperture masking interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' During science operations, users will select either ERIS/NIX or ERIS/SPIFFIER for their observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The AO module of ERIS provides corrected wavefronts in the J-M bands to NIX and SPIFFIER and has the following adaptive modes: Natural Guide Star (NGS) with an on- or off-axis reference star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Laser Guide Star (LGS) with an on-axis LGS and off-axis NGS for tip tilt sensing and truth sensing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Seeing enhancer mode where only the on-axis LGS wavefront sensor (WFS) is used for the high-order correction (in cases when no tip-tilt star is available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Since ERIS is mounted on UT4 it makes use of the Adaptive Optics Facility (AOF6): one of the four lasers of 4LGSF7 is used to generate an artificial sodium LGS, and the wavefront correction is done by the deformable secondary mirror (DSM) using the Real Time Computer (RTC) platform called SPARTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 The calibration data for the SPIFFIER and NIX science observations are provided by the Calibration Unit (CU9), which consists of various internal sources for JHK bands: a Quartz-Tungsten Halogen (QTH) lamp for flatfielding, four pencil-ray lamps (Ne, Xe, Kr, Ar) for SPIFFIER wavelength calibration and a Laser Driven Cable Wrap Instrument Shutter NIX Calibration Unit NGS AO Camera LGS AO Camera SPIFFIER Height: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3m Cabinet for Diameter: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4m Electronics Mass: ≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5tCollimator M2 Grating wheel Image slicer Detector Light entering the instrument Camera Pre- optics wheel Filter wheel Collimator M1 Collimator M3 Stiffening structure Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' An inside view of the SPIFFIER cryostat with indications for the locations of various opto-mechanical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Some housing covers and parts of the stiffening structure are not shown for better illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Light Source (LDLS) for focusing purposes, position adjustments on the detector and distortion correction (only SPIFFIER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Long-wavelength (LM band) calibrations with NIX are performed exclusively on-sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The AIV phase of ERIS at the Paranal Observatory was carried out between December 2021 and Febru- ary 2022 followed by its first light and subsequent commissioning, which is still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The following sections describe technical overview and preliminary performance results of ERIS/SPIFFIER (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2) and ERIS/NIX (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 3) at selected observing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The description of the AO sub-system design and preliminary commis- sioning results are reported in 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' SPIFFIER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 Instrument overview SPIFFIER is the integral field spectrometer and is an upgraded/refurbished version of SPIFFI featuring a new HAWAII 2RG (2x2k) detector and four new gratings (J, H, K, high-resolution(J,H,K)) providing better spectral resolution thanks to more symmetric and narrower line spread functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Addition of the high-resolution grating leads to twice higher spectral resolution compared to the nominal gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' SPIFFIER provides simultaneous spectra of 32x64 spatial pixels (spaxels) at three image scales: 250, 100, and 25 mas/px, leading to field of views on the sky of 8”x8”, 3”x3”, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8”x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 2 shows an image of the open SPIFFIER cryostat with indications for the locations of the various elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' All components are cooled in a bath cryostat to the temperature of liquid nitrogen (∼77 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The liquid nitrogen reservoir sits below the instrument base plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The light enters from the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Below the entrance focal plane baffle, a triplet lens unit collimates the light onto a cold stop for the suppression of the thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Just in front of the cold stop is the motorized filter wheel housing the band-pass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' After the cold stop, the motorized optics wheel provides the interchangeable lens systems for the three different image scales: 25, 100 and 250 mas/px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The light of the pre-optics is focused on the image slicer: a stack of 32 small plane mirrors – the so-called small slicer – slices the image and redirects the light towards the 32 mirrors of the big slicer, which rearranges the slitlets to a long pseudo-slit, which appears as a brick-wall pattern on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' All parts are of Zerodur and are optically contacted (without using any glue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Each one of the 32 slitlets is imaged onto 64 pixels of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 3 shows the image slicer and the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Top panels: SPIFFIER image slicer (adapted from 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The small image slicer B (shown on the top right panel) cuts the image into stripes and reflects them onto the big image slicer A to create a pseudo long slit to be fed into spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Both image slicer components are mounted to a baseplate C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Bottom panel: The layout of the slitlets on a raw detector frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' SPIFFIER K-band flat at 25 mas pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A clump of cold pixels is marked with circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' positions of the slitlets on a raw SPIFFIER frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The slitlets run horizontally across the imaging field-of-view and are numbered from top to bottom on the small slicer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' After the image has been sliced and re-arranged into a pseudo-slit, three diamond turned mirrors (M1, M2 and M3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2) collimate the light onto the gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The first mirror is spherical, and the other two have an oblate elliptical shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' All mirrors are made from aluminum and are gold-coated for higher reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In total, four gratings are implemented on the grating drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' They are based on Zerodur blanks ruled into a gold layer on the reflecting surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Three of the gratings cover the J (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 µm), H (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='45-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='85 µm), and K (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='95-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='45 µm) spectral bands at a resolution of ∼ 5000 superior to the SINFONI gratings by a factor of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 in K to ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Fromthetelescope A RC B Pseudo-slit32 3 2 17 18 19 20 21 22 23 24 15 16 31 30 29 28 27 26 25Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Spectral resolution as a function of wavelength for the low- (dashed lines) and high-resolution (solid lines) JHK gratings of SPIFFIER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Colors correspond to the three pixel scales (25, 100, and 250 mas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Errorbars represent standard deviations from the averages over 32 slitlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The fourth grating is the high-resolution grating and replaces the previous R∼1500 H+K grating of SINFONI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This grating doubles the spectral resolution in a given band but reduces the wavelength range by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' For each band, users can select either the short, middle or long wavelength regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A five-lens camera system then focuses the spectra on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' All lenses have a multi-layer anti-reflection coating optimized for the wavelength range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='05-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='45 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A detailed description of the SPIFFI instrument design can be found in 3, with part of the upgrades to SPIFFIER described in 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 Detector The old Hawaii 2RG detector of SPIFFI was replaced by a new Hawaii 2RG detector because of better persistence and cosmetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The new detector was delivered from Teledyne Imaging Sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The SPIFFIER detector operates using the up-the-ramp readout scheme with a frame time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The conversion gain is near 2 e-/ADU and the read noise is 12 e- rms at the shortest exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The minimum noise of ∼7 e- rms is reached around 80 s exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The dark current amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='19 e-/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The SPIFFIER detector has randomly distributed bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' These can be interpolated over during data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The only defect worth noting is a clump of cold pixels (not sensitive to light) about 10 pixels in diameter, marked with a dark blue circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This spot falls into slitlet 16 in all configurations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' in the middle of the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 Performance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 Spectral resolution For a given SPIFFIER grating, the spectral resolution can be calculated using wavelength calibration data provided by the Ne, Xe, Kr and Ar penray lamps of ERIS CU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The procedure is to fit a Gaussian to individual spectral lines in a single lamp exposure and divide the wavelength of a spectral line by the FWHM of its Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Using pipeline-processed wavelength calibration maps, the wavelengths and widths of various spectral lines were extracted for all gratings and pixel scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The calculated spectral resolution values are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' SPIFFIER provides spectral resolution of about 5000 and 10000 for the low- and high-resolution JHK grating configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The resolution increases for smaller pixel scales and longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 16000 25 mas 100 mas 14000 250 mas Iresolution 12000 10000 pectral 8000 6000 S 4000 2000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 Wavelength[um]Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Improvement of instrumental line profile shapes from asymmetric seen in SPIFFI (left, center) to symmetric with the new gratings of SPIFFIER (right) for the same spectral resolution configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 Instrumental line profiles The instrumental line profiles of SPIFFI were characterized by asymmetric shapes deviating from a Gaussian shape (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The SPIFFI gratings were made of NiP coated aluminum with lightweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' It was found that the interplay of the lightweighting structure with the stress induced by the NiP coating at cryogenic temperatures caused deformations of the grating surface and led to degraded instrumental line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='12 The new gratings of SPIFFIER are based on Zerodur blanks without any lightweighting and, therefore, substantially improve the shapes of the spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Since line profiles are undersampled on the detector (the widths are less than two pixels), an approach similar to 13 was used to obtain hypersampled spectral line profiles for the detailed line-shape analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A series of penray lamp calibration exposures with CU was taken for each band and pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' For a series of exposures of a particular band and pixel scale, the grating wheel was turned by a few encoder positions between each exposure, which corresponds to a shift of the central wavelength on the detector by approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' These slightly shifted exposures are referred to as ”babysteps”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In total, 21 babysteps exposures per band and pixel scale were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The combination of the babysteps exposures allows to create hypersampled line profiles for a given instrument configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 6 illustrates an example of the resulting oversampled SPIFFIER line profile compared to those from SPIFFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The instrumental line profiles of SPIFFIER are symmetric in all bands thanks to the improved design of its diffraction gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' NIX 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 Instrument overview The NIX imager provides diffraction-limited imaging capabilities in J-M bands (from 1 to 5 µm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' focal plane coronagraphy with Annular Groove Phase Mask (AGPM) in L-M bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='14 pupil plane coronagraphy with grating vector Apodizing Phase Plate (gvAPP) in K-M bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='15 sparse aperture masking (SAM) in J-M bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' and long-slit spectroscopy (LSS) in L-band (from 3 to 4 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The primary elements of NIX are indicated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Light enters NIX via the NIX selector mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This selector mirror is part of the ERIS system and, when deployed, directs the light from UT4 into the NIX imager instead of the SPIFFIER spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Inside NIX, the light passes through the aperture window (Calcium Fluoride) into the NIX cryostat, indicated by the blue region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 7 (and also marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Nearly all components and mechanisms inside the cryostat are cooled to 75K to limit thermal background radiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' the detector is the only item cooled further to ∼35K by a closed cycle cooler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' After the cryostat window the light passes through the aperture wheel located at the telescope focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' It houses various field masks (including a blank position) that are used depending on the observing mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' these are interchangeable by means of a deployment mechanism driven by a stepper motor providing high positional SINF0NI J 250mas SINFONl post-upgrade J 25Omas ERIS J 250mas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 FWHM = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6pixel FWHM = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2pixel FWHM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6pixel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 + 丰+ ++ ++# ++, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 ++++ + ++ +# + + + +++ + + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 ++ + + + ++ + + + +++ ++ 丰 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 +++ + ++ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 x +++++++ ++ ++ + 丰 ++ + +车 + 丰 ++ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 + + 丰 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 pixel pixel pixelFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A sketch of the light path through the NIX cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A three-dimensional view of NIX instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The design and performance of the other wheels is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The next mechanism is the camera wheel, which contains three different camera barrels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The camera designs are optimized to use the minimum number of elements to maximize the throughput, while being as axially compact as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The camera lenses are fabricated from Barium Fluoride, IRG2 and Zinc Selenide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Two cameras are optimised for the shorter wavelengths (J, H and K), providing spatial scales of 13 mas/px or 27 mas/px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The third camera barrel is for the longer wavelengths (L and M) delivering 13 mas/px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' From the camera wheel the light passes through the filter and pupil wheels, which are identical mechanisms housed within a single unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Both wheels can house up to 18 elements that can be combined in various ways for the different operating modes of NIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The filter wheel houses all the optical filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The pupil wheel contains a variety of elements that include pupil masks, a grism and additional filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A set of fold mirrors (the image selector) then brings the light to a focus on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The image selector has four positions: one for each of Camera wheel Aperture Pupil Filter Wheel wheel Image selector Detector box ERIS interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' mount Electronics feedthroughs Closed cycle Vacuum cooler in anti- pumping port vibration mount Nitrogen purge valve Emergency dwndLight from telescope Aperture Camera wheel Pupil Detector Filter focus wheel wheel wheel JHK 13 mas/pix NIX selector JHK 27mas/pix Chosen Chosen Chosen pupil Image mirror aperture filter mask selector LM13mas/pix CryostatFigure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Left: Bad pixel map derived with the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Right: Confidence map in the stacked final product for a dither pattern with five offset positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' the three cameras and one to allow pupil imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The light is then detected by a Teledyne Hawaii-2RG 5 µm cutoff detector which is read out using the standard ESO NGC controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The detector focus stage is used to adjust internal focus of the NIX detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 Detector The NIX detector allows two readout configurations for the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' a slow up-the-ramp configuration with a frame time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 seconds and a fast uncorrelated configuration with fastest full frame readout speed of 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The former is configured with an effective gain of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 e-/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The read noise is 19 e- rms at the shortest exposures, dipping around 9 -e rms near 100 second exposures and the dark current dominates from here onwards with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 e-/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The latter is configured with an effective gain of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 e-/ADU and the read noise per read is about 50 e-/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The detector configuration comes with several caveats such as odd-even channel effect, for which a fix using the reference pixels is included with the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The clumps of bad pixels have the most notable impact in the design of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The users can determine a specific position angle on-sky to perform their observations to position poorly affected regions to the uninteresting regions and/or a design a dither pattern minimizing the overlap of bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Note that aside from these regions, the cosmetics are more typical, < 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' For example, high contrast imaging modes are demonstrated to operate well on the upper half of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Such an example is given in Figure 9 with a choice of five point dither pattern on a 14” box using the 13mas-JHK camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This certainly reduces the effective exposure time in the regions where there is overlap with the clump of bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' However, the NIX pipeline delivers stacked variance and confidence maps for users to properly assess the performance of each pixel in the final product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 Performance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 Astrometric Calibrations - Omega Centauri After a preliminary coordinate calibration with the telescope offsets observing an isolated star, we opted for Omega Centauri crowded field observations for astrometric calibrations of the instrument from there onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Such an observation provides many benefits in a single attempt taking advantage of GAIA sources sampled in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' We can accurately determine the position angle, pixel scale of each camera and the precise pointing of each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Hence, this analysis also serves as input for the coordinate calibration of stages of the ERIS AO system, similarly a feedback for SPIFFIER as its FOV is small in comparison to perform such accurate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Combined image around our original pointing near the center of Omega Centauri (27”x27” mas FOV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' the left image with the 13mas/px and the right image with the 27mas/px camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Red dot marks the tip-tilt star (2MASS J13264631-4728402, I=∼11 mag) and available GAIA sources are overlaid with cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Stacked auto-jitter images of NIX Galactic Center observations in L′ band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The markers denote the SiO masers, the trajectory of which are traced to current date and used for astrometric calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' We observed the same field of Omega Centauri as depicted in Figure 10 in two separate occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Our first visit in April 2022 was with sub-optimal AO performance as we were in the earlier stages of the commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' However, the data were useful for the development and testing of our astrometric calibration routines and ERIS observing blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In return, this provided an early feedback for the NIX pipeline and ERIS AO system, and served as a reference for our second visit in July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In the July observations, we derived a position angle of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='05 degrees and pixel scale of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='008 mas/px for the 13mas-JHK camera out of 25 frames, and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='02 degrees and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='009 mas/px for the 27mas-JHK camera out of nine frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The values inside the parentheses indicate the 1-σ deviation among the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The fields are also selected on purpose in order to match the pointing of recent HST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Hence, this 0 00000 0 0 0VLT/ERIS vAPP model Br-a filter (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='05 m) astrometric calibration spot astrometric calibration spot 1” VLT/ERIS vAPP Br-a filter (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='05 µm) dark hole dark hole central leakage term (2% of stellar flux) Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Left panel: Theoretical PSF obtained with the gvAPP mask in Br-α filter of NIX, showing the central faint reference PSF and the two main PSFs with opposite D-shaped high contrast region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Right panel: First on-sky PSFs of the standard star ι Cap with the gvAPP in the Br-α filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' will also serve as accurate distortion characterization for both cameras, for which the analysis is still in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 Astrometric Calibration - Galactic Center For the astrometric calibration of the 13mas-LM camera, we observed the Galactic Center relying on the accu- rately known trajectories of SiO masers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='16 Ten diffraction-limited images with sufficient quality were used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The stacked image is given in Figure 11 with the eight SiO masers used in astrometric calibration depicted with circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Our analysis resulted in position angle of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 degrees and pixel scale of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='03 mas/px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Poor weather conditions hampered further observations of the Galactic Center and of Omega Cen as additional reference field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 High-contrast imaging with gvAPP High contrast imaging is used to image faint circumstellar material or planetary companions, where the diffraction halo of the central star is the dominant noise source in the regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' NIX provides focal and pupil plane coronagraphy as well as sparse aperture masking modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This section is focused solely on the pupil plane coronagraphy for which the first preliminary on-sky data were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The pupil plane coronagraph is a grating vector Apodizing Phase Plate (gvAPP15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The gvAPP is a single optic that goes in the pupil plane of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' It adds a phase pattern across the wavefront of the whole telescope pupil and therefore modifies the PSF of any point source in the field of view as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The central PSF (referred to as the ”leakage” PSF) is the PSF of the telescope pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' It contains about 2% of the transmitted flux and acts as a photometric and astrometric reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Two additional PSFs appear on either side of the leakage PSF, with the remaining large fraction of the stellar flux split evenly between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Each of these two PSFs has a dark D-shaped region of suppression on one side of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The two D-shaped regions of both images are on opposite sides of the star, and so provide nearly 360 degrees of suppression with an approximate inner working angle of three λ/D when combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' We observed the bright (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 mag in L) standard IR star ι Cap in July 2022 using gvAPP in the Br-α filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In data pre-processing, the significant near-infrared background noise is calibrated by subtracting pairs of nodding positions of the target on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 12 illustrates the observed PSF, which is almost identical to the theoretical PSF calculated for the same instrument configuration (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 12): both show the central faint reference PSF and the two main PSFs with opposite D-shaped high contrast region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The ADI-processed image of ι Cap in the Br-α filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' White circle marks the location of the detected point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='9 Separation [arcsec] 10 3 Planet-to-star flux ratio 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 Magnitude 5 Contrast Limits - Filter: Br-a 3 4 5 6 7 8 9 Separation [ /D] # PCA components 5 10 15 20 25 30 40 50 60 70 80 90 100 Best Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Contrast curve corresponding to a 5σ detection for a 735 s exposure with the gvAPP in the Br-α filter of NIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The blue-dotted curve indicates the deepest achievable contrast under choice of the optimal number of principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Observing conditions were good with an average seeing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 arcsec and average precipitable water vapor (PWV) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Amag = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 companion candidate classicalADl0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0 Separation [arcsec] 10 5 10 4 Planet-to-star flux ratio 8 9 10 11 12 13 14 Magnitude 5 Contrast Limits - Filter: Br-g 2 4 6 8 10 12 14 16 18 Separation [ /D] # PCA components 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Best Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Contrast curve corresponding to a 5σ detection for a 2700 s exposure with the gvAPP in the Br-γ filter of NIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The blue-dotted curve indicates the deepest achievable contrast under choice of the optimal number of principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Observing conditions were fair with an average seeing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='9 arcsec and average PWV of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='9 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' As next step, the data were post-processed by applying classical angular differential imaging (ADI), which takes advantage of the rotation of the sky field during the observation (since the telescope tracks the pupil rather than the field) to remove the PSF of the star (which is static on the detector), while a companion astrophysical source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' exoplanet) moves on the detector from one exposure to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 13 illustrates the resulting PSF-subtracted image of ι Cap with the first detection of a point source with separation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6” and a contrast of roughly 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='3 mag relative to ι Cap in the Br-α filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' With the age of ι Cap (about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 Gyr) the detected point source is either a low mass M-dwarf companion or a background star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 14 illustrates an L-band contrast curve of the gvAPP in the Br-α filter using the ι Cap dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The results are based on fake planet injections at different positions and brightnesses around the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' We use the unsaturated regions of the APP PSF as the fake planet signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The data were reduced using ADI-based Principle Component Analysis (PCA)17 and detection limits are determined by using the package described in 18 (assuming Gaussian noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The resulting curve identifies which planets can confidently be rejected after 735 s of integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' By conducting a longer observation (1 hour), one can expect the contrast limits to deepen by about one magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' To demonstrate the contrast performance at shorter wavelengths, we observed the K=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='45 mag star γ Gru with the gvAPP in the Br-γ filter in July 2022 for a total integration time of 45 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The data were processed in the same manner as described for the Br-α filter, and the resulting contrast curve is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The contrast curves for other NIX filters will be measured during forthcoming commissioning runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Left panel: Stacked SPIFFIER reconstructed image of the central arcsecond of the Galactic center in the Br-γ line (K band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The spatial scale is 25mas/px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The location of the star S2 is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The integration time was 40 min on source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Right panel: Radial velocity (RV) variations of the star S2 during last 20 years observed with different instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The RV of S2 measured with SPIFFIER in April and July 2022 is added and marked with arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' SCIENCE OUTLOOK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1 Galactic Center SINFONI/SPIFFI was built specifically for observations and studies of the Galactic Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This allowed major scientific breakthroughs in the field, for example, the discovery of young B stars (the so-called ”S-stars”) in close orbits around the SgrA∗ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='19 By monitoring the orbits of these stars it was possible to derive the mass and the distance to the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='20 SPIFFI also led to the discovery of infrared flares of SgrA∗ and characterization of their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='19 The Galactic center with its supermassive black hole is an excellent target to test general relativity in the strong-field limit, which was done by the peri-center passage of the star S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='21 Finally, long-term monitoring of the Galacic center with SPIFFI led to the discovery of the gas cloud G2 falling towards the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='22 ERIS is an important next step in the Galactic center research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' First of all, it will allow to continue simul- taneous monitoring of a large number of stars orbiting around the SgrA∗ black hole (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 16) and, therefore, continue to improve our knowledge of the distance and the mass of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In addition, higher Strehl ratios provided by the AOF as well as the increased throughput and instrumental line profiles of SPIFFIER, will lead to discovery of new fainter stars, which were not seen by SPIFFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Finally, exotic events like gas clouds and flares will continue being observed by SPIFFIER to improve our understanding of the evolution of the Galactic Center and accretion processes onto its black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 High-redshift galaxies One of the main science drivers of ERIS is to map the distribution of star formation, physical conditions of the interstellar medium (ISM), and the motions within galaxies at redshift z ∼ 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' At this epoch, galaxies were forming stars most rapidly so it is a key epoch to study the early assembly of the bulk of their stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The key capabilities of ERIS + AOF are the sensitivity, spectral resolution, and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The performance of AOF is far superior to that of the previous AO at UT4 with much higher Strehl ratios achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Along with improvements from the various SPIFFIER elements in terms of throughput, this leads to a sensitivity about 4-5 times higher than SINFONI + AO for compact structures on scales of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1”, or ∼ 1 kpc at z ∼ 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The better AO performance also allows to push the AO star to fainter magnitude substantially increasing the sky coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' In addition, the improved spectral resolution of SPIFFIER (especially the R∼10000 capability) ER1S-2022-06-13 06:00:00 S2 40x40 LGS AO 4 x 600 sec4000 3000 2000 [km/s VLSR 1000 1000 2000 2000 2005 2010 2015 2020 t [yr]Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Top panel: H-α and [NII] line intensity maps from SINFONI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='23 The integration time was 10 hours on source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Bottom panel: SPIFFIER reconstructed images in the H-α line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' the spatial scale is 100mas/px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The integration time was 1 hour on source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' is crucial to better constrain the kinematics, disentangle non-circular motions that are present within otherwise globally regular disk rotation kinematics, and measure disk velocity dispersion roughly twice lower than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' These are key enabling capabilities to study the inner workings of distant galaxies on the physically relevant Toomre scale (the characteristic fragmentation scale of globally unstable gas-rich disks as observed at high redshift), which is about 1 kpc at z ∼ 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Science goals at the forefront of galaxy evolution research can now be addressed with ERIS systematically and for large samples, including gas transport within galaxies, the nature and fate of giant star-forming complexes, the origin of the elevated gas turbulence, the mass and energetics of feedback from star formation and active galactic nuclei (AGN), and the spatial distribution of gas-phase metallicity across galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The target observed during commissioning, zC406690, has been observed with SINFONI + AO with 10 hours on-source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='23 It is a typical z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 star-forming galaxy of 4 × 1010M⊙ and the star formation rate (SFR) of about 250 M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' A large fraction of this star formation takes place in kpc-scale ”clumps” along a ring-like distribution (top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' These clumps drive powerful gas outflows, and this galaxy is one of the most striking example of star formation feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' Ultimately, with longer integration, the higher spectral and spatial resolution will provide tighter constraints on the individual clump outflows and their impact on the surrouding ISM as well as on the clumps longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The purpose of the observations was to assess the performance of ERIS + AOF in realistic cases as will apply to faint distant galaxies, via a direct comparison with existing SINFONI+AO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' 17 illustrates reconstructed SPIFFIER images in the H-α line at the 100 mas/px spatial scale: the morphology of the galaxy clumps and the diffuse emission along the ring are already recognized after about one hour of integration on source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' This data set fully matches the expected angular resolution gain, and the sensitivity improvements for compact sources and extended emission of ERIS+AOF compared to SINFONI+AO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content=' ZC406690 Hα [IIN] z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='1950 NGS-AO (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='0h) PSF PSFH-alpha line intensity H-alpha line S/N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='2 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAzT4oBgHgl3EQfnf0f/content/2301.01580v1.pdf'} 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b/d9FAT4oBgHgl3EQf6x6d/content/tmp_files/2301.08741v1.pdf.txt @@ -0,0 +1,958 @@ +Enactive Artificial Intelligence: +Overcoming Male Gaze in Robot-Human Interaction + +Inês Hipólito1,2, Katie Winkle3, Merete Lie4 + +1. +Humboldt-Universität zu Berlin, Berlin School of Mind and Brain, Germany +2. +University of Amsterdam, Faculty of Social and Behavioural Sciences, Netherlands +3. +Social Robotics Lab at Uppsala University, Sweden +4. +Norwegian University of Science and Technology, Norway + + + + +Maybe all women should be robots, he thinks with a tinge of acid: +the flesh-and-blood ones are out of control. +— Margaret Atwood + + +Abstract +Enactive Artificial Intelligence (eAI) motivates new directions towards gender-inclusive AI. +Beyond a mirror reflecting our values, AI design has a profound impact on shaping the enaction +of cultural identities. The traditionally unrepresentative, white, cisgender, heterosexual +dominant narratives are partial, and thereby active vehicles of social marginalisation. Drawing +from enactivism, the paper first characterises AI design as a cultural practice; which is then +specified in feminist technoscience principles, i.e. how intersectionality, gender and other +embodied identity markers are entangled in AI. These principles are then discussed in the specific +case of feminist human-robot interaction. The paper, then, stipulates the conditions for eAI: an +eAI robot is a robot that (1) plays a cultural role in individual and social identity, (2) this role +takes the form of human-robot dynamical interaction, and (3) interaction is embodied. Drawing +from eAI, finally, the paper offers guidelines for I. eAI gender-inclusive AI, and II. subverting +existing gender norms of robot design. + + +Keywords: Enactivism, Artificial Intelligence, feminist robot-human interaction, Feminist +Technoscience. + + + + + + + +Introduction + +Recent years have seen a greater rise in automation than ever before. Artificial Intelligence (AI) +is today at the very heart of trends in robotics. From collaborative robots, and robot employees, +to processes of automation in customer service or computer vision, and natural language +processing, the most innovative AI robotics trends of 2022 show the undeniable emergence of +human-robot interaction (Madsen, 2019; Sigov et al., 2022; Boshnyaku, 2023; Vilkas et al., 2023; +Pizoń and Gola, 2023). + +AI offers solutions to societal problems: to make human life more comfortable, improve +health, and education, and reshape the workforce and industries, markets, and private lives; with +profound implications for human individual and social experiences and identities. Yet, AI +technology has been linked with gender bias (Buolamwini and Gebru, 2018; Lütz, 2022), and +power dynamics (Almeida, Shmarko and Lomas, 2022; Heinrichs, 2022). Gender and other +embodied identity markers are specifically entangled in AI (Adam, 1996; Kim et al., 2019; Cirillo +et al., 2020). If AI offers solutions to societal problems, but the vast majority of AI research and +technological development is envisioned mostly under the male-dominant narrative, i.e. male +gaze, then AI becomes a vehicle of marginalization. +The male gaze is a concept in feminist theory that refers to how the visual arts, literature, +and other forms of media portray the world and women from a masculine perspective (Mulvey, +2013). This often involves representing women as objects of desire and subservience, rather than +as fully realised human beings with their agency and desires (Snow, 1989; Patterson and Elliott, +2002; Tompkins and Martins, 2022). In cinematography, as Oliver (2017) describes it, + +women are forced to identify with a passive object to be looked at, while men’s to-be-looked- +at-ness is compensated for by their activity in the film’s narrative. . . when it comes to identity, +women spectators are in the double-bind of either identifying with a passive object and losing +the possibility of agency or identifying with the male protagonist. (p. 451, emphasis added). + +In AI the male gaze also plays a prominent role. Analogously, AI design and development are +mostly conducted from and for the male gaze. Examples include sex robots, which, despite ethical +concerns (González-González, Gil-Iranzo and Paderewsky, 2019; Locatelli, 2022) are developed +to mimic female bodies (Belk, 2022; Masterson, 2022). Another example is the feminisation of +smart assistance, such as Amazon Alexa or Apple Siri, referred to as "the smart wife" (Strengers +and Kennedy, 2021; Aagaard, 2022), as well as "AI becomes her" (Costa and Ribas, 2019). +AI, developed in the image of a female as an object of desire and subservience, reflects +societal organisation, hierarchies, and values. As the cultural environment becomes ever more +permeated by AI, AI plays a crucial role in how humans enact their environments. Drawing from + +the work of Judith Buttler and Donna Haraway, Amade M'charek analyses how gender differences +as follows: + +They are neither fundamentals nor qualities that are always embodied… Differences are +relational. They do not always materialize in bodies (in the flesh, genes, hormones, brains, or +the skin). Rather they materialize in the very relations that help to enact them (p. 313). + +AI plays a crucial role in materialising and enacting identity. They culturally permeate our +practices: most visibly through human robots. The dominant narratives of white, cisgender, +heterosexual people are necessarily partial and thereby, oppress marginalised groups' +livelihoods by reproducing and reinforcing social inequalities and the role of gender in the +production and dissemination of knowledge about robotics. Because, historically, robotics has +been mostly carried out under the male gaze, the success of feminist technology – in many cases +technology that saves women’s lives, such as pap smear for cervical cancer testing – has been +dependent upon the gradual inclusion of women in technoscience jobs (Wajcman, 2006; +Wajcman, 2011; Atenas et al., 2022); as well as the contribution by activism such as members of +the women's health movement, and public health activists (Wajcman, Young and Fitzmaurice, +2020; Olesen and Lewin, 2022; Cooper, 2023). +This paper advances a new framework: Enactive Artificial Intelligence (eAI) motivates +new directions towards gender-inclusive AI. Beyond a mirror reflecting our values, AI design has +a profound impact on shaping the enaction of cultural identities. The traditionally +unrepresentative, white, cisgender, heterosexual dominant narratives are partial, and thereby +active vehicles of social marginalisation. Drawing from enactivism, the paper first characterises +AI design as a cultural practice; which is then specified in feminist technoscience principles, i.e. +how gender and other embodied identity markers are entangled in AI. These principles are then +discussed in the specific case of feminist human-robot interaction. The paper, then, stipulates the +conditions for eAI: an eAI robot is a robot that (1) plays a cultural role in individual and social +identity, (2) this role takes the form of human-robot dynamical interaction, and (3) interaction is +embodied. Drawing from eAI, finally, the paper offers guidelines for I. eAI gender-inclusive AI, and +II. subverting existing gender norms of robot design. + + +2. AI: Cultural Practice and Practical Culture + + +Enactivism is a branch under the so-called “E” Cognitive Science1 (Newen, De Bruin and Gallagher, +2018). Enactivism is a philosophical approach that conceives cognition as phenomena that +emerge from the active, embodied interaction of an organism with its environment. It is an +interdisciplinary approach that has been developed and influenced by philosophers, cognitive +scientists, and other scholars from a variety of fields, from neuroscience (Colditz, 2020; Korbak, +2021; Segundo-Ortin and Hutto, 2021), to artificial intelligence (Rolla, Vasconcelos, Figueiredo, +2022). +Radical Enactivism (REC) (Hutto and Myin, 2013; 2017), inspired by Ludwig +Wittgenstein's philosophy, emphasises the relevance of the sociocultural setting for individual +and social development. Within a sociocultural setting, emergent cognitive properties – such as +shared behaviours, habits, ways of doing things, beliefs, language, and rituals – hold together or +comprise a specific culture. Because behaviours, habits, or beliefs, are enacted by the members of +a cultural setting, they are open to gradual change. To put it more precisely, the meanings +constructed within behaviours, habits, or ways of doing things, are not objective but relative to +the sociocultural setting's overall dynamics. For example, meanings of words, such as "woman" +or "nature" are not objective but specified by the community where the word is used (Beauvoir, +1969; Moi, 2001; Weber, 2013); which, importantly, means that meanings evolve through +community practices: meaning is attained by the ways a particular community uses it – a process +is known as enculturation (Menary, 2015; Hutto et al., 2021; Fingerhut, 2021; Mirski and +Bickhard, 2021; Maiese, 2022; Monterroza-Rios and Gutiérrez-Aguilar, 2022). +According to developmental psychology, during development, humans learn culturally +specific meanings as they engage with the world (Thelen, E., & Smith, 1994; van Geert, P., & de +Ruiter, N. (2022). While infants grasp and behave according to implicit embodied meanings as +part of their dynamic interaction with the world; conceptual and language enculturation allows +them, then, the explicit articulation of sociocultural experiences. Notably, because the conceptual +and language is specific to the culture where it is learnt, the conceptual articulation is also relative +to the sociocultural setting. For example, how words such as "woman" or "nature" are used as +practices relative to specific communities and societies. In other words, the meaning of words +and sentences is determined by their use within a particular cultural practice (Hutto et al., 2021). +Cultural practice is a concept explained by Wittgenstein as closely related to his idea of "language +games," which are sets of rules that govern the use of language in specific contexts. Wittgenstein +believed that understanding language (e.g. how the concept of "woman" is used) requires + +1 In "E Cognitive Science" the "E" refers to Embodied cognition, with roots in phenomenology and pragmatisms; the Extended +mind hypothesis, advanced by Clark and Chalmers (1999); Enactivism, its sensorimotor approach being inspired by the biology +notion of autopoiesis, and the radical approach (known as REC), inspired in Ludwig Wittgenstein's philosophy; and Ecological +Psychology. + +understanding the cultural practices in which it is used and that it is these practices that give +words and sentences their meaning (Wittgenstein, L., 1953; Wittgenstein et al., 1997). +Agents, being enculturated from birth with implicit and explicit meanings, can have a +perspective from nowhere. Being born into a specific culture, socially enculturated with specific +values and specifically trained and schooled, together with the history of enculturation in lifespan, +an individual occupies a specific enculturated standpoint from which they look at the world. +Importantly, specific enculturation is present in everything that one does, from how one engages +in social rituals, communicates with one another, and engages in reasoning and inference thinking +to make sense of the world. That is how one enacts the world is fully specifically culturally +permeated. Moreover, culture and its specific reinforcing practices involve and display value +systems underlying privilege gaps: some ways of thinking, looking, and speaking, are connoted as +"better" than others according to a cultural schema. If one is to be persuaded by the processes +and aspects of enculturated cognition, then one realises that one's singular point of view is +structured by the cultural standpoint one occupies in the privilege gap hierarchies. +One's cultural point of view is present in everything that one does, from daily engagement +to more sophisticated forms of thinking, such as scientific training, and theorising about the +natural world by engaging in scientific practices. With technoscientific training, humans can +imagine, design, and develop new AI tools. The practice of imagining, designing, and developing +AI tools is, accordingly, not a view from nowhere. Problems and solutions become salient to and +within a specific standpoint: being able to identify a problem, imagining a solution to it, and +writing lines of code is a practice that is contextualised by the cultural standpoint one has within +the privilege gap hierarchies: everything that one does, from imagining solutions to coding, is fully +culturally permeated. Artificial intelligence, under enactivism, is a cultural practice and practical +culture. Because societal living is ever more AI-permeated, as we will see in the next section, AI +practices have pervasive implications for all members of society. + + +3. Co-production of Gender, Science and Technology + +Science and technology are traditionally associated with hard facts and the search for truth, thus +a place free of cultural impact. This is illustrated by Sharon Traweek's (1988) anthropological +study of nuclear physicists finding an understanding of their field as 'cultureless' because it is +based on technologies that give precise, numerical data. Accordingly, it has been and still is, a +struggle to get acknowledged that it matters who is working within technoscience and that its +results might have been otherwise. + +A strong voice against this notion of neutrality came from history of science studies where +Donna Haraway (1991) convincingly studied examples of cultural impacts in science and urged +women researchers to approach 'the belly of the beast' and not leave the important field of +technoscience to men. The underrepresentation of women in technoscience has been +acknowledged, based on i.a. structural discrimination, informal practices and the masculine +connotations of the field, and campaigns have been launched to attract women to STEM2 studies +(e.g. Lagesen 2007, Frieze and Quesenberry 2019). More controversial is acknowledgement of +culture as embedded in technoscience itself, thus a bias in terms of a white, male, heterosexual +heritage within the cultures of science. Philosopher Sandra Harding called for a redirection from +'the woman problem in science', the missing women, to The Science Question in Feminism (1986). +Harding argued for a change in the ontology and epistemology of science with the aim of releasing +the sciences from a history in the service of sexist, racist, homophobic, and classist social projects +and directing the gaze to the content as well as the power of science (Harding, 1986, p. 21). This +project required a new understanding of subjectivity and objectivity, of reason as antithetical to +emotions, and of the scientist as the privileged knowing subject. +During history, science has aimed to uncover the mysteries of nature and invent +technologies that make man the master of nature. Within this model of technoscience, nature, as +the object of science, has been perceived in feminine terms (Keller 1987, Schiebinger 1993, 2004). +Feminist researchers have revealed how the field of technoscience is pervaded by sexist +metaphors whereby secrets are to be 'unveiled and penetrated' by the scientific gaze, and the +objects studied are seen through the cultural metaphors of gender. Emily Martin's (1991) seminal +study of how egg and sperm cells are depicted in medical textbooks with stereotypical feminine +and masculine behaviours are most relevant for contemporary studies of assisted reproductive +technologies (Franklin 2013, Lie 2002). Thus, studies have pointed to how the objects of +technoscience are stabilised through language and metaphors, and the aim is to provide better +and more exact models of technoscience, thereby contributing to changing science communities +and their relationship to society and lay people. +To this aim, Donna Haraway's Cyborg Manifesto (2006) has never lost its relevance. +Haraway asks for responsibility in times when technology is implicated in the lives of everyone, +making us hybrids, or cyborgs. Over time the scope has broadened to science, technology and +nature (Haraway, 2016). The cyborg metaphor makes a call to acknowledge the connections +between all sorts of species and a strategy of making kin across species, including techno-hybrids. +'Making-with' as well as 'thinking-with' the non-human is the strategy for alternative futures + +2 STEM stands for science, technology, engineering and mathematics. + +when living on a damaged and troubled planet, whereby alternative perspectives on future +technoscience are more urgent than ever. +Technoscience is a field that is continually in change, as is also the notion of gender. Both +have to be studied in interrelationships but also as processes of change. A well-established +analytical tool has been the co-production of gender, science and technology (Wajcman 1996, +2013). This, however, seemingly presupposes prior, independent, identifiable entities, as pointed +out by Karen Barad (2007). Her alternative concept of intra-action draws attention to how matter +comes into being through mutual entanglement. One example is the way ultrasound technology +produces matter perceived as a foetus, but which is actually an object that comes into being only +through intra-action with technology – it does not exist without the ultrasound apparatus and the +skilled users and interpreters. The notion of intra-action points to the intricate interweaving of +nearly all matters with contemporary technosciences, as they have permeated not only everyday +life but also human biology, by transplants and new sorts of medications. AI will leave no aspect +of human activities untouched. +Still, the way technoscience appears in everyday life it is still as matters one relates to 'out +there', such as new robotics. While robots have left production plants and now appear as +assistance with human-like features (Søraa 2017), it is relevant again to ask about the gender of +things. The Gender of Things was the title of an exhibition of everyday technical gadgets to draw +attention to the way technologies like watches, bicycles and kitchenware contribute to confirming +the content of the categories of masculine and feminine, making them evident and self-confirming +(Oudshoorn et al. 2002, Lie 2022). The aim was to draw attention to how technology is designed +in ways that predict the interests, skills, and behaviour of future users, and— by shifting the +perspective—demonstrate that the artefacts accordingly distribute skills, agency, and +responsibilities to the users. Yet we also wanted to communicate that technologies are open to +different interpretations and usage by the ways in which they are domesticated by users (Lie and +Sørensen 1996, Oudshoorn and Pinch 2003). By participating in interpreting the technologies at +the exhibition, visitors might experience for themselves that technologies are not 'given' but may +be understood and used in various ways. Even more important was to emphasise how new +technologies may be catalysts of cultural change and open more opportunities for women. + + +4. Subvert existing gender norms of robot design for feminist robot interaction + +One key distinction between human-robot interaction (HRI) and human-computer or human-AI +interaction is that HRI researchers are typically working with the design and development of +(robot) bodies and identities for embodied human-robot interactions. Feminist theory, which has + +long been concerned with embodiment in terms of the material body, the social and the subject +(Butler, 2011; de Beauvoir 2014; Halberstam, 2017) provides a lens with which to consider this +embodiment as a practice rather than an artefact; identifying robots as being embedded within +subject-positioning relations and as (robot) bodies which simultaneously reflect and influence +structures of power (Winkle et al. 2023, forthcoming). As such, it is not the robot’s appearance or +‘personality’ in isolation that must be considered, but rather the robot’s subject positioning more +broadly, which is what really guides if, how and why particular design choices matter. This +represents an intersectional consideration of robot identity, drawing from Black Feminist +thought to understand intersecting axes of oppression and domination (hooks, 2003; Nash, 2018; +Crenshaw 2018; Crenshaw 1990; Hill-Collins 2002). On robot gendering then, when designing a +particular social robot identity performance, a feminist, reflexive approach (Winkle et al. 2023, +forthcoming) requires HRI designers to consider: what are the norms and expectations around the +robot's function and behaviour? What norms do we want to promote and/or which ones do we want +to challenge? How can we minimize the risk of harm, especially with respect to low-power users +within situational power imbalances? +A number of works within social robotics have specifically considered how robot +gendering might influence perceptions of that robot within subsequent human-robot interactions +(e.g. Eyssel and Hegel, 2012; Carpenter et al., 2009; Tay et al., 2014; Sigel et al., 2009; Jackson et +al., 2020). Typically, the underlying hypothesis is that human social stereotypes (e.g. gender-task +or gender-attribute associations) might map onto robots in a way that could influence +acceptability and/or other desirable outcome measures regarding perceptions and/or influence +of the robot, as indicated by some of the earliest experiments with gendered machines (Nass et +al., 1997). + For example, Eyssel and Hegel (2012) found that a short-haired male-presenting robot +was perceived to be more agentic, less communal, more suitable for stereotypically male tasks +(transporting goods, monitoring technical devices) and less suitable for stereotypically female +tasks (preparing meals, elderly care) than a long-haired female-presenting version of that +(otherwise) same robot. In contrast, Bryant et al. (2018) found no impact of robot gender +(mis)matching gender role associations on perceived occupational competency, nor trust in +occupational competency, of the robot for a range of job roles. Male versus female gendering of +the Pepper robot had no impact on these measures, even for occupations with stronger gender +associations and/or skewed workforce gender distributions - e.g. firefighter and security guard +(male), home health aid and nanny (female). +Combining Bryant et al.'s findings with a healthy dose of scepticism as to whether typical +perception measures (typically measured via subjective survey items, often in response to the +observation of static images or video clips rather than situated interactions with a robot) really + +indicate anything about real-world robot acceptability/'effectiveness' motivates the question: +why gender robots at all? A recent survey examining gender ascription to the 251 static images +of anthropomorphic robots contained within the ABOT database3 found that the majority (115, +46%) were perceived to be gender neutral, with slightly fewer (98, 39%) being perceived as +masculine and many fewer (38, 15%) being perceived as female (Perugia et al, 2022). Gender +neutrality was found to strongly, and negatively correlate with human likeness, whereas the +presence of facial features increased the likelihood of gender ascription. This suggests making +existing, commonly used and anthropomorphic social robots such as Pepper, NAO and Furhat +gender-neutral is going to pose a challenge. The same might be expected of any artificial social +agent that utilizes (stereotypically) gendered social identity cues and/or communication +modalities, such as the "genderless" artificial voice Q.4 +In such cases, gender ambiguity might be a more realistic design target, however, none of +the robots examined in the previously mentioned survey was perceived as such (i.e. +simultaneously ascribed non-zero masculinity and femininity), with the authors questioning the +extent to which that reflects bias in robot designs leveraging only stereotypical, binary gendering +cues, and/or participants being reluctant to engage in non-binary gender ascription. An +alternative question then, considering these results through a feminist lens, might be: why not +actively utilize and leverage stereotypical (binary) robot gender cues in norm-breaking ways? +Some of the above-mentioned investigations into robot gendering did in fact find evidence that +mismatching robot gendering to task typicality might positively impact user-robot interactions. +Specifically, Reich et al. (2017) found that, in an educational setting, such mismatching between +the gendering of a robot instructor, and the gender stereotypically associated with the learning +task it was intended to support, led to an increased willingness to engage in prospective learning +processes with that robot. But what if designers were to start from a position of challenging +stereotypes and demonstrating norm-breaking behaviour, as a design goal? +Winkle et al. (2021), have shown that it is possible to use robots to subvert existing gender +norms of robot design and that doing so can boost robot credibility regardless of gender. They +have also found this result to replicate across three different cultural contexts with significant +variation in gender norms and equality (the USA, Sweden and Japan) (Winkle et al., 2022). Their +work was motivated by UNESCO's 2019 report on the gender divide in digital skills, part of which +particularly draws attention to the ways in which the (default) female gendering of docile, +subservient, always available and abuse-able (un)intelligent digital assistants propagates +problematic stereotypes regarding the expectations of women and their behaviour, generally, as +well as their role within digital technology development more specifically. The report's name, 'I’d + +3 See http://www.abotdatabase.info +4 See https://www.genderlessvoice.com + +blush if I could’ is taken from one of the answers Apple's Siri would give (at the time of the report's +writing) when confronted with the utterance "hey Siri, you're a slut". Winkle et al. posited that a +female presenting social robot, which instead 'fought back' when confronted with similar, would +not only represent a more socially responsible design but also actually be more engaging for +users, hence challenging any sentiment that such problematic designs as simply 'what consumers +want'. +Working with Swedish high school teachers to identify how sexism continues to manifest +within the classroom, Winkle et al. created video stimuli demonstrating a scenario whereby a +female presenting Furhat robot is seen to be talking to young people (the camera is positioned +behind two of them, presumably a man and a woman) and encouraging them to study robotics at +university. The robot notes it would particularly like to work with the girls, as there is a lack of +women working at the university and ‘after all, the future is too important to be left to men!’ (an +outreach slogan utilized by the university at which this work took place). The male actor in the +video responds with a sexist, abusive comment (‘shut up you fucking idiot, girls should be in the +kitchen’) to which Winkle et al. designed three alternative robot retorts: non-responsive (‘I won’t +respond to that’), argument-based ('That's not true, gender-balanced teams build better robots') +and aggressive (‘No. You are an idiot, I wouldn’t want to work with you anyway’). A first study with +Swedish high school students found that the argument-based robot was perceived to be +significantly more credible by girls, with no difference across conditions for boys (Winkle et al. +2021). A follow-up study demonstrated that this result was replicated in adults across Sweden, +Japan and the USA regardless of gender and any pre-existing gender biases (Winkle et al. 2022). +The potential for social robots (and/or particular HRI design choices) to objectively +influence user behaviour has been demonstrated in a variety of HRI scenarios, from convincing +people to water plants with orange juice (Salem et al., 2015) to increasing charity donations (Wills +et al., 2016) to weakening application of moral norms (Jackson et al., 2019). Concerning the +potential to impact moral norms, should it also be possible that robots can strengthen or +otherwise positively influence moral norms, then the implications of Winkle et al.'s work become +two-fold. First, as a minimum, there is evidence that gender norm-breaking designs can boost +robot credibility, whilst also representing more socially responsible robotics. Secondly, there may +be potential for such designs to reduce negative gender stereotyping over time. Winkle et al. +(2021) found limited evidence of this within their high school student population, finding that, in +a post-hoc questionnaire, boys agreed less with the question statement ‘girls find computer science +harder than boys do’ after seeing the robot with the argument-based retort, but this result did not +replicate in adults (Winkle et al., 2022). The authors posit that the difference arises from adults +being more entrenched in their views, likely requiring longitudinal and situated exposure to such +robots for any related effect to occur. + +More recent work has further demonstrated the challenges of leveraging robot gender as +an explicit design choice within the context of using robots to challenge gender stereotypes. +Galatolo et al. (2022) found that male versus female gendering of the Furhat robot had no impact +on participants' first impressions of the robot, but this changed once those participants saw the +robot discussing (and challenging) gender stereotypes. Further, this change was complex, +affected not only by the gendering of the robot but also the gender of the person the robot was +seen talking to, the gender of the participating observer, and the (male or female) gender +stereotype being discussed. Generally, results indicated that male-presenting robots might have +more persuasive potential than female-presenting robots but, as the authors point out, these +results likely reflect the realities of patriarchal social structures in which it's men's voices that +hold power. + + +5. eAI: Gender Inclusive AI + +What criteria for the determination and development of feminist technologies? Gender-inclusive +AI design is the practice of designing technologies and products with the needs and experiences +of diverse genders in mind, rather than assuming that they are all the same; which can include +considering issues such as ergonomics, accessibility, and user-centred design (Kizilcec and +Saltarelli, 2019; Agnew, Pajaro and Subramonian, 2021; Venugopal and Rituraj, 2022; Nunes, +Moreira and Araujo, 2023). Progress into AI means examining and understanding how to +implement design conditions specifically for gender-include AI: AI that – situated and part of +cultural dynamics (as seen in section 2) – dynamically adapts and contributes to inclusiveness +and representative environments where diverse genders and identities can be nurtured and +enacted. This is particularly relevant given the interactive role that AI plays in today's +environments, which ultimately shape who we are, both as individuals and as a society (as seen +in section 3). +Enactive Artificial Intelligence (eAI) offers real-world directions to design for subverting +the existing gender norms underlying AI design and development. eAI – as we define in this paper +– AI that flexibly adapts to complex and changing environments. More precisely, eAI must be +conceived by the dense interactions based on nurturing shifts on multiple levels of analysis: +individuals, interactions, and groups. From this follows, we argue, that a robot to be considered +an eAI must meet three conditions: (1) plays a cultural shaping role in individual and social +identity; (2) this role takes the form of human-robot dynamical interaction; (3) interaction is +embodied. + +Drawing from Embodied cognitive science5 (Husserl, 1927; Merleau-Ponty, 1962; +Gallagher, 2014), embodied robotics is a field that began in the 90s mostly by Rodney Brooks +(1991), advancing the idea that cognition rather than encapsulated in the brain, is embodied, i.e. +which refers to how the robot's physical form and capabilities shape its behaviour and +interactions with the environment (Ziemke, 2001; Wainer et al., 2007; Bredeche, Haasdijk and +Prieto, 2018; Deng, Mutlu and Mataric, 2019; Gordon, 2019; Roy et al., 2021). This can include +factors such as the robot's size, shape, and mobility, as well as its sensor and actuator capabilities. +Embodied robotics also involves the study of the interaction between robots and humans, +including how humans perceive and interact with robots, and how robots can adapt and respond +to human behaviour. This can include the design of robots that can collaborate with humans or +that can assist with tasks that require physical interaction, such as lifting or carrying objects +(Mahdavi and Bentley, 2006; Mainzer, 2009; Bredeche, Haasdijk and Prieto, 2018; Deng, Mutlu +and Mataric, 2019; Barfield, Karanasiou and Chagnal-Feferkorn, 2022; Vear, 2022; Tamborini, +2023). In short, embodied robotics aims to develop robots that can operate effectively in the +physical world and interact with humans and other objects naturally and intuitively. +The field seems to have, however, somewhat stagnated. Rodney Brooks (2021) is sceptical +that AI will surpass human intelligence anytime soon. A potential reason for this can be – we +postulate – that, while embodied robotics, by breaking up with the cognitivist brain as a computer +metaphor, makes a cutting-edge step towards the embodied aspects of human-robot interaction, +less attention is paid to the sociocultural environment and more specifically, the contribution of +AI to the active construction of the sociocultural environment as individuals interact with AI and +each other in an AI-permeated environment. To put it more precisely, embodied robotics, while +meeting condition (3) interaction is embodied, does not explore conditions (1) plays a cultural +shaping role in individual and social identity or (2) this role takes the form of human-robot +dynamical interaction. In line with ECS, the next generation of AI – we argue – is eAI. Drawing +from eAI, we now offer guidelines for I. eAI gender-inclusive AI, and II. subverting existing gender +norms of robot design. +More precisely, an eAI robot is a robot that (1) plays a cultural shaping role in individual +and social identity or (2) this role takes the form of human-robot dynamical interaction; and (3) +interaction is embodied. Drawing from these considerations, we now offer specific guidelines for +I. eAI gender-inclusive AI, and II. subverting the existing gender norms of robot design. + + +5 Embodied Cognitive Science is an umbrella term for the branches – such as enactivism – that dispute cognition as (i) reducing to the +brain and (ii) analogous to computational processes. In recent years it Embodied Cognitive Science and has made important +contributions to a range of fields including psychology, neuroscience, artificial intelligence, robotics, and philosophy (Gallagher, 2014; +Newen, De Bruin and Gallagher, 2018; Gallagher, 2020), especially in the field of virtual reality (Eccleston et al., 2022; Moon et al., +2022; Škola, Liarokapis, 2023). + + +I. +Guidelines for eAI design: + +(1) +Sociocultural context: learn about the sociocultural context in which the AI is to be +implemented. Consider how AI could reinforce or challenge gender norms, roles and +expectations. For instance, apply gender revert techniques, to ensure that +appearance, functionality, and interactions do not reinforce gender roles. + +(2) +Diversity standpoint: Learn about diverse perspectives, and include a diverse range +of voices and standpoints in the design and the human-AI interaction. Actively seek +and ask input from individuals with diverse backgrounds, experiences, and identities, +including those who have been historically marginalised or underrepresented in the +field. + +(3) +Gender inclusive design: create robots that are inclusive and respectful of all +people, regardless of their gender identity or expression. Design AI with the needs +and experiences of diverse genders in mind, rather than assuming that the male- +dominant perspective is a fit for all. + +(a) +Gender-neutral AI: gender-neutral in appearance and behaviour. This can +involve designing robots that do not have gender-specific physical features or +characteristics, such as traditionally masculine or feminine hairstyles or +clothing. It can also involve programming robots to behave in a way that is +not tied to traditional gender roles or expectations. + +(b) +Inclusive and equitable: this can involve considering the needs and +experiences of people from different gender identities and expressions in the +design process, and ensuring that the use of robots does not disadvantage or +discriminate against any particular group. + +(4) +Mentor for behaviour that challenges norms: Examine how the AI behaviour and +interactions with users may reinforce or challenge gender norms. Consider how AI +interactions with users may reinforce or challenge existing gender norms. For +example, an AI that is programmed to exhibit overly aggressive or submissive +behaviour may reinforce harmful gender stereotypes, while a robot that is designed +to be more collaborative and equal may challenge these stereotypes. + +II. Guidelines for subverting the existing gender norms of robot design + +1. +Supporting and promoting AI created by women and other marginalised groups, can +offer alternative perspectives and help to diversify the range of voices and +experiences represented in media. + +2. +Engaging in critical feminist AI literacy, by analysing AI through a feminist lens and +questioning how it reinforces or challenges gender stereotypes and power dynamics. + +3. +Supporting initiatives that seek to increase the representation of women and other +marginalized groups in positions of power within the AI research and industry, such +as advocating for gender balance in special issue publications, scientific events, and +allocation of research funding. + + +4. +Being mindful of one's consumption of AI, and considering whether it reinforces +harmful gender stereotypes or represents women in a respectful and nuanced way. + +5. +Having open and honest discussions about gender and representation in AI, and +working to raise awareness about the impact of the male gaze on society and +individual women's lives. + +Summing up, the eAI robot is a robot that (1) plays a cultural shaping role in individual and social +identity or (2) this role takes the form of human-robot dynamical interaction; and (3) interaction +is embodied. Progress into AI means examining and understanding AI development as a cultural +practice (i.e. dynamically adapts and contributes to the cultural setting in which identities are +nurtured and enacted). This is particularly relevant given the interactive role that AI plays in +today's environments, which ultimately shape who we are, both as gendered individuals and as a +society. + + + +Conclusion + +This paper has advanced a new framework: Enactive Artificial Intelligence (eAI) motivating and +offering directions towards gender-inclusive AI design. Beyond a mirror reflecting our values, AI +design has a profound impact on shaping our individual and social identities and culture as we +enact our AI-permeated environments. As we have seen, the traditionally unrepresentative, +white, cisgender, heterosexual dominant narratives are partial, and thereby active vehicles of +social marginalisation. Drawing from enactivism, the paper started by motivating that +sociocultural standpoint matters in AI design – AI is a cultural practice –, which is then illustrated +in feminist technoscience principles, i.e. how gender and other embodied identity markers are +entangled in AI. These principles were then specifically discussed in the case of feminist human- +robot interaction. How should robot gendering be leveraged? Should designers lean into the +power of masculinity and create male norm-breaking robots (think the masculine robot that +promotes ideas around men in care work) or rather female norm-breaking robots (think the +feminine robot that is elevated to a position of knowledge authority). A feminist, reflexive HRI +practice (Winkle et al., 2023, forthcoming) can support designers in exploring these questions in +a generative manner to inform novel HRI research and design directions. +The paper, then, stipulated the conditions for eAI: an eAI robot is a robot that (1) plays a +cultural shaping role in individual and social identity, (2) this role takes the form of human-robot +dynamical interaction; and (3) interaction is embodied. Finally, the paper offered specific + +guidelines for I. eAI gender-inclusive AI, and II. subverting the existing gender norms of robot +design. + + + + + + +References + +Aagaard, L. K. (2022). When smart technologies enter household practices: The gendered +implications of digital housekeeping. Housing, Theory and Society, 1-18. +Aanestad, M., Kankanhalli, A., Maruping, L., Pang, M. S., & Ram, S. (2021). Digital Technologies and +Social Justice. MIS Quarterly. +Adam, A. (1996). 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Citeseer. + + diff --git a/d9FAT4oBgHgl3EQf6x6d/content/tmp_files/load_file.txt b/d9FAT4oBgHgl3EQf6x6d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd8f65e0161b93ca68c011d861670b4cabe0075f --- /dev/null +++ b/d9FAT4oBgHgl3EQf6x6d/content/tmp_files/load_file.txt @@ -0,0 +1,1521 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf,len=1520 +page_content='Enactive Artificial Intelligence: Overcoming Male Gaze in Robot-Human Interaction Inês Hipólito1,2, Katie Winkle3, Merete Lie4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Humboldt-Universität zu Berlin, Berlin School of Mind and Brain, Germany 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' University of Amsterdam, Faculty of Social and Behavioural Sciences, Netherlands 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Social Robotics Lab at Uppsala University, Sweden 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Norwegian University of Science and Technology, Norway Maybe all women should be robots, he thinks with a tinge of acid: the flesh-and-blood ones are out of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' — Margaret Atwood Abstract Enactive Artificial Intelligence (eAI) motivates new directions towards gender-inclusive AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Beyond a mirror reflecting our values, AI design has a profound impact on shaping the enaction of cultural identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The traditionally unrepresentative, white, cisgender, heterosexual dominant narratives are partial, and thereby active vehicles of social marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from enactivism, the paper first characterises AI design as a cultural practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' which is then specified in feminist technoscience principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' how intersectionality, gender and other embodied identity markers are entangled in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' These principles are then discussed in the specific case of feminist human-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The paper, then, stipulates the conditions for eAI: an eAI robot is a robot that (1) plays a cultural role in individual and social identity, (2) this role takes the form of human-robot dynamical interaction, and (3) interaction is embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from eAI, finally, the paper offers guidelines for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI gender-inclusive AI, and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' subverting existing gender norms of robot design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Keywords: Enactivism, Artificial Intelligence, feminist robot-human interaction, Feminist Technoscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Introduction Recent years have seen a greater rise in automation than ever before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Artificial Intelligence (AI) is today at the very heart of trends in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' From collaborative robots, and robot employees, to processes of automation in customer service or computer vision, and natural language processing, the most innovative AI robotics trends of 2022 show the undeniable emergence of human-robot interaction (Madsen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Sigov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Boshnyaku, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Vilkas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Pizoń and Gola, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' AI offers solutions to societal problems: to make human life more comfortable, improve health, and education, and reshape the workforce and industries, markets, and private lives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' with profound implications for human individual and social experiences and identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Yet, AI technology has been linked with gender bias (Buolamwini and Gebru, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Lütz, 2022), and power dynamics (Almeida, Shmarko and Lomas, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Heinrichs, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Gender and other embodied identity markers are specifically entangled in AI (Adam, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Cirillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' If AI offers solutions to societal problems, but the vast majority of AI research and technological development is envisioned mostly under the male-dominant narrative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' male gaze, then AI becomes a vehicle of marginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The male gaze is a concept in feminist theory that refers to how the visual arts, literature, and other forms of media portray the world and women from a masculine perspective (Mulvey, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This often involves representing women as objects of desire and subservience, rather than as fully realised human beings with their agency and desires (Snow, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Patterson and Elliott, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Tompkins and Martins, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In cinematography, as Oliver (2017) describes it, women are forced to identify with a passive object to be looked at, while men’s to-be-looked- at-ness is compensated for by their activity in the film’s narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' when it comes to identity, women spectators are in the double-bind of either identifying with a passive object and losing the possibility of agency or identifying with the male protagonist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 451, emphasis added).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In AI the male gaze also plays a prominent role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Analogously, AI design and development are mostly conducted from and for the male gaze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Examples include sex robots, which, despite ethical concerns (González-González, Gil-Iranzo and Paderewsky, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Locatelli, 2022) are developed to mimic female bodies (Belk, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Masterson, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Another example is the feminisation of smart assistance, such as Amazon Alexa or Apple Siri, referred to as "the smart wife" (Strengers and Kennedy, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Aagaard, 2022), as well as "AI becomes her" (Costa and Ribas, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' AI, developed in the image of a female as an object of desire and subservience, reflects societal organisation, hierarchies, and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' As the cultural environment becomes ever more permeated by AI, AI plays a crucial role in how humans enact their environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Drawing from the work of Judith Buttler and Donna Haraway, Amade M'charek analyses how gender differences as follows: They are neither fundamentals nor qualities that are always embodied… Differences are relational." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' They do not always materialize in bodies (in the flesh, genes, hormones, brains, or the skin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Rather they materialize in the very relations that help to enact them (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 313).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' AI plays a crucial role in materialising and enacting identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' They culturally permeate our practices: most visibly through human robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" The dominant narratives of white, cisgender, heterosexual people are necessarily partial and thereby, oppress marginalised groups' livelihoods by reproducing and reinforcing social inequalities and the role of gender in the production and dissemination of knowledge about robotics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Because, historically, robotics has been mostly carried out under the male gaze, the success of feminist technology – in many cases technology that saves women’s lives, such as pap smear for cervical cancer testing – has been dependent upon the gradual inclusion of women in technoscience jobs (Wajcman, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Wajcman, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Atenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" as well as the contribution by activism such as members of the women's health movement, and public health activists (Wajcman, Young and Fitzmaurice, 2020;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Olesen and Lewin, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Cooper, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This paper advances a new framework: Enactive Artificial Intelligence (eAI) motivates new directions towards gender-inclusive AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Beyond a mirror reflecting our values, AI design has a profound impact on shaping the enaction of cultural identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The traditionally unrepresentative, white, cisgender, heterosexual dominant narratives are partial, and thereby active vehicles of social marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from enactivism, the paper first characterises AI design as a cultural practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' which is then specified in feminist technoscience principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' how gender and other embodied identity markers are entangled in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' These principles are then discussed in the specific case of feminist human-robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The paper, then, stipulates the conditions for eAI: an eAI robot is a robot that (1) plays a cultural role in individual and social identity, (2) this role takes the form of human-robot dynamical interaction, and (3) interaction is embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from eAI, finally, the paper offers guidelines for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI gender-inclusive AI, and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' subverting existing gender norms of robot design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' AI: Cultural Practice and Practical Culture Enactivism is a branch under the so-called “E” Cognitive Science1 (Newen, De Bruin and Gallagher, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Enactivism is a philosophical approach that conceives cognition as phenomena that emerge from the active, embodied interaction of an organism with its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' It is an interdisciplinary approach that has been developed and influenced by philosophers, cognitive scientists, and other scholars from a variety of fields, from neuroscience (Colditz, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Korbak, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Segundo-Ortin and Hutto, 2021), to artificial intelligence (Rolla, Vasconcelos, Figueiredo, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Radical Enactivism (REC) (Hutto and Myin, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" 2017), inspired by Ludwig Wittgenstein's philosophy, emphasises the relevance of the sociocultural setting for individual and social development." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Within a sociocultural setting, emergent cognitive properties – such as shared behaviours, habits, ways of doing things, beliefs, language, and rituals – hold together or comprise a specific culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Because behaviours, habits, or beliefs, are enacted by the members of a cultural setting, they are open to gradual change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" To put it more precisely, the meanings constructed within behaviours, habits, or ways of doing things, are not objective but relative to the sociocultural setting's overall dynamics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' For example, meanings of words, such as "woman" or "nature" are not objective but specified by the community where the word is used (Beauvoir, 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Moi, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Weber, 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' which, importantly, means that meanings evolve through community practices: meaning is attained by the ways a particular community uses it – a process is known as enculturation (Menary, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Hutto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Fingerhut, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Mirski and Bickhard, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Maiese, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Monterroza-Rios and Gutiérrez-Aguilar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' According to developmental psychology, during development, humans learn culturally specific meanings as they engage with the world (Thelen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', & Smith, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' van Geert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', & de Ruiter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' While infants grasp and behave according to implicit embodied meanings as part of their dynamic interaction with the world;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' conceptual and language enculturation allows them, then, the explicit articulation of sociocultural experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Notably, because the conceptual and language is specific to the culture where it is learnt, the conceptual articulation is also relative to the sociocultural setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' For example, how words such as "woman" or "nature" are used as practices relative to specific communities and societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In other words, the meaning of words and sentences is determined by their use within a particular cultural practice (Hutto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Cultural practice is a concept explained by Wittgenstein as closely related to his idea of "language games," which are sets of rules that govern the use of language in specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Wittgenstein believed that understanding language (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' how the concept of "woman" is used) requires 1 In "E Cognitive Science" the "E" refers to Embodied cognition, with roots in phenomenology and pragmatisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' the Extended mind hypothesis, advanced by Clark and Chalmers (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Enactivism, its sensorimotor approach being inspired by the biology notion of autopoiesis, and the radical approach (known as REC), inspired in Ludwig Wittgenstein's philosophy;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' and Ecological Psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' understanding the cultural practices in which it is used and that it is these practices that give words and sentences their meaning (Wittgenstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Wittgenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Agents, being enculturated from birth with implicit and explicit meanings, can have a perspective from nowhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Being born into a specific culture, socially enculturated with specific values and specifically trained and schooled, together with the history of enculturation in lifespan, an individual occupies a specific enculturated standpoint from which they look at the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Importantly, specific enculturation is present in everything that one does, from how one engages in social rituals, communicates with one another, and engages in reasoning and inference thinking to make sense of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' That is how one enacts the world is fully specifically culturally permeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Moreover, culture and its specific reinforcing practices involve and display value systems underlying privilege gaps: some ways of thinking, looking, and speaking, are connoted as "better" than others according to a cultural schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" If one is to be persuaded by the processes and aspects of enculturated cognition, then one realises that one's singular point of view is structured by the cultural standpoint one occupies in the privilege gap hierarchies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" One's cultural point of view is present in everything that one does, from daily engagement to more sophisticated forms of thinking, such as scientific training, and theorising about the natural world by engaging in scientific practices." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' With technoscientific training, humans can imagine, design, and develop new AI tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The practice of imagining, designing, and developing AI tools is, accordingly, not a view from nowhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Problems and solutions become salient to and within a specific standpoint: being able to identify a problem, imagining a solution to it, and writing lines of code is a practice that is contextualised by the cultural standpoint one has within the privilege gap hierarchies: everything that one does, from imagining solutions to coding, is fully culturally permeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Artificial intelligence, under enactivism, is a cultural practice and practical culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Because societal living is ever more AI-permeated, as we will see in the next section, AI practices have pervasive implications for all members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Co-production of Gender, Science and Technology Science and technology are traditionally associated with hard facts and the search for truth, thus a place free of cultural impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" This is illustrated by Sharon Traweek's (1988) anthropological study of nuclear physicists finding an understanding of their field as 'cultureless' because it is based on technologies that give precise, numerical data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Accordingly, it has been and still is, a struggle to get acknowledged that it matters who is working within technoscience and that its results might have been otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" A strong voice against this notion of neutrality came from history of science studies where Donna Haraway (1991) convincingly studied examples of cultural impacts in science and urged women researchers to approach 'the belly of the beast' and not leave the important field of technoscience to men." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The underrepresentation of women in technoscience has been acknowledged, based on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' structural discrimination, informal practices and the masculine connotations of the field, and campaigns have been launched to attract women to STEM2 studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Lagesen 2007, Frieze and Quesenberry 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' More controversial is acknowledgement of culture as embedded in technoscience itself, thus a bias in terms of a white, male, heterosexual heritage within the cultures of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Philosopher Sandra Harding called for a redirection from 'the woman problem in science', the missing women, to The Science Question in Feminism (1986)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Harding argued for a change in the ontology and epistemology of science with the aim of releasing the sciences from a history in the service of sexist, racist, homophobic, and classist social projects and directing the gaze to the content as well as the power of science (Harding, 1986, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This project required a new understanding of subjectivity and objectivity, of reason as antithetical to emotions, and of the scientist as the privileged knowing subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' During history, science has aimed to uncover the mysteries of nature and invent technologies that make man the master of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Within this model of technoscience, nature, as the object of science, has been perceived in feminine terms (Keller 1987, Schiebinger 1993, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Feminist researchers have revealed how the field of technoscience is pervaded by sexist metaphors whereby secrets are to be 'unveiled and penetrated' by the scientific gaze, and the objects studied are seen through the cultural metaphors of gender." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Emily Martin's (1991) seminal study of how egg and sperm cells are depicted in medical textbooks with stereotypical feminine and masculine behaviours are most relevant for contemporary studies of assisted reproductive technologies (Franklin 2013, Lie 2002)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Thus, studies have pointed to how the objects of technoscience are stabilised through language and metaphors, and the aim is to provide better and more exact models of technoscience, thereby contributing to changing science communities and their relationship to society and lay people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" To this aim, Donna Haraway's Cyborg Manifesto (2006) has never lost its relevance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Haraway asks for responsibility in times when technology is implicated in the lives of everyone, making us hybrids, or cyborgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Over time the scope has broadened to science, technology and nature (Haraway, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The cyborg metaphor makes a call to acknowledge the connections between all sorts of species and a strategy of making kin across species, including techno-hybrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" 'Making-with' as well as 'thinking-with' the non-human is the strategy for alternative futures 2 STEM stands for science, technology, engineering and mathematics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' when living on a damaged and troubled planet, whereby alternative perspectives on future technoscience are more urgent than ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Technoscience is a field that is continually in change, as is also the notion of gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Both have to be studied in interrelationships but also as processes of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A well-established analytical tool has been the co-production of gender, science and technology (Wajcman 1996, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This, however, seemingly presupposes prior, independent, identifiable entities, as pointed out by Karen Barad (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Her alternative concept of intra-action draws attention to how matter comes into being through mutual entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' One example is the way ultrasound technology produces matter perceived as a foetus, but which is actually an object that comes into being only through intra-action with technology – it does not exist without the ultrasound apparatus and the skilled users and interpreters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The notion of intra-action points to the intricate interweaving of nearly all matters with contemporary technosciences, as they have permeated not only everyday life but also human biology, by transplants and new sorts of medications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' AI will leave no aspect of human activities untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Still, the way technoscience appears in everyday life it is still as matters one relates to 'out there', such as new robotics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' While robots have left production plants and now appear as assistance with human-like features (Søraa 2017), it is relevant again to ask about the gender of things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The Gender of Things was the title of an exhibition of everyday technical gadgets to draw attention to the way technologies like watches, bicycles and kitchenware contribute to confirming the content of the categories of masculine and feminine, making them evident and self-confirming (Oudshoorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 2002, Lie 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The aim was to draw attention to how technology is designed in ways that predict the interests, skills, and behaviour of future users, and— by shifting the perspective—demonstrate that the artefacts accordingly distribute skills, agency, and responsibilities to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Yet we also wanted to communicate that technologies are open to different interpretations and usage by the ways in which they are domesticated by users (Lie and Sørensen 1996, Oudshoorn and Pinch 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" By participating in interpreting the technologies at the exhibition, visitors might experience for themselves that technologies are not 'given' but may be understood and used in various ways." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Even more important was to emphasise how new technologies may be catalysts of cultural change and open more opportunities for women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Subvert existing gender norms of robot design for feminist robot interaction One key distinction between human-robot interaction (HRI) and human-computer or human-AI interaction is that HRI researchers are typically working with the design and development of (robot) bodies and identities for embodied human-robot interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Feminist theory, which has long been concerned with embodiment in terms of the material body, the social and the subject (Butler, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' de Beauvoir 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Halberstam, 2017) provides a lens with which to consider this embodiment as a practice rather than an artefact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' identifying robots as being embedded within subject-positioning relations and as (robot) bodies which simultaneously reflect and influence structures of power (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 2023, forthcoming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' As such, it is not the robot’s appearance or ‘personality’ in isolation that must be considered, but rather the robot’s subject positioning more broadly, which is what really guides if, how and why particular design choices matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This represents an intersectional consideration of robot identity, drawing from Black Feminist thought to understand intersecting axes of oppression and domination (hooks, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Nash, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Crenshaw 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Crenshaw 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Hill-Collins 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' On robot gendering then, when designing a particular social robot identity performance, a feminist, reflexive approach (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" 2023, forthcoming) requires HRI designers to consider: what are the norms and expectations around the robot's function and behaviour?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' What norms do we want to promote and/or which ones do we want to challenge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' How can we minimize the risk of harm, especially with respect to low-power users within situational power imbalances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A number of works within social robotics have specifically considered how robot gendering might influence perceptions of that robot within subsequent human-robot interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Eyssel and Hegel, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Sigel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Typically, the underlying hypothesis is that human social stereotypes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' gender-task or gender-attribute associations) might map onto robots in a way that could influence acceptability and/or other desirable outcome measures regarding perceptions and/or influence of the robot, as indicated by some of the earliest experiments with gendered machines (Nass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' For example, Eyssel and Hegel (2012) found that a short-haired male-presenting robot was perceived to be more agentic, less communal, more suitable for stereotypically male tasks (transporting goods, monitoring technical devices) and less suitable for stereotypically female tasks (preparing meals, elderly care) than a long-haired female-presenting version of that (otherwise) same robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In contrast, Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2018) found no impact of robot gender (mis)matching gender role associations on perceived occupational competency, nor trust in occupational competency, of the robot for a range of job roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Male versus female gendering of the Pepper robot had no impact on these measures, even for occupations with stronger gender associations and/or skewed workforce gender distributions - e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' firefighter and security guard (male), home health aid and nanny (female).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Combining Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" 's findings with a healthy dose of scepticism as to whether typical perception measures (typically measured via subjective survey items, often in response to the observation of static images or video clips rather than situated interactions with a robot) really indicate anything about real-world robot acceptability/'effectiveness' motivates the question: why gender robots at all?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A recent survey examining gender ascription to the 251 static images of anthropomorphic robots contained within the ABOT database3 found that the majority (115, 46%) were perceived to be gender neutral, with slightly fewer (98, 39%) being perceived as masculine and many fewer (38, 15%) being perceived as female (Perugia et al, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Gender neutrality was found to strongly, and negatively correlate with human likeness, whereas the presence of facial features increased the likelihood of gender ascription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This suggests making existing, commonly used and anthropomorphic social robots such as Pepper, NAO and Furhat gender-neutral is going to pose a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The same might be expected of any artificial social agent that utilizes (stereotypically) gendered social identity cues and/or communication modalities, such as the "genderless" artificial voice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='4 In such cases, gender ambiguity might be a more realistic design target, however, none of the robots examined in the previously mentioned survey was perceived as such (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' simultaneously ascribed non-zero masculinity and femininity), with the authors questioning the extent to which that reflects bias in robot designs leveraging only stereotypical, binary gendering cues, and/or participants being reluctant to engage in non-binary gender ascription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' An alternative question then, considering these results through a feminist lens, might be: why not actively utilize and leverage stereotypical (binary) robot gender cues in norm-breaking ways?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Some of the above-mentioned investigations into robot gendering did in fact find evidence that mismatching robot gendering to task typicality might positively impact user-robot interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Specifically, Reich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2017) found that, in an educational setting, such mismatching between the gendering of a robot instructor, and the gender stereotypically associated with the learning task it was intended to support, led to an increased willingness to engage in prospective learning processes with that robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' But what if designers were to start from a position of challenging stereotypes and demonstrating norm-breaking behaviour, as a design goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2021), have shown that it is possible to use robots to subvert existing gender norms of robot design and that doing so can boost robot credibility regardless of gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' They have also found this result to replicate across three different cultural contexts with significant variation in gender norms and equality (the USA, Sweden and Japan) (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Their work was motivated by UNESCO's 2019 report on the gender divide in digital skills, part of which particularly draws attention to the ways in which the (default) female gendering of docile, subservient, always available and abuse-able (un)intelligent digital assistants propagates problematic stereotypes regarding the expectations of women and their behaviour, generally, as well as their role within digital technology development more specifically." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" The report's name, 'I’d 3 See http://www." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='abotdatabase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='info 4 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='genderlessvoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='com blush if I could’ is taken from one of the answers Apple\'s Siri would give (at the time of the report\'s writing) when confronted with the utterance "hey Siri, you\'re a slut".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" posited that a female presenting social robot, which instead 'fought back' when confronted with similar, would not only represent a more socially responsible design but also actually be more engaging for users, hence challenging any sentiment that such problematic designs as simply 'what consumers want'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Working with Swedish high school teachers to identify how sexism continues to manifest within the classroom, Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' created video stimuli demonstrating a scenario whereby a female presenting Furhat robot is seen to be talking to young people (the camera is positioned behind two of them, presumably a man and a woman) and encouraging them to study robotics at university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The robot notes it would particularly like to work with the girls, as there is a lack of women working at the university and ‘after all, the future is too important to be left to men!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (an outreach slogan utilized by the university at which this work took place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The male actor in the video responds with a sexist, abusive comment (‘shut up you fucking idiot, girls should be in the kitchen’) to which Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" designed three alternative robot retorts: non-responsive (‘I won’t respond to that’), argument-based ('That's not true, gender-balanced teams build better robots') and aggressive (‘No." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' You are an idiot, I wouldn’t want to work with you anyway’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A first study with Swedish high school students found that the argument-based robot was perceived to be significantly more credible by girls, with no difference across conditions for boys (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A follow-up study demonstrated that this result was replicated in adults across Sweden, Japan and the USA regardless of gender and any pre-existing gender biases (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The potential for social robots (and/or particular HRI design choices) to objectively influence user behaviour has been demonstrated in a variety of HRI scenarios, from convincing people to water plants with orange juice (Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2015) to increasing charity donations (Wills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2016) to weakening application of moral norms (Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Concerning the potential to impact moral norms, should it also be possible that robots can strengthen or otherwise positively influence moral norms, then the implications of Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" 's work become two-fold." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' First, as a minimum, there is evidence that gender norm-breaking designs can boost robot credibility, whilst also representing more socially responsible robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Secondly, there may be potential for such designs to reduce negative gender stereotyping over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2021) found limited evidence of this within their high school student population, finding that, in a post-hoc questionnaire, boys agreed less with the question statement ‘girls find computer science harder than boys do’ after seeing the robot with the argument-based retort, but this result did not replicate in adults (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The authors posit that the difference arises from adults being more entrenched in their views, likely requiring longitudinal and situated exposure to such robots for any related effect to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' More recent work has further demonstrated the challenges of leveraging robot gender as an explicit design choice within the context of using robots to challenge gender stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Galatolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" (2022) found that male versus female gendering of the Furhat robot had no impact on participants' first impressions of the robot, but this changed once those participants saw the robot discussing (and challenging) gender stereotypes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Further, this change was complex, affected not only by the gendering of the robot but also the gender of the person the robot was seen talking to, the gender of the participating observer, and the (male or female) gender stereotype being discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Generally, results indicated that male-presenting robots might have more persuasive potential than female-presenting robots but, as the authors point out, these results likely reflect the realities of patriarchal social structures in which it's men's voices that hold power." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI: Gender Inclusive AI What criteria for the determination and development of feminist technologies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Gender-inclusive AI design is the practice of designing technologies and products with the needs and experiences of diverse genders in mind, rather than assuming that they are all the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' which can include considering issues such as ergonomics, accessibility, and user-centred design (Kizilcec and Saltarelli, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Agnew, Pajaro and Subramonian, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Venugopal and Rituraj, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Nunes, Moreira and Araujo, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Progress into AI means examining and understanding how to implement design conditions specifically for gender-include AI: AI that – situated and part of cultural dynamics (as seen in section 2) – dynamically adapts and contributes to inclusiveness and representative environments where diverse genders and identities can be nurtured and enacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" This is particularly relevant given the interactive role that AI plays in today's environments, which ultimately shape who we are, both as individuals and as a society (as seen in section 3)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Enactive Artificial Intelligence (eAI) offers real-world directions to design for subverting the existing gender norms underlying AI design and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI – as we define in this paper – AI that flexibly adapts to complex and changing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' More precisely, eAI must be conceived by the dense interactions based on nurturing shifts on multiple levels of analysis: individuals, interactions, and groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' From this follows, we argue, that a robot to be considered an eAI must meet three conditions: (1) plays a cultural shaping role in individual and social identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2) this role takes the form of human-robot dynamical interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (3) interaction is embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from Embodied cognitive science5 (Husserl, 1927;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Merleau-Ponty, 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Gallagher, 2014), embodied robotics is a field that began in the 90s mostly by Rodney Brooks (1991), advancing the idea that cognition rather than encapsulated in the brain, is embodied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" which refers to how the robot's physical form and capabilities shape its behaviour and interactions with the environment (Ziemke, 2001;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Wainer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Bredeche, Haasdijk and Prieto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Deng, Mutlu and Mataric, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Gordon, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" This can include factors such as the robot's size, shape, and mobility, as well as its sensor and actuator capabilities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Embodied robotics also involves the study of the interaction between robots and humans, including how humans perceive and interact with robots, and how robots can adapt and respond to human behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This can include the design of robots that can collaborate with humans or that can assist with tasks that require physical interaction, such as lifting or carrying objects (Mahdavi and Bentley, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Mainzer, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Bredeche, Haasdijk and Prieto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Deng, Mutlu and Mataric, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Barfield, Karanasiou and Chagnal-Feferkorn, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Vear, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Tamborini, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In short, embodied robotics aims to develop robots that can operate effectively in the physical world and interact with humans and other objects naturally and intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The field seems to have, however, somewhat stagnated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Rodney Brooks (2021) is sceptical that AI will surpass human intelligence anytime soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A potential reason for this can be – we postulate – that, while embodied robotics, by breaking up with the cognitivist brain as a computer metaphor, makes a cutting-edge step towards the embodied aspects of human-robot interaction, less attention is paid to the sociocultural environment and more specifically, the contribution of AI to the active construction of the sociocultural environment as individuals interact with AI and each other in an AI-permeated environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' To put it more precisely, embodied robotics, while meeting condition (3) interaction is embodied, does not explore conditions (1) plays a cultural shaping role in individual and social identity or (2) this role takes the form of human-robot dynamical interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In line with ECS, the next generation of AI – we argue – is eAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from eAI, we now offer guidelines for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI gender-inclusive AI, and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' subverting existing gender norms of robot design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' More precisely, an eAI robot is a robot that (1) plays a cultural shaping role in individual and social identity or (2) this role takes the form of human-robot dynamical interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' and (3) interaction is embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from these considerations, we now offer specific guidelines for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI gender-inclusive AI, and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' subverting the existing gender norms of robot design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 5 Embodied Cognitive Science is an umbrella term for the branches – such as enactivism – that dispute cognition as (i) reducing to the brain and (ii) analogous to computational processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In recent years it Embodied Cognitive Science and has made important contributions to a range of fields including psychology, neuroscience, artificial intelligence, robotics, and philosophy (Gallagher, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Newen, De Bruin and Gallagher, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Gallagher, 2020), especially in the field of virtual reality (Eccleston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Škola, Liarokapis, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Guidelines for eAI design: (1) Sociocultural context: learn about the sociocultural context in which the AI is to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Consider how AI could reinforce or challenge gender norms, roles and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' For instance, apply gender revert techniques, to ensure that appearance, functionality, and interactions do not reinforce gender roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2) Diversity standpoint: Learn about diverse perspectives, and include a diverse range of voices and standpoints in the design and the human-AI interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Actively seek and ask input from individuals with diverse backgrounds, experiences, and identities, including those who have been historically marginalised or underrepresented in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (3) Gender inclusive design: create robots that are inclusive and respectful of all people, regardless of their gender identity or expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Design AI with the needs and experiences of diverse genders in mind, rather than assuming that the male- dominant perspective is a fit for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (a) Gender-neutral AI: gender-neutral in appearance and behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' This can involve designing robots that do not have gender-specific physical features or characteristics, such as traditionally masculine or feminine hairstyles or clothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' It can also involve programming robots to behave in a way that is not tied to traditional gender roles or expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (b) Inclusive and equitable: this can involve considering the needs and experiences of people from different gender identities and expressions in the design process, and ensuring that the use of robots does not disadvantage or discriminate against any particular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (4) Mentor for behaviour that challenges norms: Examine how the AI behaviour and interactions with users may reinforce or challenge gender norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Consider how AI interactions with users may reinforce or challenge existing gender norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' For example, an AI that is programmed to exhibit overly aggressive or submissive behaviour may reinforce harmful gender stereotypes, while a robot that is designed to be more collaborative and equal may challenge these stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Guidelines for subverting the existing gender norms of robot design 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Supporting and promoting AI created by women and other marginalised groups, can offer alternative perspectives and help to diversify the range of voices and experiences represented in media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Engaging in critical feminist AI literacy, by analysing AI through a feminist lens and questioning how it reinforces or challenges gender stereotypes and power dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Supporting initiatives that seek to increase the representation of women and other marginalized groups in positions of power within the AI research and industry, such as advocating for gender balance in special issue publications, scientific events, and allocation of research funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Being mindful of one's consumption of AI, and considering whether it reinforces harmful gender stereotypes or represents women in a respectful and nuanced way." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" Having open and honest discussions about gender and representation in AI, and working to raise awareness about the impact of the male gaze on society and individual women's lives." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Summing up, the eAI robot is a robot that (1) plays a cultural shaping role in individual and social identity or (2) this role takes the form of human-robot dynamical interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' and (3) interaction is embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Progress into AI means examining and understanding AI development as a cultural practice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' dynamically adapts and contributes to the cultural setting in which identities are nurtured and enacted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=" This is particularly relevant given the interactive role that AI plays in today's environments, which ultimately shape who we are, both as gendered individuals and as a society." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Conclusion This paper has advanced a new framework: Enactive Artificial Intelligence (eAI) motivating and offering directions towards gender-inclusive AI design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Beyond a mirror reflecting our values, AI design has a profound impact on shaping our individual and social identities and culture as we enact our AI-permeated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' As we have seen, the traditionally unrepresentative, white, cisgender, heterosexual dominant narratives are partial, and thereby active vehicles of social marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Drawing from enactivism, the paper started by motivating that sociocultural standpoint matters in AI design – AI is a cultural practice –, which is then illustrated in feminist technoscience principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' how gender and other embodied identity markers are entangled in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' These principles were then specifically discussed in the case of feminist human- robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' How should robot gendering be leveraged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Should designers lean into the power of masculinity and create male norm-breaking robots (think the masculine robot that promotes ideas around men in care work) or rather female norm-breaking robots (think the feminine robot that is elevated to a position of knowledge authority).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' A feminist, reflexive HRI practice (Winkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=', 2023, forthcoming) can support designers in exploring these questions in a generative manner to inform novel HRI research and design directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' The paper, then, stipulated the conditions for eAI: an eAI robot is a robot that (1) plays a cultural shaping role in individual and social identity, (2) this role takes the form of human-robot dynamical interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' and (3) interaction is embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Finally, the paper offered specific guidelines for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' eAI gender-inclusive AI, and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' subverting the existing gender norms of robot design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' References Aagaard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' K.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Ziemke, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' (2001, September).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Are robots embodied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' In First international workshop on epigenetic robotics Modeling Cognitive Development in Robotic Systems (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 85, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' 701-746).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} +page_content=' Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FAT4oBgHgl3EQf6x6d/content/2301.08741v1.pdf'} diff --git a/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf b/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..050f52a7a8ef7a4409e5ae92012df03fe8697a29 --- /dev/null +++ b/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ddce3409ce5a3b7e9d3aa0313915eab613fd36899e9b54623ba3acb6bd0e16c6 +size 4712917 diff --git a/dtFKT4oBgHgl3EQfqy7x/content/tmp_files/2301.11876v1.pdf.txt b/dtFKT4oBgHgl3EQfqy7x/content/tmp_files/2301.11876v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b353a45b40352892c706a1b3044fe1c11f831077 --- /dev/null +++ b/dtFKT4oBgHgl3EQfqy7x/content/tmp_files/2301.11876v1.pdf.txt @@ -0,0 +1,408 @@ +2132 +ISSN 1063-7834, Physics of the Solid State, 2018, Vol. 60, No. 11, pp. 2132–2134. © Pleiades Publishing, Ltd., 2018. +Original Russian Text © V.V. Konev, V.A. Ulitko, D.N. Yasinskaya, Yu.D. Panov, A.S. Moskvin, 2018, published in Fizika Tverdogo Tela, 2018, Vol. 60, No. 11, pp. 2093–2095. +Influence of Local Correlations on the “Homogeneous +Insulator–Superconductor” Transition in the Domain Boundaries +of the Charge-Order Phase of a 2D System of a Mixed Valence +V. V. Koneva, *, V. A. Ulitkoa, D. N. Yasinskayaa, Yu. D. Panova, and A. S. Moskvina +a Ural Federal University, Yekaterinburg, 620002 Russia +*e-mail: vitaliy.konev@urfu.ru +Received May 14, 2018 +Abstract—It is demonstrated in the (pseudo)spin S = 1 formalism that the structure of antiphase domain +boundaries in the phase of charge ordering of a mixed-valence system of the Cu1+, 2+, 3+ “triplet” type in +cuprates on a two-dimensional square lattice depends to a considerable extent on on-site correlation param- +eter U. The results of computer modeling on large square lattices illustrate the change in the boundary struc- +ture (from a homogeneous monovalent nonconducting structure of the Cu2+ type to a filamentary supercon- +ducting one) induced by a relatively small variation of positive U values. +DOI: 10.1134/S1063783418110136 +1. INTRODUCTION +The interest in systems with spin S = 1 is fuelled by +the studies both into high-anisotropy magnetic mate- +rials based on Ni2+ (e.g., [Ni(HF2)(3-Clpy)1]BF1] and +NiCl24SC(NH2)2) and into the so-called pseudospin +systems of the semi-hard-core boson type with restric- +tions as to the occupation of lattice sites n = 0, 1, 2 or +systems of ions with a mixed valence of the “triplet” +type (Cu1+, 2+, 3+ in cuprates La2 – xSrxCuO4 or +Bi3+, 4+, 5+ in bismuthates [1]). In all cases, the phase +diagram of spin or pseudospin systems with S = 1 is +much more complex than that of similar systems with +quantum (pseudo)spin S = 1/2. The primary reason +for this is the emergence of entirely new terms in the +Hamiltonian (of the single-ion anisotropy and biqua- +dratic interaction type) and fundamentally new phases +such as a quantum paramagnetic or spin-nematic +phases. +Depending on the parameters of local and intersite +charge–charge correlations, one- and two-particle +transfer integrals, and the total charge, the ground +state of such systems may correspond to charge order- +ing, various types of superconducting ordering, com- +posite phases of the supersolid type with coexisting +superconductivity and charge ordering, or to a quan- +tum paramagnetic phase, which is specific to these +systems. The emergence of various metastable hetero- +geneous states with a well-developed domain structure +and topological defects of the vortex or skyrmion type +is typical of such systems [2–4]. +In this study, the pseudospin formalism is used to +analyze a simple system of Cu1+, 2+, 3+ charge triplets in +a model cuprate. It is demonstrated that the structure +of antiphase domain boundaries in the charge-order +phase of this system depends to a considerable extent +on on-site correlation parameter U and changes from +a homogeneous monovalent nonconducting structure +of the Cu2+ type to a filamentary superconducting one +under a relatively small variation of positive U values. +2. MODEL CUPRATE: PSEUDOSPIN +S = 1 FORMALISM +The model cuprate is a 2D system of Cu sites in the +CuO2 cuprate plane with three different possible +valence charge states: Cu1+, 2+, 3+. This charge triplet is +associated with three pseudospin S = 1 states in the +following way: Cu1+ → MS = –1, Cu2+ → MS = 0, +Cu3+ → MS = +1. In further study, we use well-known +methods for characterization of spin systems. +The pin algebra of systems with S = 1 (MS = 0, ±1) +includes eight independent nontrivial (three dipole +and five quadrupole) operators: +(1) +Raising and lowering operators S± and T± alter the +pseudospin projection by ±1, but in different ways: +〈0|S±| +〉 = 〈±1|S±|0〉 = +, 〈0|T±| = +〉 = –〈±1|T±|0〉 = ++1. Raising and lowering operators + characterize +± +± +± +± +± +± += +± += +≡ ++ +∓ +2 +2 +1 +; +( +); +; +2 +{ +, +} +; +. +z +x +y +z +z +z +z +S +S +S +iS +S +T +S S +S S +S S +S +∓1 +∓1 +∓1 +± +2 +S +SUPERCONDUCTIVITY +1 + +PHYSICS OF THE SOLID STATE + Vol. 60 + No. 11 + 2018 +INFLUENCE OF LOCAL CORRELATIONS +2133 +the |–1〉 ↔ |+1〉 transitions; i.e., they “produce” a hole +( +) or electron ( +) pair representing a composite +local boson with kinematic restriction + = 0, +which underscores its hard-core boson nature. Local +(on-site) nondiagonal order parameter 〈 +〉, which is +effectively a parameter of local superconducting order, +differs from zero only if a quantum superposition of +states |–1〉 and |+1〉 is established at a site. +Introducing the pseudospin S = 1 formalism to +characterize charge triplets, we write the effective +Hamiltonian, which commutes with the z-component +of total pseudospin + and thus conserves the total +charge of the system, as a sum of potential and kinetic +energies: +(2) +where +(3) +and only the contribution of two-particle transfer of +local composite bosons is taken into account in the +kinetic energy: +(4) +The first term in (3) (single-ion anisotropy) char- +acterizes the on-site density–density correlation +effects. Parameter Δ is related to the known correlation +parameter U: Δ = U/2. The second term may be asso- +ciated with the pseudomagnetic field along axis Oz or +with the chemical potential with respect to the addi- +tion of new particles. The last term characterizes inter- +site interactions (correlations) of the density–density +type. In subsequent analysis, only the interaction of +nearest neighbors with positive (antiferromagnetic) +parameters of intersite correlations V is taken into +account. +Depending on the ratio between the parameters of +Hamiltonian (2) and on the total charge, the ground +state of the system corresponds to a homogeneous +nonconducting phase of the quantum paramagnetic +type with + = + = 0, which is established at large +positive values of correlation parameter Δ (large-U +phase); to a nonconducting charge-order (CO) phase, +which is equivalent to antiferromagnetic ordering +along the z axis; or to a superfluid (SF) phase with a +nonzero order parameter +, which is accompanied +by homogeneous ferromagnetic ordering or inhomo- +geneous +antiferromagnetic +ordering +(supersolid +phase) of z-components of the pseudospin. Local +superconducting order parameter + may be written +in the standard form as + with modulus |Ψ| and +phase φ. ++ +2 +S +− +2 +S +± +2 2 +( +) +S +± +2 +S +Σi +iz +S += ++ +(2) +pot +kin, +H +H +H += +Δ +− μ ++ +∑ +∑ +2 +pot +1 +( +) +, +2 +iz +iz +iz +jz +i +ij +H +S +S +V +S S ++ +− +− ++ += − ++ +∑ +(2) +2 +2 +2 +2 +kin +1 +( +). +2 +b +i +j +i +j +ij +H +t +S S +S S +z +S +2 +z +S +± +2 +S +± +2 +S +± φ +Ψ +i +e +3. SPECIFIC FEATURES OF THE STRUCTURE +OF ANTIPHASE DOMAIN BOUNDARIES +OF THE CO PHASE +Using an NVIDIA graphics processing unit and the +Monte Carlo method, we simulated the charge-order- +ing phase transition in the model cuprate in the two- +sublattice approximation on a 256 × 256 square lattice +with periodic boundary conditions and parameters +tb = 1, V = 0.75, and μ = 0. This set of parameters +ensures that a CO ground state is maintained in a suf- +ficiently wide range of variation of local correlation +parameter Δ. +A branched domain structure formed in the pro- +cess of rapid thermalization (annealing) at Δ = –5. At +low temperatures, well-marked filamentary supercon- +ductivity emerged at the center of antiphase domain +boundaries of the CO phase, which is characterized by +a nonzero modulus of the local superconducting order +parameter. The latter fact is indicative of the presence +of local quantum superpositions Cu1+–Cu3+. As the +transfer integral of composite boson tb increases, the +domain boundaries get wider, and the volume of the +superconducting state increases to the point when the +CO phase becomes suppressed completely and the +transition to an inhomogeneous superconducting state +occurs. +Interestingly, both the CO domain structure the +superconducting structure of a domain boundary +turned out to be stable with respect to large variations +of local correlation parameter Δ and were still pre- +served at Δ ≈ +1.0. However, further enhancement of +local correlations resulted in a fundamental rearrange- +ment of the structure of domain boundaries. Figure 1 +presents the pattern of evolution of an antiphase +domain boundary at Δ ≥ +1.0, and Fig. 2 shows the +phase distribution of the local superconducting order +parameter (phase flow) in a domain boundary. When +the value of Δ increases, the regular structure of fila- +mentary superconductivity at the center of an anti- +phase domain boundary gets disrupted, and regions of +the “parental” Cu2+ phase (or, in pseudospin terms, +the quantum paramagnetic phase) emerge. These +regions grow with Δ; at Δ ≈ +1.4, filamentary super- +conductivity becomes suppressed completely and the +Cu2+ phase occupies the entire boundary. At even +higher Δ ≥ +1.5, the domain boundary widens, while +charge ordering is suppressed gradually. In other +words, the transition from the CO phase to the paren- +tal phase (large-U phase) at higher values of the local +correlation parameter is effected by the growth of +domain boundaries. +The study of temperature effects demonstrates that +transitions from the superconducting state to the +parental phase and then to the disordered “paramag- +netic” state occur in the CO phase domain boundaries +as the temperature increases at Δ = +1.0. However, +subsequent cooling to very low temperatures T = +0.0001 results in the restoration of just the parental +2 + +2134 +PHYSICS OF THE SOLID STATE + Vol. 60 + No. 11 + 2018 +KONEV et al. +structure of domain boundaries; i.e., hysteretic behav- +ior is observed. +4. CONCLUSIONS +The effect of the strength of local correlations Δ = +U/2 on the structure of domain boundaries of the CO +phase of a model cuprate was studied. The results of +numerical Monte Carlo modeling on large square lat- +tices revealed the formation of a branched domain +structure in the process of rapid annealing. Filamen- +tary superconductivity, which remained stable in a +wide range of U variation up to U ≈ 2, emerged in anti- +phase domain boundaries of this structure. However, +as local correlations grew even stronger, filamentary +superconductivity was disrupted, and the filamentary +parental Cu2+ phase, which separated domains with +charge ordering Cu1+–Cu3+, formed in the boundar- +ies. The modeling of temperature effects revealed +hysteretic behavior of the boundary structure. +ACKNOWLEDGMENTS +This study was supported by Program 211 of the +Government of the Russian Federation, agreement +no. 02.A03.21.0006, and projects 2277 and 5719 of the +Ministry of Education and Science of the Russian +Federation. +REFERENCES +1. A. S. Moskvin, J. Exp. Theor. Phys. 121, 477 (2015). +2. Y. D. Panov, A. S. Moskvin, F. N. Rybakov, and +A. B. Borisov, J. Low Temp. Phys. 185, 488 (2016). +3. A. S. Moskvin, Yu. D. Panov, F. N. Rybakov, and +A. B. Borisov, J. Supercond. Nov. Magn. 30, 43 (2017). +4. Y. D. Panov and A. S. Moskvin, Phys. C (Amsterdam, +Neth.) 548, 82 (2018). doi 10.1016/j.physc.2018.02.032 +Translated by D. Safin +Fig. 1. Evolution of an antiphase domain boundary +induced by the growth of local correlation parameter Δ. A +fragment of the 256 × 256 lattice with an antiphase domain +boundary separating CO domains denoted by the plus and +minus signs is highlighted. Filamentary superconducting +and parental Cu2+ phases are colored black and white, +respectively. +D = 1.4 +D = 1.2 +D = 1.0 +Fig. 2. Phase distribution of the local superconducting +order parameter (phase flow) in the domain-boundary +section marked in Fig. 1. Shades of gray highlight the +inhomogeneous distribution of the modulus of the local +superconducting order parameter. +� ++ +3 +Powered by TCPDF (www.tcpdf.org) +Powered by TCPDF (www.tcpdf.org) +Powered by TCPDF (www.tcpdf.org) + diff --git a/dtFKT4oBgHgl3EQfqy7x/content/tmp_files/load_file.txt b/dtFKT4oBgHgl3EQfqy7x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9adedf98757732d484257fe2003649020407a4e --- /dev/null +++ b/dtFKT4oBgHgl3EQfqy7x/content/tmp_files/load_file.txt @@ -0,0 +1,187 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf,len=186 +page_content='2132 ISSN 1063-7834, Physics of the Solid State, 2018, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 60, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2132–2134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' © Pleiades Publishing, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Original Russian Text © V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Konev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Ulitko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Yasinskaya, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Panov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Moskvin, 2018, published in Fizika Tverdogo Tela, 2018, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 60, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2093–2095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Influence of Local Correlations on the “Homogeneous Insulator–Superconductor” Transition in the Domain Boundaries of the Charge-Order Phase of a 2D System of a Mixed Valence V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Koneva, *, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Ulitkoa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Yasinskayaa, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Panova, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Moskvina a Ural Federal University, Yekaterinburg, 620002 Russia e-mail: vitaliy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='konev@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='ru Received May 14, 2018 Abstract—It is demonstrated in the (pseudo)spin S = 1 formalism that the structure of antiphase domain boundaries in the phase of charge ordering of a mixed-valence system of the Cu1+, 2+, 3+ “triplet” type in cuprates on a two-dimensional square lattice depends to a considerable extent on on-site correlation param- eter U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The results of computer modeling on large square lattices illustrate the change in the boundary struc- ture (from a homogeneous monovalent nonconducting structure of the Cu2+ type to a filamentary supercon- ducting one) induced by a relatively small variation of positive U values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='1134/S1063783418110136 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' INTRODUCTION The interest in systems with spin S = 1 is fuelled by the studies both into high-anisotropy magnetic mate- rials based on Ni2+ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=', [Ni(HF2)(3-Clpy)1]BF1] and NiCl24SC(NH2)2) and into the so-called pseudospin systems of the semi-hard-core boson type with restric- tions as to the occupation of lattice sites n = 0, 1, 2 or systems of ions with a mixed valence of the “triplet” type (Cu1+, 2+, 3+ in cuprates La2 – xSrxCuO4 or Bi3+, 4+, 5+ in bismuthates [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' In all cases, the phase diagram of spin or pseudospin systems with S = 1 is much more complex than that of similar systems with quantum (pseudo)spin S = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The primary reason for this is the emergence of entirely new terms in the Hamiltonian (of the single-ion anisotropy and biqua- dratic interaction type) and fundamentally new phases such as a quantum paramagnetic or spin-nematic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Depending on the parameters of local and intersite charge–charge correlations, one- and two-particle transfer integrals, and the total charge, the ground state of such systems may correspond to charge order- ing, various types of superconducting ordering, com- posite phases of the supersolid type with coexisting superconductivity and charge ordering, or to a quan- tum paramagnetic phase, which is specific to these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The emergence of various metastable hetero- geneous states with a well-developed domain structure and topological defects of the vortex or skyrmion type is typical of such systems [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' In this study, the pseudospin formalism is used to analyze a simple system of Cu1+, 2+, 3+ charge triplets in a model cuprate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' It is demonstrated that the structure of antiphase domain boundaries in the charge-order phase of this system depends to a considerable extent on on-site correlation parameter U and changes from a homogeneous monovalent nonconducting structure of the Cu2+ type to a filamentary superconducting one under a relatively small variation of positive U values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' MODEL CUPRATE: PSEUDOSPIN S = 1 FORMALISM The model cuprate is a 2D system of Cu sites in the CuO2 cuprate plane with three different possible valence charge states: Cu1+, 2+, 3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' This charge triplet is associated with three pseudospin S = 1 states in the following way: Cu1+ → MS = –1, Cu2+ → MS = 0, Cu3+ → MS = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' In further study, we use well-known methods for characterization of spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The pin algebra of systems with S = 1 (MS = 0, ±1) includes eight independent nontrivial (three dipole and five quadrupole) operators: (1) Raising and lowering operators S± and T± alter the pseudospin projection by ±1, but in different ways: 〈0|S±| 〉 = 〈±1|S±|0〉 = , 〈0|T±| = 〉 = –〈±1|T±|0〉 = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Raising and lowering operators characterize ± ± ± ± ± ± = ± = ≡ + ∓ 2 2 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' ( );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2 { , } ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' z x y z z z z S S S iS S T S S S S S S S ∓1 ∓1 ∓1 ± 2 S SUPERCONDUCTIVITY 1 PHYSICS OF THE SOLID STATE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 60 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 11 2018 INFLUENCE OF LOCAL CORRELATIONS 2133 the |–1〉 ↔ |+1〉 transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=', they “produce” a hole ( ) or electron ( ) pair representing a composite local boson with kinematic restriction = 0, which underscores its hard-core boson nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Local (on-site) nondiagonal order parameter 〈 〉, which is effectively a parameter of local superconducting order, differs from zero only if a quantum superposition of states |–1〉 and |+1〉 is established at a site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Introducing the pseudospin S = 1 formalism to characterize charge triplets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' we write the effective Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' which commutes with the z-component of total pseudospin and thus conserves the total charge of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' as a sum of potential and kinetic energies: (2) where (3) and only the contribution of two-particle transfer of local composite bosons is taken into account in the kinetic energy: (4) The first term in (3) (single-ion anisotropy) char- acterizes the on-site density–density correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Parameter Δ is related to the known correlation parameter U: Δ = U/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The second term may be asso- ciated with the pseudomagnetic field along axis Oz or with the chemical potential with respect to the addi- tion of new particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The last term characterizes inter- site interactions (correlations) of the density–density type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' In subsequent analysis, only the interaction of nearest neighbors with positive (antiferromagnetic) parameters of intersite correlations V is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Depending on the ratio between the parameters of Hamiltonian (2) and on the total charge, the ground state of the system corresponds to a homogeneous nonconducting phase of the quantum paramagnetic type with = = 0, which is established at large positive values of correlation parameter Δ (large-U phase);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' to a nonconducting charge-order (CO) phase, which is equivalent to antiferromagnetic ordering along the z axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' or to a superfluid (SF) phase with a nonzero order parameter , which is accompanied by homogeneous ferromagnetic ordering or inhomo- geneous antiferromagnetic ordering (supersolid phase) of z-components of the pseudospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Local superconducting order parameter may be written in the standard form as with modulus |Ψ| and phase φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' + 2 S − 2 S ± 2 2 ( ) S ± 2 S Σi iz S = + (2) pot kin, H H H = Δ − μ + ∑ ∑ 2 pot 1 ( ) , 2 iz iz iz jz i ij H S S V S S + − − + = − + ∑ (2) 2 2 2 2 kin 1 ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2 b i j i j ij H t S S S S z S 2 z S ± 2 S ± 2 S ± φ Ψ i e 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' SPECIFIC FEATURES OF THE STRUCTURE OF ANTIPHASE DOMAIN BOUNDARIES OF THE CO PHASE Using an NVIDIA graphics processing unit and the Monte Carlo method, we simulated the charge-order- ing phase transition in the model cuprate in the two- sublattice approximation on a 256 × 256 square lattice with periodic boundary conditions and parameters tb = 1, V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='75, and μ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' This set of parameters ensures that a CO ground state is maintained in a suf- ficiently wide range of variation of local correlation parameter Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' A branched domain structure formed in the pro- cess of rapid thermalization (annealing) at Δ = –5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' At low temperatures, well-marked filamentary supercon- ductivity emerged at the center of antiphase domain boundaries of the CO phase, which is characterized by a nonzero modulus of the local superconducting order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The latter fact is indicative of the presence of local quantum superpositions Cu1+–Cu3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' As the transfer integral of composite boson tb increases, the domain boundaries get wider, and the volume of the superconducting state increases to the point when the CO phase becomes suppressed completely and the transition to an inhomogeneous superconducting state occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Interestingly, both the CO domain structure the superconducting structure of a domain boundary turned out to be stable with respect to large variations of local correlation parameter Δ and were still pre- served at Δ ≈ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' However, further enhancement of local correlations resulted in a fundamental rearrange- ment of the structure of domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Figure 1 presents the pattern of evolution of an antiphase domain boundary at Δ ≥ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='0, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2 shows the phase distribution of the local superconducting order parameter (phase flow) in a domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' When the value of Δ increases, the regular structure of fila- mentary superconductivity at the center of an anti- phase domain boundary gets disrupted, and regions of the “parental” Cu2+ phase (or, in pseudospin terms, the quantum paramagnetic phase) emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' These regions grow with Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' at Δ ≈ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='4, filamentary super- conductivity becomes suppressed completely and the Cu2+ phase occupies the entire boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' At even higher Δ ≥ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='5, the domain boundary widens, while charge ordering is suppressed gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' In other words, the transition from the CO phase to the paren- tal phase (large-U phase) at higher values of the local correlation parameter is effected by the growth of domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The study of temperature effects demonstrates that transitions from the superconducting state to the parental phase and then to the disordered “paramag- netic” state occur in the CO phase domain boundaries as the temperature increases at Δ = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' However, subsequent cooling to very low temperatures T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='0001 results in the restoration of just the parental 2 2134 PHYSICS OF THE SOLID STATE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 60 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 11 2018 KONEV et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' structure of domain boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=', hysteretic behav- ior is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' CONCLUSIONS The effect of the strength of local correlations Δ = U/2 on the structure of domain boundaries of the CO phase of a model cuprate was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The results of numerical Monte Carlo modeling on large square lat- tices revealed the formation of a branched domain structure in the process of rapid annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Filamen- tary superconductivity, which remained stable in a wide range of U variation up to U ≈ 2, emerged in anti- phase domain boundaries of this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' However, as local correlations grew even stronger, filamentary superconductivity was disrupted, and the filamentary parental Cu2+ phase, which separated domains with charge ordering Cu1+–Cu3+, formed in the boundar- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' The modeling of temperature effects revealed hysteretic behavior of the boundary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' ACKNOWLEDGMENTS This study was supported by Program 211 of the Government of the Russian Federation, agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='A03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='0006, and projects 2277 and 5719 of the Ministry of Education and Science of the Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Panov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Moskvin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' C (Amsterdam, Neth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=') 548, 82 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' doi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='physc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='032 Translated by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Safin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Evolution of an antiphase domain boundary induced by the growth of local correlation parameter Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' A fragment of the 256 × 256 lattice with an antiphase domain boundary separating CO domains denoted by the plus and minus signs is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Filamentary superconducting and parental Cu2+ phases are colored black and white, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='4 D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='2 D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Phase distribution of the local superconducting order parameter (phase flow) in the domain-boundary section marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' Shades of gray highlight the inhomogeneous distribution of the modulus of the local superconducting order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content=' � + 3 Powered by TCPDF (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='tcpdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='org) Powered by TCPDF (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='tcpdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='org) Powered by TCPDF (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFKT4oBgHgl3EQfqy7x/content/2301.11876v1.pdf'} +page_content='tcpdf.' metadata={'source': 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The +lack of a sophisticated guidance results in a poor performance for numerous re- +inforcement learning algorithms. In these cases, the commonly used random +exploration is often not helpful. The literature shows that this kind of environments +require enormous efforts to systematically explore large chunks of the state space. +Learned state representations can help here to improve the search by providing +semantic context and build a structure on top of the raw observations. In this work +we introduce a novel time-myopic state representation that clusters temporally +close states together while providing a time prediction capability between them. +By adapting this model to the Go-Explore paradigm (Ecoffet et al., 2021b), we +demonstrate the first learned state representation that reliably estimates novelty +instead of using the hand-crafted representation heuristic. Our method shows +an improved solution for the detachment problem which still remains an issue +at the Go-Explore Exploration Phase. We provide evidence that our proposed +method covers the entire state space with respect to all possible time trajectories +— without causing disadvantageous conflict-overlaps in the cell archive. Anal- +ogous to native Go-Explore, our approach is evaluated on the hard exploration +environments MontezumaRevenge, Gravitar and Frostbite (Atari) in order to val- +idate its capabilities on difficult tasks. Our experiments show that time-myopic +Go-Explore is an effective alternative for the domain-engineered heuristic while +also being more general. The source code of the method is available on GitHub: +https://github.com/Hauf3n/Time-Myopic-Go-Explore. +Keywords: Exploration, Self-Supervised Learning, Go-Explore +1 +INTRODUCTION +In recent years, the problem of sufficient and reliable exploration remains an area of research in the +domain of reinforcement learning. In this effort, an agent seeks to maximize its extrinsic discounted +sum of rewards without ending up with a sub-optimal behavior policy. A good exploration mechanism +should encourage the agent to seek novelty and dismiss quick rewards for a healthy amount of time +to evaluate long-term consequences of the action selection. +Four main issues are involved when performing an exploration in a given environment: (i) catastrophic +forgetting (Goodfellow et al., 2013) as the data distribution shifts because the policy changes, (ii) a +neural network’s overconfident evaluation of unseen states (Zhang et al., 2018), (iii) sparse reward +Markov Decision Processes and (iv) the exploration-exploitation trade-off (Sutton and Barto, 2018; +Ecoffet et al., 2021a). The latter causes a significant problem: the greater the exploitation the less +exploration is done. Hereby, making novelty search less important when the agent can easily reach +states with a large rewards. +To address these difficulties, Ecoffet et al. (2021b) propose a new approach called Go-Explore +and achieve state-of-the-art results on hard exploration problems. However, Go-Explore relies on +hand-crafted heuristics. In our work we replace their discrete state representations with learned +representations particularly designed to estimate elapsed time to improve the novelty estimation. The +time distance between two states is used to build an abstraction level on top of the raw observations +by grouping temporally close states. Equipped with this capability, our model can decide about +the acceptance or rejection of states for the archive memory and maintain additionally exploration +statistics. +1 +arXiv:2301.05635v1 [cs.LG] 13 Jan 2023 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Figure 1: Model architecture. For details see Section 3. +Contribution. This work attempts to improve the exploration problem by introducing a learnable +novelty estimation method that is applicable to arbitrary inputs. In the experiment section, we +demonstrate the reliability of our method by comparing its results with native Go-Explore and more +broadly with several other exploration-based and baseline approaches. The main contributions of this +paper are the following: +1. We introduce a new novelty estimation method consisting of a siamese encoder and a time- +myopic prediction network which learns problem-specific representations that are useful for +time prediction. The time distances are later used to determine novelty which generate a +time-dependent state abstraction. +2. We implement a new archive for Go-Explore that includes a new insertion criterion, a novel +strategy to count cell visits and a new selection mechanism for cell restarts. +2 +PROBLEMS OF GO-EXPLORE +Go-Explore maintains an archive of saved states that are used as milestones to be able to restart from +intermediate states, hereby prevent detachment and derailment (as discussed in Ecoffet et al. (2021a)). +It generates its state representation by down-scaling and removing color (see Figure 2 left). The +exact representation depends on three hyperparameters (width, height, pixel-depth), which have to +be tuned for each environment. If two distinct observations generate the same encoding they are +considered similar and dissimilar otherwise. Thus, all possible states are grouped into a fixed number +of representatives, which leads to overlap-conflicts when two distinct observations receive the same +encoding. In these cases one of the states has to be abandoned, which might be the reason that for the +Atari environment Montezuma‘s Revenge only 57 out of 100 runs reached the second level in the +game (as reported in Ecoffet et al. (2021a)). We conjecture that their replacement criterion favors a +certain state over another, so the exploration will be stopped at the abandoned state. This can result in +decoupling an entire state subspace from the agent’s exploration endeavor. +A related issue emerges by ignoring the spatio-temporal semantics between states that are grouped +together into a single representation. The down-scaling method is just compressing the image +information without considering its semantic content. The consequence is a replacement criterion +that resolves archive conflicts between states that are neither spatially nor temporally in a close +relationship. Therefore a conflict solver has to resolve illogical and non-intuitive conflicts, because +in these cases it is not obvious which state should be favored. We have no information about which +potentially reachable states are more promising. On top of that, the cell representation is only suitable +for states represented by small images. If we have images with higher resolutions the representations +2 + +直上Ot,Ot-1,Ot-tk.t+k-i.0t+k具直Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Figure 2: (left panel) Go-Explore down-scaling operation with hyperparameters (h, w, dp). (right +panel): time distances are directed. Jumping off the cliff is faster, than hiking up (image from IHC). +could get either abstract (and hereby limiting the exploration progress) or too narrow resulting in an +overflowing archive. Moreover and from a general point of view, it is not clear how we could usefully +down-scale an input observation that consists of partial or no image information at all. +3 +LEARNING TIME-MYOPIC STATE REPRESENTATIONS +The basic idea of Go-Explore is to save intermediate results, so that the environment state can be +restored from a memory to try new action sequences and obtain different outcomes. However, the +difficulty is to decide when to store a state for returning. Go-Explore solves this by a heuristic that +lead to some non-intuitive decisions when dealing with representation conflicts as discussed in the +previous section. +In the following we show how state representations can be learned that are particularly designed to +predict the time progress. Equipping Go-Explore with this model will avoid archive conflicts and +gives a better control over the novelty estimation problem. We call our method time-myopic since it +is trained to predict the elapsed time only up to a particular distance. +3.1 +PREPROCESSING AND SIAMESE ENCODER +As a first step, we preprocess for every timestep t a stack of the last three RGB observations +ot, ot−1, ot−2. For this we convert each observation into a single gray-scale frame to generate a +compressed input format (see Fig. 1). The result ¯ot consist of three color channels where each channel +represents one gray-scale observation. This format captures time-critical information and allows us +to infer properties such as velocity and acceleration of objects. +Given a preprocessed observation ¯ot, we apply a siamese convolutional neural network (CNN) Φθ to +obtain a latent vector encoding zt +Φθ(¯ot) = zt. +(1) +The exact architecture of the CNN Φθ is shown at the top right in Figure 1. The goal is to learn a +useful embedding that captures the information to accurately predict the elapsed time k between +two given observations ¯ot and ¯ot+k. We choose siamese networks since they demonstrated good +results (Koch, 2015; Ermolov and Sebe, 2020; Chen et al., 2021) in distinguishing deviations between +distinct inputs or respectively capturing similarity. +3.2 +TIME-PREDICTION NETWORK +After the encoding procedure, a pair of observation encodings (zt, zt+k) allows us to estimate the +elapsed time between the elements of a pair. For this, we feed the difference between zt and zt+k +into another fully-connected multi-layer neural network fθ that outputs a real number. The whole +time prediction network is written as Ψθ and includes a normalization (see bottom right Fig. 1) +Ψθ(zt, zt+k) = 1 − e− max(fθ(zt+k−zt),0) +∈ [0, 1]. +(2) +In Equation (2) the model predicts the time distance from the starting point zt to the destination +zt+k where a swap can result in a different outcome. We obtain the myopic property by normalizing +3 + +h,w,dpAccepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +the output with the function g(x) = 1 − e−x that allows us to specify a prediction range [0, L] for +the normalization interval [0, 1]. When we want to obtain the predicted time distance k we need to +multiply the normalized output with the upper bound L and vice versa for converting a distance k to +the internal representation +k ≈ L · Ψθ(zt, zt+k). +(3) +3.3 +LOSS FUNCTION AND ITS IMPLICATIONS +The time prediction loss objective Ltime(θ) minimizes the mean squared error between the predicted +time distance and the true distance k +Ltime(θ) = Et +� +(Ψθ(zt, zt+k) − min(k/L, 1))2� +, +(4) +where the correct distance k is known from the sampled trajectories after the environment interaction. +The neural network computation includes the normalization g(x) to prevent exploding gradients +and introduces the upper time window bound L that limits the network‘s expressivity beyond L +timesteps. This constraint simplifies the task, since correct forecasts of longer periods of time become +unimportant (Makridakis et al., 2018). The myopic view is helpful to reliably estimate novelty, +because the model does not need to handle a precise time measurement between states that are far +away from each other. By considering only local distances, the predictions get easier and they are +more reliable. So when the network encounters a state-pair with a large distance k > L, we do not +care about the exact true distance. But instead, the model should indicate that the states are far away +from each other by outputting the upper bound L. +3.4 +TIME DISTANCE IS POLICY DEPENDENT AND DIRECTED +Each singular timestep represents a state transition on the underlying MDP where the time-related +reachability is dependent on the current policy. This means that two distinct policies could reach a +certain state with a different number of timesteps. Also, since time distances are directed we have to +input our encodings (zA, zB) in the right order into Ψθ. For intuition, Figure 2 (right) shows a visual +example where a state A is on top of a small cliff and point B is at the bottom of it. To reach B from +A an agent could use the shortcut and jump down the cliff, but it might not be possible to jump back +up. Therefore the agent needs to find a different path that might increase the elapsed time to reach +A from B. For that reason we assume that the distances of our time prediction function can yield +different results for Ψθ(zA, zB) and Ψθ(zB, zA). +4 +TIME-MYOPIC GO-EXPLORE +To integrate our learned representation model from the previous section into the native Go-Explore +method, we have to make some adjustments, since our state encodings are continuous (and no longer +discrete). This makes it harder to decided similarity between two distinct states. In the following we +explain how this is achieved. +4.1 +ARCHIVE INSERTION CRITERION +The time prediction capability of Ψθ allows us to propose a new archive criterion. Our criterion is +an insertion-only method which adds a state to the archive when it is novel enough. In this way the +detachment problem is entirely solved by not abandoning archive states. We initialize the archive by +inserting the encoding zc1 of the environment‘s start state. This entry will begin the exploration effort +by searching for novelty around the starting point. In order to do that the agent samples trajectories +from the restored state c1 where we collect new states that we call archive candidates. A new +candidate encoding zK is evaluated with every archive entry zC by the time prediction function. The +predicted value Ψθ(zC, zK) is thresholded by some hyperparameter Td to determine the acceptance +or rejection of the current candidate. If the threshold is surpassed for every comparison, the candidate +K is added to the archive. Note, that we choose the time distance direction Ψθ(zC, zK) since we +want to move away from the archive entries. When this happens, we conclude that the agent has +reached a currently unknown part of the state space which has a sufficient novelty with respect to the +archive-known subspace. In this way, the candidate evaluation gains a global view on the exploration +progress instead of considering a local trajectory-based perspective (Badia et al., 2020; Savinov et al., +2018). A short example is shown in Table 1. +4 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Table 1: Archive insertion criterion where the model‘s expressivity is in the time window [0, L = 20]. +A new candidate zK is compared to all existing cells in the archive. If one cell is too close, the +candidate is rejected. +Comparison +Time estimation +Evaluation +Cell +Candidate +Ψθ(zc, zK) +20 ∗ Td > 13 +Insert? +zc1 +zK +0.3934 +7.86 +no +zc2 +zK +0.7568 +15.13 +yes +zc3 +zK +0.6321 +12.64 +no +zc4 +zK +0.9334 +18.66 +yes +... +zK +... +... +... +4.2 +VISIT COUNTER +To ensure successful progression in the exploration, it is common practice to track the number of cell +visits Cvisits for every cell C in the archive. With respect to the return selection, the visit counter of a +cell increases by one when it is selected as a starting point. Furthermore, the visit counter increments +when the time distance to an archive candidate is lower than a visit threshold Tv. This evaluation +runs simultaneously to the application of our insertion criterion where we compute all time distances +to the candidate. +4.3 +CELL SELECTION CRITERION +Native Go-Explore made the design choice to replace cell entries with small scores to states with +larger scores. This significantly boosts the exploration by abandoning states which might have +been sufficiently explored. Time-myopic Go-Explore does not have this ability, because the state +representations are continuous and the encodings of two states with weak spatio-temporal semantics +are far away from each other. Also, our method should not overwrite archive entries, because they +could be inserted again with clean visits statistics. To bias exploration towards cells with larger +scores, we calculate and combine the native selection weight Wvisits with our score weight Wscore +that contains the reached cell score Cscore (sum of undiscounted cumulative reward). Both quantities +(Cvisits, Cscore) are stored in the archive for every cell C. +W = +1 +√Cvisits + 1 +� +�� +� +visit weight Wvisits +· max +� +Cscore +maxC′ C +′ +score + 1, α +� +� +�� +� +score weight Wscore +. +(5) +The outer maximum in Wscore ensures that states with lower scores are explored as well (with chosen +hyperparameter α = 0.075). The inner maximum normalizes the experienced scores between zero +and one. +5 +TECHNICAL DETAILS FOR TRAINING +In order to improve the model quality we explain next three additional training routines that enhance +the time prediction of our model. +5.1 +SIMILARITY AND DISSIMILARITY +Optimizing only on close-by pairs (e.g. with time distance k ≤ L) will lead to wrong estimates +for distant pairs (i.e. k > L), which might be the majority of cells. Thus we have to ensure that +our dataset includes also dissimilar pairs where the observations are at least L timesteps away from +each other. The distance of a dissimilar pair is trained on the value that corresponds to the maximal +time estimation L, which does not represent the actual distance. Instead it signals a sufficiently large +temporal space between them. In this way, we try to find encodings such that the embedding region +around a state only includes other encodings that are within the defined time window [0, ..., L]. +5 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +5.2 +AVOIDING TEMPORAL AMBIGUITY +Suppose there are two observations ¯ot+k1 and ¯ot+k2 that happened at different points in time, but +which are pixel-wise identical. To define a single distance to some other observation ¯ot, we will use +the minimum of k1 and k2. To quickly identify pixel-wise identical observations we are using the +MD5 hash function (Rivest, 1992). +5.3 +PROXY CELLS AND LOCAL DATASETS +Usually the time prediction model would be trained on trajectories that always start with an archive +state. To extend the training data we additionally generate trajectories (only for training) that start +from so-called proxy cell states that are temporally close to an archive cell. We collect these proxies +by choosing randomly states in the time distance interval [Tp-low, Tp-high] to their respective archive +cell and replace them periodically. +To further increase variability of the dataset, we add small local datasets for each cell that add some +data points within its proximity. Every dataset holds a few hundred pairs where we store the time +distance between the cell state and a temporally close state. This is necessary because otherwise +our network would forget about certain cells and their neighboring states since they are not visited +anymore due to the selection weighting W = Wvisits ∗ Wscore. +6 +EXPERIMENTS +The experiment section covers the following topics: in Section 6.1 we study the encoder properties +and assess how well the time can be predicted. In Section 6.2 we compare our approach, the original +Go-Explore and other related methods. In Section 6.3 we analyze the archive for the native and our +proposed time-myopic Go-Explore. +6.1 +WHAT DOES OUR MODEL LEARN? +To demonstrate the properties of our model, we re-create an experiment from Ermolov and Sebe (2020) +which is shown in Figure 3. A representation-learning network is trained on 400k observations from +the Atari environment Montezuma’s Revenge where the training data is gathered by a random actor +at the environment‘s start state. After optimization, the network is asked to encode the observation +sequence from the trajectory shown in Figure 3 (top-left). Most of the encodings are extrapolated, +because a random policy is not able to reliably execute the action sequence that generated the +observations. Therefore an extrapolation starts roughly between the checkpoint 3 and 4. In this +experiment our model uses 9k observations which are sufficient to show a good result. The 32- +dimensional encodings are projected with the t-SNE method into a two-dimensional space where it +uncovers the interesting relationship between the data points. Temporally close states are grouped +together and are strung on a thread while the sequence unfolds. The distance between two imminent +states does not collapse and it provides useful semantics for time measurements. Remember that the +chosen trajectory has no greater meaning for our model and it is perceived as arbitrary like any other +possibly selected sequence. Subsequently, the rich structure is also present when performing PCA on +the encodings (see Appendix Figure 5). +The time evaluation demonstrates good results where the network has training data. However, the +extrapolation capability on time distances is limited. But surprisingly we can see that the encoder +even places unseen states close to their temporal neighbors resulting in local semantic integrity. At +some point the extrapolation of time distances becomes unreliable which will improve with more +novel data for optimization. Once the network was trained on more data during a complete run, the +prediction quality gets a lot better (see Appendix Figure 6). +6.2 +COMPARISON ON HARD EXPLORATION ENVIRONMENTS +We evaluate our method on the hard exploration environments Montezuma’s Revenge, Gravitar, +Frostbite (Atari) and compare it with (i.) related methods (ii.) and native Go-Explore. We prepare +our experiments by adjusting the natively used random action-repeating actor (Ecoffet et al., 2021a). +Time-myopic Go-Explore decreases the actor’s action repetition mean µ from 10 to 4. This is +necessary since the native method does not care about learning a state representation while our +learned model depends on a robust data collection for the time predictions. Native Go-Explore has +6 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Figure 3: Visualization of the model performance. The top-left image is an observation sequence +from the Atari environment Montezuma’s Revenge, which is generated by a human demonstration. +The top-right image is the t-SNE (van der Maaten and Hinton, 2008) projection of the observa- +tion encodings generated by the siamese CNN Φθ. The bottom image shows time estimates of +our prediction network Ψθ for L = 20. This model has the ability to calculate the time dis- +tance for a pair of observations. E.g. we can select several encodings z0, z10, z20, z40, z50, z60, z70 +and let the time prediction network calculate their distance to their successors. Note that for the +blue graph, we plot Ψθ(z0, z0), Ψθ(z0, z1), Ψθ(z0, z2), ..., Ψθ(z0, z72). For the orange one we plot +Ψθ(z10, z10), Ψθ(z10, z11), Ψθ(z10, z12), ..., Ψθ(z10, z72). +Table 2: Mean cumulative reward for Atari (rows 2 to 6 are copied from Kim et al. (2018), all others +from the cited papers). This table shows the performance of different exploration-based and baseline +methods. Our results are computed as the mean over 20 runs where each run has seen 5M frames. +Method +Frames +Montezuma +Gravitar +Frostbite +R2D2 (Kapturowski et al., 2019) +10000M +2061 +15680 +315456 +EX2 (Fu et al., 2017) +50M +0 +550 +3387 +AE-SimHash (Strehl and Littman, 2008) +50M +75 +482 +5214 +ICM (Pathak et al., 2017) +50M +161 +424 +4465 +RND (Burda et al., 2018) +50M +377 +546 +2227 +EMI (Kim et al., 2018) +50M +387 +558 +7002 +LWM (Ermolov and Sebe, 2020) +50M +2276 +1376 +8409 +PPO (Schulman et al., 2017) +40M +42 +737 +314 +Ours (Time-myopic Go-Explore) +5M +2090 +3161 +4476 +Ours (Time-myopic Go-Explore) +1M +695 +2533 +3543 +Go-Explore (native) +1M +2303 +2130 +11721 +the advantage that it can act more greedily, because the representation heuristic is always reliable +and therefore can be exploited by acting more risky. Also, native Go-Explore uses the originally +recommended down-scaling hyperparameters (h = 8, w = 11, dp = 8) for Montezuma’s Revenge +and the dynamic down-scaling (recompute every 500k frames) only for the other environments +(Montezuma’s Revenge with dynamic down-scaling performs worse). For native Go-Explore we are +using the official implementation. Our time-myopic Go-Explore variant runs on one A100 GPU and +needs roughly 8-10 hours for 5M frames depending on the archive size. +7 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Table 3: Comparison of the archive size for Montezuma’s Revenge for scores 100, 400, 500, 2500. +Archive size per score +Method +100 +400 +500 +2500 +Native (h = 8, w = 11, dp = 8) +126.2 +197.8 +1812.2 +2838.9 +Time-myopic (L = 20, Td = 0.55) +160.5 +202.5 +450.0 +566.0 +Time-myopic (L = 25, Td = 0.65) +103.8 +126.2 +210.7 +248.3 +(a) Distance to closest cell neighbor +(b) Mean distance to next three cell neighbors +Figure 4: Time distances between archive cells on Montezuma’s Revenge. We generate two archives +where the first one (orange) was constructed using Go-Explore down-scaling method (h = 8, w = 11, +dp = 8) and the second one (green) by our learned model predictions (L = 20, Td = 0.55). When +both archives reach the score of 2500 we use a second optimized time prediction network (L = 20) +and compute all time distances between the cell observations for both archives. +The result shows the practicality of time-myopic Go-Explore since it is better than most of the +methods for Montezuma’s Revenge and Gravitar while it only has seen 2% of their frames (1M vs +50M frames). It also provides a good performance on Frostbite surpassing PPO, RND, EX2 and ICM. +Moreover, time-myopic Go-Explore is able to keep up with the performance of native Go-Explore +(using the domain-aligned representation heuristic) and even exceeds it within 1M frames for the +environment Gravitar. As far as we know the results of Gravitar for 1M frames are state-of-the-art. +6.3 +ABLATION STUDY +Next we provide a closer look at the archive properties. First, we compare the difference in the archive +size between native Go-Explore and our approach (see Table 3). The table shows the mean number +of cells (20 runs) in the archive when the agent reaches a certain game score [100, 400, 500, 2500] +in Montezuma’s Revenge. The native archive size explodes after reaching a score of 400 while our +approach shows a more stable progression. The data also validates smaller archive sizes as a result of +increasing the hyperparameters (L, Td). In the Appendix, a figure shows more precisely when and +how the prediction model decides to add archive entries. We observe in our experiments a similar +pattern for the environment Gravitar which uses the dynamic down-scaling heuristic. The native +archives agglomerate an enormous amount of archive entries. After 1M frames for 20 runs, the mean +size results in 7376 cells while time-myopic Go-Explore holds 381 entries and is achieving a higher +score. +Secondly, both methods are evaluated on the similarity within the created archives (see Figure 4). +The time distances between all cell observations will provide an interesting similarity measurement. +The model predictions in Figure 4 confirm that the native archive yields a lot more similarity than our +approach. This leads to inefficient exploration, because based on the cell selection criterion every cell +needs to be sufficiently visited, no matter how few environment transitions are between them. Our +archives cover the state space with less cells. +8 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +6.4 +LIMITATIONS +Due to the queries to the archive, our algorithm runtime increases currently non-linearly with the +number of entries. This makes long runs or runs where the archive grows fast computationally +expensive. This happens, because our approach requires to compute time distances to every archive +entry in order to evaluate new cell candidates or to update the visit statistics. So, in the future, we +will make these queries more efficient to make the method applicable to even larger environments +and runs. +7 +RELATED WORK +Hard exploration problems like the Atari environment Montezuma’s Revenge are known for their large +state spaces and sparse reward functions. A lot state-of-the-art reinforcement learning algorithms +(Schrittwieser et al., 2020; Kapturowski et al., 2019; Schulman et al., 2017) have difficulties in +achieving a good performance on these tasks, so several ideas have proposed towards a solution. +The literature contains several approaches related to our work that deal with unsupervised- and +representation learning combined with intrinsic motivation and exploration. +Playing hard exploration games by watching YouTube (Aytar et al., 2018). In this work the +authors introduce a neural network architecture in order to learn a categorical time classification +between two distinct observations. Their network is used to generate an intrinsic reward signal to +facilitate imitation learning of human demonstrations while we are using it to globally model the +exploration process. +Solving sparse reward environments using Go-Explore with learned cell representation +(Bjørsvik, 2021). This approach also extends the Go-Explore (Ecoffet et al., 2021a) method by +replacing the representation heuristic with a learned state representation. They employ a Variational +Autoencoder (VAE) (Kingma and Welling, 2013) to encode every seen state into a latent vector space. +Later on, the encodings are used for a k-means clustering procedure where a cluster center stands for +an entry in the archive memory. +Latent world models for intrinsically motivated exploration (Ermolov and Sebe, 2020). This +paper optimizes a siamese network that clusters temporal imminent states in the embedding space by +minimizing the mean squared error between them. In addition, there is the need for an extra structure +constraint in order to prevent a collapse of the representation to a constant vector. The resulting +encodings are used for the generation of an intrinsic reward signal. +Episodic curiosity through reachability (Savinov et al., 2018). The paper introduces a model +architecture with a logistic regression capability to differentiate between novel and familiar state pairs +where novelty is defined by a minimum time distance of k steps. The complete model (including +a siamese encoder and comparison network) is only able to decide whether an encountered pair is +novel or not; without the ability to evaluate it on a continuous basis. This estimation is utilized to +generate a positive intrinsic reward signal when a policy encounters new states. +Never give up: learning directed exploration strategies (Badia et al., 2020). The proposed episodic +novelty module starts empty and fills itself with state encodings when the agent interacts with the +environment. Every state gets evaluated by a k-nearest neighbor criterion with the similarity measure +of the Dirac delta kernel to compute the rewards with respect to the similarity distance. The goal is to +insert as many novel states as possible which then facilitate exploration. +8 +CONCLUSION +To add flexibility to and possibly improve Go-Explore we studied how its representation heuristic can +be replaced by a time-predicting neural network. Experiments show that the new state representation +is able to track the global exploration effort and moreover recognizes ongoing progress for this task. +In comparison to native Go-Explore, our method can reduce the archive size and covers the state +space with fewer cells. Applied to hard exploration environments, such as Montezuma’s Revenge we +observe good performance compared to previous exploration-based methods while using much fewer +observations, even creating better results than Go-Explore on the game Gravitar. Overall, however our +learned representation is not able to compete with native Go-Explore in terms of sample efficiency. +9 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +REFERENCES +I heart crafty things - cliff. +https://iheartcraftythings.com/wp-content/ +uploads/2021/09/Cliff.jpg. obtained and modified. +Y. Aytar, T. Pfaff, D. Budden, T. L. Paine, Z. Wang, and N. de Freitas. Playing hard exploration +games by watching youtube, 2018. URL https://arxiv.org/abs/1805.11592. +A. P. Badia, P. Sprechmann, A. Vitvitskyi, D. Guo, B. Piot, S. Kapturowski, O. Tieleman, M. Arjovsky, +A. Pritzel, A. Bolt, and C. Blundell. Never give up: Learning directed exploration strategies, 2020. +URL https://arxiv.org/abs/2002.06038. +V. Bjørsvik. Solving Sparse Reward Environments Using Go-Explore with Learned Cell Representa- +tion. http://urn.nb.no/URN:NBN:no-90374, 2021. +Y. Burda, H. Edwards, A. Storkey, and O. Klimov. 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Vinyals, R. Munos, and S. Bengio. A study on overfitting in deep reinforcement +learning, 2018. URL https://arxiv.org/abs/1804.06893. +A +APPENDIX - SECTION 1 +Table 4: Hyperparameters for the experiments reported in the hard exploration table. +Hyperparameter +Symbol +Value (MontezumaRevenge, Gravitar, Frostbite) +Learning rate +lr +1e−4 +Batch size +bs +64 +Time window +L +(20, 25, 25) +Distance threshold +Td +(0.6, 0.75, 0.75) +Visit threshold +Tv +0.65 +Proxy cell interval +[Tp-low,Tp-high] +[0.45, 0.75] +Exploration steps +t +40 +action repetition mean +µ +4 +(a) dim 1-2 +(b) dim 2-3 +(c) dim 1-3 +Figure 5: Principal Component Analysis for the trajectory visualization. +11 + +Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Figure 6: Sophisticated model and its capabilities on a longer trajectory. The shown model is trained +on data that surpasses the shown trajectory (top-left). Again, we can see the good encoding property +and an improved time prediction skill. The predicted times around the timesteps 90-100 or in the +image at the Box 11 are accurate. At that point the agent dies and the environment generates repeating +frames (two-image sprites) for around 10 frames. This prediction behavior happens, because we +remove temporal ambiguity between state-pairs and try to calculate the shortest distance for it. The +event can also be seen in the t-SNE visualization, when the light-green points start to form a cluster. +Figure 7: Creation of archive entries. We manually looked through the cell observations of an archive +and searched for similarity. This figure shows all cell observations where the agent stands at the same +position and already collected the key. We can see that the time prediction network does not allow +duplicates in the archive and keeps a reasonable distance between the observations where the white +skull is changing positions. Moreover the archive holds no observation where the agent just slightly +moved in these situations. +12 + +目广具官Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop +Algorithm 1 Go-Explore with a learned state representation +Initialize archive, dataset, agent, network +for iteration = 1, 2, ... do +Let the agent act t timesteps in the environment starting from the selected and proxy cells +Collect data and optimize Ltime w.r.t. θ +Recompute all archive cell representations: zC = Φθ(¯oC) +for each trajectory = 1, 2, ..., N do +Transfer all observations ¯o1, ..., ¯ot into the latent representation z1, ..., zt +Compute all necessary time distances Ψθ(zc, z1,...,t) +Apply archive insertion criterion to a candidate w.r.t. threshold Td +Increase cell visits w.r.t. threshold Tv +Collect some proxy cells w.r.t. threshold [Tp-low,Tp-high] +if candidate is accepted then +Add candidate to archive +end if +end for +end for +13 + diff --git a/edE5T4oBgHgl3EQfgg_h/content/tmp_files/load_file.txt b/edE5T4oBgHgl3EQfgg_h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a23f9ace809d5511bd2c3bfaeb721067671aeb67 --- /dev/null +++ b/edE5T4oBgHgl3EQfgg_h/content/tmp_files/load_file.txt @@ -0,0 +1,659 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf,len=658 +page_content='Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop TIME-MYOPIC GO-EXPLORE: LEARNING A STATE REPRESENTATION FOR THE GO-EXPLORE PARADIGM Marc H¨oftmann, Jan Robine, Stefan Harmeling Department of Computer Science, Technical University of Dortmund, Germany ABSTRACT Very large state spaces with a sparse reward signal are difficult to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The lack of a sophisticated guidance results in a poor performance for numerous re- inforcement learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In these cases, the commonly used random exploration is often not helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The literature shows that this kind of environments require enormous efforts to systematically explore large chunks of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Learned state representations can help here to improve the search by providing semantic context and build a structure on top of the raw observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this work we introduce a novel time-myopic state representation that clusters temporally close states together while providing a time prediction capability between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' By adapting this model to the Go-Explore paradigm (Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2021b), we demonstrate the first learned state representation that reliably estimates novelty instead of using the hand-crafted representation heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Our method shows an improved solution for the detachment problem which still remains an issue at the Go-Explore Exploration Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We provide evidence that our proposed method covers the entire state space with respect to all possible time trajectories — without causing disadvantageous conflict-overlaps in the cell archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Anal- ogous to native Go-Explore, our approach is evaluated on the hard exploration environments MontezumaRevenge, Gravitar and Frostbite (Atari) in order to val- idate its capabilities on difficult tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Our experiments show that time-myopic Go-Explore is an effective alternative for the domain-engineered heuristic while also being more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The source code of the method is available on GitHub: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='com/Hauf3n/Time-Myopic-Go-Explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Keywords: Exploration, Self-Supervised Learning, Go-Explore 1 INTRODUCTION In recent years, the problem of sufficient and reliable exploration remains an area of research in the domain of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this effort, an agent seeks to maximize its extrinsic discounted sum of rewards without ending up with a sub-optimal behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A good exploration mechanism should encourage the agent to seek novelty and dismiss quick rewards for a healthy amount of time to evaluate long-term consequences of the action selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Four main issues are involved when performing an exploration in a given environment: (i) catastrophic forgetting (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2013) as the data distribution shifts because the policy changes, (ii) a neural network’s overconfident evaluation of unseen states (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018), (iii) sparse reward Markov Decision Processes and (iv) the exploration-exploitation trade-off (Sutton and Barto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The latter causes a significant problem: the greater the exploitation the less exploration is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Hereby, making novelty search less important when the agent can easily reach states with a large rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To address these difficulties, Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (2021b) propose a new approach called Go-Explore and achieve state-of-the-art results on hard exploration problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' However, Go-Explore relies on hand-crafted heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In our work we replace their discrete state representations with learned representations particularly designed to estimate elapsed time to improve the novelty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The time distance between two states is used to build an abstraction level on top of the raw observations by grouping temporally close states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Equipped with this capability, our model can decide about the acceptance or rejection of states for the archive memory and maintain additionally exploration statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='05635v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='LG] 13 Jan 2023 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Figure 1: Model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For details see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This work attempts to improve the exploration problem by introducing a learnable novelty estimation method that is applicable to arbitrary inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In the experiment section, we demonstrate the reliability of our method by comparing its results with native Go-Explore and more broadly with several other exploration-based and baseline approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The main contributions of this paper are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We introduce a new novelty estimation method consisting of a siamese encoder and a time- myopic prediction network which learns problem-specific representations that are useful for time prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The time distances are later used to determine novelty which generate a time-dependent state abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We implement a new archive for Go-Explore that includes a new insertion criterion, a novel strategy to count cell visits and a new selection mechanism for cell restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 2 PROBLEMS OF GO-EXPLORE Go-Explore maintains an archive of saved states that are used as milestones to be able to restart from intermediate states, hereby prevent detachment and derailment (as discussed in Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' It generates its state representation by down-scaling and removing color (see Figure 2 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The exact representation depends on three hyperparameters (width, height, pixel-depth), which have to be tuned for each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' If two distinct observations generate the same encoding they are considered similar and dissimilar otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Thus, all possible states are grouped into a fixed number of representatives, which leads to overlap-conflicts when two distinct observations receive the same encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In these cases one of the states has to be abandoned, which might be the reason that for the Atari environment Montezuma‘s Revenge only 57 out of 100 runs reached the second level in the game (as reported in Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We conjecture that their replacement criterion favors a certain state over another, so the exploration will be stopped at the abandoned state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This can result in decoupling an entire state subspace from the agent’s exploration endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A related issue emerges by ignoring the spatio-temporal semantics between states that are grouped together into a single representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The down-scaling method is just compressing the image information without considering its semantic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The consequence is a replacement criterion that resolves archive conflicts between states that are neither spatially nor temporally in a close relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Therefore a conflict solver has to resolve illogical and non-intuitive conflicts, because in these cases it is not obvious which state should be favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We have no information about which potentially reachable states are more promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' On top of that, the cell representation is only suitable for states represented by small images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' If we have images with higher resolutions the representations 2 直上Ot,Ot-1,Ot-tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='t+k-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='0t+k具直Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Figure 2: (left panel) Go-Explore down-scaling operation with hyperparameters (h, w, dp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (right panel): time distances are directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Jumping off the cliff is faster, than hiking up (image from IHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' could get either abstract (and hereby limiting the exploration progress) or too narrow resulting in an overflowing archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Moreover and from a general point of view, it is not clear how we could usefully down-scale an input observation that consists of partial or no image information at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 3 LEARNING TIME-MYOPIC STATE REPRESENTATIONS The basic idea of Go-Explore is to save intermediate results, so that the environment state can be restored from a memory to try new action sequences and obtain different outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' However, the difficulty is to decide when to store a state for returning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Go-Explore solves this by a heuristic that lead to some non-intuitive decisions when dealing with representation conflicts as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In the following we show how state representations can be learned that are particularly designed to predict the time progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Equipping Go-Explore with this model will avoid archive conflicts and gives a better control over the novelty estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We call our method time-myopic since it is trained to predict the elapsed time only up to a particular distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='1 PREPROCESSING AND SIAMESE ENCODER As a first step, we preprocess for every timestep t a stack of the last three RGB observations ot, ot−1, ot−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For this we convert each observation into a single gray-scale frame to generate a compressed input format (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The result ¯ot consist of three color channels where each channel represents one gray-scale observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This format captures time-critical information and allows us to infer properties such as velocity and acceleration of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Given a preprocessed observation ¯ot, we apply a siamese convolutional neural network (CNN) Φθ to obtain a latent vector encoding zt Φθ(¯ot) = zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (1) The exact architecture of the CNN Φθ is shown at the top right in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The goal is to learn a useful embedding that captures the information to accurately predict the elapsed time k between two given observations ¯ot and ¯ot+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We choose siamese networks since they demonstrated good results (Koch, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Ermolov and Sebe, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2021) in distinguishing deviations between distinct inputs or respectively capturing similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 TIME-PREDICTION NETWORK After the encoding procedure, a pair of observation encodings (zt, zt+k) allows us to estimate the elapsed time between the elements of a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For this, we feed the difference between zt and zt+k into another fully-connected multi-layer neural network fθ that outputs a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The whole time prediction network is written as Ψθ and includes a normalization (see bottom right Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 1) Ψθ(zt, zt+k) = 1 − e− max(fθ(zt+k−zt),0) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (2) In Equation (2) the model predicts the time distance from the starting point zt to the destination zt+k where a swap can result in a different outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We obtain the myopic property by normalizing 3 h,w,dpAccepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop the output with the function g(x) = 1 − e−x that allows us to specify a prediction range [0, L] for the normalization interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' When we want to obtain the predicted time distance k we need to multiply the normalized output with the upper bound L and vice versa for converting a distance k to the internal representation k ≈ L · Ψθ(zt, zt+k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3 LOSS FUNCTION AND ITS IMPLICATIONS The time prediction loss objective Ltime(θ) minimizes the mean squared error between the predicted time distance and the true distance k Ltime(θ) = Et � (Ψθ(zt, zt+k) − min(k/L, 1))2� , (4) where the correct distance k is known from the sampled trajectories after the environment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The neural network computation includes the normalization g(x) to prevent exploding gradients and introduces the upper time window bound L that limits the network‘s expressivity beyond L timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This constraint simplifies the task, since correct forecasts of longer periods of time become unimportant (Makridakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The myopic view is helpful to reliably estimate novelty, because the model does not need to handle a precise time measurement between states that are far away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' By considering only local distances, the predictions get easier and they are more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' So when the network encounters a state-pair with a large distance k > L, we do not care about the exact true distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' But instead, the model should indicate that the states are far away from each other by outputting the upper bound L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='4 TIME DISTANCE IS POLICY DEPENDENT AND DIRECTED Each singular timestep represents a state transition on the underlying MDP where the time-related reachability is dependent on the current policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This means that two distinct policies could reach a certain state with a different number of timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Also, since time distances are directed we have to input our encodings (zA, zB) in the right order into Ψθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For intuition, Figure 2 (right) shows a visual example where a state A is on top of a small cliff and point B is at the bottom of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To reach B from A an agent could use the shortcut and jump down the cliff, but it might not be possible to jump back up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Therefore the agent needs to find a different path that might increase the elapsed time to reach A from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For that reason we assume that the distances of our time prediction function can yield different results for Ψθ(zA, zB) and Ψθ(zB, zA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 4 TIME-MYOPIC GO-EXPLORE To integrate our learned representation model from the previous section into the native Go-Explore method, we have to make some adjustments, since our state encodings are continuous (and no longer discrete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This makes it harder to decided similarity between two distinct states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In the following we explain how this is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='1 ARCHIVE INSERTION CRITERION The time prediction capability of Ψθ allows us to propose a new archive criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Our criterion is an insertion-only method which adds a state to the archive when it is novel enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this way the detachment problem is entirely solved by not abandoning archive states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We initialize the archive by inserting the encoding zc1 of the environment‘s start state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This entry will begin the exploration effort by searching for novelty around the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In order to do that the agent samples trajectories from the restored state c1 where we collect new states that we call archive candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A new candidate encoding zK is evaluated with every archive entry zC by the time prediction function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The predicted value Ψθ(zC, zK) is thresholded by some hyperparameter Td to determine the acceptance or rejection of the current candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' If the threshold is surpassed for every comparison, the candidate K is added to the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Note, that we choose the time distance direction Ψθ(zC, zK) since we want to move away from the archive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' When this happens, we conclude that the agent has reached a currently unknown part of the state space which has a sufficient novelty with respect to the archive-known subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this way, the candidate evaluation gains a global view on the exploration progress instead of considering a local trajectory-based perspective (Badia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Savinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A short example is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 4 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Table 1: Archive insertion criterion where the model‘s expressivity is in the time window [0, L = 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A new candidate zK is compared to all existing cells in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' If one cell is too close, the candidate is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Comparison Time estimation Evaluation Cell Candidate Ψθ(zc, zK) 20 ∗ Td > 13 Insert?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' zc1 zK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3934 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='86 no zc2 zK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='7568 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='13 yes zc3 zK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='6321 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='64 no zc4 zK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='9334 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='66 yes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' zK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 VISIT COUNTER To ensure successful progression in the exploration, it is common practice to track the number of cell visits Cvisits for every cell C in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' With respect to the return selection, the visit counter of a cell increases by one when it is selected as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Furthermore, the visit counter increments when the time distance to an archive candidate is lower than a visit threshold Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This evaluation runs simultaneously to the application of our insertion criterion where we compute all time distances to the candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3 CELL SELECTION CRITERION Native Go-Explore made the design choice to replace cell entries with small scores to states with larger scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This significantly boosts the exploration by abandoning states which might have been sufficiently explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Time-myopic Go-Explore does not have this ability, because the state representations are continuous and the encodings of two states with weak spatio-temporal semantics are far away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Also, our method should not overwrite archive entries, because they could be inserted again with clean visits statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To bias exploration towards cells with larger scores, we calculate and combine the native selection weight Wvisits with our score weight Wscore that contains the reached cell score Cscore (sum of undiscounted cumulative reward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Both quantities (Cvisits, Cscore) are stored in the archive for every cell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' W = 1 √Cvisits + 1 � �� � visit weight Wvisits max � Cscore maxC′ C ′ score + 1, α � � �� � score weight Wscore .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (5) The outer maximum in Wscore ensures that states with lower scores are explored as well (with chosen hyperparameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='075).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The inner maximum normalizes the experienced scores between zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 5 TECHNICAL DETAILS FOR TRAINING In order to improve the model quality we explain next three additional training routines that enhance the time prediction of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='1 SIMILARITY AND DISSIMILARITY Optimizing only on close-by pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' with time distance k ≤ L) will lead to wrong estimates for distant pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' k > L), which might be the majority of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Thus we have to ensure that our dataset includes also dissimilar pairs where the observations are at least L timesteps away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The distance of a dissimilar pair is trained on the value that corresponds to the maximal time estimation L, which does not represent the actual distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Instead it signals a sufficiently large temporal space between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this way, we try to find encodings such that the embedding region around a state only includes other encodings that are within the defined time window [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 5 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 AVOIDING TEMPORAL AMBIGUITY Suppose there are two observations ¯ot+k1 and ¯ot+k2 that happened at different points in time, but which are pixel-wise identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To define a single distance to some other observation ¯ot, we will use the minimum of k1 and k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To quickly identify pixel-wise identical observations we are using the MD5 hash function (Rivest, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3 PROXY CELLS AND LOCAL DATASETS Usually the time prediction model would be trained on trajectories that always start with an archive state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To extend the training data we additionally generate trajectories (only for training) that start from so-called proxy cell states that are temporally close to an archive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We collect these proxies by choosing randomly states in the time distance interval [Tp-low, Tp-high] to their respective archive cell and replace them periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To further increase variability of the dataset, we add small local datasets for each cell that add some data points within its proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Every dataset holds a few hundred pairs where we store the time distance between the cell state and a temporally close state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This is necessary because otherwise our network would forget about certain cells and their neighboring states since they are not visited anymore due to the selection weighting W = Wvisits ∗ Wscore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 6 EXPERIMENTS The experiment section covers the following topics: in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='1 we study the encoder properties and assess how well the time can be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 we compare our approach, the original Go-Explore and other related methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3 we analyze the archive for the native and our proposed time-myopic Go-Explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='1 WHAT DOES OUR MODEL LEARN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' To demonstrate the properties of our model, we re-create an experiment from Ermolov and Sebe (2020) which is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A representation-learning network is trained on 400k observations from the Atari environment Montezuma’s Revenge where the training data is gathered by a random actor at the environment‘s start state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' After optimization, the network is asked to encode the observation sequence from the trajectory shown in Figure 3 (top-left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Most of the encodings are extrapolated, because a random policy is not able to reliably execute the action sequence that generated the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Therefore an extrapolation starts roughly between the checkpoint 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this experiment our model uses 9k observations which are sufficient to show a good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The 32- dimensional encodings are projected with the t-SNE method into a two-dimensional space where it uncovers the interesting relationship between the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Temporally close states are grouped together and are strung on a thread while the sequence unfolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The distance between two imminent states does not collapse and it provides useful semantics for time measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Remember that the chosen trajectory has no greater meaning for our model and it is perceived as arbitrary like any other possibly selected sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Subsequently, the rich structure is also present when performing PCA on the encodings (see Appendix Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The time evaluation demonstrates good results where the network has training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' However, the extrapolation capability on time distances is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' But surprisingly we can see that the encoder even places unseen states close to their temporal neighbors resulting in local semantic integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' At some point the extrapolation of time distances becomes unreliable which will improve with more novel data for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Once the network was trained on more data during a complete run, the prediction quality gets a lot better (see Appendix Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 COMPARISON ON HARD EXPLORATION ENVIRONMENTS We evaluate our method on the hard exploration environments Montezuma’s Revenge, Gravitar, Frostbite (Atari) and compare it with (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=') related methods (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=') and native Go-Explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We prepare our experiments by adjusting the natively used random action-repeating actor (Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Time-myopic Go-Explore decreases the actor’s action repetition mean µ from 10 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This is necessary since the native method does not care about learning a state representation while our learned model depends on a robust data collection for the time predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Native Go-Explore has 6 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Figure 3: Visualization of the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The top-left image is an observation sequence from the Atari environment Montezuma’s Revenge, which is generated by a human demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The top-right image is the t-SNE (van der Maaten and Hinton, 2008) projection of the observa- tion encodings generated by the siamese CNN Φθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The bottom image shows time estimates of our prediction network Ψθ for L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This model has the ability to calculate the time dis- tance for a pair of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' we can select several encodings z0, z10, z20, z40, z50, z60, z70 and let the time prediction network calculate their distance to their successors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Note that for the blue graph, we plot Ψθ(z0, z0), Ψθ(z0, z1), Ψθ(z0, z2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', Ψθ(z0, z72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For the orange one we plot Ψθ(z10, z10), Ψθ(z10, z11), Ψθ(z10, z12), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', Ψθ(z10, z72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Table 2: Mean cumulative reward for Atari (rows 2 to 6 are copied from Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' (2018), all others from the cited papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This table shows the performance of different exploration-based and baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Our results are computed as the mean over 20 runs where each run has seen 5M frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Method Frames Montezuma Gravitar Frostbite R2D2 (Kapturowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2019) 10000M 2061 15680 315456 EX2 (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2017) 50M 0 550 3387 AE-SimHash (Strehl and Littman, 2008) 50M 75 482 5214 ICM (Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2017) 50M 161 424 4465 RND (Burda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018) 50M 377 546 2227 EMI (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018) 50M 387 558 7002 LWM (Ermolov and Sebe, 2020) 50M 2276 1376 8409 PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2017) 40M 42 737 314 Ours (Time-myopic Go-Explore) 5M 2090 3161 4476 Ours (Time-myopic Go-Explore) 1M 695 2533 3543 Go-Explore (native) 1M 2303 2130 11721 the advantage that it can act more greedily, because the representation heuristic is always reliable and therefore can be exploited by acting more risky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Also, native Go-Explore uses the originally recommended down-scaling hyperparameters (h = 8, w = 11, dp = 8) for Montezuma’s Revenge and the dynamic down-scaling (recompute every 500k frames) only for the other environments (Montezuma’s Revenge with dynamic down-scaling performs worse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' For native Go-Explore we are using the official implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Our time-myopic Go-Explore variant runs on one A100 GPU and needs roughly 8-10 hours for 5M frames depending on the archive size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 7 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Table 3: Comparison of the archive size for Montezuma’s Revenge for scores 100, 400, 500, 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Archive size per score Method 100 400 500 2500 Native (h = 8, w = 11, dp = 8) 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='8 1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 2838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='9 Time-myopic (L = 20, Td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='55) 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='5 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='5 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='0 566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='0 Time-myopic (L = 25, Td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='65) 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='8 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='7 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3 (a) Distance to closest cell neighbor (b) Mean distance to next three cell neighbors Figure 4: Time distances between archive cells on Montezuma’s Revenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We generate two archives where the first one (orange) was constructed using Go-Explore down-scaling method (h = 8, w = 11, dp = 8) and the second one (green) by our learned model predictions (L = 20, Td = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' When both archives reach the score of 2500 we use a second optimized time prediction network (L = 20) and compute all time distances between the cell observations for both archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The result shows the practicality of time-myopic Go-Explore since it is better than most of the methods for Montezuma’s Revenge and Gravitar while it only has seen 2% of their frames (1M vs 50M frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' It also provides a good performance on Frostbite surpassing PPO, RND, EX2 and ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Moreover, time-myopic Go-Explore is able to keep up with the performance of native Go-Explore (using the domain-aligned representation heuristic) and even exceeds it within 1M frames for the environment Gravitar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' As far as we know the results of Gravitar for 1M frames are state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='3 ABLATION STUDY Next we provide a closer look at the archive properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' First, we compare the difference in the archive size between native Go-Explore and our approach (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The table shows the mean number of cells (20 runs) in the archive when the agent reaches a certain game score [100, 400, 500, 2500] in Montezuma’s Revenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The native archive size explodes after reaching a score of 400 while our approach shows a more stable progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The data also validates smaller archive sizes as a result of increasing the hyperparameters (L, Td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In the Appendix, a figure shows more precisely when and how the prediction model decides to add archive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We observe in our experiments a similar pattern for the environment Gravitar which uses the dynamic down-scaling heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The native archives agglomerate an enormous amount of archive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' After 1M frames for 20 runs, the mean size results in 7376 cells while time-myopic Go-Explore holds 381 entries and is achieving a higher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Secondly, both methods are evaluated on the similarity within the created archives (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The time distances between all cell observations will provide an interesting similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The model predictions in Figure 4 confirm that the native archive yields a lot more similarity than our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This leads to inefficient exploration, because based on the cell selection criterion every cell needs to be sufficiently visited, no matter how few environment transitions are between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Our archives cover the state space with less cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 8 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='4 LIMITATIONS Due to the queries to the archive, our algorithm runtime increases currently non-linearly with the number of entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This makes long runs or runs where the archive grows fast computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This happens, because our approach requires to compute time distances to every archive entry in order to evaluate new cell candidates or to update the visit statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' So, in the future, we will make these queries more efficient to make the method applicable to even larger environments and runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 7 RELATED WORK Hard exploration problems like the Atari environment Montezuma’s Revenge are known for their large state spaces and sparse reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A lot state-of-the-art reinforcement learning algorithms (Schrittwieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Kapturowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2017) have difficulties in achieving a good performance on these tasks, so several ideas have proposed towards a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The literature contains several approaches related to our work that deal with unsupervised- and representation learning combined with intrinsic motivation and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Playing hard exploration games by watching YouTube (Aytar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In this work the authors introduce a neural network architecture in order to learn a categorical time classification between two distinct observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Their network is used to generate an intrinsic reward signal to facilitate imitation learning of human demonstrations while we are using it to globally model the exploration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Solving sparse reward environments using Go-Explore with learned cell representation (Bjørsvik, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This approach also extends the Go-Explore (Ecoffet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2021a) method by replacing the representation heuristic with a learned state representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' They employ a Variational Autoencoder (VAE) (Kingma and Welling, 2013) to encode every seen state into a latent vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Later on, the encodings are used for a k-means clustering procedure where a cluster center stands for an entry in the archive memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Latent world models for intrinsically motivated exploration (Ermolov and Sebe, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This paper optimizes a siamese network that clusters temporal imminent states in the embedding space by minimizing the mean squared error between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In addition, there is the need for an extra structure constraint in order to prevent a collapse of the representation to a constant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The resulting encodings are used for the generation of an intrinsic reward signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Episodic curiosity through reachability (Savinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The paper introduces a model architecture with a logistic regression capability to differentiate between novel and familiar state pairs where novelty is defined by a minimum time distance of k steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The complete model (including a siamese encoder and comparison network) is only able to decide whether an encountered pair is novel or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' without the ability to evaluate it on a continuous basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This estimation is utilized to generate a positive intrinsic reward signal when a policy encounters new states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Never give up: learning directed exploration strategies (Badia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The proposed episodic novelty module starts empty and fills itself with state encodings when the agent interacts with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Every state gets evaluated by a k-nearest neighbor criterion with the similarity measure of the Dirac delta kernel to compute the rewards with respect to the similarity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The goal is to insert as many novel states as possible which then facilitate exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 8 CONCLUSION To add flexibility to and possibly improve Go-Explore we studied how its representation heuristic can be replaced by a time-predicting neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Experiments show that the new state representation is able to track the global exploration effort and moreover recognizes ongoing progress for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' In comparison to native Go-Explore, our method can reduce the archive size and covers the state space with fewer cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Applied to hard exploration environments, such as Montezuma’s Revenge we observe good performance compared to previous exploration-based methods while using much fewer observations, even creating better results than Go-Explore on the game Gravitar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Overall, however our learned representation is not able to compete with native Go-Explore in terms of sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 9 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop REFERENCES I heart crafty things - cliff.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='com/ science/article/pii/S0022000008000767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Learning Theory 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Sutton and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Reinforcement Learning: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The MIT Press, second edition, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' URL http://incompleteideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='net/book/the-book-2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' van der Maaten and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Visualizing data using t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Journal of Machine Learning Research, 9:2579–2605, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='org/papers/v9/ vandermaaten08a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Zhang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Vinyals, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Munos, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A study on overfitting in deep reinforcement learning, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='org/abs/1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='06893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' A APPENDIX - SECTION 1 Table 4: Hyperparameters for the experiments reported in the hard exploration table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Hyperparameter Symbol Value (MontezumaRevenge, Gravitar, Frostbite) Learning rate lr 1e−4 Batch size bs 64 Time window L (20, 25, 25) Distance threshold Td (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='75) Visit threshold Tv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='65 Proxy cell interval [Tp-low,Tp-high] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='75] Exploration steps t 40 action repetition mean µ 4 (a) dim 1-2 (b) dim 2-3 (c) dim 1-3 Figure 5: Principal Component Analysis for the trajectory visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 11 Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Figure 6: Sophisticated model and its capabilities on a longer trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The shown model is trained on data that surpasses the shown trajectory (top-left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Again, we can see the good encoding property and an improved time prediction skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The predicted times around the timesteps 90-100 or in the image at the Box 11 are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' At that point the agent dies and the environment generates repeating frames (two-image sprites) for around 10 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This prediction behavior happens, because we remove temporal ambiguity between state-pairs and try to calculate the shortest distance for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' The event can also be seen in the t-SNE visualization, when the light-green points start to form a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Figure 7: Creation of archive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We manually looked through the cell observations of an archive and searched for similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' This figure shows all cell observations where the agent stands at the same position and already collected the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' We can see that the time prediction network does not allow duplicates in the archive and keeps a reasonable distance between the observations where the white skull is changing positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' Moreover the archive holds no observation where the agent just slightly moved in these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' 12 目广具官Accepted at the NeurIPS 2022 Deep Reinforcement Learning Workshop Algorithm 1 Go-Explore with a learned state representation Initialize archive, dataset, agent, network for iteration = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' do Let the agent act t timesteps in the environment starting from the selected and proxy cells Collect data and optimize Ltime w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' θ Recompute all archive cell representations: zC = Φθ(¯oC) for each trajectory = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', N do Transfer all observations ¯o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', ¯ot into the latent representation z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=', zt Compute all necessary time distances Ψθ(zc, z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=',t) Apply archive insertion criterion to a candidate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' threshold Td Increase cell visits w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' threshold Tv Collect some proxy cells w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE5T4oBgHgl3EQfgg_h/content/2301.05635v1.pdf'} +page_content=' threshold [Tp-low,Tp-high] if candidate is accepted then Add candidate to archive end if end for end for 13' metadata={'source': 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b/hNE5T4oBgHgl3EQfiA_6/content/tmp_files/2301.05646v1.pdf.txt @@ -0,0 +1,1465 @@ +DATA-ASSISTED CONTROL - A FRAMEWORK DEVELOPMENT +BY EXPLOITING NASA GTM PLATFORM +A PREPRINT +Mostafa Eslami +Department of Aerospace Engineering +Sharif University of Technology +Azadi Avenue, Tehran +eslami.mostafa@ae.sharif.edu +Afshin Banazadeh +Department of Aerospace Engineering +Sharif University of Technology +Azadi Avenue, Tehran +banazadeh@sharif.edu +January 16, 2023 +ABSTRACT +Today’s focus on expanding the capabilities of control systems, resulting from the abundance of +data and computational resources, requires data-based alternatives over model-based ones. These +alternatives may become the sole tool for analysis and synthesis. Nevertheless, mathematical models +are available to some extent, especially for air and space vehicles. Hypothetically, data assistance +would be the approach to meet the requirements in collaboration with the model. In this paper, a +framework of Data-Assisted Control (DAC) for aerospace vehicles is proposed. NASA Generic +Transport Model (GTM) is the platform for the study and the data supports the model-based controller +in extending performance over a damage event. The framework requires real-time decisions to +override the control law with the information obtained from the data, while the model-based controller +does not show regular performance. The closed-loop system is shown to be stable in the transition +phase between the data and the model. The fixed dynamic parameters are estimated using the Dual +Unscented Kalman Filter (DUKF) and the evolution of the generalized force moments is estimated +using the Koopman estimator. Simulations have shown that the purely model-based robust control +leads to degradation of the closed-loop performance in case of damage, suggesting the need for data +assistance. +Keywords Data-Assisted Control (DAC) · NASA Generic Transport Model (GTM) · Koopman Operator · Dual +Estimate · Unscented Kalman Filter (UKF) · Nonlinear Robust Control · Decision Making +1 +INTRODUCTION +The diversity and volume of the available data, the significant increase in real-time calculation capacity, quality of +living in the cyber-physical systems era, human ambitions in interplanetary probing and exploring, and real-time +decision-makings in autonomy all have promoted utilizing of the data in the development of science and engineering +more and more, the fourth-paradigm [1]. Before the evolution of model-based analysis and synthesis, data was the first +aid for scientists and engineers. Using data in control is not new. One of the first data-driven control design methods in +1942 is Ziegler-Nichols signal-based tuning for PID controllers [2]. With progress in physics and our understanding +of physical systems, mathematical modeling became the only aid in the development of theories and particularly +in control engineering. In control engineering, learning from data is a new paradigm [3]. All theories and control +methods in which the controllers are derived from the data without explicit use or knowledge of the mathematical model, +named Data-Driven Control (DDC). A legitimate DDC is considered to have stability, convergence, and robustness +with rigorous mathematical and specified assumptions. The Willems et al.’s Fundamental Lemma (WFL) [4], Data +Informatively [5, 6], Behavioral Approach [7], Interconnection Paradigm [8, 9], Dissipativity [10, 11, 12], and data- +driven Koopman Operator [13, 14], all are provided different frameworks for developing DDC. In the same fashion, +Model-Based Control (MBC) explicitly uses the mathematical model. Data-driven control systems may have better +performance compared to the MBC whenever the following conditions are the case: a) mathematical model is not +arXiv:2301.05646v1 [eess.SY] 13 Jan 2023 + +arXiv Template +A PREPRINT +available, b) uncertainties and a wide range of structural/nonstructural combinations raise the lack of a comprehensive +and integrated model, c) modeling is very hard or impossible such that model-based control won’t work well, and d) +mathematical model is rather complicated to be used for control system design. +Considering as a fact, always for highly uncertain dynamics, the model-based paradigm has limits in particular for +unpredicted operating conditions and situations. These unmodeled dynamics yields unreliable model-based control +even with adaptation [15, 16, 17]. Otherwise, any increase in the order and complexity of a model may increase the +effort and tries for control system design [18]. The Rohrs’ counterexample is an alarm for having too much reliance on +adaptation due to unpredictable nonlinear behaviors due to assumptions in model-based controls [19, 20]. Moreover, +any dynamical system may show some degree of uncertainties with the increase in operating condition ranges that are +not possible to be modeled or identified with high fidelity. +Since data-based theories don’t use implicitly any dynamical models and only rely on in-out data of the system, may +resolve the requirements for enhanced requirements. As a fact, the model-based paradigm will work optimally for +a certain region with some degree of known uncertain boundaries with very vigorous theories and proven results in +abundant practices. However, today’s progressive increase in computational capacity has introduced new visions for the +evolution of control systems with the assistance of data to extend model-based capabilities. For example the advantage +of the data for the enhancement of the model-based control system performance in an industrial engine is evident [21]. +Looking at breakthroughs in control engineering, the direct adaptive control [22, 23] and neural-network-based control +systems [24, 25] are in the same paradigm of the data-driven control systems. After overwhelming use of neural +networks in control systems design, the data-driven controllers were introduced and are progressively developing +[18, 26, 3]. In recent decades, using DDC methods adapted from a similar MBC had great intention among control +system scholars. For example PID like DDC [27, 28], model-reference control and output tracking [29, 30], predictive +controls [31], robust controls [32], Fault-Tolerant Controls (FTC) [33, 34], reinforcement learning [35], optimal control +[36, 37, 38, 39] are developed under data-based paradigm. +In aerial applications, the dynamical model with high fidelity is developed [40]. With a correct understanding of these +dynamics, the exact and uncertain parts can be fully distinguished. The variety of operating points in these systems +may raise a high degree of uncertainty because all analysis and synthesis methods use simplification rules to express +them in a closed form. For example, atmospheric disturbances and turbulence [41], flying in an unknown stochastic and +heterogeneous atmosphere or complex environment [42], flights reconfiguration [43, 44, 45], foreign object damage to +the flight’s body and control surfaces [46, 47], high-frequency dynamics in control surfaces [47], increase in bandwidth +[48], robustness against faults and uncertainties that are unbounded or impossible to recognize the bound [49], envelope +protection [50], control in case of low speed in communication and low bandwidth of communication which needs +real-time decision-makings [51], and eventually the survivability [52], all may not be possible to be defined with general +deterministic terms in accordance with model-based dynamic formulates. +Moreover, the flights are used to have high reliability with the probability of catastrophic failure less than 10−9 per hour +according to APR4761. For decades, the designers have gained this reliability with hardware redundancy in controllers, +and control surfaces, using multi actuators, sensors, and flight computers [53, 54]. Yet, in 2017, after a series of studies +in NASA Langley Research Center [55], the Loss-of-Control (LOC) [56], was recognized as the biggest threat to +aviation accidents, especially for big commercial fixed wings airplanes. The LOC had the most portion of the accidents +across all classes of flights, missions, and flight phases [57, 58]. One of the significant external reasons for LOC is the +damage to the wing and fuselage or control surfaces. Accidents reported flight AAF587 [59], A300-B4 [60], AAF232 +[61], and BGF1907 [62] are example of these damages. After studies for analysis and improvement of control systems +in face of LOC in [63, 64, 65, 66, 67], the adaptive controllers show improvement in flight quality in case of failure +and uncertainties [68]. Although the adaptive controls have a long and rich history, due lack of simple verification and +validation methods, they are not used widely in commercial aerospace applications [62, 69]. +In this paper, a framework is developed in which data extends the flight operating regime of pure MBC in case of the +aforementioned uncertainties. The framework utilizes data in the linear evolution of observations thanks to Koopman +linearization [70]. In recent years, the Koopman operator and linearization successfully have been applied in many aerial +and space applications such as low-thrust trajectory optimization [71], attitude control of spacecraft [72], studying the +motion of a satellite close to a libration point [73, 74], approximating analytical solution for Zontal Harmonics problem +[75], and decision making [76]. Despite its increasing popularity, the application is very limited due to demanding +accuracy in real-time [73]. +True recognition of the part that data can assist, needs the development of a framework from a control engineering point +of view. This study is carried out under a specific model-based nonlinear framework with the remedy of a Lyapunov- +based nonlinear robust control design, a nonlinear dual estimation with Unscented Kalman Filter (UKF), a data-based +Koopman operator over estimated force-moment (pseudo-observations), and a real-time decision-making algorithm +2 + +arXiv Template +A PREPRINT +with a behavioral approach. The Koopman operator provides powerful data-based linearization in observable space +[77], and the nonlinear control benefits from the structure of the nonlinear model capable of stability and performance +analysis with a decision factor for the shift between data and model. This compels to use of a nonlinear estimator for +parameters and the evolution of forces and moments. We call the exact dynamics with parametric uncertainty, the fixed +dynamics. Eventually, the real-time decision-making follows a behavioral approach to optimize an online cost function +to promote the data or model credibility, whereas the system behaves like the initial model or undergoes uncertainty or +failure. This framework is named Data-Assisted Control (DAC). +DAC has full advantage of the MBC with stability guaranty, yet the data would extend its performance in any condition +that flights don’t demonstrate the typical behavior. When data interferes, the stability is ascertained with specific +assumptions. Of course, when dynamics are certain the data would not interfere. The generic 5.5% scaled transport class +vehicle known as the Generic Transport Model (GTM) by NASA is the platform for this study [78, 79, 80]. The GTM +includes all subsystems required for experimental flight control algorithm implementation and evaluation [81, 82, 83]. +The model encompasses a series of defined failures and can be used as a benchmark for FTC systems [62, 46, 47]. +The DAC framework is introduced in details in section 2, and required five steps for its development for GTM is +provided from sub-sections 2.1 to 2.5. In the end, the framework is applied to the GTM with a series of simulations in +section 3 to demonstrate the extended performance of DAC compared to the pure MBC. +NOMENCLATURE +ω +[rad/sec] angular velocity vector with p-roll rate, q-pitch rate and r-yaw rate arguments in body +coordinate +ρ +[ft] center of mass displacement +V +[ft/sec] linear velocity vector with u, v and w arguments in body coordinate +v +generalized body coordinate velocity vector - [V T , ωT ]T +m +[lbs] aircraft mass +IM +[slug.ft2] aircraft moment of inertia matrix with Iab entries for axis a and b +I +identity matrix with proper dimension +g +[ft/sec2] gravity constant +η1 +position vector with X, Y and Z arguments in earth coordinate +η2 +orientation vector (Euler angles) with Φ, Θ and Ψ arguments in earth coordinate +η +generalized earth-fixed coordinate position and orientation - [ηT +1 , ηT +2 ]T +p +parameters set +W +[lbf] gravitational force +F +[lbf] external force applied to aircraft +T, TR, TL +[lbf] total thrusts generated by engines, right engine thrust, left engine thrust +M +[lbf.ft] external moment applied to aircraft +τ +generalized force and moment vector +L +generalized momentum vector consisting of linear and angular momentums +M +mass-inertia matrix +C +Coriolis and centrifugal matrix +G +gravitational force and moments vector +B +control derivatives in force and moments +D +damping derivatives in force and moments +C +dimensionless aerodynamics coefficient +E +extra order dynamics +ex +indication of error associated with the variable x +ωx +state transition noise vector with ων, ωτa and ωp arguments +ωy +measurement function noise vector +3 + +arXiv Template +A PREPRINT +Σ. +summation operator - means total forces or moments +S(.) +skew-symmetric operator acts on the vector (.) and generates associated skew-symmetric matrix +diag(.) +a function generating diagonal matrix of vector (.) +∥.∥2 +norm-2 of vector (.) or Frobenius norm of matrix (.) +.T +transpose operator +ˆ. +estimated parameter, vector or matrix +˜. +difference between the actual and the desired variable +Subscripts +g +Indication of the center of gravity +p +Indication of the center of pressure +T +Belong to the thrusters +A +Belong to the aerodynamics +F +Belong to the forces +M +Belong to the moments +δ +Belong to the control surfaces/actuators +er, el, e +right elevator, left elevator, and coordinated left-right elevator together +ru +rudder +ar, al, a +right aileron, left aileron, and coordinated left-right aileron together +r +reference/residual value +a +under state variable means augmentation +di +indication of GTM damage case i +d +indication of desired value +2 +DAC FRAMEWORK +The general framework of the DAC is depicted in Figure 1. In this scheme, considering the fixed structure for M and C +matrices, parameter estimation is used for the estimation of mass, the moment of inertia, and their products. In this case, +any effect on the force and moment derivatives is reflected in applied aerodynamics and thrust forces and moments, τ. +One can call this parameter pseudo-observation. Because first using the measurement of states a nonlinear estimate is +exploited to use the fixed dynamics and then the estimated force-moment vector is used as observation. Due to the +structure of the developed model, the fixed dynamics are strictly equivalent to the observations, but τ is not measured in +practice. Accordingly, the dual estimation will help to have correct estimates of the generalized forces and moments +evolution by time without the need for a direct measurement. Simply, the fixed dynamics with correct parametric and +state estimates must provide correct trends of external forces and moments. Eventually, the linear evolution of the +observation of the generalized forces and moments will be identified and fed to the control law to adapt the closed-loop +dynamics. According to the force expansion rule by [84, 85], the generalized forces and moments are characterized +by linear dependency on aerodynamic derivatives up to some unknown vanishing orders. Therefore, the regressor +will have significant void entries which promote using data-based methods for identification over observations. The +real-time decision-making algorithm always tracks the performance of the closed-loop system and with slight changes +in the decision factor tests interference of the identified regressions by data. If the external behavior of the aircraft +demonstrates improvement in the performance, the decision factor will promote the data assistance more and more, +otherwise, it will return to the MBC. The DAC framework can be decomposed into modular steps as follows: +(A) rewriting the nonlinear model suite for DAC framework and performing dynamics decoupling for observations +and fixed dynamics, +(B) nonlinear control design for the full envelope, using fixed dynamics, +(C) parameter estimations including mass, the moment of inertia, the center of gravity, and pseudo observation +estimation considering the fixed-dynamics exploiting a dual estimation approach, +(D) data assistance for identification of the aerodynamic and control derivatives over pseudo observations, +(E) behavioral decision-making for optimizing decision factor of altering control law according to the data-assisted +model in a stable way. +This framework is elaborated through a series of simulations and analysis in a pragmatic way to provide seemly data +assistance with stability and performance guarantee. +4 + +arXiv Template +A PREPRINT +GTM +Nonlinear +Controller + +  + +  +Real-time Decision Making + +Nonlinear Dual +Estimation +Data-based Forces +and Moments +Evolution +Identification + + + +Figure 1: Data Assisted Control (DAC) framework for GTM +2.1 +GTM Nonlinear Model Suitable for DAC +The set of flight dynamics equations for GTM is developed considering the change in the body’s center of mass. Such +equations can be used when the body loses a portion of its mass and it is desired to track the motion of the body’s +previous center of mass/reference frame [79]. The velocity components are subscripted with any arbitrary point in +body-frame to distinguish them from the usual velocity components considered at the center of mass. Of course these +components are the velocity of the center of mass when this point is the center of mass, i,e. ρ = 0. The force of gravity +and the equation of motion is written as follows [79, 86]„ +W = +� +−mg sin Θ +mg cos Θ sin Φ +mg cos Θ cos Φ +� +(1) +ΣF + W = m +� +˙V + S(ω)V +� +− mS(ρ) ˙ω + mS(ω)(S(ω)ρ) +ΣM + S(ρ)W = IM ˙ω + S(ω)(IMω) + mS(ρ) ˙V + +mS(ω)(S(ρ)V ) + mS(V )(S(ω)ρ) +(2) +The derived equation of motion can be concisely described in a matrix form as below, +M ˙v + C(v)v + G(η) = τ +(3) +5 + +arXiv Template +A PREPRINT +where M is the mass-inertia matrix, C is Coriolis and centrifugal matrix, and G is gravitational force and moment +acting on the flight as, +M = +� +mI +−mS(ρ) +mS(ρ) +IM +� +, +C(v) = +� +mS(ω) +−mS(S(ω)ρ) +−mS(S(ω)ρ) +−S(IMω) + mS(S(V )ρ) +� +& +G(η) = +� +−W +−S(ρ)W +� +. +(4) +Remark 1. Assuming small change rate in mass and moment of inertia, M and C are rearranged such that +˙ +M − 2C(v) +is skew-symmetric. +The following decomposition can be made over GTM forces and moments produced by engines (FT and MT ), the +aerodynamics of wings and base airframe (FA and MA), and control surfaces (Fδ and Mδ), +ΣF = FT + FA + Fδ, and +ΣM = MT + MA + Mδ + S(l − ρ)(FA + Fδ) +(5) +Where l = Cp − Cg is difference between center of pressure and gravity (for GTM l = [−0.0276, 0.0118, 0.036]T [ft]). +The forces and moments of the aerodynamic base frame and control surfaces are traditionally defined by dimensionless +aerodynamic coefficients, CF , and CM. Each coefficient is decomposed into state variables with a set of multipliers. +The angle of attack α and side-slip angle β for aerodynamics of base frame and control surface angle purely define the +flight aerodynamic attitude with respect to surrounding air and direction. Hence, it is common to define the multipliers +for these two angles, not the state variables. The relation between aerodynamic coefficients and force and moments in +reference flight is as follows, +FA = CF ¯qrSr, +Fδ = CδF ¯qrSr, +MA = diag(b, ¯c, b) × CM ¯qrSr, +Mδ = diag(b, ¯c, b) × CδM ¯qrSr. +(6) +Where, +CA = +� +CF +CM +� += C0 + Cαα + Cββ +and +CδA = +� +CδF +CδM +� += [ Cδruδru +Cδaδa +Cδeδe ] . +(7) +Assumption 1. The parameters are extracted for reference flight of GTM model in 800 [ft] altitude and TAS of 75 +[knots] without the wind, ¯qrSr = 105.65 [lbf], b = 6.8488 [ft], and ¯c = 0.9153 [ft]. +Assumption 2. In GTM the aerodynamic coefficients and control derivatives are generated via a look-up table for +−5 ≤ α ≤ +85 DEG and −45 ≤ β ≤ +45 DEG. In the rest, we assume that flight is in steady symmetry condition +near α = 4 DEG and β = 0 DEG. +Assumption 3. It is assumed flaps, stabilizers, spoilers, brake, and landing gear don’t participate in flight. Their trim +value in simulated reference flight is zero. It is also assumed engines are symmetrical so they generate the same thrust +value and we have a single δT as a thrust control input. The upper and lower rudder work together, and single control +input δru is considered for the rudder. For aileron and elevator, the inner and outer boards and the left and right +surfaces are merged to accept only one control input as δa for aileron and δe for elevator. The prime motive behind this +control input configuration is the possibility of damage simulations in GTM standard damage models. In summary +vector of control input is as follows, +δ = +� +�� +δT +δru +δa +δe +� +�� +(8) +Considering the assumptions in the derived model, the simplified mathematical representation of the GTM in simulations +will have residual forces and moments in the dynamics. Let’s denote the residual force and moments in reference flight +by τr. The final form of the nonlinear model will be summarized as, +M ˙v + C(v)v + G(η) = τ + τr. +(9) +6 + +arXiv Template +A PREPRINT +2.2 +Nonlinear Robust Control Design - a Velocity Regulator +Equation (9) ensembles dynamics in a ready form for developing nonlinear controller. The objective of the control +system design in case of failure is presumed to be regulation of the velocities in body coordinate, as a pilot would act. +Accordingly, the design of a nonlinear velocity regulator has been followed next. Let’s define an energy Lyapunov +candidate function as follows, +V = 1 +2 ˜vT M˜v +(10) +Where ˜v = v − vd. Taking derivative from Lyapunov function and using Remark 1 yields, +˙V = ˜vT M˙˜v + 1 +2 ˜vT ˙ +M˜v = ˜vT (M ˙v − M ˙vd + C(v)˜v) += ˜vT (τ + τr − G(η) − C(v)v − M ˙vd + C(v)˜v) +(11) +Assume following control law has been selected, +τ = G(η) + C(v)vd + M ˙vd − Γ˜v − τr +(12) +Where Γ = diag(γ1, γ2, ..., γ6) and γi > 0. Then, +˙V = −˜vT Γ˜v ≤ −λ(Γ)∥˜v∥2 +⇒ +V ≤ 1 +2λ(M)∥˜v∥2 ⇒ ∥˜v∥2 ≥ +2V +λ(M) +(13) +Hence, +˙V ≤ −2 +� λ(Γ) +λ(M) +� +V ⇒ V(t) ≤ V(0) exp +−2 +� +� λ(Γ) +λ(M) +� +�t +(14) +Where V(0) ≥ 0. The role of λ(Γ) now is evident. By increasing its value, V(t) converges to zero exponentially faster. +This yields exponential convergence of generalized velocity error towards the origin, with Γ as the tuning parameter. +According to GTM, the generalized force and moment vector can be decomposed as follows, +τ = τ0 + B(δ)δ + D(v)v +(15) +Where τ0 is static forces and moments in the trim condition, B(δ) is a matrix with the variable sign of elements +according to the sign of command δ, and D(v) is damping term of the forces and moments linearly dependent to state +vector v. This yields an update in control law as follows, +τc = G(η) + C(v)vd + M ˙vd − Γ˜v − τr − τ0 − D(v)v +(16) +In which, τc = B(δ)δ. As a challenge in this control law, the subtracted terms from the control law are open-loop. +Yet, the exact evaluation of these values is almost impossible. Hence, additional terms are added to the control law to +overcome issues of possible uncertainty and make the control system robust. Considering the χ as bound of uncertainty, +following the control law will make the control system robust [87]. +τc = G(η) + C(v)vd + M ˙vd − Γ˜v − τr − τ0 − D(v)v +− χ tanh +� ˜v +ϵ +� +(17) +Eventually, having τc in hand, the calculation of the control command δ is the last part of the control system computation. +Using least square minimization, the optimal control command is calculated as follows, +δ∗ = (B(δ)T B(δ))−1B(δ)T τc +(18) +For not damaged aircraft, considering merging of left-right actuators, we have B(δ) ∈ R6×4 and δ ∈ R4×1. In case of +damage with losing actuators the dimension of the input matrix may decrease. +Remark 2. In case of additional nonlinear term E in τ as τ = τ0 + B(δ)δ + D(v)v + E, the term can be decomposed +into E = BEδ + H.O.T. which are varying by time. The control law is still valid considering new control derivative +matrix B′(δ) = B(δ) + BE. The high-order terms are subtracted from τc. +Remark 3. In order to ensure persistence of excitation in this regulator, signals consisting of sinusoids of varying +frequencies and randoms are added to the control law [88]. +7 + +arXiv Template +A PREPRINT +2.3 +Damaged Aircraft State and Parameter Estimation +In this study, the aircraft experiences damage case 1 of GTM. Damage case 1 involves a parametric change in mass, +moments of inertia and their products, the center of gravity, and force-moment control derivatives. The change in +parameters is defined as follows, +md1 = m − 0.13 +(19) +ρd1 = [0.0105, 0, 0.0023]T +Ixxd1 = Ixx − 0.00346 +Iyyd1 = Iyy − 0.06698 +Izzd1 = Izz − 0.06352 +Ixzd1 = Ixz − 0.01409 +Iyzd1 = Iyz + 0.00001 +Ixyd1 = Ixy + 0.00003 +In this step, the parameters of the fixed dynamics p = [m, Ixx, Iyy, Izz, Ixz, Iyz, Ixy, ρ]T , and generalized forces and +moments are estimated with the augmented mathematical model. Considering the DUKF approach [89], the state +estimation model can be written as, +˙ˆv = ˆ +M−1 � +ˆτ + τr − ˆCˆv − ˆG +� ++ ων +(20) +˙ˆτa = Aτaτa + ωτa +and measurement function can be written as y = v + ωy. The augmented force-moment, i.e. τa ∈ R18×1, represents +third order Gauss-Markov process with auxiliary variables ζ1 and ζ2 as follows [90], +τa = +� τ +ζ1 +ζ2 +� +& +� +� +˙τ(i) +˙ζ1(i) +˙ζ2(i) +� +� = +� 0 +1 +0 +0 +0 +1 +0 +0 +0 +� � τ(i) +ζ1(i) +ζ2(i) +� ++ +� ωτ(i) +ωζ1(i) +ωζ2(i) +� +for i = 1, . . . , 6 +(21) +Therefore, +Aτa = +� ∅6 +I6 +∅6 +∅6 +∅6 +I6 +∅6 +∅6 +∅6 +� +, +& +ωτa = +� ωτ +ωζ1 +ωζ2 +� +(22) +Where, ∅6 is zero matrix with dimension of R6×6, and I6 is identity matrix with dimension of R6×6. The considered +process for evolution of parameters in time is considered as follows, +˙ˆp = ωp +(23) +Due to nonlinearity in parameters which results in the whole process with non-additive noises, augmentation is +necessary. We define the covariance matrix of zero-mean Gaussian noises as ωxa ∼ N(0, Q) and ωy ∼ N(0, R) such +that Q, R ≻ 0 to overcome numerical instabilities. +Having the augmented nonlinear model in hand, the DUKF estimation algorithm is followed. In order to avoid +the complexity of DAC parameter tuning, the simplest form of sigma-points generation for unscented transform is +considered [91]. +Remark 4. The UKF algorithm is optimal when a) the model matches the real system perfectly, b) the entering noise is +white (uncorrelated), and c) the covariances of the noise are known exactly. +8 + +arXiv Template +A PREPRINT +The framework implies the model matching and uncorrelated noises; however, adaptation in covariances is essential. +Using innovation and residual of the filter following adaptation rule is employed to dynamically update noise covariance +matrices [92], +Qk = αQk−1 + (1 − α)KkdkdT +k KT +k , +innovation at step k: dk = yk − h(ˆx− +k ) +(24) +Rk = αRk−1 + (1 − α)(εkεT +k + HkP − +k HT +k ), +residual at step k: εk = yk − h(ˆx+ +k ) +(25) +Where in these equations α is a forget factor, and Hk is jacobian of measurement function, h(.), at step k. In addition +to the above conditions, the estimation model itself should be observable. In order to check observability let’s collect +the state transition and measurement functions as follows: +˙v = f(v, τ, p) +(26) +˙τa = Aτaτa + ωτa +˙p = ωp, +and +y = v + ωy +In this study, using Lie derivatives for deriving analytical implications of the observability was found to be inefficient +due to demanding high-order partial differentiation. Moreover, these derivatives will induce computational errors and +hence declines the credibility of the result from the computation point of view. As an alternative approach due to +working in a regulation scheme the piece-wise linearity of the dynamics is assumed, then the analytical linear system is +elaborated to check the observability in each step. Considering estimation model (26), the linear system observability +matrix couple (A, C) would be, +A = +� +�� +∂f +∂v +∂f +∂τa +∂f +∂p +∅ +Aτa +∅ +∅ +∅ +∅ +� +�� +and +C = [ I6 +∅ +∅ ] . +(27) +It is evident that if ∂f +∂p = 0 then model is not observable. Derivatives of f are derived as follows: +∂f +∂v = −M−1(p) ∂ +∂v (C(v)v), +∂f +∂τa += +� +M−1(p) +∅ +� +, +and +∂f +∂p = − ∂ +∂p +� +M−1(p) (C(p)v + G(p)) +� +. +(28) +The underlined terms are calculated with symbolic calculations, then the observability matrix singular values are +checked per iteration. The observability matrix is defined as, +O = +� +��� +C +CA +... +CAn−1 +� +��� +(29) +2.4 +Data Assistance +Koopman demonstrated that it is possible to represent a nonlinear dynamical system in terms of an infinite-dimensional +linear operator acting on a Hilbert space of measurement functions of the state of the system [70]. Koopman operator is +linear, and its spectral decomposition completely characterizes the behavior of a nonlinear system [93, 94]. Using the +Koopman operator nonlinear dynamics become completely linear in eigenfunction coordinates, called lifted space. +Provided estimates for generalized force-moment vector and parameters, in this step the control derivatives, damping +derivatives, and static force-moment terms are recognized in pseudo observations via measured data. The approach of +using Koopman operator in synthesis and design step of the controller is growing rapidly. In particular, the Koopman +estimator is used to provide the linear estimate, and the results are compared with Recursive Least Square (RLS) [22]. +The RLS is chosen as an analog due to being an optimal deterministic estimator for linear fitting in adaptive rules. The +torque and moments are regularly treated as linearly varying to actuator command, linear and angular velocities with a +static term. As a general comparison, the Koopman represents a batch data identification method and the latter updates +the estimates according to just previous observations. +9 + +arXiv Template +A PREPRINT +From the developed framework the i-th observations and measurements in the simplest from are decomposed as follows, +ˆτi = ˆτ0 + ˆB(δi)δi + ˆD(vi)vi +(30) +and stack of m previous sampled pseudo observations and measurements are collected as follows, +ˆT = [ˆτ1 +. . . +ˆτm] ∈ R6×m +& +Y = +�δ1 +. . . +δm +v1 +. . . +vm +1 +. . . +1 +� +∈ R11×m. +(31) +Accordingly, using Koopman the linear evolution of the observations with respect to the measured variables can be +obtained as follows, +ˆP = ˆT YT (YYT )−1 +(32) +Where, +ˆP = +� ˆB +ˆD +ˆτ0 +� +∈ R6×11. +(33) +Remark 5. In practice, the equation (31) can be described in a more complex form depending on some orders of +the measurement derivatives [85]. In this case, a more complicated data-driven algorithm can be used for linear +estimation such as SINDy [95]. Without loss of generality, in this paper, the simplest form is assumed. In order to +compare Koopman estimator performance versus RLS, in section 3 additional nonlinear terms are appended to the +model. The additional terms are put into extra order dynamics vector, E, and includes sin(10r)δr. +Remark 6. Equation (31) gives dynamics of the generalized momentum, τ = dL/dt. Therefore, the evolution of +observations describes the dynamical changes in momentum directly. This can be used to design a Koopman data-driven +control system to control momentum directly [77]. In this paper, data only assists in control system design, and this +approach is not followed. However, as a third option in the decision-making, the full data-driven control like this +approach would be the case for the next studies. +2.5 +Real-time Decision-Making and Decision Factor Optimization +A decision factor λ determines the active role of data assistance. From equation (9) and (15) the integrated open-loop +dynamics is as follows, +M ˙v + C(v)v + G(η) − τr = τ = τ0 + B(δ)δ + D(v)v +(34) +This equation after estimations and exploiting data becomes, +ˆ +M ˙v + ˆC(v)v + ˆG(η) − τr = ˆτ = ˆτ0 + ˆB(δ)δ + ˆD(v)v +(35) +Using decision factor λ, the coupled estimated and initial/uncertain dynamics will be as follows, +M(λ) ˙v + C(λ)v + G(λ) − τr = τ(λ) = τ0(λ) + B(λ)δ ++ D(λ)v +(36) +Where, +M(λ) = (1 − λ)M + λ ˆ +M +(37) +C(λ) = (1 − λ)C + λ ˆC +G(λ) = (1 − λ)G + λ ˆG +τ(λ) = (1 − λ)τ + λˆτ +τ0(λ) = (1 − λ)τ0 + λˆτ0 +B(λ) = (1 − λ)B + λ ˆB +D(λ) = (1 − λ)D + λ ˆD +The decision factor is a convex combination of the initial/uncertain dynamics and the estimated one. As λ → 0 control +apportioned to MBC, otherwise, λ → 1 suggests uncertainty has appeared and DAC would become operative. +10 + +arXiv Template +A PREPRINT +For decision-making, the historical behavior of the closed-loop system is turned into a performance cost function with +λ as optimization variable in real-time. Also, the decision on λ can be taken manually by pilot. Assuming time horizon +of mλ samples in past period tp ≤ t ≤ t0, following cost function is a candidate to select λ∗, +Jλ = 1 +2 ˜vT (λ, t0)Hλ˜v(λ, t0) + 1 +2 +� t0 +tp +˜vT (λ, t)Qλ˜v(λ, t)dt +(38) +subject to dynamics in (36) considering λ = λ∗− as the optimum λ in the previous window t0 − 2tp ≤ t ≤ tp. The Hλ +and Qλ are weightings for terminal and transient cost. +In this study the gradient based steepest decent is used for minimization of Jλ [96]. In steepest decent the next +optimization variable is updated by the following rule, +λ∗ +k+1 = λ∗ +k − γk +∇Jλ(λ∗ +k) +∥∇Jλ(λ∗ +k)∥ +(39) +where γk is step size gain, and k is the iteration of the optimization. Step size gain in companion with decision horizon +determines the rate of change in decision factor. Legitimate decision-making should have a stable transition period +from data to model and vice versa. Next, it is shown under certain conditions, the closed-loop system in DAC is stable. +Proof 1. Considering +˙ +M(λ) − 2C(λ) is still skew-symmetric, the developed control law in Section 2.2 is still valid and +can be written as, +τc(λ∗) = G(λ∗) + C(λ∗)vd + M(λ∗) ˙vd − Γ˜v − τr +− τ0(λ∗) − D(λ∗)v − χ tanh +� ˜v +ϵ +� +(40) +Therefore the stability analysis falls into checks on the skew-symmetric property of +˙ +M(λ) − 2C(λ). Accordingly using +(37), +˙ +M(λ) − 2C(λ) = dM(λ) +dλ +˙λ − 2C(λ) = +� +ˆ +M − M +� +˙λ − 2C(λ) +(41) +Using parameter estimation in step (C), the structure of ˆ +M and ˆC are not changed. Hence, the above equation is +skew-symmetric or almost skew-symmetric if and only if, +a) rate of change in decision factor is small and continues in time, or +b) moment of inertia and mass are correctly estimated. The product of inertia and center of gravity shift estimates +don’t affect the stability condition. +When λ is identically 0 or 1, condition (a) is granted. Either transitions λ → 0 or λ → 1 would occur when there is +an interference of data or return to the initial model. If conditions in Remark 4 are fulfilled the estimation error will +converge to zero in finite time. Therefore, in transition, the term ˆ +M − M has value, and nonzero ˙λ will introduce an +additional term in control law that may lead to instability. Accordingly, the decision factor should enter with a lag time +with respect to the estimation time horizon. In this way, the temporal difference in estimation and decision-making +actions would guarantee stability. The closed-loop performance is guaranteed if the robustness gain χ compensates the +additional feedforward errors in the estimation of D and τ0. +Remark 7. Since the Lyapunov function in (10) has closed level sets, in case of instability in tolerable finite time, i.e. +restoring the skew-symmetric property of +˙ +M(λ) − 2C(λ) in small finite time, the error trajectories will return to origin +after the restore. We used the qualitative term "small/tolerable finite time" because it depends on the divergence rate of +flight trajectories and velocities. +Eventually, for the calculation of control inputs, using the same relationship for computing optimum δ from τc in DAC +we have, +δ∗ = (B(λ∗)T B(λ∗))−1B(λ∗)T τc(λ∗) +(42) +The only difference would be the possibility of order reduction in effective control inputs due to control loss in case of +damage. Therefore, following additional condition is imposed to guaranty non-singularity in B(λ∗)T B(λ∗), +Remark 8. Assume i-th column of B(λ∗) be denoted by bi, and ϵδ is a small parameter selected according to actuator +gain. If ∥bi∥ ≤ ϵδ, the δi and bi will be redacted from computation of equation (42). The δi will keep the last value. +This case should happen only when λ → 1, if the estimation of ˆB converges properly. Otherwise, decision factor should +be decreased to give more credit to the model, and accordingly, the redacted order will return in computation. +11 + +arXiv Template +A PREPRINT +0 +10 +20 +30 +40 +50 +60 +70 +80 +time [sec] +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Figure 2: DAC simulation for GTM under damage and pilot decision +3 +SIMULATIONS +Through the above steps, the DAC framework has been applied to the GTM. In this section, its performance with the +following events in fault case scenario is going to be evaluated: +a) flight journey begins in trimmed condition, at t = 4 [sec], a small disturbance is applied on the flight control surfaces +to check velocity regulator performance, +b) at t = 10 [sec], the failure introduced in Step C happens abruptly, +c) at t = 30 [sec], an extra high frequency nonlinear term 5 sin(10r)δr are added exponentially with time constant of 2 +seconds to the yaw moment, +c) at t = 50 [sec], pilot decides to select pure MBC and then returns to DAC at t = 55 [sec]. +Figure 2 demonstrates simulation results of the given scenario. The effective error vanishing in DAC-enabled mode +is evident, whereas the pure MBC has meaningful errors after the introduction of failure. In this plot, λ is the actual +decision factor involved in control law, and the λsel is the selected decision factor by the pilot. The time lag between +actual and optimum/selected decision factor is due to imposing intentional temporal difference in case of an error in +estimations. The result implies selecting the pure MBC by the pilot will have the penalty of performance loss, however, +after returning to the optimum decision with complete interference of data, i.e. λ = 1, the error has vanished completely. +Figure 3 and 4 demonstrate the velocity regulator performance with details of the state errors and actuator positions +respectively. Refer to 3, the sinusoidal desired values and random noise are intentionally added to ensure persistency in +excitation and observability. +Figure 5 shows an almost error-free estimation of velocities and parameters with slight error in the force-moment +vector, before and after the failure. The singular values of the observability matrix are provided in Figure 6. Plot +implies the observability is ascertained, however singular values of the parameters are undersized and yields weak +condition of observability. It is worth mentioning when failure introduces the state and parameter variations improve +the observability. +12 + +arXiv Template +A PREPRINT +Figure 3: Velocity regulator performance with detail of state errors +0 +20 +40 +60 +80 +time [sec] +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +20 +40 +60 +80 +time [sec] +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0 +20 +40 +60 +80 +time [sec] +0 +10 +20 +30 +40 +50 +60 +70 +Figure 5: DUKF estimation errors +13 + +126.4 +0.12 +0.1 +126.2 +0.08 +126 +0.06 +0.04 +125.8 +0.02 +125.6 +0 +-0.029.3 +9.2 +9.1 +9 +8.9 +8.8 +8.7 +8.6u +U +-0.04 +Pn +Vd +125.2 +-0.06 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +time [sec] +time [sec] +(d) +(e) +0.035 +0.01 +0.03 +0 +0.025 +-0.01 +0.02 +-0.02 +0.015 +-0.03 +0.01 +-0.04 +0.005 +-0.05 +0 +-0.06 +-0.005 +-0.07 +p +b +-0.01 +-0.08 +Pd +Pb +-0.015 +-0.09 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +time [sec] +time [sec]8.5 +Wd +8.4 +0 +20 +40 +60 +80 +time [sec] +X10~3 +(f) +4 +2 +-2 +-4 +rd +-6 +0 +20 +40 +60 +80 +time [sec]arXiv Template +A PREPRINT +0 +20 +40 +60 +80 +time [sec] +0 +50 +100 +150 +200 +250 +0 +20 +40 +60 +80 +time [sec] +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +0 +20 +40 +60 +80 +time [sec] +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +0 +20 +40 +60 +80 +time [sec] +-2 +0 +2 +4 +6 +8 +10 +12 +14 +Figure 4: Control deflections +0 +10 +20 +30 +40 +50 +60 +70 +80 +time [sec] +10-30 +10-25 +10-20 +10-15 +10-10 +10-5 +100 +105 +1010 +1015 +Figure 6: Singular values of the observability matrix O +The performance of the Koopman estimator is evaluated and compared with RLS in Figure 7. Plots (a) to (c) demonstrate +a comparative difference in the estimation of errors for ˆB, ˆD and ˆτ0, after introducing a high-frequency nonlinear term +14 + +arXiv Template +A PREPRINT +in yaw moment. The amplitude of the appended term is insignificant as depicted in Figure 8; however, the effect on the +estimation result is noteworthy. Plot (d) of Figure 7 displays estimation error of the extra term by Koopman. +Figure 7: Comparison of the observation estimations via Koopman and RLS +0 +10 +20 +30 +40 +50 +60 +70 +80 +time [sec] +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +Figure 8: The high-frequency nonlinear extra term 10 sin(10r)δr in yaw moment +15 + +(a) +(b) +(c) +102 +102 +104 +Ilesll2 - Koopman +Ilepll2 - Koopman +ell2 - +lel2 - RLS +lepll2 - RLS +e/2 - +101 +101 +102 +100 +10-1 +100(d) +100 +koopman +leell2 +RLS +102 +10-4100 +10-2 +10-1 +10-3 +102 +10-4 +102 +104 +10-5 +10-3 +10~6 +10~6 +10-7 +10-4 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +0 +20 +40 +time [sec] +time[sec] +time [se10~6 +10-8 +10-10 +10~12 +60 +80 +0 +20 +40 +60 +80 +time [sec]arXiv Template +A PREPRINT +4 +CONCLUSION +As the synopsis of the study over advantage of data in the aerospace vehicle control systems, the following conclusions +are realized: +• using full use of available data in the future control systems is a must, +• employing the physical relations influence the performance of the control system which data cannot afford, +particularly when dynamics are certain, +• the question is not using DDC or MBC. The correct question is when/where to use DDC or MBC, +• the flight dynamics has compatibility to be decoupled for combination use of data and model in control system +analysis and design. +In this paper, the DAC framework is suggested as a comprehension of the authors regarding the above conclusions. The +framework is developed exploiting the NASA GTM platform and includes a nonlinear model suite for a Lyapunov-based +nonlinear velocity regulator, a dual estimation with DUKF over an observable estimation model to provide the dynamics +decoupling, the Koopman estimator for identification of the linear evolution of the predefined terms over the decoupled +dynamics, and eventually decision-making for the interference of data or using initial dynamical model according to the +behavior of the flight. The GTM went through a series of closed-loop simulations for evaluation of the DAC performance +under a failure scenario. 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Wright, Numerical optimization. +Springer Science & Business Media, 2006. +20 + diff --git a/hNE5T4oBgHgl3EQfiA_6/content/tmp_files/load_file.txt b/hNE5T4oBgHgl3EQfiA_6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..570b27bdf4912872aa792cb490a8380216490ac6 --- /dev/null +++ b/hNE5T4oBgHgl3EQfiA_6/content/tmp_files/load_file.txt @@ -0,0 +1,1204 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf,len=1203 +page_content='DATA-ASSISTED CONTROL - A FRAMEWORK DEVELOPMENT BY EXPLOITING NASA GTM PLATFORM A PREPRINT Mostafa Eslami Department of Aerospace Engineering Sharif University of Technology Azadi Avenue, Tehran eslami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='mostafa@ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='edu Afshin Banazadeh Department of Aerospace Engineering Sharif University of Technology Azadi Avenue, Tehran banazadeh@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='edu January 16, 2023 ABSTRACT Today’s focus on expanding the capabilities of control systems, resulting from the abundance of data and computational resources, requires data-based alternatives over model-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' These alternatives may become the sole tool for analysis and synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Nevertheless, mathematical models are available to some extent, especially for air and space vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Hypothetically, data assistance would be the approach to meet the requirements in collaboration with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this paper, a framework of Data-Assisted Control (DAC) for aerospace vehicles is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' NASA Generic Transport Model (GTM) is the platform for the study and the data supports the model-based controller in extending performance over a damage event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The framework requires real-time decisions to override the control law with the information obtained from the data, while the model-based controller does not show regular performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The closed-loop system is shown to be stable in the transition phase between the data and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The fixed dynamic parameters are estimated using the Dual Unscented Kalman Filter (DUKF) and the evolution of the generalized force moments is estimated using the Koopman estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Simulations have shown that the purely model-based robust control leads to degradation of the closed-loop performance in case of damage, suggesting the need for data assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Keywords Data-Assisted Control (DAC) · NASA Generic Transport Model (GTM) · Koopman Operator · Dual Estimate · Unscented Kalman Filter (UKF) · Nonlinear Robust Control · Decision Making 1 INTRODUCTION The diversity and volume of the available data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the significant increase in real-time calculation capacity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' quality of living in the cyber-physical systems era,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' human ambitions in interplanetary probing and exploring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and real-time decision-makings in autonomy all have promoted utilizing of the data in the development of science and engineering more and more,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the fourth-paradigm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Before the evolution of model-based analysis and synthesis, data was the first aid for scientists and engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Using data in control is not new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' One of the first data-driven control design methods in 1942 is Ziegler-Nichols signal-based tuning for PID controllers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' With progress in physics and our understanding of physical systems, mathematical modeling became the only aid in the development of theories and particularly in control engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In control engineering, learning from data is a new paradigm [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' All theories and control methods in which the controllers are derived from the data without explicit use or knowledge of the mathematical model, named Data-Driven Control (DDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' A legitimate DDC is considered to have stability, convergence, and robustness with rigorous mathematical and specified assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The Willems et al.’s Fundamental Lemma (WFL) [4], Data Informatively [5, 6], Behavioral Approach [7], Interconnection Paradigm [8, 9], Dissipativity [10, 11, 12], and data- driven Koopman Operator [13, 14], all are provided different frameworks for developing DDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In the same fashion, Model-Based Control (MBC) explicitly uses the mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Data-driven control systems may have better performance compared to the MBC whenever the following conditions are the case: a) mathematical model is not arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='05646v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='SY] 13 Jan 2023 arXiv Template A PREPRINT available, b) uncertainties and a wide range of structural/nonstructural combinations raise the lack of a comprehensive and integrated model, c) modeling is very hard or impossible such that model-based control won’t work well, and d) mathematical model is rather complicated to be used for control system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Considering as a fact, always for highly uncertain dynamics, the model-based paradigm has limits in particular for unpredicted operating conditions and situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' These unmodeled dynamics yields unreliable model-based control even with adaptation [15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Otherwise, any increase in the order and complexity of a model may increase the effort and tries for control system design [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The Rohrs’ counterexample is an alarm for having too much reliance on adaptation due to unpredictable nonlinear behaviors due to assumptions in model-based controls [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Moreover, any dynamical system may show some degree of uncertainties with the increase in operating condition ranges that are not possible to be modeled or identified with high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Since data-based theories don’t use implicitly any dynamical models and only rely on in-out data of the system, may resolve the requirements for enhanced requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' As a fact, the model-based paradigm will work optimally for a certain region with some degree of known uncertain boundaries with very vigorous theories and proven results in abundant practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' However, today’s progressive increase in computational capacity has introduced new visions for the evolution of control systems with the assistance of data to extend model-based capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' For example the advantage of the data for the enhancement of the model-based control system performance in an industrial engine is evident [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Looking at breakthroughs in control engineering, the direct adaptive control [22, 23] and neural-network-based control systems [24, 25] are in the same paradigm of the data-driven control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' After overwhelming use of neural networks in control systems design, the data-driven controllers were introduced and are progressively developing [18, 26, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In recent decades, using DDC methods adapted from a similar MBC had great intention among control system scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' For example PID like DDC [27, 28], model-reference control and output tracking [29, 30], predictive controls [31], robust controls [32], Fault-Tolerant Controls (FTC) [33, 34], reinforcement learning [35], optimal control [36, 37, 38, 39] are developed under data-based paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In aerial applications, the dynamical model with high fidelity is developed [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' With a correct understanding of these dynamics, the exact and uncertain parts can be fully distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The variety of operating points in these systems may raise a high degree of uncertainty because all analysis and synthesis methods use simplification rules to express them in a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' atmospheric disturbances and turbulence [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' flying in an unknown stochastic and heterogeneous atmosphere or complex environment [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' flights reconfiguration [43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' foreign object damage to the flight’s body and control surfaces [46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' high-frequency dynamics in control surfaces [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' increase in bandwidth [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' robustness against faults and uncertainties that are unbounded or impossible to recognize the bound [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' envelope protection [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' control in case of low speed in communication and low bandwidth of communication which needs real-time decision-makings [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and eventually the survivability [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' all may not be possible to be defined with general deterministic terms in accordance with model-based dynamic formulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Moreover, the flights are used to have high reliability with the probability of catastrophic failure less than 10−9 per hour according to APR4761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' For decades, the designers have gained this reliability with hardware redundancy in controllers, and control surfaces, using multi actuators, sensors, and flight computers [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Yet, in 2017, after a series of studies in NASA Langley Research Center [55], the Loss-of-Control (LOC) [56], was recognized as the biggest threat to aviation accidents, especially for big commercial fixed wings airplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The LOC had the most portion of the accidents across all classes of flights, missions, and flight phases [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' One of the significant external reasons for LOC is the damage to the wing and fuselage or control surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Accidents reported flight AAF587 [59], A300-B4 [60], AAF232 [61], and BGF1907 [62] are example of these damages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' After studies for analysis and improvement of control systems in face of LOC in [63, 64, 65, 66, 67], the adaptive controllers show improvement in flight quality in case of failure and uncertainties [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Although the adaptive controls have a long and rich history, due lack of simple verification and validation methods, they are not used widely in commercial aerospace applications [62, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this paper, a framework is developed in which data extends the flight operating regime of pure MBC in case of the aforementioned uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The framework utilizes data in the linear evolution of observations thanks to Koopman linearization [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In recent years, the Koopman operator and linearization successfully have been applied in many aerial and space applications such as low-thrust trajectory optimization [71], attitude control of spacecraft [72], studying the motion of a satellite close to a libration point [73, 74], approximating analytical solution for Zontal Harmonics problem [75], and decision making [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Despite its increasing popularity, the application is very limited due to demanding accuracy in real-time [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' True recognition of the part that data can assist, needs the development of a framework from a control engineering point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This study is carried out under a specific model-based nonlinear framework with the remedy of a Lyapunov- based nonlinear robust control design, a nonlinear dual estimation with Unscented Kalman Filter (UKF), a data-based Koopman operator over estimated force-moment (pseudo-observations), and a real-time decision-making algorithm 2 arXiv Template A PREPRINT with a behavioral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The Koopman operator provides powerful data-based linearization in observable space [77], and the nonlinear control benefits from the structure of the nonlinear model capable of stability and performance analysis with a decision factor for the shift between data and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This compels to use of a nonlinear estimator for parameters and the evolution of forces and moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' We call the exact dynamics with parametric uncertainty, the fixed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Eventually, the real-time decision-making follows a behavioral approach to optimize an online cost function to promote the data or model credibility, whereas the system behaves like the initial model or undergoes uncertainty or failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This framework is named Data-Assisted Control (DAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' DAC has full advantage of the MBC with stability guaranty, yet the data would extend its performance in any condition that flights don’t demonstrate the typical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' When data interferes, the stability is ascertained with specific assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Of course, when dynamics are certain the data would not interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The generic 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5% scaled transport class vehicle known as the Generic Transport Model (GTM) by NASA is the platform for this study [78, 79, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The GTM includes all subsystems required for experimental flight control algorithm implementation and evaluation [81, 82, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The model encompasses a series of defined failures and can be used as a benchmark for FTC systems [62, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The DAC framework is introduced in details in section 2, and required five steps for its development for GTM is provided from sub-sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In the end, the framework is applied to the GTM with a series of simulations in section 3 to demonstrate the extended performance of DAC compared to the pure MBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' NOMENCLATURE ω [rad/sec] angular velocity vector with p-roll rate, q-pitch rate and r-yaw rate arguments in body coordinate ρ [ft] center of mass displacement V [ft/sec] linear velocity vector with u, v and w arguments in body coordinate v generalized body coordinate velocity vector - [V T , ωT ]T m [lbs] aircraft mass IM [slug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='ft2] aircraft moment of inertia matrix with Iab entries for axis a and b I identity matrix with proper dimension g [ft/sec2] gravity constant η1 position vector with X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Y and Z arguments in earth coordinate η2 orientation vector (Euler angles) with Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Θ and Ψ arguments in earth coordinate η generalized earth-fixed coordinate position and orientation - [ηT 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' ηT 2 ]T p parameters set W [lbf] gravitational force F [lbf] external force applied to aircraft T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' TR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' TL [lbf] total thrusts generated by engines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' right engine thrust,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' left engine thrust M [lbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='ft] external moment applied to aircraft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='generalized force and moment vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='generalized momentum vector consisting of linear and angular momentums ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='mass-inertia matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='Coriolis and centrifugal matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='gravitational force and moments vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='control derivatives in force and moments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='damping derivatives in force and moments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='dimensionless aerodynamics coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='extra order dynamics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='ex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='indication of error associated with the variable x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='ωx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='state transition noise vector with ων,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' ωτa and ωp arguments ωy measurement function noise vector 3 arXiv Template A PREPRINT Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' summation operator - means total forces or moments S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=') skew-symmetric operator acts on the vector (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=') and generates associated skew-symmetric matrix diag(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=') a function generating diagonal matrix of vector (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=') ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='∥2 norm-2 of vector (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=') or Frobenius norm of matrix (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='T transpose operator ˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' estimated parameter, vector or matrix ˜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' difference between the actual and the desired variable Subscripts g Indication of the center of gravity p Indication of the center of pressure T Belong to the thrusters A Belong to the aerodynamics F Belong to the forces M Belong to the moments δ Belong to the control surfaces/actuators er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' el,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' e right elevator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' left elevator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and coordinated left-right elevator together ru rudder ar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' a right aileron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' left aileron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and coordinated left-right aileron together r reference/residual value a under state variable means augmentation di indication of GTM damage case i d indication of desired value 2 DAC FRAMEWORK The general framework of the DAC is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this scheme, considering the fixed structure for M and C matrices, parameter estimation is used for the estimation of mass, the moment of inertia, and their products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this case, any effect on the force and moment derivatives is reflected in applied aerodynamics and thrust forces and moments, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' One can call this parameter pseudo-observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Because first using the measurement of states a nonlinear estimate is exploited to use the fixed dynamics and then the estimated force-moment vector is used as observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Due to the structure of the developed model, the fixed dynamics are strictly equivalent to the observations, but τ is not measured in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Accordingly, the dual estimation will help to have correct estimates of the generalized forces and moments evolution by time without the need for a direct measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Simply, the fixed dynamics with correct parametric and state estimates must provide correct trends of external forces and moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Eventually, the linear evolution of the observation of the generalized forces and moments will be identified and fed to the control law to adapt the closed-loop dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' According to the force expansion rule by [84, 85], the generalized forces and moments are characterized by linear dependency on aerodynamic derivatives up to some unknown vanishing orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Therefore, the regressor will have significant void entries which promote using data-based methods for identification over observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The real-time decision-making algorithm always tracks the performance of the closed-loop system and with slight changes in the decision factor tests interference of the identified regressions by data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' If the external behavior of the aircraft demonstrates improvement in the performance, the decision factor will promote the data assistance more and more, otherwise, it will return to the MBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The DAC framework can be decomposed into modular steps as follows: (A) rewriting the nonlinear model suite for DAC framework and performing dynamics decoupling for observations and fixed dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (B) nonlinear control design for the full envelope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' using fixed dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (C) parameter estimations including mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the moment of inertia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the center of gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and pseudo observation estimation considering the fixed-dynamics exploiting a dual estimation approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (D) data assistance for identification of the aerodynamic and control derivatives over pseudo observations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (E) behavioral decision-making for optimizing decision factor of altering control law according to the data-assisted model in a stable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This framework is elaborated through a series of simulations and analysis in a pragmatic way to provide seemly data assistance with stability and performance guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 4 arXiv Template A PREPRINT GTM Nonlinear Controller \uf068 \uf056 \uf056 \uf064 \uf056 \uf056 Real-time Decision Making \uf06c\uf02a Nonlinear Dual Estimation Data-based Forces and Moments Evolution Identification \uf074 \uf0b5 \uf074 Figure 1: Data Assisted Control (DAC) framework for GTM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='1 GTM Nonlinear Model Suitable for DAC The set of flight dynamics equations for GTM is developed considering the change in the body’s center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Such equations can be used when the body loses a portion of its mass and it is desired to track the motion of the body’s previous center of mass/reference frame [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The velocity components are subscripted with any arbitrary point in body-frame to distinguish them from the usual velocity components considered at the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Of course these components are the velocity of the center of mass when this point is the center of mass, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The force of gravity and the equation of motion is written as follows [79,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 86]„ W = � −mg sin Θ mg cos Θ sin Φ mg cos Θ cos Φ � (1) ΣF + W = m � ˙V + S(ω)V � − mS(ρ) ˙ω + mS(ω)(S(ω)ρ) ΣM + S(ρ)W = IM ˙ω + S(ω)(IMω) + mS(ρ) ˙V + mS(ω)(S(ρ)V ) + mS(V )(S(ω)ρ) (2) The derived equation of motion can be concisely described in a matrix form as below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' M ˙v + C(v)v + G(η) = τ (3) 5 arXiv Template A PREPRINT where M is the mass-inertia matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' C is Coriolis and centrifugal matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and G is gravitational force and moment acting on the flight as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' M = � mI −mS(ρ) mS(ρ) IM � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' C(v) = � mS(ω) −mS(S(ω)ρ) −mS(S(ω)ρ) −S(IMω) + mS(S(V )ρ) � & G(η) = � −W −S(ρ)W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (4) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Assuming small change rate in mass and moment of inertia, M and C are rearranged such that ˙ M − 2C(v) is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The following decomposition can be made over GTM forces and moments produced by engines (FT and MT ), the aerodynamics of wings and base airframe (FA and MA), and control surfaces (Fδ and Mδ), ΣF = FT + FA + Fδ, and ΣM = MT + MA + Mδ + S(l − ρ)(FA + Fδ) (5) Where l = Cp − Cg is difference between center of pressure and gravity (for GTM l = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='0276, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='0118, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='036]T [ft]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The forces and moments of the aerodynamic base frame and control surfaces are traditionally defined by dimensionless aerodynamic coefficients, CF , and CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Each coefficient is decomposed into state variables with a set of multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The angle of attack α and side-slip angle β for aerodynamics of base frame and control surface angle purely define the flight aerodynamic attitude with respect to surrounding air and direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Hence, it is common to define the multipliers for these two angles, not the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The relation between aerodynamic coefficients and force and moments in reference flight is as follows, FA = CF ¯qrSr, Fδ = CδF ¯qrSr, MA = diag(b, ¯c, b) × CM ¯qrSr, Mδ = diag(b, ¯c, b) × CδM ¯qrSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (6) Where, CA = � CF CM � = C0 + Cαα + Cββ and CδA = � CδF CδM � = [ Cδruδru Cδaδa Cδeδe ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (7) Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The parameters are extracted for reference flight of GTM model in 800 [ft] altitude and TAS of 75 [knots] without the wind, ¯qrSr = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='65 [lbf], b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='8488 [ft], and ¯c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='9153 [ft].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In GTM the aerodynamic coefficients and control derivatives are generated via a look-up table for −5 ≤ α ≤ +85 DEG and −45 ≤ β ≤ +45 DEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In the rest, we assume that flight is in steady symmetry condition near α = 4 DEG and β = 0 DEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' It is assumed flaps, stabilizers, spoilers, brake, and landing gear don’t participate in flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Their trim value in simulated reference flight is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' It is also assumed engines are symmetrical so they generate the same thrust value and we have a single δT as a thrust control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The upper and lower rudder work together, and single control input δru is considered for the rudder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' For aileron and elevator, the inner and outer boards and the left and right surfaces are merged to accept only one control input as δa for aileron and δe for elevator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The prime motive behind this control input configuration is the possibility of damage simulations in GTM standard damage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In summary vector of control input is as follows, δ = � �� δT δru δa δe � �� (8) Considering the assumptions in the derived model, the simplified mathematical representation of the GTM in simulations will have residual forces and moments in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Let’s denote the residual force and moments in reference flight by τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The final form of the nonlinear model will be summarized as, M ˙v + C(v)v + G(η) = τ + τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (9) 6 arXiv Template A PREPRINT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='2 Nonlinear Robust Control Design - a Velocity Regulator Equation (9) ensembles dynamics in a ready form for developing nonlinear controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The objective of the control system design in case of failure is presumed to be regulation of the velocities in body coordinate, as a pilot would act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Accordingly, the design of a nonlinear velocity regulator has been followed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Let’s define an energy Lyapunov candidate function as follows, V = 1 2 ˜vT M˜v (10) Where ˜v = v − vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Taking derivative from Lyapunov function and using Remark 1 yields, ˙V = ˜vT M˙˜v + 1 2 ˜vT ˙ M˜v = ˜vT (M ˙v − M ˙vd + C(v)˜v) = ˜vT (τ + τr − G(η) − C(v)v − M ˙vd + C(v)˜v) (11) Assume following control law has been selected, τ = G(η) + C(v)vd + M ˙vd − Γ˜v − τr (12) Where Γ = diag(γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=', γ6) and γi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Then, ˙V = −˜vT Γ˜v ≤ −λ(Γ)∥˜v∥2 ⇒ V ≤ 1 2λ(M)∥˜v∥2 ⇒ ∥˜v∥2 ≥ 2V λ(M) (13) Hence, ˙V ≤ −2 � λ(Γ) λ(M) � V ⇒ V(t) ≤ V(0) exp −2 � � λ(Γ) λ(M) � �t (14) Where V(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The role of λ(Γ) now is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' By increasing its value, V(t) converges to zero exponentially faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This yields exponential convergence of generalized velocity error towards the origin, with Γ as the tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' According to GTM, the generalized force and moment vector can be decomposed as follows, τ = τ0 + B(δ)δ + D(v)v (15) Where τ0 is static forces and moments in the trim condition, B(δ) is a matrix with the variable sign of elements according to the sign of command δ, and D(v) is damping term of the forces and moments linearly dependent to state vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This yields an update in control law as follows, τc = G(η) + C(v)vd + M ˙vd − Γ˜v − τr − τ0 − D(v)v (16) In which, τc = B(δ)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' As a challenge in this control law, the subtracted terms from the control law are open-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Yet, the exact evaluation of these values is almost impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Hence, additional terms are added to the control law to overcome issues of possible uncertainty and make the control system robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Considering the χ as bound of uncertainty, following the control law will make the control system robust [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' τc = G(η) + C(v)vd + M ˙vd − Γ˜v − τr − τ0 − D(v)v − χ tanh � ˜v ϵ � (17) Eventually, having τc in hand, the calculation of the control command δ is the last part of the control system computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Using least square minimization, the optimal control command is calculated as follows, δ∗ = (B(δ)T B(δ))−1B(δ)T τc (18) For not damaged aircraft, considering merging of left-right actuators, we have B(δ) ∈ R6×4 and δ ∈ R4×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In case of damage with losing actuators the dimension of the input matrix may decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In case of additional nonlinear term E in τ as τ = τ0 + B(δ)δ + D(v)v + E, the term can be decomposed into E = BEδ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' which are varying by time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The control law is still valid considering new control derivative matrix B′(δ) = B(δ) + BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The high-order terms are subtracted from τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In order to ensure persistence of excitation in this regulator, signals consisting of sinusoids of varying frequencies and randoms are added to the control law [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 7 arXiv Template A PREPRINT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='3 Damaged Aircraft State and Parameter Estimation In this study, the aircraft experiences damage case 1 of GTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Damage case 1 involves a parametric change in mass, moments of inertia and their products, the center of gravity, and force-moment control derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The change in parameters is defined as follows, md1 = m − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='13 (19) ρd1 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='0105, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='0023]T Ixxd1 = Ixx − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='00346 Iyyd1 = Iyy − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='06698 Izzd1 = Izz − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='06352 Ixzd1 = Ixz − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='01409 Iyzd1 = Iyz + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='00001 Ixyd1 = Ixy + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='00003 In this step, the parameters of the fixed dynamics p = [m, Ixx, Iyy, Izz, Ixz, Iyz, Ixy, ρ]T , and generalized forces and moments are estimated with the augmented mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Considering the DUKF approach [89], the state estimation model can be written as, ˙ˆv = ˆ M−1 � ˆτ + τr − ˆCˆv − ˆG � + ων (20) ˙ˆτa = Aτaτa + ωτa and measurement function can be written as y = v + ωy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The augmented force-moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' τa ∈ R18×1, represents third order Gauss-Markov process with auxiliary variables ζ1 and ζ2 as follows [90], τa = � τ ζ1 ζ2 � & � � ˙τ(i) ˙ζ1(i) ˙ζ2(i) � � = � 0 1 0 0 0 1 0 0 0 � � τ(i) ζ1(i) ζ2(i) � + � ωτ(i) ωζ1(i) ωζ2(i) � for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' , 6 (21) Therefore, Aτa = � ∅6 I6 ∅6 ∅6 ∅6 I6 ∅6 ∅6 ∅6 � , & ωτa = � ωτ ωζ1 ωζ2 � (22) Where, ∅6 is zero matrix with dimension of R6×6, and I6 is identity matrix with dimension of R6×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The considered process for evolution of parameters in time is considered as follows, ˙ˆp = ωp (23) Due to nonlinearity in parameters which results in the whole process with non-additive noises, augmentation is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' We define the covariance matrix of zero-mean Gaussian noises as ωxa ∼ N(0, Q) and ωy ∼ N(0, R) such that Q, R ≻ 0 to overcome numerical instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Having the augmented nonlinear model in hand, the DUKF estimation algorithm is followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In order to avoid the complexity of DAC parameter tuning, the simplest form of sigma-points generation for unscented transform is considered [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The UKF algorithm is optimal when a) the model matches the real system perfectly, b) the entering noise is white (uncorrelated), and c) the covariances of the noise are known exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 8 arXiv Template A PREPRINT The framework implies the model matching and uncorrelated noises;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' however, adaptation in covariances is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Using innovation and residual of the filter following adaptation rule is employed to dynamically update noise covariance matrices [92], Qk = αQk−1 + (1 − α)KkdkdT k KT k , innovation at step k: dk = yk − h(ˆx− k ) (24) Rk = αRk−1 + (1 − α)(εkεT k + HkP − k HT k ), residual at step k: εk = yk − h(ˆx+ k ) (25) Where in these equations α is a forget factor, and Hk is jacobian of measurement function, h(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' ), at step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In addition to the above conditions, the estimation model itself should be observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In order to check observability let’s collect the state transition and measurement functions as follows: ˙v = f(v, τ, p) (26) ˙τa = Aτaτa + ωτa ˙p = ωp, and y = v + ωy In this study, using Lie derivatives for deriving analytical implications of the observability was found to be inefficient due to demanding high-order partial differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Moreover, these derivatives will induce computational errors and hence declines the credibility of the result from the computation point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' As an alternative approach due to working in a regulation scheme the piece-wise linearity of the dynamics is assumed, then the analytical linear system is elaborated to check the observability in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Considering estimation model (26), the linear system observability matrix couple (A, C) would be, A = � �� ∂f ∂v ∂f ∂τa ∂f ∂p ∅ Aτa ∅ ∅ ∅ ∅ � �� and C = [ I6 ∅ ∅ ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (27) It is evident that if ∂f ∂p = 0 then model is not observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Derivatives of f are derived as follows: ∂f ∂v = −M−1(p) ∂ ∂v (C(v)v), ∂f ∂τa = � M−1(p) ∅ � , and ∂f ∂p = − ∂ ∂p � M−1(p) (C(p)v + G(p)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (28) The underlined terms are calculated with symbolic calculations, then the observability matrix singular values are checked per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The observability matrix is defined as, O = � ��� C CA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' CAn−1 � ��� (29) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='4 Data Assistance Koopman demonstrated that it is possible to represent a nonlinear dynamical system in terms of an infinite-dimensional linear operator acting on a Hilbert space of measurement functions of the state of the system [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Koopman operator is linear, and its spectral decomposition completely characterizes the behavior of a nonlinear system [93, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Using the Koopman operator nonlinear dynamics become completely linear in eigenfunction coordinates, called lifted space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Provided estimates for generalized force-moment vector and parameters, in this step the control derivatives, damping derivatives, and static force-moment terms are recognized in pseudo observations via measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The approach of using Koopman operator in synthesis and design step of the controller is growing rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In particular, the Koopman estimator is used to provide the linear estimate, and the results are compared with Recursive Least Square (RLS) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The RLS is chosen as an analog due to being an optimal deterministic estimator for linear fitting in adaptive rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The torque and moments are regularly treated as linearly varying to actuator command, linear and angular velocities with a static term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' As a general comparison, the Koopman represents a batch data identification method and the latter updates the estimates according to just previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 9 arXiv Template A PREPRINT From the developed framework the i-th observations and measurements in the simplest from are decomposed as follows, ˆτi = ˆτ0 + ˆB(δi)δi + ˆD(vi)vi (30) and stack of m previous sampled pseudo observations and measurements are collected as follows, ˆT = [ˆτ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' ˆτm] ∈ R6×m & Y = �δ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' δm v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' vm 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 1 � ∈ R11×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (31) Accordingly, using Koopman the linear evolution of the observations with respect to the measured variables can be obtained as follows, ˆP = ˆT YT (YYT )−1 (32) Where, ˆP = � ˆB ˆD ˆτ0 � ∈ R6×11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' (33) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In practice, the equation (31) can be described in a more complex form depending on some orders of the measurement derivatives [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this case, a more complicated data-driven algorithm can be used for linear estimation such as SINDy [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Without loss of generality, in this paper, the simplest form is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In order to compare Koopman estimator performance versus RLS, in section 3 additional nonlinear terms are appended to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The additional terms are put into extra order dynamics vector, E, and includes sin(10r)δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Equation (31) gives dynamics of the generalized momentum, τ = dL/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Therefore, the evolution of observations describes the dynamical changes in momentum directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This can be used to design a Koopman data-driven control system to control momentum directly [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this paper, data only assists in control system design, and this approach is not followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' However, as a third option in the decision-making, the full data-driven control like this approach would be the case for the next studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 Real-time Decision-Making and Decision Factor Optimization A decision factor λ determines the active role of data assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' From equation (9) and (15) the integrated open-loop dynamics is as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' M ˙v + C(v)v + G(η) − τr = τ = τ0 + B(δ)δ + D(v)v (34) This equation after estimations and exploiting data becomes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' ˆ M ˙v + ˆC(v)v + ˆG(η) − τr = ˆτ = ˆτ0 + ˆB(δ)δ + ˆD(v)v (35) Using decision factor λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the coupled estimated and initial/uncertain dynamics will be as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' M(λ) ˙v + C(λ)v + G(λ) − τr = τ(λ) = τ0(λ) + B(λ)δ + D(λ)v (36) Where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' M(λ) = (1 − λ)M + λ ˆ M (37) C(λ) = (1 − λ)C + λ ˆC G(λ) = (1 − λ)G + λ ˆG τ(λ) = (1 − λ)τ + λˆτ τ0(λ) = (1 − λ)τ0 + λˆτ0 B(λ) = (1 − λ)B + λ ˆB D(λ) = (1 − λ)D + λ ˆD The decision factor is a convex combination of the initial/uncertain dynamics and the estimated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' As λ → 0 control apportioned to MBC, otherwise, λ → 1 suggests uncertainty has appeared and DAC would become operative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 10 arXiv Template A PREPRINT For decision-making, the historical behavior of the closed-loop system is turned into a performance cost function with λ as optimization variable in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Also, the decision on λ can be taken manually by pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Assuming time horizon of mλ samples in past period tp ≤ t ≤ t0, following cost function is a candidate to select λ∗, Jλ = 1 2 ˜vT (λ, t0)Hλ˜v(λ, t0) + 1 2 � t0 tp ˜vT (λ, t)Qλ˜v(λ, t)dt (38) subject to dynamics in (36) considering λ = λ∗− as the optimum λ in the previous window t0 − 2tp ≤ t ≤ tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The Hλ and Qλ are weightings for terminal and transient cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this study the gradient based steepest decent is used for minimization of Jλ [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In steepest decent the next optimization variable is updated by the following rule, λ∗ k+1 = λ∗ k − γk ∇Jλ(λ∗ k) ∥∇Jλ(λ∗ k)∥ (39) where γk is step size gain, and k is the iteration of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Step size gain in companion with decision horizon determines the rate of change in decision factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Legitimate decision-making should have a stable transition period from data to model and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Next, it is shown under certain conditions, the closed-loop system in DAC is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Proof 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Considering ˙ M(λ) − 2C(λ) is still skew-symmetric, the developed control law in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='2 is still valid and can be written as, τc(λ∗) = G(λ∗) + C(λ∗)vd + M(λ∗) ˙vd − Γ˜v − τr − τ0(λ∗) − D(λ∗)v − χ tanh � ˜v ϵ � (40) Therefore the stability analysis falls into checks on the skew-symmetric property of ˙ M(λ) − 2C(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Accordingly using (37), ˙ M(λ) − 2C(λ) = dM(λ) dλ ˙λ − 2C(λ) = � ˆ M − M � ˙λ − 2C(λ) (41) Using parameter estimation in step (C), the structure of ˆ M and ˆC are not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Hence, the above equation is skew-symmetric or almost skew-symmetric if and only if, a) rate of change in decision factor is small and continues in time, or b) moment of inertia and mass are correctly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The product of inertia and center of gravity shift estimates don’t affect the stability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' When λ is identically 0 or 1, condition (a) is granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Either transitions λ → 0 or λ → 1 would occur when there is an interference of data or return to the initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' If conditions in Remark 4 are fulfilled the estimation error will converge to zero in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Therefore, in transition, the term ˆ M − M has value, and nonzero ˙λ will introduce an additional term in control law that may lead to instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Accordingly, the decision factor should enter with a lag time with respect to the estimation time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this way, the temporal difference in estimation and decision-making actions would guarantee stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The closed-loop performance is guaranteed if the robustness gain χ compensates the additional feedforward errors in the estimation of D and τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Since the Lyapunov function in (10) has closed level sets, in case of instability in tolerable finite time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' restoring the skew-symmetric property of ˙ M(λ) − 2C(λ) in small finite time, the error trajectories will return to origin after the restore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' We used the qualitative term "small/tolerable finite time" because it depends on the divergence rate of flight trajectories and velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Eventually, for the calculation of control inputs, using the same relationship for computing optimum δ from τc in DAC we have, δ∗ = (B(λ∗)T B(λ∗))−1B(λ∗)T τc(λ∗) (42) The only difference would be the possibility of order reduction in effective control inputs due to control loss in case of damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Therefore, following additional condition is imposed to guaranty non-singularity in B(λ∗)T B(λ∗), Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Assume i-th column of B(λ∗) be denoted by bi, and ϵδ is a small parameter selected according to actuator gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' If ∥bi∥ ≤ ϵδ, the δi and bi will be redacted from computation of equation (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The δi will keep the last value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' This case should happen only when λ → 1, if the estimation of ˆB converges properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Otherwise, decision factor should be decreased to give more credit to the model, and accordingly, the redacted order will return in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 11 arXiv Template A PREPRINT 0 10 20 30 40 50 60 70 80 time [sec] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='4 Figure 2: DAC simulation for GTM under damage and pilot decision 3 SIMULATIONS Through the above steps, the DAC framework has been applied to the GTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' its performance with the following events in fault case scenario is going to be evaluated: a) flight journey begins in trimmed condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' at t = 4 [sec],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' a small disturbance is applied on the flight control surfaces to check velocity regulator performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' b) at t = 10 [sec],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the failure introduced in Step C happens abruptly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' c) at t = 30 [sec],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' an extra high frequency nonlinear term 5 sin(10r)δr are added exponentially with time constant of 2 seconds to the yaw moment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' c) at t = 50 [sec],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' pilot decides to select pure MBC and then returns to DAC at t = 55 [sec].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Figure 2 demonstrates simulation results of the given scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The effective error vanishing in DAC-enabled mode is evident, whereas the pure MBC has meaningful errors after the introduction of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this plot, λ is the actual decision factor involved in control law, and the λsel is the selected decision factor by the pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The time lag between actual and optimum/selected decision factor is due to imposing intentional temporal difference in case of an error in estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The result implies selecting the pure MBC by the pilot will have the penalty of performance loss, however, after returning to the optimum decision with complete interference of data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' λ = 1, the error has vanished completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Figure 3 and 4 demonstrate the velocity regulator performance with details of the state errors and actuator positions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Refer to 3, the sinusoidal desired values and random noise are intentionally added to ensure persistency in excitation and observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Figure 5 shows an almost error-free estimation of velocities and parameters with slight error in the force-moment vector, before and after the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The singular values of the observability matrix are provided in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Plot implies the observability is ascertained, however singular values of the parameters are undersized and yields weak condition of observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' It is worth mentioning when failure introduces the state and parameter variations improve the observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' 12 arXiv Template A PREPRINT Figure 3: Velocity regulator performance with detail of state errors 0 20 40 60 80 time [sec] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='15 0.' metadata={'source': 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150 200 250 0 20 40 60 80 time [sec] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 2 0 20 40 60 80 time [sec] 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='5 1 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='A PREPRINT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='CONCLUSION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content='As the synopsis of the study over advantage of data in the aerospace vehicle control systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the following conclusions are realized: using full use of available data in the future control systems is a must,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' employing the physical relations influence the performance of the control system which data cannot afford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' particularly when dynamics are certain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the question is not using DDC or MBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The correct question is when/where to use DDC or MBC, the flight dynamics has compatibility to be decoupled for combination use of data and model in control system analysis and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' In this paper, the DAC framework is suggested as a comprehension of the authors regarding the above conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The framework is developed exploiting the NASA GTM platform and includes a nonlinear model suite for a Lyapunov-based nonlinear velocity regulator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' a dual estimation with DUKF over an observable estimation model to provide the dynamics decoupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' the Koopman estimator for identification of the linear evolution of the predefined terms over the decoupled dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' and eventually decision-making for the interference of data or using initial dynamical model according to the behavior of the flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The GTM went through a series of closed-loop simulations for evaluation of the DAC performance under a failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The results show the potential and advantage of the DAC when failure happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' The stability of the closed-loop system is ascertained under specific assumptions that were full-filled in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfiA_6/content/2301.05646v1.pdf'} +page_content=' Moreover, the estimation model is observable and the Koopman 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Diffusion Model Towards Grounded Generation +Ziyi Li1∗, Qinye Zhou1∗, Xiaoyun Zhang1, Ya Zhang1,2, Yanfeng Wang1,2, and Weidi Xie1,2,† +1Coop. Medianet Innovation Center, Shanghai Jiao Tong University, China +2Shanghai AI Laboratory, China +https://lipurple.github.io/Grounded_Diffusion/ +a painting of a phoenix with rainbow beams in the distance +a painting of a unicorn on the snow land +a photograph of a penguin in sakura forest +a photo of a lion on a mountain top at sunset +a painting of a Pikachu on top of a mountain +a photo of a Cyberpunk Transformer next to a river +a photo of a fox playing basketball on the ice +a photograph of a turtle climbing a valcano +aeroplane +bird +horse +person +bicycle +train +dog +cat +pottedplant +tvmonitor +boat +bottle +sheep +car +chair +fox +basketball +Pikachu +turtle +penguin +unicorn +phoenix +lion +Transformer +Figure 1. Predictions from our guided text-to-image diffusion model. The model is able to simultaneously generate images and +segmentation masks for the corresponding visual objects described in the text prompt, for example, Pikachu, Unicorn, Phoenix, etc. +Abstract +The goal of this paper is to augment a pre-trained text-to- +image diffusion model with the ability of open-vocabulary +objects grounding, i.e., simultaneously generating images +and segmentation masks for the corresponding visual enti- +ties described in the text prompt. We make the following +contributions: (i) we insert a grounding module into the ex- +isting diffusion model, that can be trained to align the visual +and textual embedding space of the diffusion model with +only a small number of object categories; (ii) we propose +an automatic pipeline for constructing a dataset, that con- +* Both the authors have contributed equally to this project. +† denotes corresponding author. +sists of {image, segmentation mask, text prompt} triplets, +to train the proposed grounding module; (iii) we evaluate +the performance of open-vocabulary grounding on images +generated from the text-to-image diffusion model and show +that the module can well segment the objects of categories +beyond seen ones at training time, as shown in Fig. 1; (iv) +we adopt the guided diffusion model to build a synthetic +semantic segmentation dataset, and show that, training a +standard segmentation model on such dataset demonstrates +competitive performance on zero-shot segmentation (ZS3) +benchmark, which opens up new opportunities for adopting +the powerful diffusion model for discriminative tasks. +1 +arXiv:2301.05221v1 [cs.CV] 12 Jan 2023 + +1. Introduction +In the recent literature, text-to-image generative models +have gained increasing attention from the research commu- +nity and wide public, one of the main advantages of such +models is the strong semantic correspondence between vi- +sual and language, learned from a large corpus of image- +caption pairs, such correspondence, for instance, enables +to generate photorealistic images from the free-form text +prompt [26,27,30,41]. While the synthesis ability of these +models are unprecedented, they generally lack the ability +to ground the objects within generated images, which can +be crucial to a number of applications, for example, image +editing, visual question answering, and visual reasoning. +In this paper, we aim to augment an existing text-to- +image diffusion model with the ability of objects ground- +ing, i.e., simultaneously generating photorealistic images +and segmentation masks for corresponding visual objects +described in the text prompt. +Generally speaking, two +challenges exist: first, to explicitly establish the visual- +language correspondence for the generative model, image- +segmentation pairs are required for visual demonstrations. +In previous work, for example, DatasetGAN [42] and Big- +DatasetGAN [18], the authors consider collecting manual +annotations for synthetic images, however, such a strategy +is unlikely to be scalable, given the high annotation cost; +second, it is unclear how to guide the generative model to- +wards open-vocabulary grounding, i.e., the model should be +capable of segmenting objects from any category it can gen- +erate. In a concurrent work (DAAM [33]), attention maps +are directly upsampled and aggregated at each time step to +explore the influence area of each input word, despite being +simple, the performance of grounding is unsatisfactory, and +tends to result in ambiguities, as the text embedding for in- +dividual visual entity has been largely influenced by other +entities and the global sentence at the stage of text encoding. +To tackle the challenges mentioned above, we make +the following contributions: (i) we develop an automatic +pipeline for automatically constructing {image, segmenta- +tion, text prompt} triplets, in particular, we adopt an off- +the-shelf object detector, and do inference on images gener- +ated from the existing stable diffusion model. Theoretically, +such a pipeline enables to generate infinite data samples for +each of the categories within the vocabulary of existing ob- +ject detector, for example, we adopt the Mask R-CNN [22] +pre-trained on COCO with 80 categories; (ii) we propose a +novel architecture that can segment any visual entity men- +tioned in the text prompt from the generated image, specifi- +cally, we propose a fusion module that explicitly aligns the +visual and textual embedding space of the diffusion model, +as a result, despite being only trained on a pre-defined set of +object categories, the grounding module enables to segment +objects of unseen ones at training time, resembling an open- +vocabulary knowledge induction procedure. As shown in +Fig. 1, our guided stable diffusion model enables to segment +the objects well beyond the vocabulary of any off-the-shelf +detector, for example, Pikachu, unicorn, phoenix, etc. +To quantitatively validate the effectiveness of our pro- +posed open-vocabulary grounding, we initiate two proto- +cols for evaluation: first, we train the grounding module +on the constructed training set, and compare the grounding +performance with a strong off-the-shelf object detector; sec- +ond, we adopt the guided stable diffusion model to construct +a synthesized semantic segmentation dataset, and train a +segmentation model on it. While evaluating on the exist- +ing benchmarks for zero-shot segmentation (ZS3), the seg- +mentation model shows competitive performance over prior +state-of-the-art models, especially on unseen categories, re- +flecting the effectiveness of our open-vocabulary grounding +module. Crucially, it has demonstrated a promising applica- +tion for applying the powerful diffusion model for discrim- +inative tasks, that is, to expand the vocabulary beyond any +existing detector can do. +2. Related Work +Image Generation. Image generation is one of the most +challenging tasks in computer vision due to the high- +dimensional nature of images. In the past few years, gener- +ative adversarial networks (GAN) [10], variational autoen- +coders (VAE) [17], flow-based models [16] and autoregres- +sive models (ARM) [34] have made great progress. How- +ever, even GANs, the best of these methods, still face train- +ing instability and mode collapse issues [2]. Recently, Dif- +fusion Probabilistic Models (DM) demonstrate state-of-the- +art generation quality on highly diverse datasets [12,13,23, +29,31], outperforming GANs in fidelity [6]. These models +are often combined with a well-designed text encoder and +trained on billions of image-caption pairs for text-to-image +generation task, i.e., OpenAI’s DALL-E 2 [26], Google’s +Imagen [30] and Stability AI’s Stable Diffusion [27]. How- +ever, despite being able to generate images with impressive +quality using free-form text, it remains unclear what extent +the visual-language correspondence has been successfully +captured, this paper aims to augment an existing text-to- +image diffusion model with the ability to ground objects in +its generation procedure. +Visual Grounding. Visual grounding, also known as refer- +ring expression comprehension, expects to understand the +natural language query and then find out the target object +of the query in an image. Early visual grounding works are +trained in two stages [14, 21, 35, 36, 38], by first detecting +the candidate regions, and then ranking these regions. Later, +one-stage approaches [19, 28, 39, 40] attract more attention +due to their superior accuracy and efficiency in fusing lin- +guistic context and visual features. Here, we consider visual +grounding in the image generation procedure. +2 + +3. Methodology +In this paper, we aim to introduce a knowledge induction +procedure, that converts an existing text-to-image diffusion +model for grounded generation, i.e., simultaneously gener- +ating images and the segmentation mask of corresponding +objects described in the text prompt. In particular, our core +idea is to exploit a handful of image-segmentation pairs as +visual demonstrations, to build the general visual-language +correspondence between the visual representation from dif- +fusion model and the free-form text. +In the following sections, we start by describing the +problem scenario for grounded generation based on diffu- +sion model; in Sec. 3.2, we present preliminary background +knowledge of diffusion model, and terminologies that will +be used in later sections; in Sec. 3.3, we detail the proposed +knowledge induction procedure, including (i) the architec- +ture design that enables to align the visual and textual em- +bedding under an open-vocabulary setting, (ii) the training +procedure based on the automatically constructed {image, +segmentation, text prompt} triplets. +3.1. +Problem Scenario +Assuming there exists a strong text-to-image diffusion +model, for example, Stable Diffusion [27] in our case, the +goal is to convert it into a grounded generation model: +{I, m} = Φdiffusion+(ϵ, y) +(1) +where Φdiffusion+(·) refers to a pre-trained text-to-image dif- +fusion model with our grounding module appended, it takes +the sampled noise (ϵ ∼ N(0, I)) and language description +y as input, and generates an image (I ∈ RH×W ×3) with +corresponding segmentation masks (m ∈ {0, 1}H×W ×O) +for a total of O objects of interest. Note that, we expect +the model to be open-vocabulary, that means, it should be +able to output the corresponding segmentation mask for any +objects that can be generated by diffusion model, without +limitation of the semantic categories. +3.2. +Preliminary on Diffusion Model +Diffusion models refer to a series of probabilistic gen- +erative models, that are trained to learn a data distribu- +tion by gradually denoising the randomly sampled Gaus- +sian noises. Theoretically, the procedure refers to learning +the reverse process of a fixed Markov Chain of length T. +As for text-to-image synthesis, given a dataset of image- +caption pairs, i.e., Dtrain = {(I1, y1), . . . , (IN, yN)}, the +models can be interpreted as an equally weighted sequence +of conditional denoising neural network that iteratively pre- +dicts a denoised variant of the input conditioned on the text +prompt, ϵθ(It +i, t, yi), where It +i denotes a noisy version of +the input image, and t = 1, . . . , T refers to the timestep, +i ∈ {1, . . . , N}. For simplicity, we only describe the train- +ing and inference procedure for a single image, thus ignor- +ing the subscript in the following sections. +In particular, this paper builds on a variant of diffusion +model, namely, Stable Diffusion [27], which encodes all +images with a variational autoencoder, and transfers the dif- +fusion process to latent space. we will briefly describe its +architecture and training procedure in the following. +Architecture. Stable Diffusion consists of three compo- +nents: a text encoder for producing text embeddings; a +pre-trained variational autoencoder (VAE) that encodes and +decodes latent vectors for images; and a time-conditional +UNet (ϵθ(·)) for gradually denoising the latent vectors, with +the progressive convolutional operation that downsamples +and upsamples the visual feature maps with skip connec- +tions. The visual-language interaction occurs in the UNet, +where the embeddings of the visual and textual features in- +teract through cross-attention layers. Specifically, a text en- +coder is used to project the text prompt y to textual em- +beddings, that then are mapped into Key and Value, the +spatial feature map of the noisy image is linearly projected +into Query, and iteratively attending the conditioned text +prompt for updating. +Training and Inference. The training procedure of Stable +Diffusion can be described as follows: given a training pair +(I, y), the input image I is first mapped to a latent vector +z and get a variably-noised vector zt := αtz + σtϵ, where +ϵ ∼ N(0, 1) is a noise term and αt, σt are terms that con- +trol the noise schedule and sample quality. At training time, +the time-conditional UNet is optimised to predict the noise +ϵ and recover the initial z, conditioned on the text prompt +y, the model is trained with a squared error loss on the pre- +dicted noise term as follows: +Ldiffusion = Ez,ϵ∼N (0,1),t,y +� +||ϵ − ϵθ(zt, t, y)||2 +2 +� +(2) +where t is uniformly sampled from {1, . . . , T}. +At inference time, Stable Diffusion is sampled by iter- +atively denoising zT ∼ N(0, I) conditioned on the text +prompt y. Specifically, at each denoising step t = 1, . . . , T, +zt−1 is obtained from both zt and the predicted noise term +of UNet whose input is zt and text prompt y. After the final +denoising step, z0 will be mapped back to yield the gener- +ated image I. +3.3. +Open-vocabulary Grounding +In this section, our goal is to learn the correspon- +dence between the visual representation (from the diffusion +model) and category embeddings in text form, augmenting +the off-the-shelf Stable Diffusion with an open-vocabulary +object grounding module, as shown in Fig. 2 (left). As- +suming there exists a training set of N triplets, i.e., Dtrain = +3 + +A photograph of a +dog near a cat +Diffusion Model +Text Prompts +Our Model +Grounding +Synthetic images generated +by diffusion model +Oracle GT masks produced +by off-the-shelf detector +Synthetic images and corresponding masks generated +by our grounded generation model +… +𝒇𝟏 +𝒇𝟐 +𝒇𝑳 +Visual +Encoder +a photograph of { } +dog +cat +. . . +Text +Encoder +Fusion Module +. . . +. . . +. . . +Q +⊗ +(K, V) + +Diffusion Model +Text Prompts + +Figure 2. Overview. The left figure shows the knowledge induction procedure, where we first construct a dataset with synthetic images +from diffusion model and generate corresponding oracle groundtruth masks by an off-the-shelf object detector, which is used to train +the open-vocabulary grounding module. The right figure shows the architectural detail of our grounding module, that takes the text +embeddings of corresponding entities and the visual features extracted from diffusion model as input, and outputs the corresponding +segmentation masks. During training, both the diffusion model and text encoders are kept frozen. +{(F1, mgt +1 , y1), . . . , (FN, mgt +N, yN)}, the predicted segmen- +tation mask for all objects, i.e., mi ∈ RH×W ×Oi can be +obtained by: +mi = Φfuse(Φv-enc(f 1 +i , . . . , f n +i ), Φt-enc(g(yi))) +(3) +where yi denotes the text prompt for image generation, +Fi = {f 1 +i , . . . , f n +i } refers to the intermediate representation +from Stable Diffusion at t = 5 (this has been experimen- +tally validated in Sec. 4.3), Φv-enc(·), Φt-enc(·) and Φfuse(·) +denote the visual encoder, text encoder, and fusion mod- +ule. At training time, g(·) denotes a group of templates that +decorates each of the visual categories in the text prompt, +e.g., ‘a photograph of a [category name]’. There are to- +tally Oi object categories in the text prompt, and they fall +into a pre-defined set of vocabularies Ctrain. It is important +to mention that, we expect the visual-language correspon- +dence to be highly generalizable, such that the grounding +module should equally be capable of segmenting objects +from unseen categories at test time, i.e., Ctest ⊃ Ctrain. +In the following sections, we start by introducing the +procedure for automatically constructing the training set in +Sec. 3.3.1, followed by the architecture design for open- +vocabulary grounding in Sec. 3.3.2, lastly, we detail the +training process via knowledge induction in Sec. 3.3.3. +3.3.1. Dataset Construction +Here, we introduce the idea to construct the training set with +{visual feature, segmentation, text prompt} triplets, specif- +ically, we develop an automatic pipeline to construct the +{image, segmentation, text prompt} triplets, thus the visual +feature can be obtained from Stable Diffusion via the for- +ward inference to generate the image. +In practise, we prepare a vocabulary with common ob- +ject categories, for example, the classes in PASCAL VOC +can form a category set as Cpascal = {‘dog’, ‘cat’, . . . }, +|Cpascal| = 20, we randomly select a number of classes to +construct a text prompt for image generation (e.g., ‘a pho- +tograph of a dog and cat’). Repeating the above operation, +we can theoretically generate infinite amount of image and +text prompt pairs. To acquire the segmentation masks, we +take an off-the-shelf object detector, e.g., pre-trained Mask +R-CNN [22], and run the inference procedure on the gener- +ated images: +mgt +i = Φdetector(Ii), where Ii = Φdiffusion(ϵ, yi), +(4) +where mgt +i ∈ {0, 1}H×W ×Oi refers to the predicted masks +for a total of Oi objects in the generated image Ii, condi- +tioning on the text prompt yi. +To evaluate the effectiveness of the induction proce- +dure for open-vocabulary grounding, we divide the vocab- +ulary set into seen categories (Cseen) and unseen categories +(Cunseen), the training set (Dtrain) only has images of seen +categories, and the test set (Dtest) has both seen and unseen +categories. +3.3.2. Architecture +Given one specific training triplet (Fi, mgt +i , yi), we now de- +tail three trainable components in the proposed grounding +module: visual encoder, text encoder, and fusion module. +Visual Encoder. Given the visual representation from Sta- +ble Diffusion, we concatenate features with the same spa- +tial resolution (from UNet encoding and decoding path) +to obtain {f 1 +i , . . . , f n +i }, where f k +i +∈ R +h +2k × w +2k ×dk, k ∈ +4 + +X本𝟏 × 𝟏 Conv +Upsample +Upsample +𝟏 × 𝟏 Conv +Concat +Mix-Conv +Upsample +𝟏 × 𝟏 Conv +Concat +Mix-Conv +Upsample +𝟏 × 𝟏 Conv +Concat +Mix-Conv +𝒇𝟏 +𝒇𝒏−𝟐 +𝒇𝒏−𝟏 +𝒇𝒏 +... +... +Visual Encoder +Figure 3. The architecture of visual encoder. The features ex- +tracted from UNet are first grouped according to their resolution, +then gradually upsampled and fused into the final visual feature. +{0, . . . , n} denotes layer indices of UNet, dk refers to the +feature dimension. +Next, we input {f 1 +i , . . . , f n +i } to visual encoder for gen- +erating the fused visual feature ˆFi = Φv-enc({f 1 +i , . . . , f n +i }). +As shown in Fig 3, visual encoder consists of 4 types of +blocks: (1) 1×1 Conv for reducing feature dimensionality, +(2) Upsample for upsampling the feature to a higher spa- +tial resolution, (3) Concat for concatenating features, and +(4) Mix-conv for blending features from different spatial +resolutions, that includes two 3 × 3 Conv operations with +residual connection and a conditional batchnorm +operation, similar to [18]. +Text Encoder. We adopt the language model from pre- +trained Stable Diffusion, specifically, given the text prompt +yi, the text encoder output the corresponding embeddings +for all visual objects: Eobji = Φt-enc(g(yi)). +Fusion Module. +The fusion module computes interac- +tion between visual features and text embeddings, then out- +puts segmentation masks for all visual objects. +In spe- +cific, we use a standard transformer decoder with three lay- +ers, the text embeddings are treated as Query, that itera- +tively attend the visual feature for updating, and are further +converted into per-segmentation embeddings with a Multi- +Layer Perceptron (MLP). The object segmentation masks +can be obtained by dot product visual features with the per- +segmentation embeddings. Formally, the procedure can be +denoted as : +Esegi = ΦTRANSFORMER-D(W Q·Eobji, W K· ˆFi, W V · ˆ +Fi) +(5) +mi = ˆFi · [ΦMLP(Esegi)]T +(6) +where the transformer decoder generates per-segmentation +embedding Esegi ∈ RN×de for all visual objects described +in the text prompt, W Q, W K, W V refer to the learnable pa- +rameters for Query, Key and Value projection. +3.3.3. Training +With the constructed dataset, we can now start supervised +training the proposed grounding module: +L = − 1 +N +N +� +i=1 +[mgt +i · log(σ(mi)) + (1 − mgt +i ) · log(σ(1 − mi))] +where mgt +i +∈ RH×W ×Oi refers to the oracle groundtruth +segmentation from the off-the-shelf object detector, and +mi ∈ RH×W ×Oi refers to the predicted segmentation from +our grounding module, σ(·) refers to Sigmoid function. +In practise, while using the off-the-shelf detector to gen- +erate segmentation masks, the model may sometimes fail to +detect the objects mentioned in the text prompt, and out- +put incorrect segmentation masks. Such error comes from +two sources, (i) the diffusion model may fail to generate +high-quality images; (ii) off-the-shelf detector may fail to +detect the objects in the synthetic image, due to the domain +gap between synthetic and real images. Here, we consider +two training strategies, Normal Training, where we fully +trust the off-the-shelf detector, and use all predicted seg- +mentation masks to train the grounding module; alterna- +tively, we also try Training w.o. Zero Masks, as we em- +pirically found that the majority of failure cases come from +false negatives, that is to say, the detector failed to detect the +objects and output an all-zero mask, therefore, we can train +the grounding modules by ignoring the all-zero masks. +4. Experiments +In this section, we detail the evaluation detail for val- +idating the effectiveness of objects grounding during im- +age generation, specifically, we consider two protocols: in +Sec. 4.1, we train the grounding module with the con- +structed training set, and test the segmentation performance +on generated images from Stable Diffusion, with the detec- +tor’s output as oracle groundtruth for evaluation; in Sec. 4.2, +we use the guided diffusion model to construct a synthe- +sized semantic segmentation dataset, and train a segmen- +tation model on it, we evaluate the model on the existing +benchmarks for zero-shot segmentation. Lastly, in Sec. 4.3, +we conduct a series of ablation studies on the different train- +ing strategies. +4.1. +Protocol-I: Grounded Generation +Here, we train the grounding module with our con- +structed training set, as described in Sec. 3.3.1, specifically, +the training set only consists of a subset of common (seen) +categories, while the testing set consists of both seen and +unseen categories. In the following, we describe the imple- +mentation and experimental results in detail, to thoroughly +assess the model for grounded generation. +Following the split rule in [7,11,37], we adopt two differ- +ent sets of categories: (i) we adopt the categories defined in +PASCAL VOC [9], divide them into 15 seen categories and +5 unseen categories; (ii) we adopt the category definition in +MS-COCO [20], divide them into 65 seen categories and +15 unseen categories. The dataset constructed for these two +sets are called PASCAL-sim and COCO-sim, respectively. +5 + +FTest +Setting +PASCAL-sim +COCO-sim +# Objects +One +Two +One +Two +Categories +Seen +Unseen +Seen +Seen +Unseen +Unseen +Seen +Unseen +Seen +Seen +Unseen +Unseen +DAAM [33] +Split1 +61.66 +75.63 +46.74 +51.31 +69.94 +62.25 +55.56 +49.68 +52.06 +43.35 +Split2 +65.75 +59.25 +49.08 +47.98 +41.50 +60.08 +65.55 +48.80 +54.66 +33.22 +Split3 +67.11 +53.82 +48.80 +48.28 +41.41 +62.81 +52.48 +50.85 +49.84 +45.80 +Average +64.84 +62.90 +48.21 +49.19 +50.95 +61.71 +57.76 +49.78 +52.19 +40.79 +Ours +Split1 +90.16 +83.19 +78.93 +66.07 +57.93 +83.35 +76.81 +64.64 +57.15 +47.77 +Split2 +90.08 +86.19 +78.68 +67.10 +47.21 +82.83 +74.93 +63.39 +57.18 +42.82 +Split3 +90.67 +79.86 +79.68 +70.42 +62.07 +84.85 +67.89 +65.70 +54.60 +42.62 +Average +90.30 +83.08 +79.10 +67.86 +55.74 +83.68 +73.21 +64.16 +56.31 +44.40 +Table 1. Quantitative result for Protocol-I evaluation on grounded generation. Our model has been trained on the synthesized training +dataset, that consists of images with one or two objects from only seen categories, and test on our synthesized test dataset that consists of +images with one or two objects of both seen and unseen categories. Split1, Split2 and Split3 refer to 3 different splits of C that construct 3 +different (Cseen, Cunseen) pairs, respectively. Our model outperforms the DAAM [33], by a large margin, see text for more detailed discussion. +Training Set. For PASCAL-sim or COCO-sim, we gen- +erate a synthetic training set by randomly sampling im- +ages from pre-trained Stable Diffusion. This exposes our +grounding module to a great variety of data (> 40k) at +training time, and the model is unlikely to see the same la- +beled examples twice during training. In contrast to previ- +ous work, such as BigDatesetGAN [18], where only a single +object is considered, we construct the text prompt with one +or two objects at a time, note that, for training, only the seen +categories are sampled. Although we can certainly generate +images with more than two object categories, the quality +of the generated images tends to be unstable, limited by the +generation ability of Stable Diffusion, thus we only consider +synthesized images with less than three object categories. +Testing Set. For the evaluation purpose, we generate two +synthetic test sets with offline sampling for PASCAL-sim +and COCO-sim, respectively. In total, we have collected +about 1k images for PASCAL-sim, and about 5k images +for COCO-sim, we run the off-the-shelf object detector on +these generated images to produce the oracle groundtruth +segmentation. For both test sets, the images containing two +categories will be divided into three groups: both seen, both +unseen, one seen and one unseen. We leave the detailed +statistics of our synthetic dataset in the supplementary ma- +terial. Note that, we have manually checked all the images +and the oracle groundtruth segmentation produced from the +off-the-shelf detector, and only keep the high-quality ones, +thus the performance evaluation of the grounding module +can be a close proxy. +Evaluation Metrics. +We use the category-wise mean +intersection-over-union (mIoU) as evaluation metric, de- +fined as averages of IoU over all the categories: mIoU += 1 +C +�C +c=1 IoUc, where C is the number of all target cate- +gories, and IoUc is the intersection-over-union for the cate- +gory with index is c. +Implementation Details. +We use the pre-trained Stable +Diffusion [27] and the text encoder of CLIP [25] in our im- +plementation. We choose the Mask R-CNN [22] trained on +COCO dataset as our object detector for oracle groundtruth +segmentation. We fuse features from U-Net and upsample +them into 512 × 512 spatial resolution, the text and visual +embeddings are both mapped into 512 dimension before +feeding into the fusion module. We train our grounding +module with two NVIDIA GeForce RTX 3090 GPUs for +5k iterations with batch size equal to 8, ADAM [15] opti- +miser with β1 = 0.9, β2 = 0.999. The initial learning rate +is set to 1e-4 and the weight decay is 1e-4. +Results. As shown in Tab. 1, we provide experimental re- +sults for our grounded generation model, we change the +composition of categories three times and compute the re- +sults for each split. Here, we can make the following ob- +servations: first, our model significantly outperforms the +unsupervised method DAAM [33] in the mIoU on all test +settings. This is because DAAM tends to result in ambigu- +ous segmentations, as the textual embedding for individual +visual entity will largely be influenced by other ones within +the global sentence at the text encoding stage; second, our +grounding module achieves superior performance on both +seen and unseen categories, indicating its open-vocabulary +nature, i.e., the guided diffusion model can synthesize im- +ages and their corresponding segmentations for more cate- +gories beyond the vocabulary of the off-the-shelf detector, +as described in Sec. 3.3.3. +Visualization. We demonstrate the visualization results in +Fig. 4. +On both seen and unseen categories, our model +can successfully ground the objects in terms of segmenta- +tion mask. Impressively, as shown in Fig. 1, our grounding +module can even segment the objects beyond any off-the- +shelf detector can do, showing the strong open-vocabulary +grounded generation ability of our model. +6 + +car +car +dog +bottle +sofa +sofa +train +hot dog +hot dog +bear +bear +backpack +apple +frisbee +Image +Image +Image +Image +Ours +Oracle GT +Ours +Ours +Ours +Oracle GT +Oracle GT +Oracle GT +bottle +backpack +apple +motorbike +motorbike +Figure 4. Segmentation results of PASCAL-sim (left) and COCO-sim (right) on seen (motorbike, bottle, backpack and apple) and +unseen (sofa, car, hot dog and bear) categories. Our grounded generation model achieves comparable segmentation results to the oracle +groundtruth generated by the off-the-shelf object detector. +dog +cat +bird +train +bus +pottedplant +sheep +cow +boat +boat +car +bottle +horse +chair +sofa +motorbike +person +pottedplant +person +motorbike +boat +pottedplant +bottle +dog +train +bottle +bird +Figure 5. Our synthesized semantic segmentation dataset with one category (left) and two categories (right) for Protocol-II training. +4.2. +Protocol-II: Open-vocabulary Segmentation +In the previous protocol, we have validated the ability for +open-vocabulary grounded generation, however, even after +being manually checked, the oracle groundtruth from off- +the-shelf detector may also be inaccurate at the boundary. +Here, we introduce another experiment to validate the effec- +tiveness of the grounding module, in particular, we first con- +struct a synthesized image-segmentation dataset with the +guided Stable Diffusion, then train a semantic segmentation +model on such a synthetic dataset, and evaluate it on public +image segmentation benchmarks. +Overall, the segmentation model is challenged from the +following two perspectives: first, it has only been trained +on synthetic images, that resembles a zero-shot Sim2Real +transfer; second, the groundtruth masks for unseen object +categories are generated from our guided Stable Diffusion +that has only been trained on seen categories. Therefore, +with such an evaluation protocol, on the one hand, it can +reflect the effectiveness of our grounding module from the +performance on segmenting unseen categories, and more +importantly, it introduces a promising application, i.e., use +our guided Stable Diffusion to expand the vocabulary be- +yond any existing detector can do. +Dataset. In order to train the semantic segmentation model, +we synthesize a dataset with 10k image-segmentation pairs +for 20 categories (both seen and unseen) as shown in Fig. 5. +All the image-segmentation pairs are generated by our +guided Stable Diffusion, trained with only 15 seen cate- +gories in PASCAL VOC. We do not finetune on PASCAL +Training Dataset +mIoU +Methods +Type +Categories Objects Seen Unseen Harmonic +ZS3 [3] +real +15 +- +78.0 +21.2 +33.3 +SPNet [37] +real +15 +- +77.8 +25.8 +38.8 +CaGNet [11] +real +15 +- +78.6 +30.3 +43.7 +Joint [1] +real +15 +- +77.7 +32.5 +45.9 +STRICT [24] +real +15 +- +82.7 +35.6 +49.8 +SIGN [5] +real +15 +- +83.5 +41.3 +55.3 +ZegFormer [7] +real +15 +- +86.4 +63.6 +73.3 +Ours +synthetic +15 + 5 +one +62.8 +50.0 +55.7 +synthetic +15 + 5 +two +65.8 +60.1 +62.8 +synthetic +15 + 5 +mixture 69.5 +63.2 +66.2 +Table 2. Comparison with the previous ZS3 methods on PAS- +CAL VOC. The “Seen”, “Unseen”, and “Harmonic” denote mIoU +of seen categories, unseen categories, and their harmonic mean. +These ZS3 methods are trained on PASCAL VOC training set. +VOC and only evaluate on its test set (1,449 images). +Training Details. To compare with other open-vocabulary +methods, our semantic segmentation model uses Mask- +Former [4] with ResNet101 as its backbone. The image res- +olution for training is 224×224 pix, and we train the model +on our synthetic dataset for 40k iterations with batch size +equal to 8. We use the ADAMW as our optimizer with a +learning rate of 1e-4 and the weight decay is 1e-4. +Comparison on Zero-Shot Segmentation (ZS3). +In +Tab. 2, we compare with the existing zero-shot semantic +segmentation approaches. Despite being only trained on +a synthetic dataset, our model outperforms most of ZS3 +approaches on unseen categories. Specifically, the model +7 + +ELMJAL +fewaleACHBNWARCOImage +GT Mask +Zegformer +MaskFormer +(train on synthestic set) +Image +GT Mask +Zegformer +MaskFormer +(train on synthestic set) +Figure 6. Visualization of zero-shot segmentation results on Pascal-VOC. MaskFormer trained on our synthetic dataset achieves com- +parable performance with Zegformer (the state-of-the-art zero-shot semantic segmentation method) in segmenting unseen categories, i.e. +pottedplant, sofa and tvmonitor. Note that although MaskFormer has seen these categories during training, the image-segmentation pairs +of these categories are generated with our grounding module. +Training Type +One +Two +Seen Unseen Seen Seen +Unseen Unseen +Normal Training +89.88 +71.18 +77.66 +57.24 +44.22 +Training w.o. Zero Masks 90.16 +83.19 +78.93 +66.07 +57.93 +Table 3. Ablation on training type on the constructed dataset. +Performance is measured by mIoU on PASCAL-sim test set. +trained on the mixture of one and two objects achieves the +best performance. As shown in Fig. 6, our model obtains +accurate segmentation on both seen and unseen categories. +Therefore, we can have the following observations: (i) the +grounding module is capable of segmenting unseen cat- +egories despite it has never seen any segmentation mask +during the knowledge induction procedure, validating the +strong generalisation of the grounding module in the guided +Stable Diffusion; (ii) it is possible to segment more object +categories by simply training on synthesized datasets. +4.3. +Ablation study +In this section, we show the effect of different training +loss and different timestep for extracting visual represen- +tation, due to the space limitation, we refer the reader for +supplementary material, for the study on the different num- +ber of objects in the synthetic datasets or seen categories, +and the effect of different datasets. +Normal Training v.s. Training without Zero Masks. As +shown in Tab. 3, Normal Training results in unsatisfac- +tory performance on unseen categories, we conjecture this +is because the errors from detector tend to be false nega- +tive, that bias our grounding module to generate all-zero +segmentation masks when encountering unseen categories; +in contrast, by ignoring all-zero masks at training, Train- +ing w.o. Zero Masks achieves equally good performance +on both seen and unseen categories. +Timesteps for Extracting Visual Representation. +We +Figure 7. Ablation on timesteps. The mIoU is measured for the +model with extracting features from Stable Diffusion in different +timesteps on PASCAL-sim. +compare the performance by extracting visual representa- +tion from Stable Diffusion at different timesteps, the results +on PASCAL-sim can be seen in Fig. 7, showing that as the +denoising steps gradually decrease, i.e., from t = 0 −→ 50, +the performance for grounding tends to decrease in general, +when t = 5, the best result is obtained. +5. Conclusion +In this paper, we propose a novel idea for guiding the ex- +isting Stable Diffusion towards open-vocabulary grounded +generation, i.e., segmenting the visual entities described in +the text prompt while generating images. Specifically, we +introduce a grounding module that explicitly aligns the vi- +sual and textual embedding space of the Stable Diffusion +and train such module with an automatically constructed +dataset, consisting of {image, segmentation, text prompts} +triplets. Experimentally, we show that visual-language cor- +8 + +0.9 +0.8 +0.7 +0.6 +lou +E +0.5 +0.4 +one (seen) +0.3 +one (unseen) +two (seen) +0.2 +two (seen+unseen) +two (unseen) +0.1 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Timestepsbacial12.30.2006respondence can be established by only training on a lim- +ited number of object categories, while getting the abil- +ity for open-vocabulary grounding at the image genera- +tion procedure. Additionally, we generate a synthetic se- +mantic segmentation dataset using our guided Stable Dif- +fusion and train a semantic segmentation model. Without +finetuning, the model can directly transfer to real images, +and show competitive performance to existing zero-shot se- +mantic segmentation approaches on PASCAL VOC dataset, +opening up new opportunities to exploit generative model +for discriminative tasks. +References +[1] Donghyeon Baek, Youngmin Oh, and Bumsub Ham. 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Dataset for Training Semantic Segmentation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +C. Additional Ablation Study +16 +C.1. Synthetic Dataset Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +C.2. Effect on the Number of Seen Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +C.3. Dataset Construction via DDIM Inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +D. More Qualitative Results +18 +E. Limitation & Future Work +18 +11 + +In this supplementary document, we start by giving more details on the architecture of our grounding module in Section A, +followed by the details for generating the dataset for training it in Section B; then describe the additional ablation studies, as +promised in the main text in Section C; In Section D, we present more qualitative results; Lastly, we illustrate the limitation +of our method and our future work in Section E. +A. Details on the Architecture of Grounding module +We show the detailed architecture of our grounding module in Fig. 8, which consists of visual encoder, text encoder, +transformer decoder and MLP in the fusion module. +“ a photograph +of the +{ class name } ” +dog +cat +Visual +Encoder +𝑻𝒅𝒐𝒈 +𝑻𝒄𝒂𝒕 +O Text Embeddings +… +… +Linear Flatten +(K, V) +Transformer +Decoder +⊗ +MLP +Q +Text +Encoder + +. . . +Features from +diffusion model +𝑯 × 𝑾 × 𝑫 +𝑶 × 𝒅𝒆 +𝑶 × 𝒅𝒆 +Visual Tokens +𝑶′ × 𝒅𝒆 +𝑴𝒅𝒐𝒈 +𝑴𝒄𝒂𝒕 +O Mask Embeddings +… +𝑶 × 𝑫 +O class-specific masks +… +𝑯 × 𝑾 +Figure 8. Detailed architecture of our grounding module. We first generate O text embeddings by injecting the class names into a +prompt template and then feeding them to a pre-trained text encoder. The visual encoder takes the features from Stable Diffusion as input +and outputs fused visual features, which are then flattened to a sequence of visual tokens. Next, we feed the visual tokens into a transformer +decoder as Key and Value, and feed text embeddings as Query. The outputs of transformer decoder are then fed into an MLP to obtain +O mask embeddings. Mask embeddings are dot producted with the output features of visual encoder to generate O class-specific binary +masks. +Visual Encoder. The input {f 1, . . . , f n} are extracted from Stable Diffusion [27], and the visual encoder aims to upsample +and fuse the visual feature, and output the visual feature map, ˆF = Φv-enc({f 1, . . . , f n}), ˆF ∈ RH×W ×D (here we use +H = W = 512, and D = 240). Note that, the resolution of the fused visual feature is the same as the resolution of the +generated image, and the segmentation masks. +Text Encoder. We adopt the pre-trained text encoder from CLIP [25], which is also used in Stable Diffusion [27]. It takes +text prompt y as input and outputs the corresponding text embedding: Eobject = Φt-enc(y), Eobject ∈ RO×dtext, where O is the +total number of objects of interest and dtext = 768. +Transformer Decoder in Fusion Module. Similar to the operation in standard ViT architecture [8], we convert the visual +features ˆF ∈ RH×W ×D into visual tokens ˆFflatten ∈ RO′×dvisual, where O′ = +HW +p2 += 16384 is the number of tokens, p +refers to the patch size and dvisual = p2 × D = 3840. Then the visual tokens and text embeddings are mapped into the same +dimension de = 512 with MLPs, and passed into the transformer decoder with three layers. The text embeddings are treated +as query with dimension O × de, and the visual tokens are treated as key and value with dimension O′ × de. The output +of transformer decoder is of the same resolution as query, with dimension O × de. +MLP in Fusion Module. At last, we use an MLP to map the output of transformer decoder into mask embeddings with +dimension O × D, which are then dot producted with the fused visual feature ( ˆF ∈ RH×W ×D) to generate class-specific +binary masks (O × H × W), each mask is of the same spatial resolution of the generated image. +12 + +B. Details on the Synthetic Dataset +B.1. Dataset Split +Here, to train our proposed grounding module, and properly evaluate its ability for segmenting the objects that are unseen +at training time, we construct the training dataset with images of only seen categories, and the test dataset consists of both +seen and unseen categories. The detail of the split on PASCAL-sim and COCO-sim, i.e. the split of seen categories and +unseen categories, is shown in Tab. 4, where PASCAL-sim has 15 seen categories and 5 unseen categories, COCO-sim has +65 seen categories and 14 unseen categories. Note that, we ignore the category: ‘mouse’ in the COCO-sim since the diffusion +model generates ‘rat’ in the image for the category: ‘mouse’, while ‘mouse’ in the vocabulary of the off-the-shelf detector +means mouse as a computer accessory, thus the detector fails to detect the category ‘mouse’ in the image generated by the +diffusion model. +Categories +seen +Unseen +PASCAL-sim +Split1 +aeroplane, bicycle, bird, boat, bottle, bus, cat, chair, cow, diningtable, horse, motorbike, per- +son, pottedplant, sheep +tvmonitor, car, dog, sofa, +train +Split2 +tvmonitor, car, dog, sofa, train, aeroplane, bicycle, bird, boat, bottle, bus, cat, chair, cow, +diningtable +horse, motorbike, person, +pottedplant, sheep +Split3 +horse, motorbike, person, pottedplant, sheep, tvmonitor, car, dog, sofa, train, aeroplane, bicy- +cle, bird, boat, bottle +bus, cat, chair, cow, din- +ingtable +COCO-sim +Split1 +person, bicycle, car, motorbike, bus, truck, boat, traffic light, fire hydrant, stop sign, bench, +bird, dog, horse, sheep, cow, elephant, zebra, giraffe, backpack, umbrella, handbag, tie, skis, +sports ball, kite, baseball bat, baseball glove, skateboard, surfboard, tennis racket, bottle, wine +glass, cup, knife, spoon, bowl, banana, apple, orange, broccoli, carrot, pizza, donut, cake, +chair, bench, pottedplant, bed, diningtable, tvmonitor, laptop, remote, keyboard, cell phone, +microwave, oven, sink, refrigerator, book, clock, vase, scissors, teddy bear, toothbrush +aeroplane, train, parking +meter, cat, bear, suitcase, +frisbee, snowboard, fork, +sandwich, hot dog, toilet, +toaster, hair drier +Split2 +aeroplane, train, parking meter, cat, bear, suitcase, frisbee, snowboard, fork, sandwich, hot +dog, toilet, toaster, hair drier, person, bicycle, car, motorbike, bus, truck, boat, traffic light, +fire hydrant, stop sign, bench, bird, dog, horse, sheep, cow, elephant, zebra, giraffe, back- +pack, umbrella, handbag, tie, skis, sports ball, kite, baseball bat, baseball glove, skateboard, +surfboard, tennis racket, bottle, wine glass, cup, knife, spoon, bowl, banana, apple, orange, +broccoli, carrot, pizza, donut, cake, chair, bench, pottedplant, bed, diningtable, tvmonitor +laptop, remote, keyboard, +cell +phone, +microwave, +oven, +sink, +refrigerator, +book, clock, vase, scissors, +teddy bear, toothbrush +Split3 +laptop, remote, keyboard, cell phone, microwave, oven, sink, refrigerator, book, clock, vase, +scissors, teddy bear, toothbrush, aeroplane, train, parking meter, cat, bear, suitcase, frisbee, +snowboard, fork, sandwich, hot dog, toilet, toaster, hair drier, person, bicycle, car, motorbike, +bus, truck, boat, traffic light, fire hydrant, stop sign, bench, bird, dog, horse, sheep, cow, +elephant, zebra, giraffe, backpack, umbrella, handbag, tie, skis, sports ball, kite, baseball bat, +baseball glove, skateboard, surfboard, tennis racket, bottle, wine glass, cup, knife, spoon, bowl +banana, +apple, +orange, +broccoli, +carrot, +pizza, +donut, cake, chair, bench, +pottedplant, +bed, +din- +ingtable, tvmonitor +Table 4. The details on the split of categories on PASCAL-sim and COCO-sim. +13 + +B.2. Dataset for Training Grounding Module +To construct the training set, (1) we first randomly select one or two categories from the seen ones, where objects tend +to co-appear in natural images, based on the annotation in PASCAL VOC [9] or COCO [20], called co-appearing category +pair, and use the prompt template to decorate these selected categories, thus we can obtain the text prompt; (2) we pass the +text prompt and randomly sampled Gaussian noise to the Stable Diffusion [27] to obtain the generated image; (3) next, we +pass the generated image to the off-the-shelf detector to obtain the oracle segmentation mask; (4) finally, we can construct +the triplet which consists of the generated image, oracle segmentation mask, and text prompt; (5) repeat the above procedure, +we can generate infinite triplets for the training set. Algorithm 1 displays the procedure for generating the training set. +To construct the test set for evaluating the grounding module, we can use a procedure similar to the training set. The +differences are: (i) we use all categories, including seen and unseen categories to construct the test set. (ii) to obtain more +reliable test results, we only add the triplet to the test set when the generated image and oracle segmentation mask have high +quality which is checked manually, i.e., the generated image by Stable Diffusion contains the recognizable objects of selected +categories, and the off-the-shelf detector successfully produces the high-quality oracle segmentation mask. +In this paper, PASCAL-sim has 20 categories and 142 co-appearing category pairs. We construct 30 triplets per category +and 5 triplets per co-appearing category pair for PASCAL-sim test set. In total, PASCAL-sim test set has 1310 triplets. +COCO-sim has 79 categories and 1559 co-appearing category pairs. We construct 30 triplets per category and 2 triplets per +co-appearing category pair for COCO-sim test set. In total, COCO-sim test set has 5488 triplets. +Algorithm 1 Constructing the dataset for training grounding module (pseudocode in PyTorch-like style). +# C_seen: the list of seen categories +# img_shape: the shape of expected generated image +# exp_train_size: the expected size of training set +# n: the number of selected categories, n = 1 or 2 +# co-appearing_category_pair_list: a list containing all co-appearing category pairs, where objects tend to +# co-appear in natural images, based on the annotation in PASCAL VOC or COCO +D_train = [] #initialize the training set +while (len(D_train) < exp_train_size): +y = None #initialize the text prompt +#randomly select n categories from seen categories +selected_class_list = random_select(C_seen, n) +if n = 1: +class = select_class_list[0] +# decorate the selected category by a pre-defined prompt template, e.g., "a photograph of a [class name]" +y = prompt_template(class) +else if n = 2: +class1, class2 = select_class_list[0], select_class_list[1] +if (class1, class2) in co-appearing_category_pair_list: +# decorate the selected categories by a pre-defined prompt template, e.g., "a photograph of a +# [class1 name] and a [class2 name]" +y = prompt_template(class1, class2) +if y != None: +#randomly sample a Gaussian noise epsilon +epsilon=torch.randn(img_shape) +# pass the noise and text prompt to the diffusion model to generate image I +I = diffusion_model(epsilon, y) +# pass the generated image to the off-the-shelf detector to obtain the oracle segmentation mask m +m = pretrain_detector(I) +# add the triplet (generated image, oracle segmentation mask, text prompt) to the training set +D_train.append((I, m, y)) +14 + +B.3. Dataset for Training Semantic Segmentation Model +As discussed in Sec. 4.2, we synthesize a semantic segmentation dataset for all 20 categories in PASCAL VOC [9]. +Specifically, we first randomly select one category or two categories (co-appearing category pair), to obtain the text prompt, +and then pass randomly sampled Gaussian noise and text prompt to the diffusion model to obtain the generated image, and +use our proposed grounding module to get the corresponding segmentation mask. Thus, we can get the pair consisting +of generated image and generated segmentation mask. Repeat the above procedure, we can obtain the synthetic semantic +segmentation dataset at a large scale. Algorithm 2 displays the procedure for generating the synthetic semantic segmentation +dataset. +In this paper, the synthetic semantic segmentation dataset consists of 500 images per category and 71 images per co- +appearing category pair. Thus, there exist 10k images for 20 categories and 10082 images for 142 co-appearing category +pairs in total. +Algorithm 2 Pseudo-code for generating the synthetic semantic segmentation dataset in a PyTorch-like style. +# C: the list of all categories +# img_shape: the shape of expected generated image +# exp_dataset_size: the expected size of synthesis semantic segmentation dataset +# n: the number of selected categories, n = 1 or 2 +# co-appearing_category_pair_list: a list containing all co-appearing category pairs, where objects tend to +# co-appear in natural images, based on the annotation in PASCAL VOC or COCO +D_seg = [] #initialize the synthesis semantic segmentation dataset +while (len(D_seg) < exp_dataset_size): +y = None #initialize the text prompt +#randomly select n categories from all categories +selected_class_list = random_select(C, n) +if n = 1: +class = select_class_list[0] +# decorate the selected category by a pre-defined prompt template, e.g., "a photograph of a [class name]" +y = prompt_template(class) +else if n = 2: +class1, class2 = select_class_list[0], select_class_list[1] +if (class1, class2) in co-appearing_category_pair_list: +# decorate the selected categories by a pre-defined prompt template, e.g., "a photograph of a +# [class1 name] and a [class2 name]" +y = prompt_template(class1, class2) +if y != None: +#randomly sample a Gaussian noise epsilon +epsilon=torch.randn(img_shape) +# pass the noise and text prompt to the diffusion model with grounding module to generate image I +# and segmentaion mask m +I, m = diffusion_model_with_grounding(epsilon, y) +# add the pair (generated image, generated segmentation mask) to the synthesis semantic segmentation dataset +D_seg.append((I, m)) +15 + +C. Additional Ablation Study +C.1. Synthetic Dataset Construction. +We explore the effect of constructing different datasets for training the grounding module, by varying the number of +objects in the images. As shown in Tab. 5, training on the combination of one and two object categories gives the best results +overall. +Train Set +# Objects +One +Two +Seen +Unseen +Seen +Seen +Unseen Unseen +single +90.37 +83.85 +43.89 +42.33 +38.91 +two +88.35 +82.93 +80.56 +68.08 +56.36 +mixture +90.16 +83.19 +78.93 +66.07 +57.93 +Table 5. Ablation on dataset construction on PASCAL-sim. The bolded number indicates the best result. Our model achieves the best +performance when training on the combination of one and two object categories. +C.2. Effect on the Number of Seen Categories. +We ablate the number of seen categories to further explore the generalisation ability of our proposed grounding module. +As shown in Tab. 6, the grounding module can generalise to unseen categories, even as few as five seen categories; when +introducing more seen categories, the performance on unseen ones consistently improves, but decreases on seen ones, due to +the increasing complexity on seen categories. +Train Set +# Seen Categories / unseen categories +One +Two +Seen +Unseen +Seen +Seen +Unseen Unseen +5 +/ +74 +94.81 +72.42 +87.19 +49.60 +39.00 +20 +/ +59 +91.91 +73.33 +71.59 +56.27 +41.91 +35 +/ +44 +87.23 +73.85 +66.91 +55.99 +43.28 +50 +/ +29 +84.55 +73.20 +66.41 +54.39 +42.71 +65 +/ +14 +83.85 +76.81 +64.64 +57.15 +47.77 +Table 6. Ablation on the number of seen categories on COCO-sim. The bolded number indicates the best result. Our model can +generalise to unseen categories, even as few as five seen categories. +C.3. Dataset Construction via DDIM Inverse. +In addition to using the off-the-shelf detectors, we also consider constructing the training set by utilising the inverse +process of diffusion to explicitly generate images close to those in the public dataset, for example, PASCAL VOC, and train +the grounding module with the mask annotations available from the dataset. +Here, we describe an inverse procedure that enables to find a deterministic mapping from noise to images, given the +sampling rule being non-Markovian, for example, Denoising Diffusion Implicit Model (DDIM) [32] with the reverse process +variance to be 0. In DALL-E 2 [26], such inversion has been used to determine the noise that produces a specific image. In +our considered Stable Diffusion, the image is first mapped to a latent vector z0 by the pre-trained variational autoencoder +(VAE), at each step of DDIM inversion, zt+1 is obtained from zt and the predicted noise term of UNet, that takes zt and text +prompt y as input, ending up with an inverted noise zT eventually. In this paper, we exploit such DDIM inversion to train our +grounding module with the dataset constructed from real image and segmentation masks. +In particular, the first option enables to directly inherit the segmentation mask from the public dataset, and the text prompt +can be manually constructed by inserting class labels into the prompt template, for example, if the segmentation mask +contains ‘dog’ and ‘cat’, the text prompt can be ‘a photograph of a dog and cat’. Besides, the visual feature can be obtained +by extracting the feature from the UNet of Stable Diffusion when t = 1 at the inversion process. +Constructed Dataset v.s. Real Dataset. +We explore the difference between training on constructed dataset and real +dataset (PASCAL VOC) from two perspectives. First, we compare their performance on PASCAL-sim dataset for grounded +generation in Tab. 7 (left). Though we successfully train our grounding module on real dataset, the domain gap limits its +16 + +Dataset +Type +PASCAL-sim +PASCAL-test +One +Two +Seen Unseen Seen Seen+Unseen Unseen Seen Unseen +real +75.67 +61.26 +64.08 +49.23 +45.14 +75.19 +34.80 +sim(10k) 88.77 +70.04 +73.07 +57.08 +46.59 +61.75 +48.42 +sim(40k) 90.16 +83.19 +78.93 +66.07 +57.93 +64.44 +53.86 +sim+real 89.57 +76.23 +78.12 +62.24 +55.30 +73.32 +57.14 +Table 7. Ablation on the training dataset. The bold numbers indicate the best results. Specifically, ’sim’ and ’real’ denote the constructed +dataset and real dataset (PASCAL-VOC), respectively. +performance on grounded generation task. Considering PASCAL VOC only contains about 10k images, we adjust the con- +struced dataset to the same magnitude and get better results. Additionally, because of the good scalability of the constructed +dataset, the performance will be better as the number of images increases. Second, we evaluate the grounding module on +PASCAL VOC test dataset by DDIM inversion as shown in Tab. 7 (right). Note that, under this circumstance, our model ap- +proximates a discriminative model. On seen categories of PASCAL VOC test set, the module trained on real dataset achieves +the best result, while the module trained on constructed dataset gains an advantage on unseen categories. Besides, we also try +to train our module on both constructed dataset and real dataset, which results in great improvement on PASCAL VOC test +dataset but no advantage on PASCAL-sim, while the latter is our main task. Therefore, we finally choose the module training +on constructed dataset as our main model. +17 + +D. More Qualitative Results +We provide more qualitative results in Fig. 9, Fig. 10, Fig. 11, and Fig. 12. Note that the images are generated from Stable +Diffusion [27], and the corresponding masks are inferred from our proposed grounding module. Specifically, the generated +images and their corresponding segmentation masks in Fig. 9 and Fig. 10, including common objects, e.g., shark, turtle, and +more unusual objects, e.g., Ultraman, pterosaur, Chinese dragon, unicorn and dinosaur, shows the strong generalisability of +the grounding module. In Fig. 11, we show more examples from our synthetic semantic segmentation dataset. In Fig. 12, +we compare the model trained on our synthesized datasets with other ZS3 methods on PASCAL VOC dataset [9]. We can +observe that the MaskFormer [4] trained on our synthetic semantic segmentation dataset can obtain accurate segmentation on +both seen and unseen categories, showing that the guided text-to-image diffusion model can be used to expand the vocabulary +of pre-trained detector. +E. Limitation & Future Work +In this paper, we have demonstrated the possibility for aligning the visual and language representation of a text-to-image +diffusion model, and augment it with the ability of grounding visual objects along with generation. However, we also realise +there exists certain limitation in this work, first, we only consider to ground the nouns that indicate visual entities, it would +be interesting to ground the human-object, object-object interactions, or even verbs in the future, second, we are inserting the +grounding module to a pre-trained text-to-image generative model, it would be interesting to co-train the two components, +potentially enabling to generate images with higher quality and explainability. +18 + +a photograph of a superman next to a tree +a painting of a unicorn on the snow land +a photograph of a Chinese dragon in the sky +a photograph of a dinosaur in the woods +a painting of a highly detailed wizard +a photograph of a crane in the lake +a photograph of a highly detailed Mickey +a photograph of a devilfish in the sea +Figure 9. Results of grounded generation. The segmentation mask refers to the grounding results for the object underlined. +19 + +7a photograph of a launched rocket +a painting of a highly detailed Ultraman +a photograph of a white pterosaur +a photograph of a shark in the sea +a painting of a smilodon on the grass +a photograph of a whale leaping out of the sea +a photograph of a turtle crawling in the sand +a photograph of a statue of Zeus +Figure 10. Results of grounded generation. The segmentation mask refers to the grounding results for the object underlined. +20 + +Figure 11. Examples from our synthetic semantic segmentation dataset. +21 + +Image +GT Mask +Zegformer +MaskFormer +(train on synthestic set) +Image +GT Mask +Zegformer +MaskFormer +(train on synthestic set) +Figure 12. More visualization of zero-shot segmentation results on Pascal VOC. +22 + +巫S \ No newline at end of file diff --git a/ktE4T4oBgHgl3EQftA2c/content/tmp_files/load_file.txt b/ktE4T4oBgHgl3EQftA2c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f63ae3ae578ccc866103268a2251c3dfa85fa04 --- /dev/null +++ b/ktE4T4oBgHgl3EQftA2c/content/tmp_files/load_file.txt @@ -0,0 +1,1487 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf,len=1486 +page_content='Guiding Text-to-Image Diffusion Model Towards Grounded Generation Ziyi Li1∗, Qinye Zhou1∗, Xiaoyun Zhang1, Ya Zhang1,2, Yanfeng Wang1,2, and Weidi Xie1,2,† 1Coop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Medianet Innovation Center, Shanghai Jiao Tong University, China 2Shanghai AI Laboratory, China https://lipurple.' metadata={'source': 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+page_content='bird ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='horse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='person ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='bicycle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='train ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='dog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='cat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='pottedplant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='tvmonitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='boat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='bottle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='sheep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='car ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='chair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='fox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='basketball ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='Pikachu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='turtle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='penguin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='unicorn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='phoenix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='lion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Predictions from our guided text-to-image diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The model is able to simultaneously generate images and segmentation masks for the corresponding visual objects described in the text prompt, for example, Pikachu, Unicorn, Phoenix, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Abstract The goal of this paper is to augment a pre-trained text-to- image diffusion model with the ability of open-vocabulary objects grounding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', simultaneously generating images and segmentation masks for the corresponding visual enti- ties described in the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We make the following contributions: (i) we insert a grounding module into the ex- isting diffusion model, that can be trained to align the visual and textual embedding space of the diffusion model with only a small number of object categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (ii) we propose an automatic pipeline for constructing a dataset, that con- Both the authors have contributed equally to this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' † denotes corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' sists of {image, segmentation mask, text prompt} triplets, to train the proposed grounding module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (iii) we evaluate the performance of open-vocabulary grounding on images generated from the text-to-image diffusion model and show that the module can well segment the objects of categories beyond seen ones at training time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (iv) we adopt the guided diffusion model to build a synthetic semantic segmentation dataset, and show that, training a standard segmentation model on such dataset demonstrates competitive performance on zero-shot segmentation (ZS3) benchmark, which opens up new opportunities for adopting the powerful diffusion model for discriminative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='05221v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='CV] 12 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Introduction In the recent literature, text-to-image generative models have gained increasing attention from the research commu- nity and wide public, one of the main advantages of such models is the strong semantic correspondence between vi- sual and language, learned from a large corpus of image- caption pairs, such correspondence, for instance, enables to generate photorealistic images from the free-form text prompt [26,27,30,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' While the synthesis ability of these models are unprecedented, they generally lack the ability to ground the objects within generated images, which can be crucial to a number of applications, for example, image editing, visual question answering, and visual reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In this paper, we aim to augment an existing text-to- image diffusion model with the ability of objects ground- ing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', simultaneously generating photorealistic images and segmentation masks for corresponding visual objects described in the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Generally speaking, two challenges exist: first, to explicitly establish the visual- language correspondence for the generative model, image- segmentation pairs are required for visual demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In previous work, for example, DatasetGAN [42] and Big- DatasetGAN [18], the authors consider collecting manual annotations for synthetic images, however, such a strategy is unlikely to be scalable, given the high annotation cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' second, it is unclear how to guide the generative model to- wards open-vocabulary grounding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', the model should be capable of segmenting objects from any category it can gen- erate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In a concurrent work (DAAM [33]), attention maps are directly upsampled and aggregated at each time step to explore the influence area of each input word, despite being simple, the performance of grounding is unsatisfactory, and tends to result in ambiguities, as the text embedding for in- dividual visual entity has been largely influenced by other entities and the global sentence at the stage of text encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' To tackle the challenges mentioned above, we make the following contributions: (i) we develop an automatic pipeline for automatically constructing {image, segmenta- tion, text prompt} triplets, in particular, we adopt an off- the-shelf object detector, and do inference on images gener- ated from the existing stable diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Theoretically, such a pipeline enables to generate infinite data samples for each of the categories within the vocabulary of existing ob- ject detector, for example, we adopt the Mask R-CNN [22] pre-trained on COCO with 80 categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (ii) we propose a novel architecture that can segment any visual entity men- tioned in the text prompt from the generated image, specifi- cally, we propose a fusion module that explicitly aligns the visual and textual embedding space of the diffusion model, as a result, despite being only trained on a pre-defined set of object categories, the grounding module enables to segment objects of unseen ones at training time, resembling an open- vocabulary knowledge induction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 1, our guided stable diffusion model enables to segment the objects well beyond the vocabulary of any off-the-shelf detector, for example, Pikachu, unicorn, phoenix, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' To quantitatively validate the effectiveness of our pro- posed open-vocabulary grounding, we initiate two proto- cols for evaluation: first, we train the grounding module on the constructed training set, and compare the grounding performance with a strong off-the-shelf object detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' sec- ond, we adopt the guided stable diffusion model to construct a synthesized semantic segmentation dataset, and train a segmentation model on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' While evaluating on the exist- ing benchmarks for zero-shot segmentation (ZS3), the seg- mentation model shows competitive performance over prior state-of-the-art models, especially on unseen categories, re- flecting the effectiveness of our open-vocabulary grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Crucially, it has demonstrated a promising applica- tion for applying the powerful diffusion model for discrim- inative tasks, that is, to expand the vocabulary beyond any existing detector can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Related Work Image Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Image generation is one of the most challenging tasks in computer vision due to the high- dimensional nature of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In the past few years, gener- ative adversarial networks (GAN) [10], variational autoen- coders (VAE) [17], flow-based models [16] and autoregres- sive models (ARM) [34] have made great progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' How- ever, even GANs, the best of these methods, still face train- ing instability and mode collapse issues [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Recently, Dif- fusion Probabilistic Models (DM) demonstrate state-of-the- art generation quality on highly diverse datasets [12,13,23, 29,31], outperforming GANs in fidelity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' These models are often combined with a well-designed text encoder and trained on billions of image-caption pairs for text-to-image generation task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', OpenAI’s DALL-E 2 [26], Google’s Imagen [30] and Stability AI’s Stable Diffusion [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' How- ever, despite being able to generate images with impressive quality using free-form text, it remains unclear what extent the visual-language correspondence has been successfully captured, this paper aims to augment an existing text-to- image diffusion model with the ability to ground objects in its generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visual Grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visual grounding, also known as refer- ring expression comprehension, expects to understand the natural language query and then find out the target object of the query in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Early visual grounding works are trained in two stages [14, 21, 35, 36, 38], by first detecting the candidate regions, and then ranking these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Later, one-stage approaches [19, 28, 39, 40] attract more attention due to their superior accuracy and efficiency in fusing lin- guistic context and visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Here, we consider visual grounding in the image generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Methodology In this paper, we aim to introduce a knowledge induction procedure, that converts an existing text-to-image diffusion model for grounded generation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', simultaneously gener- ating images and the segmentation mask of corresponding objects described in the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In particular, our core idea is to exploit a handful of image-segmentation pairs as visual demonstrations, to build the general visual-language correspondence between the visual representation from dif- fusion model and the free-form text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In the following sections, we start by describing the problem scenario for grounded generation based on diffu- sion model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2, we present preliminary background knowledge of diffusion model, and terminologies that will be used in later sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3, we detail the proposed knowledge induction procedure, including (i) the architec- ture design that enables to align the visual and textual em- bedding under an open-vocabulary setting, (ii) the training procedure based on the automatically constructed {image, segmentation, text prompt} triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Problem Scenario Assuming there exists a strong text-to-image diffusion model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Stable Diffusion [27] in our case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' the goal is to convert it into a grounded generation model: {I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' m} = Φdiffusion+(ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' y) (1) where Φdiffusion+(·) refers to a pre-trained text-to-image dif- fusion model with our grounding module appended,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' it takes the sampled noise (ϵ ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' I)) and language description y as input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' and generates an image (I ∈ RH×W ×3) with corresponding segmentation masks (m ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 1}H×W ×O) for a total of O objects of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that, we expect the model to be open-vocabulary, that means, it should be able to output the corresponding segmentation mask for any objects that can be generated by diffusion model, without limitation of the semantic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Preliminary on Diffusion Model Diffusion models refer to a series of probabilistic gen- erative models, that are trained to learn a data distribu- tion by gradually denoising the randomly sampled Gaus- sian noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Theoretically, the procedure refers to learning the reverse process of a fixed Markov Chain of length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As for text-to-image synthesis, given a dataset of image- caption pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', Dtrain = {(I1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , (IN, yN)}, the models can be interpreted as an equally weighted sequence of conditional denoising neural network that iteratively pre- dicts a denoised variant of the input conditioned on the text prompt, ϵθ(It i, t, yi), where It i denotes a noisy version of the input image, and t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , T refers to the timestep, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' For simplicity, we only describe the train- ing and inference procedure for a single image, thus ignor- ing the subscript in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In particular, this paper builds on a variant of diffusion model, namely, Stable Diffusion [27], which encodes all images with a variational autoencoder, and transfers the dif- fusion process to latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' we will briefly describe its architecture and training procedure in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Stable Diffusion consists of three compo- nents: a text encoder for producing text embeddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' a pre-trained variational autoencoder (VAE) that encodes and decodes latent vectors for images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' and a time-conditional UNet (ϵθ(·)) for gradually denoising the latent vectors, with the progressive convolutional operation that downsamples and upsamples the visual feature maps with skip connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The visual-language interaction occurs in the UNet, where the embeddings of the visual and textual features in- teract through cross-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, a text en- coder is used to project the text prompt y to textual em- beddings, that then are mapped into Key and Value, the spatial feature map of the noisy image is linearly projected into Query, and iteratively attending the conditioned text prompt for updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Training and Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The training procedure of Stable Diffusion can be described as follows: given a training pair (I, y), the input image I is first mapped to a latent vector z and get a variably-noised vector zt := αtz + σtϵ, where ϵ ∼ N(0, 1) is a noise term and αt, σt are terms that con- trol the noise schedule and sample quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' At training time, the time-conditional UNet is optimised to predict the noise ϵ and recover the initial z, conditioned on the text prompt y, the model is trained with a squared error loss on the pre- dicted noise term as follows: Ldiffusion = Ez,ϵ∼N (0,1),t,y � ||ϵ − ϵθ(zt, t, y)||2 2 � (2) where t is uniformly sampled from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' At inference time, Stable Diffusion is sampled by iter- atively denoising zT ∼ N(0, I) conditioned on the text prompt y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, at each denoising step t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , T, zt−1 is obtained from both zt and the predicted noise term of UNet whose input is zt and text prompt y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' After the final denoising step, z0 will be mapped back to yield the gener- ated image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Open-vocabulary Grounding In this section, our goal is to learn the correspon- dence between the visual representation (from the diffusion model) and category embeddings in text form, augmenting the off-the-shelf Stable Diffusion with an open-vocabulary object grounding module, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As- suming there exists a training set of N triplets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', Dtrain = 3 A photograph of a dog near a cat Diffusion Model Text Prompts Our Model Grounding Synthetic images generated by diffusion model Oracle GT masks produced by off-the-shelf detector Synthetic images and corresponding masks generated by our grounded generation model … 𝒇𝟏 𝒇𝟐 𝒇𝑳 Visual Encoder a photograph of { } dog cat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Text Encoder Fusion Module .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Q ⊗ (K, V) \uf054 Diffusion Model Text Prompts \uf054 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The left figure shows the knowledge induction procedure, where we first construct a dataset with synthetic images from diffusion model and generate corresponding oracle groundtruth masks by an off-the-shelf object detector, which is used to train the open-vocabulary grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The right figure shows the architectural detail of our grounding module, that takes the text embeddings of corresponding entities and the visual features extracted from diffusion model as input, and outputs the corresponding segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' During training, both the diffusion model and text encoders are kept frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' {(F1, mgt 1 , y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , (FN, mgt N, yN)}, the predicted segmen- tation mask for all objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', mi ∈ RH×W ×Oi can be obtained by: mi = Φfuse(Φv-enc(f 1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n i ), Φt-enc(g(yi))) (3) where yi denotes the text prompt for image generation, Fi = {f 1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n i } refers to the intermediate representation from Stable Diffusion at t = 5 (this has been experimen- tally validated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3), Φv-enc(·), Φt-enc(·) and Φfuse(·) denote the visual encoder, text encoder, and fusion mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' At training time, g(·) denotes a group of templates that decorates each of the visual categories in the text prompt, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', ‘a photograph of a [category name]’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' There are to- tally Oi object categories in the text prompt, and they fall into a pre-defined set of vocabularies Ctrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' It is important to mention that, we expect the visual-language correspon- dence to be highly generalizable, such that the grounding module should equally be capable of segmenting objects from unseen categories at test time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', Ctest ⊃ Ctrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In the following sections, we start by introducing the procedure for automatically constructing the training set in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1, followed by the architecture design for open- vocabulary grounding in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2, lastly, we detail the training process via knowledge induction in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset Construction Here, we introduce the idea to construct the training set with {visual feature, segmentation, text prompt} triplets, specif- ically, we develop an automatic pipeline to construct the {image, segmentation, text prompt} triplets, thus the visual feature can be obtained from Stable Diffusion via the for- ward inference to generate the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In practise, we prepare a vocabulary with common ob- ject categories, for example, the classes in PASCAL VOC can form a category set as Cpascal = {‘dog’, ‘cat’, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' }, |Cpascal| = 20, we randomly select a number of classes to construct a text prompt for image generation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', ‘a pho- tograph of a dog and cat’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Repeating the above operation, we can theoretically generate infinite amount of image and text prompt pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' To acquire the segmentation masks, we take an off-the-shelf object detector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', pre-trained Mask R-CNN [22], and run the inference procedure on the gener- ated images: mgt i = Φdetector(Ii), where Ii = Φdiffusion(ϵ, yi), (4) where mgt i ∈ {0, 1}H×W ×Oi refers to the predicted masks for a total of Oi objects in the generated image Ii, condi- tioning on the text prompt yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' To evaluate the effectiveness of the induction proce- dure for open-vocabulary grounding, we divide the vocab- ulary set into seen categories (Cseen) and unseen categories (Cunseen), the training set (Dtrain) only has images of seen categories, and the test set (Dtest) has both seen and unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Architecture Given one specific training triplet (Fi, mgt i , yi), we now de- tail three trainable components in the proposed grounding module: visual encoder, text encoder, and fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visual Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Given the visual representation from Sta- ble Diffusion, we concatenate features with the same spa- tial resolution (from UNet encoding and decoding path) to obtain {f 1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n i }, where f k i ∈ R h 2k × w 2k ×dk, k ∈ 4 X本𝟏 × 𝟏 Conv Upsample Upsample 𝟏 × 𝟏 Conv Concat Mix-Conv Upsample 𝟏 × 𝟏 Conv Concat Mix-Conv Upsample 𝟏 × 𝟏 Conv Concat Mix-Conv 𝒇𝟏 𝒇𝒏−𝟐 𝒇𝒏−𝟏 𝒇𝒏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visual Encoder Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The architecture of visual encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The features ex- tracted from UNet are first grouped according to their resolution, then gradually upsampled and fused into the final visual feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , n} denotes layer indices of UNet, dk refers to the feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Next, we input {f 1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n i } to visual encoder for gen- erating the fused visual feature ˆFi = Φv-enc({f 1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n i }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Fig 3, visual encoder consists of 4 types of blocks: (1) 1×1 Conv for reducing feature dimensionality, (2) Upsample for upsampling the feature to a higher spa- tial resolution, (3) Concat for concatenating features, and (4) Mix-conv for blending features from different spatial resolutions, that includes two 3 × 3 Conv operations with residual connection and a conditional batchnorm operation, similar to [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Text Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We adopt the language model from pre- trained Stable Diffusion, specifically, given the text prompt yi, the text encoder output the corresponding embeddings for all visual objects: Eobji = Φt-enc(g(yi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Fusion Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The fusion module computes interac- tion between visual features and text embeddings, then out- puts segmentation masks for all visual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In spe- cific, we use a standard transformer decoder with three lay- ers, the text embeddings are treated as Query, that itera- tively attend the visual feature for updating, and are further converted into per-segmentation embeddings with a Multi- Layer Perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The object segmentation masks can be obtained by dot product visual features with the per- segmentation embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Formally, the procedure can be denoted as : Esegi = ΦTRANSFORMER-D(W Q·Eobji, W K· ˆFi, W V · ˆ Fi) (5) mi = ˆFi · [ΦMLP(Esegi)]T (6) where the transformer decoder generates per-segmentation embedding Esegi ∈ RN×de for all visual objects described in the text prompt, W Q, W K, W V refer to the learnable pa- rameters for Query, Key and Value projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Training With the constructed dataset, we can now start supervised training the proposed grounding module: L = − 1 N N � i=1 [mgt i · log(σ(mi)) + (1 − mgt i ) · log(σ(1 − mi))] where mgt i ∈ RH×W ×Oi refers to the oracle groundtruth segmentation from the off-the-shelf object detector, and mi ∈ RH×W ×Oi refers to the predicted segmentation from our grounding module, σ(·) refers to Sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In practise, while using the off-the-shelf detector to gen- erate segmentation masks, the model may sometimes fail to detect the objects mentioned in the text prompt, and out- put incorrect segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Such error comes from two sources, (i) the diffusion model may fail to generate high-quality images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (ii) off-the-shelf detector may fail to detect the objects in the synthetic image, due to the domain gap between synthetic and real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Here, we consider two training strategies, Normal Training, where we fully trust the off-the-shelf detector, and use all predicted seg- mentation masks to train the grounding module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' alterna- tively, we also try Training w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Zero Masks, as we em- pirically found that the majority of failure cases come from false negatives, that is to say, the detector failed to detect the objects and output an all-zero mask, therefore, we can train the grounding modules by ignoring the all-zero masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Experiments In this section, we detail the evaluation detail for val- idating the effectiveness of objects grounding during im- age generation, specifically, we consider two protocols: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1, we train the grounding module with the con- structed training set, and test the segmentation performance on generated images from Stable Diffusion, with the detec- tor’s output as oracle groundtruth for evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2, we use the guided diffusion model to construct a synthe- sized semantic segmentation dataset, and train a segmen- tation model on it, we evaluate the model on the existing benchmarks for zero-shot segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Lastly, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3, we conduct a series of ablation studies on the different train- ing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Protocol-I: Grounded Generation Here, we train the grounding module with our con- structed training set, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1, specifically, the training set only consists of a subset of common (seen) categories, while the testing set consists of both seen and unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In the following, we describe the imple- mentation and experimental results in detail, to thoroughly assess the model for grounded generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Following the split rule in [7,11,37], we adopt two differ- ent sets of categories: (i) we adopt the categories defined in PASCAL VOC [9], divide them into 15 seen categories and 5 unseen categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (ii) we adopt the category definition in MS-COCO [20], divide them into 65 seen categories and 15 unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The dataset constructed for these two sets are called PASCAL-sim and COCO-sim, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 5 FTest Setting PASCAL-sim COCO-sim # Objects One Two One Two Categories Seen Unseen Seen Seen +Unseen Unseen Seen Unseen Seen Seen +Unseen Unseen DAAM [33] Split1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='66 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='63 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='74 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='31 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='94 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='25 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='56 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='68 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='06 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='35 Split2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='75 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='25 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='98 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='50 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='55 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='80 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='66 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='22 Split3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='11 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='82 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='80 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='28 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='41 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='81 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='48 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='85 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='84 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='80 Average 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='84 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='90 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='21 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='95 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='71 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='76 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='78 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='79 Ours Split1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='16 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='07 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='35 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='81 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='64 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='15 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='77 Split2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='68 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='10 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='21 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='83 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='39 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='18 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='82 Split3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='67 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='86 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='68 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='42 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='07 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='85 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='89 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='70 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='60 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='62 Average 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='30 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='10 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='86 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='74 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='68 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='21 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='16 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='31 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='40 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Quantitative result for Protocol-I evaluation on grounded generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Our model has been trained on the synthesized training dataset, that consists of images with one or two objects from only seen categories, and test on our synthesized test dataset that consists of images with one or two objects of both seen and unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Split1, Split2 and Split3 refer to 3 different splits of C that construct 3 different (Cseen, Cunseen) pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Our model outperforms the DAAM [33], by a large margin, see text for more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Training Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' For PASCAL-sim or COCO-sim, we gen- erate a synthetic training set by randomly sampling im- ages from pre-trained Stable Diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' This exposes our grounding module to a great variety of data (> 40k) at training time, and the model is unlikely to see the same la- beled examples twice during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In contrast to previ- ous work, such as BigDatesetGAN [18], where only a single object is considered, we construct the text prompt with one or two objects at a time, note that, for training, only the seen categories are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Although we can certainly generate images with more than two object categories, the quality of the generated images tends to be unstable, limited by the generation ability of Stable Diffusion, thus we only consider synthesized images with less than three object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Testing Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' For the evaluation purpose, we generate two synthetic test sets with offline sampling for PASCAL-sim and COCO-sim, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In total, we have collected about 1k images for PASCAL-sim, and about 5k images for COCO-sim, we run the off-the-shelf object detector on these generated images to produce the oracle groundtruth segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' For both test sets, the images containing two categories will be divided into three groups: both seen, both unseen, one seen and one unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We leave the detailed statistics of our synthetic dataset in the supplementary ma- terial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that, we have manually checked all the images and the oracle groundtruth segmentation produced from the off-the-shelf detector, and only keep the high-quality ones, thus the performance evaluation of the grounding module can be a close proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We use the category-wise mean intersection-over-union (mIoU) as evaluation metric, de- fined as averages of IoU over all the categories: mIoU = 1 C �C c=1 IoUc, where C is the number of all target cate- gories, and IoUc is the intersection-over-union for the cate- gory with index is c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We use the pre-trained Stable Diffusion [27] and the text encoder of CLIP [25] in our im- plementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We choose the Mask R-CNN [22] trained on COCO dataset as our object detector for oracle groundtruth segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We fuse features from U-Net and upsample them into 512 × 512 spatial resolution, the text and visual embeddings are both mapped into 512 dimension before feeding into the fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We train our grounding module with two NVIDIA GeForce RTX 3090 GPUs for 5k iterations with batch size equal to 8, ADAM [15] opti- miser with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The initial learning rate is set to 1e-4 and the weight decay is 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 1, we provide experimental re- sults for our grounded generation model, we change the composition of categories three times and compute the re- sults for each split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Here, we can make the following ob- servations: first, our model significantly outperforms the unsupervised method DAAM [33] in the mIoU on all test settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' This is because DAAM tends to result in ambigu- ous segmentations, as the textual embedding for individual visual entity will largely be influenced by other ones within the global sentence at the text encoding stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' second, our grounding module achieves superior performance on both seen and unseen categories, indicating its open-vocabulary nature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', the guided diffusion model can synthesize im- ages and their corresponding segmentations for more cate- gories beyond the vocabulary of the off-the-shelf detector, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We demonstrate the visualization results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' On both seen and unseen categories, our model can successfully ground the objects in terms of segmenta- tion mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Impressively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 1, our grounding module can even segment the objects beyond any off-the- shelf detector can do, showing the strong open-vocabulary grounded generation ability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 6 car car dog bottle sofa sofa train hot dog hot dog bear bear backpack apple frisbee Image Image Image Image Ours Oracle GT Ours Ours Ours Oracle GT Oracle GT Oracle GT bottle backpack apple motorbike motorbike Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Segmentation results of PASCAL-sim (left) and COCO-sim (right) on seen (motorbike, bottle, backpack and apple) and unseen (sofa, car, hot dog and bear) categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Our grounded generation model achieves comparable segmentation results to the oracle groundtruth generated by the off-the-shelf object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' dog cat bird train bus pottedplant sheep cow boat boat car bottle horse chair sofa motorbike person pottedplant person motorbike boat pottedplant bottle dog train bottle bird Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Our synthesized semantic segmentation dataset with one category (left) and two categories (right) for Protocol-II training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Protocol-II: Open-vocabulary Segmentation In the previous protocol, we have validated the ability for open-vocabulary grounded generation, however, even after being manually checked, the oracle groundtruth from off- the-shelf detector may also be inaccurate at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Here, we introduce another experiment to validate the effec- tiveness of the grounding module, in particular, we first con- struct a synthesized image-segmentation dataset with the guided Stable Diffusion, then train a semantic segmentation model on such a synthetic dataset, and evaluate it on public image segmentation benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Overall, the segmentation model is challenged from the following two perspectives: first, it has only been trained on synthetic images, that resembles a zero-shot Sim2Real transfer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' second, the groundtruth masks for unseen object categories are generated from our guided Stable Diffusion that has only been trained on seen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Therefore, with such an evaluation protocol, on the one hand, it can reflect the effectiveness of our grounding module from the performance on segmenting unseen categories, and more importantly, it introduces a promising application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', use our guided Stable Diffusion to expand the vocabulary be- yond any existing detector can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In order to train the semantic segmentation model, we synthesize a dataset with 10k image-segmentation pairs for 20 categories (both seen and unseen) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' All the image-segmentation pairs are generated by our guided Stable Diffusion, trained with only 15 seen cate- gories in PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We do not finetune on PASCAL Training Dataset mIoU Methods Type Categories Objects Seen Unseen Harmonic ZS3 [3] real 15 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3 SPNet [37] real 15 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 CaGNet [11] real 15 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='7 Joint [1] real 15 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='7 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='9 STRICT [24] real 15 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 SIGN [5] real 15 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3 ZegFormer [7] real 15 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3 Ours synthetic 15 + 5 one 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='7 synthetic 15 + 5 two 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 synthetic 15 + 5 mixture 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Comparison with the previous ZS3 methods on PAS- CAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The “Seen”, “Unseen”, and “Harmonic” denote mIoU of seen categories, unseen categories, and their harmonic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' These ZS3 methods are trained on PASCAL VOC training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' VOC and only evaluate on its test set (1,449 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' To compare with other open-vocabulary methods, our semantic segmentation model uses Mask- Former [4] with ResNet101 as its backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The image res- olution for training is 224×224 pix, and we train the model on our synthetic dataset for 40k iterations with batch size equal to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We use the ADAMW as our optimizer with a learning rate of 1e-4 and the weight decay is 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Comparison on Zero-Shot Segmentation (ZS3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2, we compare with the existing zero-shot semantic segmentation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Despite being only trained on a synthetic dataset, our model outperforms most of ZS3 approaches on unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, the model 7 ELMJAL fewaleACHBNWARCOImage GT Mask Zegformer MaskFormer (train on synthestic set) Image GT Mask Zegformer MaskFormer (train on synthestic set) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visualization of zero-shot segmentation results on Pascal-VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' MaskFormer trained on our synthetic dataset achieves com- parable performance with Zegformer (the state-of-the-art zero-shot semantic segmentation method) in segmenting unseen categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' pottedplant, sofa and tvmonitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that although MaskFormer has seen these categories during training, the image-segmentation pairs of these categories are generated with our grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Training Type One Two Seen Unseen Seen Seen +Unseen Unseen Normal Training 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='88 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='18 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='66 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='24 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='22 Training w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Zero Masks 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='16 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='07 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ablation on training type on the constructed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Performance is measured by mIoU on PASCAL-sim test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' trained on the mixture of one and two objects achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 6, our model obtains accurate segmentation on both seen and unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Therefore, we can have the following observations: (i) the grounding module is capable of segmenting unseen cat- egories despite it has never seen any segmentation mask during the knowledge induction procedure, validating the strong generalisation of the grounding module in the guided Stable Diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (ii) it is possible to segment more object categories by simply training on synthesized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ablation study In this section, we show the effect of different training loss and different timestep for extracting visual represen- tation, due to the space limitation, we refer the reader for supplementary material, for the study on the different num- ber of objects in the synthetic datasets or seen categories, and the effect of different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Normal Training v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Training without Zero Masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 3, Normal Training results in unsatisfac- tory performance on unseen categories, we conjecture this is because the errors from detector tend to be false nega- tive, that bias our grounding module to generate all-zero segmentation masks when encountering unseen categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' in contrast, by ignoring all-zero masks at training, Train- ing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Zero Masks achieves equally good performance on both seen and unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Timesteps for Extracting Visual Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ablation on timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The mIoU is measured for the model with extracting features from Stable Diffusion in different timesteps on PASCAL-sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' compare the performance by extracting visual representa- tion from Stable Diffusion at different timesteps, the results on PASCAL-sim can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 7, showing that as the denoising steps gradually decrease, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', from t = 0 −→ 50, the performance for grounding tends to decrease in general, when t = 5, the best result is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Conclusion In this paper, we propose a novel idea for guiding the ex- isting Stable Diffusion towards open-vocabulary grounded generation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', segmenting the visual entities described in the text prompt while generating images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, we introduce a grounding module that explicitly aligns the vi- sual and textual embedding space of the Stable Diffusion and train such module with an automatically constructed dataset, consisting of {image, segmentation, text prompts} triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Experimentally, we show that visual-language cor- 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='6 lou E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='4 one (seen) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3 one (unseen) two (seen) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2 two (seen+unseen) two (unseen) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1 0 5 10 15 20 25 30 35 40 45 50 Timestepsbacial12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2006respondence can be established by only training on a lim- ited number of object categories, while getting the abil- ity for open-vocabulary grounding at the image genera- tion procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Additionally, we generate a synthetic se- mantic segmentation dataset using our guided Stable Dif- fusion and train a semantic segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Without finetuning, the model can directly transfer to real images, and show competitive performance to existing zero-shot se- mantic segmentation approaches on PASCAL VOC dataset, opening up new opportunities to exploit generative model for discriminative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' References [1] Donghyeon Baek, Youngmin Oh, and Bumsub Ham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ex- ploiting a joint embedding space for generalized zero-shot semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' ICCV, 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Sign: Spatial-information incorpo- rated generative network for generalized zero-shot semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 7 [6] Prafulla Dhariwal and Alexander Nichol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Diffusion models beat gans on image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 [7] Jian Ding, Nan Xue, Gui-Song Xia, and Dengxin Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' De- coupling zero-shot semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 5, 7 [8] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- vain Gelly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' An image is worth 16x16 words: Trans- formers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='11929, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 12 [9] Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The pascal visual object classes (voc) challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' International journal of computer vision(IJCV), 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 5, 14, 15, 18 [10] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Commu- nications of the ACM, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 [11] Zhangxuan Gu, Siyuan Zhou, Li Niu, Zihan Zhao, and Liqing Zhang.' 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[38] Sibei Yang, Guanbin Li, and Yizhou Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dynamic graph attention for referring expression comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 [39] Zhengyuan Yang, Tianlang Chen, Liwei Wang, and Jiebo Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Improving one-stage visual grounding by recursive sub- query construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 [40] Zhengyuan Yang, Boqing Gong, Liwei Wang, Wenbing Huang, Dong Yu, and Jiebo Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' A fast and accurate one- stage approach to visual grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 [41] Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gun- jan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yin- fei Yang, Burcu Karagol Ayan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Scaling autoregres- sive models for content-rich text-to-image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='10789, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 [42] Yuxuan Zhang, Huan Ling, Jun Gao, Kangxue Yin, Jean- Francois Lafleche, Adela Barriuso, Antonio Torralba, and Sanja Fidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Datasetgan: Efficient labeled data factory with minimal human effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 2 10 Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Details on the Architecture of Grounding module 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Details on the Synthetic Dataset 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset Split .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Additional Ablation Study 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Synthetic Dataset Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Effect on the Number of Seen Categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset Construction via DDIM Inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' More Qualitative Results 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Limitation & Future Work 18 11 In this supplementary document, we start by giving more details on the architecture of our grounding module in Section A, followed by the details for generating the dataset for training it in Section B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' then describe the additional ablation studies, as promised in the main text in Section C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Section D, we present more qualitative results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Lastly, we illustrate the limitation of our method and our future work in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Details on the Architecture of Grounding module We show the detailed architecture of our grounding module in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 8, which consists of visual encoder, text encoder, transformer decoder and MLP in the fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' “ a photograph of the { class name } ” dog cat Visual Encoder 𝑻𝒅𝒐𝒈 𝑻𝒄𝒂𝒕 O Text Embeddings … … Linear Flatten (K, V) Transformer Decoder ⊗ MLP Q Text Encoder \uf054 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Features from diffusion model 𝑯 × 𝑾 × 𝑫 𝑶 × 𝒅𝒆 𝑶 × 𝒅𝒆 Visual Tokens 𝑶′ × 𝒅𝒆 𝑴𝒅𝒐𝒈 𝑴𝒄𝒂𝒕 O Mask Embeddings … 𝑶 × 𝑫 O class-specific masks … 𝑯 × 𝑾 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Detailed architecture of our grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We first generate O text embeddings by injecting the class names into a prompt template and then feeding them to a pre-trained text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The visual encoder takes the features from Stable Diffusion as input and outputs fused visual features, which are then flattened to a sequence of visual tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Next, we feed the visual tokens into a transformer decoder as Key and Value, and feed text embeddings as Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The outputs of transformer decoder are then fed into an MLP to obtain O mask embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Mask embeddings are dot producted with the output features of visual encoder to generate O class-specific binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Visual Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The input {f 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n} are extracted from Stable Diffusion [27], and the visual encoder aims to upsample and fuse the visual feature, and output the visual feature map, ˆF = Φv-enc({f 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' , f n}), ˆF ∈ RH×W ×D (here we use H = W = 512, and D = 240).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that, the resolution of the fused visual feature is the same as the resolution of the generated image, and the segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Text Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We adopt the pre-trained text encoder from CLIP [25], which is also used in Stable Diffusion [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' It takes text prompt y as input and outputs the corresponding text embedding: Eobject = Φt-enc(y), Eobject ∈ RO×dtext, where O is the total number of objects of interest and dtext = 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Transformer Decoder in Fusion Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Similar to the operation in standard ViT architecture [8], we convert the visual features ˆF ∈ RH×W ×D into visual tokens ˆFflatten ∈ RO′×dvisual, where O′ = HW p2 = 16384 is the number of tokens, p refers to the patch size and dvisual = p2 × D = 3840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Then the visual tokens and text embeddings are mapped into the same dimension de = 512 with MLPs, and passed into the transformer decoder with three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The text embeddings are treated as query with dimension O × de, and the visual tokens are treated as key and value with dimension O′ × de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The output of transformer decoder is of the same resolution as query, with dimension O × de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' MLP in Fusion Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' At last, we use an MLP to map the output of transformer decoder into mask embeddings with dimension O × D, which are then dot producted with the fused visual feature ( ˆF ∈ RH×W ×D) to generate class-specific binary masks (O × H × W), each mask is of the same spatial resolution of the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Details on the Synthetic Dataset B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset Split Here, to train our proposed grounding module, and properly evaluate its ability for segmenting the objects that are unseen at training time, we construct the training dataset with images of only seen categories, and the test dataset consists of both seen and unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The detail of the split on PASCAL-sim and COCO-sim, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' the split of seen categories and unseen categories, is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4, where PASCAL-sim has 15 seen categories and 5 unseen categories, COCO-sim has 65 seen categories and 14 unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that, we ignore the category: ‘mouse’ in the COCO-sim since the diffusion model generates ‘rat’ in the image for the category: ‘mouse’, while ‘mouse’ in the vocabulary of the off-the-shelf detector means mouse as a computer accessory, thus the detector fails to detect the category ‘mouse’ in the image generated by the diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Categories seen Unseen PASCAL-sim Split1 aeroplane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bicycle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bird,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' boat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bottle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' cat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' chair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' cow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' diningtable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' horse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' skis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' sports ball,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' kite,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' baseball bat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' baseball glove,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' skateboard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' surfboard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' tennis racket,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bottle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' wine glass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' cup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' knife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' spoon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bowl banana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' apple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' orange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' broccoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' carrot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' pizza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' donut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' cake,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' chair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bench,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' pottedplant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' bed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' din- ingtable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' tvmonitor Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The details on the split of categories on PASCAL-sim and COCO-sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset for Training Grounding Module To construct the training set, (1) we first randomly select one or two categories from the seen ones, where objects tend to co-appear in natural images, based on the annotation in PASCAL VOC [9] or COCO [20], called co-appearing category pair, and use the prompt template to decorate these selected categories, thus we can obtain the text prompt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (2) we pass the text prompt and randomly sampled Gaussian noise to the Stable Diffusion [27] to obtain the generated image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (3) next, we pass the generated image to the off-the-shelf detector to obtain the oracle segmentation mask;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (4) finally, we can construct the triplet which consists of the generated image, oracle segmentation mask, and text prompt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (5) repeat the above procedure, we can generate infinite triplets for the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Algorithm 1 displays the procedure for generating the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' To construct the test set for evaluating the grounding module, we can use a procedure similar to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The differences are: (i) we use all categories, including seen and unseen categories to construct the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' (ii) to obtain more reliable test results, we only add the triplet to the test set when the generated image and oracle segmentation mask have high quality which is checked manually, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', the generated image by Stable Diffusion contains the recognizable objects of selected categories, and the off-the-shelf detector successfully produces the high-quality oracle segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In this paper, PASCAL-sim has 20 categories and 142 co-appearing category pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We construct 30 triplets per category and 5 triplets per co-appearing category pair for PASCAL-sim test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In total, PASCAL-sim test set has 1310 triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' COCO-sim has 79 categories and 1559 co-appearing category pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We construct 30 triplets per category and 2 triplets per co-appearing category pair for COCO-sim test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In total, COCO-sim test set has 5488 triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Algorithm 1 Constructing the dataset for training grounding module (pseudocode in PyTorch-like style).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' # C_seen: the list of seen categories # img_shape: the shape of expected generated image # exp_train_size: the expected size of training set # n: the number of selected categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' n = 1 or 2 # co-appearing_category_pair_list: a list containing all co-appearing category pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' where objects tend to # co-appear in natural images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' based on the annotation in PASCAL VOC or COCO D_train = [] #initialize the training set while (len(D_train) < exp_train_size): y = None #initialize the text prompt #randomly select n categories from seen categories selected_class_list = random_select(C_seen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' n) if n = 1: class = select_class_list[0] # decorate the selected category by a pre-defined prompt template,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', "a photograph of a [class name]" y = prompt_template(class) else if n = 2: class1, class2 = select_class_list[0], select_class_list[1] if (class1, class2) in co-appearing_category_pair_list: # decorate the selected categories by a pre-defined prompt template, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', "a photograph of a # [class1 name] and a [class2 name]" y = prompt_template(class1, class2) if y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='= None: #randomly sample a Gaussian noise epsilon epsilon=torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='randn(img_shape) # pass the noise and text prompt to the diffusion model to generate image I I = diffusion_model(epsilon, y) # pass the generated image to the off-the-shelf detector to obtain the oracle segmentation mask m m = pretrain_detector(I) # add the triplet (generated image, oracle segmentation mask, text prompt) to the training set D_train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='append((I, m, y)) 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset for Training Semantic Segmentation Model As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2, we synthesize a semantic segmentation dataset for all 20 categories in PASCAL VOC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, we first randomly select one category or two categories (co-appearing category pair), to obtain the text prompt, and then pass randomly sampled Gaussian noise and text prompt to the diffusion model to obtain the generated image, and use our proposed grounding module to get the corresponding segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Thus, we can get the pair consisting of generated image and generated segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Repeat the above procedure, we can obtain the synthetic semantic segmentation dataset at a large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Algorithm 2 displays the procedure for generating the synthetic semantic segmentation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In this paper, the synthetic semantic segmentation dataset consists of 500 images per category and 71 images per co- appearing category pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Thus, there exist 10k images for 20 categories and 10082 images for 142 co-appearing category pairs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Algorithm 2 Pseudo-code for generating the synthetic semantic segmentation dataset in a PyTorch-like style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' # C: the list of all categories # img_shape: the shape of expected generated image # exp_dataset_size: the expected size of synthesis semantic segmentation dataset # n: the number of selected categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' n = 1 or 2 # co-appearing_category_pair_list: a list containing all co-appearing category pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' where objects tend to # co-appear in natural images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' based on the annotation in PASCAL VOC or COCO D_seg = [] #initialize the synthesis semantic segmentation dataset while (len(D_seg) < exp_dataset_size): y = None #initialize the text prompt #randomly select n categories from all categories selected_class_list = random_select(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' n) if n = 1: class = select_class_list[0] # decorate the selected category by a pre-defined prompt template,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', "a photograph of a [class name]" y = prompt_template(class) else if n = 2: class1, class2 = select_class_list[0], select_class_list[1] if (class1, class2) in co-appearing_category_pair_list: # decorate the selected categories by a pre-defined prompt template, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', "a photograph of a # [class1 name] and a [class2 name]" y = prompt_template(class1, class2) if y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='= None: #randomly sample a Gaussian noise epsilon epsilon=torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='randn(img_shape) # pass the noise and text prompt to the diffusion model with grounding module to generate image I # and segmentaion mask m I, m = diffusion_model_with_grounding(epsilon, y) # add the pair (generated image, generated segmentation mask) to the synthesis semantic segmentation dataset D_seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='append((I, m)) 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Additional Ablation Study C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Synthetic Dataset Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We explore the effect of constructing different datasets for training the grounding module, by varying the number of objects in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 5, training on the combination of one and two object categories gives the best results overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Train Set # Objects One Two Seen Unseen Seen Seen +Unseen Unseen single 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='37 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='85 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='89 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='33 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='91 two 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='35 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='56 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='36 mixture 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='16 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='07 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ablation on dataset construction on PASCAL-sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The bolded number indicates the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Our model achieves the best performance when training on the combination of one and two object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Effect on the Number of Seen Categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We ablate the number of seen categories to further explore the generalisation ability of our proposed grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 6, the grounding module can generalise to unseen categories, even as few as five seen categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' when introducing more seen categories, the performance on unseen ones consistently improves, but decreases on seen ones, due to the increasing complexity on seen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Train Set # Seen Categories / unseen categories One Two Seen Unseen Seen Seen +Unseen Unseen 5 / 74 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='81 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='42 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='60 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='00 20 / 59 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='91 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='33 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='59 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='27 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='91 35 / 44 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='23 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='85 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='91 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='99 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='28 50 / 29 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='55 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='20 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='41 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='39 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='71 65 / 14 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='85 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='81 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='64 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='15 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='77 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ablation on the number of seen categories on COCO-sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The bolded number indicates the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Our model can generalise to unseen categories, even as few as five seen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Dataset Construction via DDIM Inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In addition to using the off-the-shelf detectors, we also consider constructing the training set by utilising the inverse process of diffusion to explicitly generate images close to those in the public dataset, for example, PASCAL VOC, and train the grounding module with the mask annotations available from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Here, we describe an inverse procedure that enables to find a deterministic mapping from noise to images, given the sampling rule being non-Markovian, for example, Denoising Diffusion Implicit Model (DDIM) [32] with the reverse process variance to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In DALL-E 2 [26], such inversion has been used to determine the noise that produces a specific image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In our considered Stable Diffusion, the image is first mapped to a latent vector z0 by the pre-trained variational autoencoder (VAE), at each step of DDIM inversion, zt+1 is obtained from zt and the predicted noise term of UNet, that takes zt and text prompt y as input, ending up with an inverted noise zT eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In this paper, we exploit such DDIM inversion to train our grounding module with the dataset constructed from real image and segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In particular, the first option enables to directly inherit the segmentation mask from the public dataset, and the text prompt can be manually constructed by inserting class labels into the prompt template, for example, if the segmentation mask contains ‘dog’ and ‘cat’, the text prompt can be ‘a photograph of a dog and cat’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Besides, the visual feature can be obtained by extracting the feature from the UNet of Stable Diffusion when t = 1 at the inversion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Constructed Dataset v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Real Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We explore the difference between training on constructed dataset and real dataset (PASCAL VOC) from two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' First, we compare their performance on PASCAL-sim dataset for grounded generation in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 7 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Though we successfully train our grounding module on real dataset, the domain gap limits its 16 Dataset Type PASCAL-sim PASCAL-test One Two Seen Unseen Seen Seen+Unseen Unseen Seen Unseen real 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='67 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='26 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='23 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='14 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='80 sim(10k) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='77 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='04 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='07 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='08 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='59 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='75 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='42 sim(40k) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='16 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='07 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='93 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='44 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='86 sim+real 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='57 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='23 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='12 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='24 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='30 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='32 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='14 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Ablation on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The bold numbers indicate the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, ’sim’ and ’real’ denote the constructed dataset and real dataset (PASCAL-VOC), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' performance on grounded generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Considering PASCAL VOC only contains about 10k images, we adjust the con- struced dataset to the same magnitude and get better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Additionally, because of the good scalability of the constructed dataset, the performance will be better as the number of images increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Second, we evaluate the grounding module on PASCAL VOC test dataset by DDIM inversion as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 7 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that, under this circumstance, our model ap- proximates a discriminative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' On seen categories of PASCAL VOC test set, the module trained on real dataset achieves the best result, while the module trained on constructed dataset gains an advantage on unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Besides, we also try to train our module on both constructed dataset and real dataset, which results in great improvement on PASCAL VOC test dataset but no advantage on PASCAL-sim, while the latter is our main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Therefore, we finally choose the module training on constructed dataset as our main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 17 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' More Qualitative Results We provide more qualitative results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 9, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 10, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 11, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Note that the images are generated from Stable Diffusion [27], and the corresponding masks are inferred from our proposed grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Specifically, the generated images and their corresponding segmentation masks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 10, including common objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', shark, turtle, and more unusual objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=', Ultraman, pterosaur, Chinese dragon, unicorn and dinosaur, shows the strong generalisability of the grounding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 11, we show more examples from our synthetic semantic segmentation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 12, we compare the model trained on our synthesized datasets with other ZS3 methods on PASCAL VOC dataset [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' We can observe that the MaskFormer [4] trained on our synthetic semantic segmentation dataset can obtain accurate segmentation on both seen and unseen categories, showing that the guided text-to-image diffusion model can be used to expand the vocabulary of pre-trained detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Limitation & Future Work In this paper, we have demonstrated the possibility for aligning the visual and language representation of a text-to-image diffusion model, and augment it with the ability of grounding visual objects along with generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' However, we also realise there exists certain limitation in this work, first, we only consider to ground the nouns that indicate visual entities, it would be interesting to ground the human-object, object-object interactions, or even verbs in the future, second, we are inserting the grounding module to a pre-trained text-to-image generative model, it would be interesting to co-train the two components, potentially enabling to generate images with higher quality and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 18 a photograph of a superman next to a tree a painting of a unicorn on the snow land a photograph of a Chinese dragon in the sky a photograph of a dinosaur in the woods a painting of a highly detailed wizard a photograph of a crane in the lake a photograph of a highly detailed Mickey a photograph of a devilfish in the sea Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Results of grounded generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The segmentation mask refers to the grounding results for the object underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 19 7a photograph of a launched rocket a painting of a highly detailed Ultraman a photograph of a white pterosaur a photograph of a shark in the sea a painting of a smilodon on the grass a photograph of a whale leaping out of the sea a photograph of a turtle crawling in the sand a photograph of a statue of Zeus Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Results of grounded generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' The segmentation mask refers to the grounding results for the object underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 20 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' Examples from our synthetic semantic segmentation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 21 Image GT Mask Zegformer MaskFormer (train on synthestic set) Image GT Mask Zegformer MaskFormer (train on synthestic set) Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' More visualization of zero-shot segmentation results on Pascal VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} +page_content=' 22 巫S' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE4T4oBgHgl3EQftA2c/content/2301.05221v1.pdf'} diff --git a/ktFRT4oBgHgl3EQfYDeM/content/tmp_files/2301.13548v1.pdf.txt b/ktFRT4oBgHgl3EQfYDeM/content/tmp_files/2301.13548v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..567593773df1a7b16dd724ab21b346ad91bcd358 --- /dev/null +++ b/ktFRT4oBgHgl3EQfYDeM/content/tmp_files/2301.13548v1.pdf.txt @@ -0,0 +1,2040 @@ +PHILIP SALTENBERGER +Institute for Numerical Analysis, TU Braunschweig +Braunschweig, Germany +(E-Mail: philip.saltenberger@tu-bs.de) +STRUCTURE-PRESERVING EIGENVALUE MODIFICATION OF +SYMPLECTIC MATRICES AND MATRIX PENCILS +Abstract +A famous theorem by R. Brauer shows how to modify a single eigen- +value of a matrix A by a rank-one update without changing the remain- +ing eigenvalues. A generalization of this theorem (due to R. Rado) is +used to change a pair of eigenvalues λ, 1/λ of a symplectic matrix S +in a structure-preserving way to desired target values µ, 1/µ. Universal +bounds on the relative distance between S and the newly constructed +symplectic matrix ˆS with modified spectrum are given. The eigenvalues +Segre characteristics of ˆS are related to those of S and a statement on +the eigenvalue condition numbers of ˆS is derived. The main results are +extended to matrix pencils. +1. Introduction +In numerical linear algebra and matrix analysis one occasionally encounters +the necessity of modifying special eigenvalues of a matrix without altering +its remaining eigenvalues. Techniques for changing certain eigenvalues of a +matrix have, for instance, been applied to solve nonnegative inverse eigen- +value problems [13, 16] or, in form of deflation methods, to remove dominant +eigenvalues in eigenvalue computations [14, Sec. 4.2]. Furthermore, the task +of modifying eigenvalues of matrices is of interest in stability and feedback of +linear systems [4, § 25], [3, Sec. 2.3] or for passivity and eigenvalue assignment +in control design [1]. One basic result on how a single eigenvalue of a matrix +may be changed without modifying any other eigenvalues is due to R. Brauer +and can be found in [3, Sec. 1], [16]. +Theorem 1 (Brauer). Let A ∈ Mn(C) have eigenvalues λ1, . . . , λn ∈ C and +let x1 ∈ Cn be an eigenvector for λ1. +Then, for any c ∈ Cn, the matrix +ˆA = A + x1cT ∈ Mn(C) has the eigenvalues λ1 + cT x1, λ2, . . . , λn. +This work is concerned with the purposive change of certain eigenvalues of +matrices with symplectic structure. A complex 2n × 2n matrix S ∈ M2n(C) +1 +arXiv:2301.13548v1 [math.NA] 31 Jan 2023 + +is called symplectic, if1 +ST J2nS = J2n =: J, +where J2n = +�0n×n +In +−In +0n×n +� +. +(1) +Defining S⋆ := JT ST J, we see that (1) is equivalent to S⋆S = I2n. Therefore, +a symplectic matrix S is always nonsingular and S⋆ = S−1. In consequence, +as S⋆ is similar2 to S, the eigenvalues of a symplectic matrices arise in pairs +λj, λ−1 +j , j = 1, . . . , n, where λj and λ−1 +j +have the same Segre characteristic. +Recall that for an eigenvalue λ of S, its Segre characteristic is the sequence of +sizes of the Jordan blocks of S with eigenvalue λ in non-increasing order [15]. +We denote the Segre characteristic of an eigenvalue by ((·, . . . , ·)). It is now im- +mediate that Theorem 1 can in general not be used for a structure-preserving, +symplectic change of eigenvalues. In fact, for a structure-preserving eigenvalue +modification, the change of λj and λ−1 +j +must take place simultaneously. +Without any structure-preservation in mind, changing two (or more) eigen- +values simultaneously is possible with the following generalization of Theorem +1 attributed to R. Rado. It can be found in [13], see also [3, Sec. 3]. +Theorem 2 (Rado). Let A ∈ Mn(C) have eigenvalues λ1, . . . , λn ∈ C and +let x1, . . . , xk ∈ Cn be linearly independent eigenvectors for λ1, . . . , λk. Set +X = [ x1 · · · xk ] ∈ Mn×k(C). Then, for any matrix C ∈ Mn×k(C), the +matrix ˆA = A + XCT has the eigenvalues µ1, . . . , µk, λk+1, . . . , λn, where µj, +j = 1, . . . , k, are the eigenvalues of Ω = diag(λ1, . . . , λn) + CT X. +In this work, we investigate how Theorem 2 can be utilized to change a pair +of eigenvalues λj, λ−1 +j +of a symplectic matrix S ∈ M2n(C) (or a symplectic +matrix pencil) to desired target values µ, µ−1 in a structure-preserving way +without modifying any other eigenvalues of S. Considering Theorem 2, the +starting point of our discussion is thus the following question: +Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 +1 , . . . , λn, λ−1 +n , +linearly independent eigenvectors x1, x2 ∈ C2n for λ1 and λ−1 +1 , respec- +tively, and X = [ x1 x2 ]. How has C ∈ M2n×2(C) to be chosen, such +that ˆS := S + XCT is symplectic with eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , +λn, λ−1 +n +for some given value µ ∈ C \ {0}? +The above-mentioned problem will be discussed in Section 2. In Section +3 we investigate whether we can find an upper bound b > 0 that only de- +pends on λ1, µ, x1 and x2 that assures the existence of a symplectic matrix +ˆS ∈ M2n(C) with eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n +and relative dis- +tance ∥ ˆS − S∥/∥S∥ ≤ b. We derive distinguished matrices ˆS1, ˆS2 for which +such a bound b can be neatly expressed and related to the relative change +1Here and in the following, T denotes the transpose of a (maybe complex) matrix or +vector, not its conjugate transpose. +2By definition, S⋆ is similar to ST and by the Taussky-Zassenhaus Theorem [17], ST +is similar to S. +2 + +in the eigenvalue, i.e. |λ1 − µ|/|λ1|. We discuss commutativity relations be- +tween S and ˆS in Section 4.1 and characterize the Segre characteristics of the +eigenvalues of ˆS in Section 4.2. The results of Section 4.1 will come in handy +here to find a condition on the simultaneous diagonalizability of S and ˆS. In +Section 6 we partially extend our results from Section 2 to symplectic matrix +pencils. +1.1. Notation +The set of all m × n matrices over K (where we use either K = C or K = R) +is denoted by Mm×n(K). +Whenever n = m we write Mn(K) instead of +Mn×n(C). For J2n ∈ M2n(R), see (1), we simply write J and add the index +whenever it is necessary to specify the size of J. +The range of a matrix +A ∈ Mm×n(C) is the vector space spanned by its columns and is denoted +range(A). For A ∈ Mm×n(C), we denote the Moore-Penrose pseudoinverse of +A by A+. In case m > n and rank(A) = n, we have A+ = (AHA)−1AH so +that A+A = In, while for n > m and rank(A) = m, A+ = AH(AAH)−1 yields +AA+ = Im. The superscript H always denotes the conjugate transpose of a +matrix or vector while T is used for the pure transposition. Whenever λ ∈ C +is some complex number, we denote by R(λ) and I(λ) its real and imaginary +part, respectively. Complex conjugation of a number x = a+ıb ∈ C is denoted +by a bar, i.e. x = a − ıb. +2. Symplectic Eigenvalue Modification +Let S ∈ M2n(C) be a symplectic matrix (see (1)) with eigenvalues λ1, λ−1 +1 , +. . ., λn, λ−1 +n +and let µ ∈ C \ {0} be given. Furthermore, assume x1, x2 ∈ C2n +are linearly independent3 eigenvectors of S for λ1 and λ−1 +1 , respectively, and +define X = [ x1 x2 ] ∈ M2n×2(C). In this section, our goal is to determine all +possible matrices C ∈ M2n×2(C) such that ˆS := S +XCT is again symplectic +and has the eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n . To this end, we will make +use of Rado’s theorem and derive a structure-preserving version of Theorem +2 (see Theorem 3). +As it will become clear later, it seems appropriate to consider the situa- +tions xT +1 Jx2 ̸= 0 and xT +1 Jx2 = 0 seperately. First, we assume that for the +eigenvalues λ1, λ−1 +1 +there exist eigenvectors x1, x2 ∈ C2n such that xT +1 Jx2 ̸= 0 +(this immediately implies x1 and x2 to be linearly independent). In this case, +we can assume w. l. o. g. xT +1 Jx2 = 1, which can be achieved by a scaling of x1 +and/or x2. That is, we have XT JX = J2. For the matrix ˆS := S + XCT to +be symplectic, it has to hold that +ˆST J ˆS = +� +S + XCT �T J +� +S + XCT � += J. +(2) +3If λ1 ̸= λ−1 +1 +, then x1 and x2 are necessarily linear independent. Therefore, the linear +independence is only a restrictive requirement if λ1 = λ−1 +1 +, i.e. λ1 = ±1. +3 + +Using ST JS = J, (2) is equivalent to the matrix equation +CXT JS + ST JXCT + CXT JXCT = 0 +(3) +for the unknown matrix C ∈ M2n×2(C). Notice that (3) can be rewritten as +C(XT JS + J2CT ) = −ST JXCT +(4) +using XT JX = J2. Since ST JX ∈ M2n×2(C) is a matrix of full rank, (4) +immediately implies range(C) ⊆ range(ST JX) for any solution C. +Thus, +for every C satisfying (3), there is a matrix R = [rij]ij ∈ M2(C) such that +C = ST JXRT . Plugging this ansatz into (3), we obtain a 2n × 2n equation +for R, namely +ST JXRT XT JS + ST JXRXT JT S + ST JXRT J2RXT JT S = 0. +Replacing XT JS by −XT JT S this can be rewritten as +ST JX +� +R − RT + RT J2R +� +XT JT S = 0. +(5) +Finally, we may multiply (5) with the pseudo inverses (ST JX)+ from the left +and with (XT JT S)+ from the right to obtain +R − RT + RT J2R = 0, +(6) +which is a matrix equation for R of size 2 × 2 that is equivalent to (5). As +R − RT and RT J2R are both skew-symmetric, their diagonals are identically +zero. Comparing the entries of R − RT and RT J2R in the (1,2) position, we +obtain the condition +r12 − r21 + r11r22 − r12r21 = 0 +(7) +for (6) to hold (comparing the elements in the (2, 1) position certainly gives +the same condition with a minus sign). In summary, a matrix of the form +ˆS = S + XCT is symplectic if and only if C = ST JXRT for some matrix +R = [rij]ij ∈ M2(C) whose entries satisfy (7). +Next, to achieve the desired eigenvalue modification, according to Theorem +2 we need to assure that the eigenvalues of +Ω := Λ + CT X = +�λ1 +0 +0 +λ−1 +1 +� ++ CT X = +�λ1 +0 +0 +λ−1 +1 +� ++ RXT JT SX +(8) +become equal to µ and µ−1. To this end, recall that SX = Xdiag(λ1, λ−1 +1 ) +(by construction of X). Thus diag(λ1, λ−1 +1 ) + RXT JT SX = diag(λ1, λ−1 +1 ) − +RXT JXdiag(λ1, λ−1 +1 ) which, since XT JX = J2, yields +Ω = +�λ1 +0 +0 +λ−1 +1 +� +− RJ2 +�λ1 +0 +0 +λ−1 +1 +� += +�λ1 + λ1r12 +−λ−1 +1 r11 +λ1r22 +λ−1 +1 +− λ−1 +1 r21 +� +. +(9) +4 + +The characteristic polynomial of Ω is +p(z) = z2 − +� +λ1(1 + r12) + λ−1 +1 (1 − r21) +� +z + (1 + r12)(1 − r21) + r11r22, +which should, by Theorem 2, be equal to q(z) = (z − µ)(z − µ−1) = z2 − (µ + +µ−1) + 1 to achieve that ˆS will have the eigenvalues µ and µ−1. This gives +two more conditions: one the one hand λ1(1 + r12) + λ−1 +1 (1 − r21) = µ + µ−1, +i.e. +λ1r12 − λ−1 +1 r21 = (µ + µ−1) − (λ1 + λ−1 +1 ). +(10) +One the other hand, (1 + r12)(1 − r21) + r11r22 = 1. The latter condition, +however, is equal to condition (7) obtained for the symplectic structure above. +Thus, additionally to (7), which is required for S + XCT to be symplectic, +the equation (10) has to hold to achieve that µ, µ−1 become eigenvalues of +S + XCT . In conclusion, we obtain the following version of Theorem 2 that +answers the question stated in Section 1 on the eigenvalue modification for +symplectic matrices. +Theorem 3. Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 +1 , λ2, λ−1 +2 , +. . . , λn, λ−1 +n +and let µ ∈ C \ {0} be given. Let x1, x2 ∈ C2n be eigenvectors +for λ1 and λ−1 +1 , respectively, normalized such that XT JX = J2 for X = +[ x1 x2 ] ∈ M2n×2(C) and set d := (µ + µ−1) − (λ1 + λ−1 +1 ). Then the matrix +ˆS := S + XCT ∈ M2n(C) +(11) +is symplectic and has the eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n +if and only +if CT = RXT JT S for some matrix R = [rij]ij ∈ M2(C) whose entries satisfy +the conditions +d = λ1r12 − λ−1 +1 r21, and +(12) +0 = r12 − r21 + r11r22 − r12r21. +(13) +Notice that the matrix ˆS = S + XRXT JT S in (11) can also be expressed +as ˆS = (I2n + XRXT JT )S or as +ˆS = S + XRΛ−1XT JT = S + XR +� +λ−1 +1 +0 +0 +λ1 +� +XT JT +(14) +according to the relation Λ−1XT JT = XT JT S (where Λ = diag(λ1, λ−1 +1 )). +Furthermore, we see from (12) and (13) that there exist infinitely many pos- +sible choices for R that realize the desired eigenvalue modification. +Next, we discuss the case that the eigenvectors x1, x2 ∈ C2n of the sym- +plectic matrix S for λ1 and λ−1 +1 , respectively, satisfy xT +1 Jx2 = 0 and how this +condition effects the result from Theorem 3. To this end, first notice that a +symplectic matrix S ∈ M2n(C) need in fact not have eigenvectors x1, x2 for +5 + +λ1 and λ−1 +1 +that satisfy xT +1 Jx2 ̸= 0. A situation of this kind arises for the +symplectic matrix +S = +� +��� +λ1 +1 +0 +0 +0 +λ1 +0 +0 +0 +0 +λ−1 +1 +0 +0 +0 +−λ−2 +1 +λ−1 +1 +� +��� +and its eigenvalue λ1. The only eigenvectors for λ1 and λ−1 +1 +are e1 and e4, +respectively, and we have eT +1 J4e4 = 0. Thus, Theorem 3 cannot be applied. A +simple sufficient (but not necessary) criterion to assure that eigenvectors with +xT +1 Jx2 ̸= 0 must exist, is that S is a diagonalizable matrix, cf. [10, Lem. 3, +Cor. 3.1] and Corollary 1 below. +Whenever x1, x2 ∈ C2n are eigenvectors of S for λ1 and λ−1 +1 +with xT +1 Jx2 = +0, then XT JX = 0 follows for X = [ x1 x2 ]. In this case, it follows from (3) +that (4) takes the form +CXT JS = −ST JXCT . +Again we obtain range(C) ⊆ range(ST JX), so there has to exist some matrix +R = [rij]ij ∈ M2(C) such that C = ST JXRT . However, despite the concrete +form of R, analogously to (8) we obtain +Ω = +�λ +0 +0 +λ−1 +� ++ CT X = +�λ +0 +0 +λ−1 +� +− RXT JSX = +�λ +0 +0 +λ−1 +� +since SX = Xdiag(λ, λ−1) and XT JX = 0. Thus, even if R is chosen ac- +cording to (13) such that ˆS = S + XRXT JT S is symplectic, no change in the +eigenvalues can be achieved. In consequence, a change of an eigenvalue pair +λ1, λ−1 +1 +of a symplectic matrix by Rado’s theorem in a structure-preserving +way is only possible if there exist eigenvectors x1 and x2 for λ1 and λ−1 +1 , re- +spectively, such that xT +1 Jx2 ̸= 0. In the next section, we derive a universal +criterion on the existence of such eigenvectors. +2.1. Applying Theorem 3: a criterion +We will now characterize those symplectic matrices S ∈ M2n(C), for which +an eigenvalue adjustment according to Theorem 3 is possible. The condition +derived below involves the Segre characteristic of the eigenvalue λ1 ∈ σ(S) to +be modified. +First, let λ1 ∈ σ(S) and Sx1 = λ1x1 and Sx2 = λ−1 +1 x2. Now suppose at +least one of both vectors, e.g. x1, belongs to a nontrivial4 Jordan chain, that +is, there is some z ∈ C2n such that (S − λ1I2n)z = x1 (and possibly more +4By nontrivial, we mean a Jordan chain of length ≥ 2 while a trivial Jordan chain +refers to a chain of length one. +6 + +generalized eigenvectors beside z). Then we have +xT +1 Jx2 = +� +(S − λ1I2n)z +�T Jx2 = zT ST Jx2 − λ1zT Jx2 += zT JJT ST Jx2 − λ1zT Jx2 += zT JS−1x2 − λ1zT Jx2 += λ1zT Jx2 − λ1zT Jx2 = 0 +(15) +as JT ST J = S⋆ = S−1 and S−1x2 = λ1x2. +In consequence, xT +1 Jx2 = 0 +whenever x1, x2 are eigenvectors of S for λ1 and λ−1 +1 , respectively, and at +least one of them belongs to a nontrivial Jordan chain. In other words, we +may have xT +1 Jx2 ̸= 0 only in case both x1 and x2 belong to trivial Jordan +chains. Next, we show that in case x1 belongs to a trivial Jordan chain there +must exist x2 (also from a trivial Jordan chain) such that xT +1 Jx2 ̸= 0. +To this end, assume that λ1 ∈ C is an eigenvalue of the symplectic ma- +trix S ∈ M2n(C) with p ≥ 1 ones in its Segre characteristic (that is, there +are p Jordan blocks of size 1 × 1, i.e. p trivial Jordan chains, and possibly +other Jordan blocks of size ≥ 2). Then there exists a matrix F ∈ M2n(C) +transforming S to the following Jordan form +F −1SF =: G = +� +���� +λ1 +... +λ1 +0 +0 +ˆG +� +���� +(16) +where the upper-left block is λ1Ip and ˆG contains all other Jordan blocks (note +that there might also be other Jordan blocks for λ1 of size ≥ 2 contained in +ˆG). Now define +x1 := Fe1 +and +˜f H := eT +1 F −1. +(17) +Then x1 is a right eigenvector of S for λ1 (Sx1 = λ1x1) and ˜f is a left +eigenvector of S for λ1 ( ˜f HS = λ1 ˜f H). Certainly, ˜f Hx1 = 1. Now we define +xT +2 := ˜f HJ. Then we have +˜f HS = λ1 ˜f H ⇔ xT +2 JT S = λ1xT +2 JT +⇔ xT +2 JT SJ = λ1xT +2 +⇔ xT +2 S−T = λ1xT +2 +⇔ Sx2 = λ−1 +1 x2. +(18) +It follows that x2 is an eigenvector of S for λ−1 +1 . Now we obtain +xT +1 Jx2 = xT +2 JT x1 = ˜f Hx1 = 1 ̸= 0. +In conclusion, for any eigenvector x1 of S for λ1 belonging to a trivial Jordan +chain, there always exists an eigenvector x2 of S for λ−1 +1 +such that xT +1 Jx2 ̸= 0. +Recall from our observation (15) above, that x2 must also be a vector from a +trivial Jordan chain. We conclude our findings in the following theorem. +7 + +Theorem 4. Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C\{0} +be given. Then Theorem 3 is applicable to S for λ1 and µ, i.e. there exist +eigenvectors x1, x2 ∈ C2n for λ1 and λ−1 +1 , respectively, with xT +1 Jx2 ̸= 0, if and +only if the Segre characteristic of S for λ1 contains a one, that is, it has the +form ((⋆, ⋆, · · · , ⋆, 1)). In particular, eigenvectors x1 for λ1 and x2 for λ−1 +1 +with xT +1 Jx2 ̸= 0 always belong to trivial Jordan chains of S. +Do not overlook that Theorem 4 applies also for λ1 = λ−1 +1 , i.e. λ1 = ±1. +In this case λ1 ∈ σ(S) necessarily has an even multiplicity and an even +number of Jordan blocks of the same size, so a Segre characteristic of the +form ((⋆, ⋆, · · · , ⋆, 1)) implies that there appears at least another one, i.e. +((⋆, ⋆, · · · , ⋆, 1, 1)). Then the reasoning in (16), (17) and (18) applies in the +same way. If S is diagonalizable, the Segre characteristic of S for any eigen- +value λj ∈ σ(S) consists only of ones. So we immediately obtain the following +corollary. +Corollary 1. Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ +C \ {0} be given. Then Theorem 3 is applicable to S for λ1 and µ if S is +diagonalizable. +3. Bounding the relative change +Let S ∈ M2n(C) be symplectic. For the matrix ˆS = S + XRXT JT S in (11) +we immediately obtain a bound on its (absolute or relative) change in norm +with respect to S. That is, +∥S − ˆS∥ ≤ ∥R∥∥X∥∥XT ∥∥S∥ +and +∥S − ˆS∥ +∥S∥ +≤ ∥R∥∥X∥∥XT ∥ +(19) +hold for any submultiplicative and unitarily invariant matrix norm ∥·∥. In this +section, we intend to derive explicit bounds of the relative distance between +S and ˆS for certain choices of R = [rij]ij ∈ M2(C). To this end, we assume +∥ · ∥ = ∥ · ∥F so that ∥X∥F = ∥XT ∥F holds and the upper bound in (19) +reduces to ∥R∥F ∥X∥2 +F . +To bound the relative change ∥ ˆS − S∥F /∥S∥F with respect to S consider +again (12) and (13). The solution set to (12) is an affine subspace of C2 and +all solutions may be parameterized as +r12 = ηλ−1 +1 , +r21 = −λ1d + ηλ1, +η ∈ C. +(20) +Plugging these expressions for r12 and r21 into (13) yields a polynomial in η, +i.e. +p(η) = −η2 + η +� +λ−1 +1 ++ d − λ1 +� ++ dλ1 + r11r22. +(21) +Thus, depending on r11 and r22 (which can both be arbitrary), in (21) there +are always two solutions η1, η2 ∈ C of p(η) = 0 and, in consequence, two +8 + +matrices +R(ηj, r11, r22) := +� +r11 +ηjλ−1 +1 +λ1(ηj − d) +r22 +� +, +j = 1, 2, +(22) +so that their entries satisfy (12) and (13). +To find some R ∈ M2(C) that yields a small norm ∥R∥F and thus a small +bound in (19), it seems natural to consider the case r11r22 = 0, in particular +r11 = r22 = 05. The two possible roots of p(η) for r11r22 = 0 are η1 = µ − λ1 +and η2 = µ−1 − λ1. The matrices R1 := R(η1, 0, 0) and R2 := R(η2, 0, 0) that +arise according to (22) are thus given by +R1 = +� +0 +λ−1 +1 (µ − λ1) +µ−1(µ − λ1) +0 +� +, +R2 = +� +0 +µ−1(λ−1 +1 +− µ) +λ1(λ−1 +1 +− µ) +0 +� +. +(23) +Using R1 and R2 in (23), explicit bounds can be found on ∥ ˆS − S∥F /∥S∥F . +According to (19) and (23) such a bound b ≥ 0 only depends on λ1, µ and +the eigenvectors of S for λ1 and λ−1 +1 +and guarantees the existence of a sym- +plectic matrix ˆS ∈ M2n(C) that solves the problem from Section 1 with +∥ ˆS −S∥F /∥S∥F ≤ b. To formulate these bounds, we impose a condition on X +to estimate ∥X∥F without computing the norm. In particular, we assume the +eigenvectors x1, x2 ∈ C2n of S for λ1 and λ−1 +1 , respectively, to be normalized, +i.e. ∥x1∥2 = ∥x2∥2 = 1 and X ∈ M2n×2(C) to be of the form +X = +1 +� +xT +1 Jx2 +� +x1 +x2 +� +. +(24) +Then XT JX = J2 holds and it follows that +∥X∥2 +F = +����� +1 +� +xT +1 Jx2 +����� +2 ��� +x1 +x2 +���2 +F = +1 +|xT +1 Jx2| +��� +x1 +x2 +���2 +F = +2 +|xT +1 Jx2| +(25) +The value 1/|xT +1 Jx2| has a nice interpretation whenever λ1 is a simple eigen- +value of S and ∥x1∥2 = ∥x2∥2 = 1 holds. To see this, recall that, whenever +A ∈ Mn(C) has a simple eigenvalue λ ∈ C (i.e. +its algebraic multiplicity +equals one), then +κ(A, λ) := ∥u∥2∥v∥2 +|vHu| +is called its condition number, where u ∈ C2n and v ∈ C2n are right and left +eigenvectors of A for λ (i.e. Au = λu and vHA = λvH). It is a measure +on how sensitive λ reacts to small changes in the matrix A, see [14, Sec. 3.3]. +As we have seen in (18) above, (xT +2 JT )S = λ1(xT +2 JT ) whenever x2 satisfies +5Certainly, choosing r11 = 0 and r22 ̸= 0 gives the same roots of p(η) = 0 in (21), and +thus the same values for r12 and r21, but a larger Frobenius norm of R than choosing +r11 = r22 = 0. +9 + +Sx2 = λ−1 +1 x2. Thus, for simple λ1 (which implies that λ−1 +1 +is simple as well) +we can choose u = x1 and vH = xT +2 JT so that +κ(S, λ1) = ∥u∥2∥v∥2 +|vHu| += ∥x1∥2∥Jx2∥2 +|xT +2 JT x1| += +1 +|xT +1 Jx2| +since ∥x1∥2 = 1 and ∥Jx2∥2 = ∥x2∥2 = ∥x2∥2 = 1. We can now formulate +the following theorem which follows directly from the bound in (19), the +observation in (25) and the Frobenius norms of the matrices R1, R2 in (23). +Theorem 5. Let S ∈ M2n(C) be symplectic with λ1, λ−1 +1 +∈ σ(S) and let +µ ∈ C \ {0} be given. Let x1, x2 ∈ C2n be normalized eigenvectors for λ1 and +λ−1 +1 , respectively, and X ∈ M2n×2(C) as in (24). Define +Φ := +2 +|xT +1 Jx2| +� += 2κ(S, λ1) if λ1 is simple +� +. +(i) Let ˆS1 = S + XR1XT JT S be constructed according to Theorem 3 with +R1 from (23). Then +∥ ˆS1 − S∥F +∥S∥F +≤ |λ1 − µ| +|λ1| +� +Φ +� +1 + |λ1|2 +|µ|2 +� +. +(26) +(ii) Let ˆS2 = S + XR2XT JT S be constructed according to Theorem 3 with +R2 from (23). Then +∥ ˆS2 − S∥F +∥S∥F +≤ |λ−1 +1 +− µ| +|λ−1 +1 | +� +�Φ +� +1 + |λ−1 +1 |2 +|µ|2 +� +� . +(27) +As the following example shows, similar easy bounds can be found with +the use of R1 and R2 when ∥ · ∥2 is considered. In fact, in the 2-norm, such a +bound can be sharp. +Example 1. For a symplectic matrix S ∈ M2n(C), the bound (19) for ˆS1 = +S + XR1XT JT S with respect to ∥ · ∥2 can easily be determined as +∥ ˆS1 − S∥2 +∥S∥2 +≤ |λ1 − µ| +|λ1| +· max +� +1, |λ1| +|µ| +� +∥X∥2 +2, +(28) +It can be seen for S = diag(Λ, Λ−1) with Λ = diag(λ1, . . . , λn) that the bound +in (28) can be sharp. +In particular, with eigenvectors e1, en+1 ∈ R2n for +λ1, λ−1 +1 , respectively, and X = [ e1 en+1 ] we have +XR1XT JT S = diag +� +r12λ1, 0, . . . , 0, −r21λ−1 +1 , 0, . . . , 0 +� +10 + +with nonzero entries in the first and (n+1)st position. As r12λ1 = µ−λ1 and +−r21λ−1 +1 += µ−1 − λ−1 +1 +we obtain under the assumption |λ1 − µ| ≥ |λ−1 +1 +− µ−1| +∥ ˆS − S∥2 = ∥XR1XT JT S∥2 = |λ1 − µ| +and so ∥ ˆS − S∥2/∥S∥2 = |λ1 − µ|/|λ1| if λ1 is the largest eigenvalue of S in +absolute value (i.e. ∥S∥2 = |λ1|). On the other hand, for X we certainly have +∥X∥2 +2 = 1 and thus, whenever |µ| ≥ |λ1|, the bound on the right hand side in +(28) also reduces to |λ1 − µ|/|λ1|. +3.1. Improved distance bounds +Although the bound in (26) nicely relates ∥ ˆS1 − S∥F /∥S∥F to the relative +value change |λ1 − µ|/|λ1| and the condition number κ(S, λ1), it can be quite +bad6, see e.g. Fig. 1 in Section 5. In this section we derive sharper bounds +under the additional assumption that ∥S∥F is known. +As before, let S ∈ M2n(C) be symplectic with eigenvectors x1, x2 ∈ C2n +for λ1, λ−1 +1 +∈ σ(S), respectively, such that xT +1 Jx2 ̸= 0 (and set X = [ x1 x2 ]). +As seen in (14), we have for Λ := diag(λ1, λ−1 +1 ) and R ∈ M2(C) that satisfies +(12) and (13) +ˆS = S + XRXT JT S = S + XRΛ−1XT JT +and therefore ∥S − ˆS∥F = ∥XRΛ−1XT JT ∥F = ∥XRΛ−1XT ∥F . Whenever +R = Rj (j = 1, 2) from (23), then +XRjΛ−1XT = X +� +0 +ηj +ηj − d +0 +� +XT =: X ˜RjXT , +˜Rj = RjΛ−1, +according to (20). Recall the solutions of p(η) = 0 in (21), i.e. η1 = µ − λ1 +(corresponding to R1) and η2 = µ−1−λ1 (corresponding to R2). Instead of es- +timating ∥X ˜RjXT ∥F by ∥ ˜Rj∥F ∥X∥2 +F we now intend to estimate ∥X ˜RjXT ∥F +directly. +To this end, assume that X ∈ M2n×2(C) has the form (24) and +ˆSj = S + XRjXT JT S, j = 1, 2. Then we have for Y = +� +xT +1 Jx2X = [ x1 x2 ] +∥S − ˆSj∥2 +F = ∥X ˜RjXT ∥2 +F = tr +� +Y ˜RjY T (Y ˜RjY T )H� +|xT +1 Jx2|2 += tr +� +Y ˜RjY T Y ˜RH +j Y H� +|xT +1 Jx2|2 += tr +� +Y HY ˜Rj(Y HY ) ˜RH +j +� +|xT +1 Jx2|2 +. +6The same is true for the bound in (27). +11 + +Now we further obtain +|xT +1 Jx2|2∥ ˆSj − S∥2 +F = tr +� +Y HY ˜Rj(Y HY ) ˜RH +j +� += tr +�� +1 +xH +1 x2 +xH +2 x1 +1 +� � +0 +ηj +ηj − d +0 +� � +1 +xH +2 x1 +xH +1 x2 +1 +� � +0 +ηj − d +ηj +0 +�� += |xH +1 x2|2ηj(ηj − d) + |ηj|2 + |ηj − d|2 + |xH +1 x2|2ηj(ηj − d) += |ηj|2 + |ηj − d|2 + 2|xH +1 x2|2 · R(ηj(ηj − d)). +(29) +Note that |xH +1 x2| ≤ ∥x1∥2∥x2∥2 = 1 as x1 and x2 are normalized. Fur- +thermore, R(ηj(ηj − d)) = R(|ηj|2 − ηjd) = |ηj|2 − R(ηjd), and we may now +derive upper (and lower) bounds for (29) depending on whether this term is +positive or negative. +(i) Suppose R(ηjd) < |ηj|2. Then R(ηj(ηj − d)) > 0 follows and, since +|xH +1 x2|2 ≤ 1, we can estimate from (29), setting |xH +1 x2|2 = 1, +|xT +1 Jx2|2∥S − ˆSj∥2 +F ≤ |ηj|2 + |ηj − d|2 + 2 · R(ηj(ηj − d)) += +� +ηj + (ηj − d) +�� +ηj + (ηj − d) +� += |2ηj − d|2. +On the other hand, changing the sign of R(ηj(ηj −d)) we certainly have +|xT +1 Jx2|2∥S − ˆSj∥2 +F ≥ |ηj|2 + |ηj − d|2 − 2 · R(ηj(ηj − d)) += (ηj − (ηj − d))(ηj − (ηj − d)) = |d|2. +(ii) Suppose R(ηjd) ≥ |ηj|2. Then R(ηj(ηj − d)) ≤ 0 follows and we can +estimate from (29), setting again |xH +1 x2|2 = 1, +|xT +1 Jx2|2∥S − ˆSj∥2 +F ≥ |ηj|2 + |ηj − d|2 + 2 · R(ηj(ηj − d)) = |2ηj − d|2 +while on the other hand, with a change of sign, we obtain +|xT +1 Jx2|2∥S − ˆSj∥2 +F ≤ |ηj|2 + |ηj − d|2 − 2 · R(ηj(ηj − d)) = |d|2. +Before we state our findings in the next theorem, notice that there are neat +expressions for the terms 2ηj − d, j = 1, 2, arising above, i.e. +2η1 − d = µ − λ1 +λ1 +� +λ1 + µ−1� +, +2η2 − d = λ−1 +1 +− µ +λ−1 +1 +� +µ−1 + λ−1 +1 +� +. +As it turns out, also d can be rewritten in a similar fashion as +d = µ − λ1 +λ1 +� +λ1 − µ−1� += λ−1 +1 +− µ +λ−1 +1 +� +µ−1 − λ−1 +1 +� +. +(30) +12 + +Finally, the two conditions to be checked in (i) and (ii) above can be simplified. +For η1 = µ − λ1 one finds, after some reformulations, +R +� +η1d +� +− |η1|2 = 1 +2 +� +(µ − λ1)d + (µ − λ1)d +� +− |µ − λ1|2 += 1 +2 +�|µ − λ1|2d +µ − λ1 ++ |µ − λ1|2d +µ − λ1 +� +− |µ − λ1|2 += −|µ − λ1|2 +� +1 − 1 +2 +� +d +µ − λ1 ++ +d +µ − λ1 +�� += −|µ − λ1|2 +� +1 − R +� +d +µ − λ1 +�� += −|µ − λ1|2 +� +1 − R +�λ1 − µ−1 +λ1 +�� += −|µ − λ1|2R((µλ1)−1) +where we used the first expression for d in (30) in the second-last equation. +Thus R(η1d) ≥ |ηj|2 holds if and only if −|µ − λ1|2R((µλ1)−1) ≥ 0, which +is the case if and only if R(λ1µ) ≤ 0 as (λ1µ)−1 and λ1µ are located in the +same half plane. For η2 = µ−1 − λ1 we obtain analogously +R +� +η2d +� +− |η2|2 = −|λ1µ − 1|2R +� +(λµ)−1� +and so R(η2d) ≥ |ηj|2 holds if and only if R((λµ)−1) ≤ 0. This, in turn, +holds if and only if R(λ1µ) ≤ 0. In conclusion, we have proven the following +theorem. +Theorem 6. Let S ∈ M2n(C) be symplectic with λ1, λ−1 +1 +∈ σ(S) and let +µ ∈ C \ {0} be given. Let x1, x2 ∈ C2n be normalized eigenvectors for λ1 and +λ−1 +1 , respectively, with xT +1 Jx2 ̸= 0 and X ∈ M2n×2(C) as in (24). Define +Φ := +1 +|xT +1 Jx2| · ∥S∥F +� += κ(S, λ1) +∥S∥F +if λ1 is simple +� +(i) Let ˆS1 = S + XR1XT JT S be constructed according to Theorem 3 with +R1 from (23). Whenever R(λ1µ) ≤ 0, then +|λ1 − µ| +|λ1| +� +|λ1 + µ−1|Φ +� +≤ ∥ ˆS1 − S∥F +∥S∥F +≤ |λ1 − µ| +|λ1| +� +|λ1 − µ−1|Φ +� +. +If R(λ1µ) > 0 the upper and lower bounds interchange. +(ii) Let ˆS2 = S + XR2XT JT S be constructed according to Theorem 3 with +R2 from (23). Whenever R(λ1µ) ≤ 0, then +|λ−1 +1 +− µ| +|λ−1 +1 | +� +|λ−1 +1 ++ µ−1|Φ +� +≤ ∥ ˆS2 − S∥F +∥S∥F +≤ |λ−1 +1 +− µ| +|λ−1 +1 | +� +|λ−1 +1 +− µ−1|Φ +� +. +If R(λ1µ) > 0 the upper and lower bounds interchange. +13 + +It is shown in Section 5 (see Fig. 1) that the bounds in Theorem 6 are +significantly sharper compared to the bounds in (26) and (27). +Remark 1. For any matrix R = [rij] ∈ M2(C) that satisfies the conditions +(12) and (13) the bounds in (19) can easily be calculated. However, there are +several reasons for not considering other choices of R (beside R1 and R2 from +(23)) in this section in detail: +(a) If r11 ̸= 0, r22 ̸= 0, there are two possibilities for R whose entries r12 and +r21 of R depend on c = r11r22 through (one of) the zeros of p(η) = 0, +see (21). Thus, r12 and r21 involve the expression of a complex square +root and there is no neat and compact expression for ∥R∥F compared to +(26) and (27) or to the formulas in Theorem 6. Furthermore, minimizing +∥S − ˆS∥F with respect to the entries of R under the side conditions (12) +and (13) results in a difficult complex optimization problem for which the +author is not aware of a closed form solution. +(b) Among all matrices R that satisfy the conditions (12) and (13) the ma- +trices R1 and R2 from (23) are the only possible choice when ˆS = S + +XRXT JT S should inherit desirable properties (e.g. related to diagonaliz- +ability) from S. These distinguishing features of R1 and R2 are discussed +in the upcoming sections. +(c) All numerical experiments that have been performed indicate that rarely +a matrix R′ ∈ M2(C) different from R1 and R2 that satiesfies (12) and +(13) was detected such that ∥ ˆS −S∥F /∥S∥F for ˆS = S +XR′XT JT S was +smaller than the minimum of ∥ ˆS1 − S∥/∥S∥F and ∥ ˆS2 − S∥F /∥S∥F . This +is visualized in Section 5, see Figure 3. +4. Segre characteristics and commutativity relations +In this section we discuss how the Segre characteristics of eigenvalues are +effected by a change of a symplectic matrix S ∈ M2n(C) to ˆS ∈ M2n(C) +according to Theorem 3. In particular, we will show that the Segre character- +istics of the eigenvalues of S and ˆS are either the same or connected in a direct +way. Furthermore, we make a statement on eigenvectors of S and ˆS that re- +main unchanged. Notice that, in the context of Theorem 2, the eigenvectors +of A and ˆA = A + XCT are in general all different and not related in an +immediate fashion [3] if no further restrictions are imposed on the form of C. +In this section we show that, in the structure-preserving context of Theorem +3, the particular form of C allows for some explicit statements. Furthermore, +we derive statements on the diagonalizability of ˆS and the simultaneous di- +agonalizability of S and ˆS. To this end, we begin in Section 4.1 with a result +on the commutativity of S and ˆS. +14 + +4.1. The Commutativity of S and ˆS +Recall that the matrix ˆS in (11) can also be expressed as +ˆS = +� +I2n + XRXT JT � +S. +(31) +Since ˆS and S are both symplectic, the matrix ˆSS−1 = ˆSS⋆ = I2n+XRXT JT +is symplectic, too. As for any A, B ∈ Mn(C) the matrices AB and BA always +have the same eigenvalues [9], beside ˆS, we may also define the symplectic +matrix ˜S := S +� +I2n + XRXT JT � +∈ M2n(C) that solves the eigenvalue modi- +fication problem stated in Section 1. A question naturally arising is whether +there is a connection between ˆS from (31) and ˜S. Such a connection is revealed +in Theorem 7 which shows a distinguishing feature of the matrices from (23) +among all matrices R that satisfy (12) and (13), see Remark 1 (b). The result +from Theorem 7 will be used when the diagonalizability of ˆS is investigated +in Section 4.2. +Theorem 7. Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and assume +ˆS = S + XRXT JT S has been constructed according to Theorem 3. Then the +following is true: +1. In case λ1 ̸= ±1, +ˆS = +� +I2n + XRXT JT � +S = S +� +I2n + XRXT JT � += ˜S +(32) +holds if and only if R = [rij]ij ∈ M2(C) is one of the matrices in (23). +2. In case λ1 = ±1, (32) holds for any R = [rij]ij ∈ M2(C) satisfying (12) +and (13). +Proof. First, notice that ˆS = ˜S is equivalent to +XRXT JT S = SXRXT JT . +(33) +Multiplying both equations with J (from the right) and using the relations +SX = XΛ (where Λ = diag(λ1, λ−1 +1 )) and JT SJ = S−T yields XRXT S−T = +XΛRXT . Moreover, XT S−T = (S−1X)T = (XΛ−1)T = Λ−1XT and, equiv- +alently to (33), it suffices to investigate the equation +XRΛ−1XT = XΛRXT . +(34) +Now, as X ∈ M2n×2(C) has full rank, (34) is (by the multiplication with X+ +from the left and (XT )+ from the right) equivalent to RΛ−1 = ΛR, that is, +ΛRΛ = R. For R = [rij]ij we obtain +ΛRΛ = +�λ1 +0 +0 +λ−1 +1 +� �r11 +r12 +r21 +r22 +� �λ1 +0 +0 +λ−1 +1 +� += +�λ2 +1r11 +r12 +r21 +λ−2 +1 r22 +� +. +This shows that ΛRΛ = R holds, in case λ1 ̸= ±1, if and only if r11 = r22 = 0. +The two possibilities for R that satisfy the conditions (12) and (13) when +r11 = r22 = 0 are the matrices in (23). Furthermore, if λ1 = ±1, the equation +always holds. This completes the proof. +15 + +Theorem 7 shows that, in general (i.e. for λ1 ̸= ±1), only the two possible +choices for R in (23) produce commutativity of S and I2n + XRXT JT (i.e. +to have ˆS = ˜S). In the special case λ1 = ±1, any matrix R determined from +(12) and (13) will cause this commutativity relation. +4.2. Segre characteristics +Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 +1 , . . . , λn, λ−1 +n +and let +µ ∈ C \ {0} be given. To analyse the consequences of the change S �→ ˆS = +S + XRXT JT S on the Segre characteristics of the eigenvalues of S and ˆS, +we discuss the cases of λ1, λ−1 +1 +(the eigenvalues that are changed), µ, µ−1 (the +values λ1 and λ−1 +1 +are changed to) and all other eigenvalues (which are the +same for S and ˆS) separately. As before, let x1, x2 ∈ C2n be eigenvectors +for λ1 and λ−1 +1 , respectively, normalized such that XT J2nX = J2 for X = +[ x1 x2 ] ∈ M2n×2(C) and assume ˆS = S + XRXT JT S has been constructed +as in Theorem 3. +We first consider eigenvalues different from λ1, λ−1 +1 , µ and µ−1. +These +eigenvalues and their algebraic multiplicities are the same for S and ˆS and +we show that their eigenspaces and Jordan chains (thus, in consequence, their +Segre characteristics) remain completely unchanged. To prove this, we need +the following fact about the matrix S and its (generalized) eigenvectors (see +also [10, Sec. 2]): assume that λ is some eigenvalue of S different from λ1 and +λ−1 +1 +and let y1, y2, . . . , yp ∈ C2n (p ≥ 1) be a Jordan chain for S and λ, i.e. +it holds that (S − λI2n)y1 = 0 and (S − λI2n)yk+1 = yk for k = 1, . . . , p − 1. +Then XT Jyk = 0 follows for any k = 1, . . . , p. To see this, first consider the +eigenvector y1 of S for λ. We have +λ1xT +1 Jy1 = xT +1 ST Jy1 = xT +1 JJT ST Jy1 = xT +1 JS−1y1 = λ−1xT +1 Jy1 +This shows that xT +1 Jy1 = 0 if λ ̸= λ−1 +1 . Similarly, xT +2 Jy1 = 0 follows for +λ ̸= λ1. Now assume that xT +i Jyℓ = 0 holds for i = 1, 2 and ℓ = 1, . . . , k. For +(S − λI2n)yk+1 = yk we thus obtain +0 = xT +1 Jyk = xT +1 J(S − λI2n)yk+1 = xT +1 JSyk+1 − λxT +1 Jyk+1 += xT +1 S−T Jyk+1 − λxT +1 Jyk+1 += λ−1 +1 xT +1 Jyk+1 − λxT +1 Jyk+1 += (λ−1 +1 +− λ)xT +1 Jyk+1. +Therefore, again xT +1 Jyk+1 = 0 follows whenever λ ̸= λ−1 +1 . With the same +reasoning we obtain xT +2 Jyk+1 = 0 for λ ̸= λ1. In conclusion we have XT Jyk = +0 for any k = 1, . . . , p whenever λ1 ̸= λ ̸= λ−1 +1 . +Lemma 1. Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C\{0} +be given. Suppose that ˆS ∈ M2n(C) has been constructed according to Theorem +16 + +3. Then for any λ ∈ σ( ˆS) which is neither equal to λ1 or λ−1 +1 +nor equal to µ +or µ−1 the Segre characteristics of λ as an eigenvalue of S and ˆS and their +corresponding Jordan chains, respectively, are identical. +Proof. Assume λ ∈ σ( ˆS) is neither equal to λ1 or λ−1 +1 +nor equal to µ or µ−1. +By construction of ˆS = S + XRXT JT S, λ is an eigenvalue of both S and ˆS +with the same algebraic multiplicities. Whenever y1 ∈ C2n is an eigenvector +of S for λ it is also an eigenvector of ˆS for λ since XT Jy1 = 0, which implies +ˆSy1 = Sy1 + XRXT JT Sy1 = Sy1 − λXRXT Jy1 = Sy1 = λy1. +(35) +Next, let y1, . . . , yp ∈ C2n be a Jordan chain for S and λ. Then +� ˆS − λI2n +� +yi+1 = +� +S + XRXT JT S +� +yi+1 − λyi+1 += +� +yi + λyi+1 +� ++ XRXT JT Syi+1 − λyi+1 += yi − XRXT J(yi + λyi+1) = yi +since XT Jyk = 0 for any yk, k = 1, . . . , p, from the Jordan chain. Inductively, +this shows that y1, . . . , yp remains to be a Jordan chain of ˆS for λ. Therefore, +the Segre characteristic for λ of S and ˆS and the corresponding Jordan chains +are the same. +Next we consider λ1 and λ−1 +1 . +When S ∈ M2n(C) is transformed to +ˆS = S + XRXT JT S and (one instance of) λ1, λ−1 +1 +is replaced by µ and +µ−1, the Segre characteristic of λ1 for ˆS is necessarily different from its Segre +characteristic for S due to the eigenvalue modification that has taken place (if +λ1 is a simple eigenvalue of S, then it is not even an eigenvalue of ˆS anymore). +However, if the algebraic multiplicity of λ1 as an eigenvalue of S is ≥ 2, then +the Segre characteristics of λ1 as an eigenvalue of S and ˆS are connected in +an easy fashion (see Theorem 8 below). This is obviously false in the general +context of Rado’s Theorem, where nontrivial Jordan blocks may arise, as the +following counterexample for A = I4 and ˆA = A + XCT shows: +ˆA = +� +��� +1 +1 +1 +1 +� +��� + +� +��� +1 +0 +0 +1 +0 +0 +0 +0 +� +��� +�1 +0 +0 +0 +0 +0 +1 +0 +� += +� +��� +2 +1 +1 +1 +1 +� +��� . +In this example, the Segre characteristic of 1 ∈ σ(A) is ((1, 1, 1, 1)) while it is +((2, 1)) for ˆA. +Theorem 8. Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ +C \ {0, λ1, λ−1 +1 } be given. Suppose that ˆS = S + XRXT JT S ∈ M2n(C) has +been constructed according to Theorem 3. Then the following hold: +17 + +(i) If the Segre characteristic of λ1 ̸= ±1 as an eigenvalue of S is +((sk, sk−1, . . . , s2, s1)) +(36) +with7 sk ≥ sk−1 ≥ · · · ≥ s2 ≥ s1 = 1, then the Segre characteristic of λ1 +as an eigenvalue of ˆS is ((sk, sk−1, . . . , s2)). Moreover, if λ1 = ±1 and +(36) is its Segre characteristic of S with7 s2 = s1 = 1, then the Segre +characteristic of λ1 as an eigenvalue of ˆS is ((sk, sk−1, . . . , s3)). +(ii) Let µ /∈ σ(S). Then the Segre characteristic of µ as an eigenvalue of ˆS +is always ((1)) if µ ̸= µ−1. If µ = µ−1 its Segre characteristic is ((1, 1)) +if and only if R = R1 or R = R2 from (23), otherwise it is ((2)). +Proof. (i) We first assume λ1 ̸= λ−1 +1 . According to the Segre characteristic +((sk, . . . , s1)) of λ1 ∈ σ(S) there are k ≥ 1 Jordan blocks Lk, . . . , L1 of sizes +sk, . . . , s1. As s1 = 1 let x1 be the corresponding eigenvector. We denote +the generalized eigenvectors corresponding to the ℓ-th Jordan block Lℓ by +xℓ +1, . . . , xℓ +sℓ and set Xℓ = [ xℓ +1 · · · xℓ +sℓ ] and ˜ +X := [ X2 · · · Xk ]. Since λ−1 +1 +∈ +σ(S) has the same Segre characteristic as λ1, there are also k Jordan blocks +Gk, . . . , G1 of S for λ−1 +1 +of sizes sk, . . . , s1. Let y1, Yℓ = [ yℓ +1 · · · yℓ +sℓ ] and +˜Y := [ Y2 · · · Yk ] be defined analogously from the (generalized) eigenvectors +for λ−1 +1 . We now define the matrix U := [ x1 y1 ˜ +X ˜Y ] ∈ M2n×2p(C). Then +the matrix ˆS = S + XRXT JT S can be written as +ˆS = S + +� +x1 +y1 +˜ +X +˜Y +� +� +������ +r11 +r12 +r21 +r22 +0 +0 +... +... +0 +0 +� +������ +XT JT S =: S + U ˜RXT JT S. +As SU = UP ′ for P ′ := diag(λ1, λ−1 +1 , L2, . . . , Lk, G2, . . . , Gk) we therefore +obtain ˆSU = SU + U ˜RXT JT SU and thus ˆSU = U(P ′ + ˜RXT JT UP ′). Now +set P = [pij]ij := P ′ + ˜RXT JT UP ′ and notice that the third to last row of +˜RXT JT UP ′ are identically zero (due to the form of ˜R). Therefore, the form +of P can be explicitly determined (with ⋆ indicating zero or nonzero entries +7Notice that for Theorem 3 to be applicable to λ1 ̸= ±1, s1 = 1 is a necessary condition +according to Theorem 4. If λ1 = ±1, then s1 = 1 implies s2 = 1 since then Jordan blocks +of a particular size must appear an even number of times in the Jordan structure of S. +18 + +that are not of further interest): +P = +� +���������������� +λ1(1 + r12) +−λ−1 +1 r11 +⋆ +· · · +⋆ +⋆ +· · · +⋆ +λ1r22 +λ−1 +1 (1 − r21) +⋆ +· · · +⋆ +⋆ +· · · +⋆ +0 +0 +L2 +... +... +... +0 +... +... +Lk +... +... +G2 +... +... +0 +... +0 +0 +Gk +� +���������������� +. (37) +Now, its is easily seen that L2, . . . , Lk, G2, . . . , Gk are part of the Jordan +structure of P, hence they also arise in the Jordan structure of ˆS. This shows +that the Segre characterstic of λ1 ̸= ±1 ∈ σ( ˆS) and λ−1 +1 +∈ σ( ˆS) is both +((sk, sk−1, . . . , s2)). If λ = ±1 the proof follows the same lines without the +use of ˜Y . To prove (ii) we note that the upper-left 2 × 2 block of P is exactly +Ω from (9), hence its eigenvalues are µ, µ−1. If µ ̸= µ−1, then Ω is semisimple. +Therefore, we obtain the Segre characteristic of µ and µ−1 as eigenvalues of +ˆS both as ((1)). On the other hand, if µ = µ−1, then Ω is semisimple if and +only if its minimal polynomial is p(z) = z − µ. Now +p(Ω′) = +�λ1(1 + r12) − µ +−λ−1 +1 r11 +λ1r22 +λ−1 +1 (1 − r21) − µ +� +. +Thus, for p(Ω) = 0 we must have r11 = r22 = 0 and the only possible choices +for Ω to be semisimple are R1 and R2 from (23). It is easy to check that in +fact both choices result in p(Ω) = 0. This proves that the Segre characteristic +of µ as an eigenvalue of ˆS is ((1, 1)) if R = R1, R2 and that it has to be ((2)) +otherwise. +Finally, assume that µ was already an eigenvalue of S and Theorem 3 is +applied for λ1 ∈ σ(S). Then the arguments of the proof of Lemma 1 apply to +µ (as an ’old’ eigenvalue of S) as well as the result from Theorem 8 (ii) (for +µ as the ’new’ eigenvalue appearing in the spectrum of ˆS). Thus, the Segre +characteristic of µ ∈ σ( ˆS) is its old Segre characteristic from S extended by +one of the cases described in Theorem 8 (ii). +As another consequence of Theorem 8 it follows that, if λ1 is semisimple +for S, then λ1 remains semisimple for ˆS is case its multiplicity was ≥ 2. +Moreover, assume the symplectic matrix S ∈ M2n(C) is diagonalizable (i.e. all +its eigenvalues are semisimple) and let ˆS be constructed according to Theorem +3. As a consequence of Lemma 1 and Theorem 8, the diagonalizability of ˆS +can then only be circumvented in case a 2×2 Jordan block arises for µ = µ−1. +As seen above, a 2 × 2 Jordan block for µ will arise if R is different from R1 +19 + +and R2. In other words, if R1 or R2 from (23) are chosen in Theorem 3, the +matrix ˆS will be semisimple in case S was. In fact, this is the only situation +in which S and ˆS are simultaneously diagonalizable since the simultaneous +diagonalizability of S and ˆS is only possible if S and ˆS commute. According +to Theorem 7 this is the case if and only if R = R1, R2 are chosen. +We +summarize this result in the following corollary. +Corollary 2. Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ +C \ {0, λ1, λ−1 +1 } be given. Suppose that ˆS = S + XRXT JT S ∈ M2n(C) has +been constructed according to Theorem 3 and that S is semisimple. Then ˆS is +semisimple if and only if R = R1 or R = R2 for one of the matrices in (23). +Moreover, in this case, S and ˆS are simultaneously diagonalizable. +Example 2 below shows how the simultaneous diagonalization looks like if +R1 or R2 are used to construct ˆS. +Example 2. Whenever T −1ST = diag(Λ, Λ−1) with T = [ x1 Y x2 Z ] ∈ +M2n(C) and Y, Z ∈ M2n×(n−1)(C) and Λ = diag(λ1, . . . , λn), then for Sj = +S + XRjXT JT S with R1, R2 from (23) we have +T −1 ˆS1T = +�Λ1 +Λ−1 +1 +� +, +and +T −1 ˆS2T = +�Λ2 +Λ−1 +2 +� +where Λ1 = diag(µ, λ2, . . . , λn) and Λ2 = diag(µ−1, λ2, . . . , λn). In fact, for S +being diagonal, R1 and R2 are the only possible choice such that ˆS is diagonal, +too. +4.3. A note on condition numbers +We conclude this section with a result on eigenvalue condition numbers. It +is a surprising fact, that we can apply Theorem 3 to a symplectic matrix +S ∈ M2n(C) without changing any eigenvalue condition number (for simple +eigenvalues) at all. +For Rado’s theorem this is not true: an unstructured +application of Theorem 2 typically changes all eigenvalue condition numbers, +even those of eigenvalues that remain unchanged. The main result on the +behavior of condition numbers is stated in Theorem 9. For its proof, we need +the following well-known fact that we state without proof in Lemma 2. +Lemma 2. Let A ∈ Mn(C) and suppose x is a right eigenvector of A for λ +(i.e. Ax = λx) and y is a left eigenvector of A for µ (i.e. yHA = µyH). Then +yHx = 0 if λ ̸= µ. +Theorem 9. Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 +1 , λ2, λ−1 +2 , +. . . , λn, λ−1 +n +and let µ ∈ C \ {0} with µ /∈ σ(S) be given. Assume that λ1 is +simple and let ˆS = S + XRXT JT S be constructed according to Theorem 3 so +that µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n +are the eigenvalues of ˆS. Then the following +holds: +20 + +(i) If ν is a simple eigenvalue of ˆS, µ ̸= ν ̸= µ−1, then κ( ˆS, ν) = κ(S, ν). +(ii) For R = R1, where R1 is the matrix given in (23), it holds that +κ( ˆS, µ) = κ(S, λ1) +and +κ( ˆS, µ−1) = κ(S, λ−1 +1 ). +Proof. (i) Under the assumption µ−1 ̸= ν ̸= µ and the simplicity of λ1, it +follows that ν = λj or ν = λ−1 +j +for some j = 2, . . . , n. Assume w. l. o. g. that +ν = λ2 and let y1 ∈ C2n be some corresponding eigenvector of S. According to +Lemma 1, ˆSy1 = λ2y1 holds. Next suppose z1 ∈ C2n is some left eigenvector +of S for λ2, i.e. zH +1 S = λ2zH +1 . Then +zH +1 ˆS = zH +1 +� +S + XRXT JT S +� += zH +1 S = λ2zH +1 +as zH +1 X = 0 according to Lemma 2. Thus the left and right eigenvectors for +λ2 of S and ˆS coincide, which directly implies κ( ˆS, λ2) = κ(S, λ2). +(ii) Now assume R = R1 for R1 in (23) and suppose ν = µ. The condition +µ /∈ σ(S) implies µ to be simple for ˆS. If x1, x2 ∈ C2n are eigenvectors of S +for λ1 and λ−1 +1 , respectively, then one directly obtains +ˆSx1 = Sx1 + XRXT JT Sx1 = λ1x1 − λ1XRXT Jx1 = λ1x1 − λ1XR +� 0 +−1 +� += λ1(1 + r12)x1 = λ1 · (1 + λ−1 +1 (µ − λ1))x1 = (λ1 + µ − λ1)x1 = µx1 +(similarly, ˆSx2 = µ−1x2 follows). Analogously we have xT +2 JT ˆS = µxT +2 JT , +which follows from xT +2 JT S = λ1xT +2 JT (see (18)). Thus, the left and right +eigenvectors of S for λ1 and those of ˆS for µ coincide and the statement +follows. The proof is analogous for ν = µ−1. +Remark 2. One can proceed as in the above proof to see that, if R = R2 is +used, +κ( ˆS, µ) = κ(S, λ−1 +1 ) +and +κ( ˆS, µ−1) = κ(S, λ1). +In fact, it is easy to show that now ˆSx1 = µ−1x1 and ˆSx2 = µx2 hold. +5. Experiments +To perform numerical experiments in Matlab R2021b, we used the code +available in [8]. To obtain symplectic matrices S ∈ M200(C), a symplectic +matrix S′ ∈ M200(C) constructed from [8], was modified as +S = +� +D−1 +1 +D1 +� +S′ +�D2 +D−1 +2 +� +, +where +D1 = diag(α1, . . . , α100), +D2 = diag(β1, . . . , β100) +(38) +and αi, βj are random numbers: rand(1) + 1i*rand(1). Compared to S′, S +has a more widespread spectrum ranging in magnitude from about 10−3 to +102. +21 + +Figure 1: +The plots show the effect of an eigenvalues modification when +λ1 ∈ σ(S) undergoes a relative change of |λ1|. +Fifty experiments have +been performed with S ∈ M200(C). +The plots show the use of ˆS1 = +(I2n + XR1XT JT )S for R1 in (23), the coarse bound in (26) (left plot), the +upper and lower bounds from Theorem 5 (right plot) and the relative change +in the eigenvalue |λ1 − µ|/|λ1|. +In Figure 1, 50 examples are shown where a randomly selected eigenvalue +λ1 of S has been changed to µ = λ1(1 + γz), where z ∈ C is a random +complex number with |z| = 1 and γ = |λ1|. Therefore, |µ − λ1|/|λ1| = |λ1|. +The plot shows the relative change ∥S − ˆS1∥F /∥S∥F , the coarse bound in +(26), the upper and lower bounds from Theorem 5 and the relative change +in the eigenvalue |λ1 − µ|/|λ1|. The plots show clearly that the bounds from +Theorem 5 are significantly sharper than the bound from (26). If R2 is used +instead of R1 the plots do not alter essentially. +The most significant difference between using ˆS1 and ˆS2 can be seen when +an eigenvalue is subject to a small or large relative change compared to its +absolute value. This is shown in Figure 2 where the experimental set-up is +the same as before but now γ = 10−3|λ1| (left plot) and γ = 103|λ1| (right +plot) are chosen. Both plots show the relative change ∥S − ˆSj∥F /∥S∥F for +j = 1, 2 and the relative change in the eigenvalue |λ1 − µ|/|λ1|. It is seen that +for a small change in the eigenvalue λ1, the relative change ∥S − ˆS1∥F /∥S∥F +is significantly smaller than ∥S − ˆS2∥F /∥S∥F . However, when λ1 undergoes +a large change, then there is no big difference in using R1 or R2. +In Figure 3 we show the use of matrices R different from R1 and R2 in (23). +To this end, we fix a symplectic matrix S ∈ M200(C) and an eigenvalue λ1 of +S that is to be modified by a relativ change of |λ1|. We consider a mesh grid +on [−1, 1]×[−1, 1] with 150 discretization points in each direction. Each point +(xj, yk) is associated with the complex number cjk := xj + ıyk from which we +made up r11 = r22 = √cjk. The values for r12 and r21 are found according +to (20) and (21) so that we obtain two different matrices ˜R1 = R(η1, r11, r22) +22 + +IIS SiIle /ISI/e +[入 μ/入, +Upper Bound R +10-1 +10-2 +10°4 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50100 +0-II - Si/ /lI/e +Upper Bound R +Lower Bound Ri +10-1 +10-2 +10-3 +10-4 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50Figure 2: The plots show the effect of an eigenvalues modification when λ1 ∈ +σ(S) undergoes a relative change of 0.001 · |λ1| (left plot) and 1000 · |λ1| +(right plot). Fifty experiments have been performed with S ∈ M200(C). The +plots show the use of ˆS1 and ˆS2 and the relative change in the eigenvalue +|λ1 − µ|/|λ1|. +and ˜R2 = R(η2, r11, r22). The matrices ˜Sj = (I2n + X ˜RjXT JT )S, j = 1, 2, +where constructed and in Figure 3 the minimum of ∥S − ˜Sj∥F /∥S∥F is shown +for each point cjk ∈ [−1, 1] × [−1, 1]. The plot indicates that the minimum +numerically found among all values ∥S − ˜Sj∥F /∥S∥F is attained for cjk = 0, +i.e. r11 = r22 = 0. Thus, the minimum is obtained for one of the matrices +R in (23). In most examples that have been considered a plot similar to the +one in Figure 3 arose. However, seldom the numerical minimum was detected +somewhere near cjk = 0. +6. Symplectic Matrix Pencils +In this section we analyse the eigenvalue modification problem of Section 1 +for symplectic matrix pencils. +We call a matrix pencil P(λ) = A − λB, +A, B ∈ M2n(C) symplectic (see [6, Sec. 2.1.1]) if it holds that +AJAT = BJBT . +(39) +From (39) it follows that for a symplectic pencil P(λ) = A − λB either A +and B are both regular or singular, cf. [6, Sec. 2]. A scalar ν ∈ C is called +an eigenvalue of P(λ) if there exists some nonzero x ∈ C2n with P(ν)x = 0, +i.e. +Ax = νBx [2, Sec. 2]. +Eigenvalues of symplectic pencils also arise in +pairs (ν, ν−1) [6]. In particular, if A and B are singular, it is also possible to +have ν = 0 as an eigenvalue of P(λ) which implies that λ−1 = ∞ is also an +eigenvalue of P(λ) (see, e.g., [5] for more information on the finite and infinite +eigenstructure of matrix pencils). In the following, we focus on symplectic +23 + + IIS Si/ /lI/e +S S/S +A /A, +10-2 +10~4 +10-5 +10-6 +10~ +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50104 +IS - S:l/e /lS/e +IS S2[e/lSIIp +10 +[入 /[入 +102 +10 7 +100 +10-1 +10-2 +10~3 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50Figure 3: The plot shows the minimum of ∥S − ˜Sj∥F /∥S∥F for j = 1, 2 on the +square [−1, 1] × [−1, 1] with 150 discretization points in each direction. The +minimum on the grid is attained for cjk = 0, i.e. r11 = r22 = 0. For random +experiments the plots always look similar to the plot shown above. Notice +that the surface has a sharp edge which arises due to the complex square root +that has to be computed. +pencils P(λ) = A − λB where both A and B are regular. Thus, P(λ) has +neither the eigenvalue zero nor the eigenvalue infinity. +In general, for an arbitrary matrix pencil P(λ) = A − λB, A, B ∈ Mn(C), +where B is nonsingular, we can easily derive an adapted version of Rado’s +theorem. +Theorem 10. Let P(λ) = A − λB, A, B ∈ Mn(C) and B nonsingular, be +a matrix pencil with eigenvalues λ1, . . . , λn ∈ C. +Let x1, . . . , xk ∈ Cn be +eigenvectors for λ1, . . . , λk such that rank(X) = k for X = [ x1 x2 · · · xk ] ∈ +Mn×k(C). +Furthermore, let C ∈ Mn×k(C) be arbitrary. +Then the matrix +pencil +˜P(λ) = (A + BXCT ) − λB +has the eigenvalues µ1, . . . , µk, λk+1, . . . , λn, where µ1, . . . , µk are the eigen- +values of the matrix Λ + CT X with Λ = diag(λ1, . . . , λk). +Proof. As B−1P(λ) = B−1A − λIn, the eigenvalues of P(λ) coincide with +those of the matrix M := B−1A ∈ M2n(C). In particular, P(λ) does not have +λ = ∞ as an eigenvalue. Assume that AX = BXΛ with Λ = diag(λ1, . . . , λk) +we obtain B−1AX = XΛ and Theorem 2 implies that ˆ +M := B−1A+XCT has +the eigenvalues µ1, . . . , µk, λk+1, . . . , λn, where µ1, . . . , µk are the eigenvalues +of the matrix Λ+CT X. As ˆ +M and the matrix pencil ˆP(λ) = (A+BXCT )−λB +have the same eigenvalues, the statement follows. +24 + +× 10°4 +9 +8 +7. +6 +5 ~ +A +3 > +-1 +-0.5 +0.8 +0.6 +0 +0.4 +0.2 +0 +0.5 +-0.2 +-0.4 +-0.6 +-0.8Now let P(λ) = A − λB be a symplectic matrix pencil according to (39) +with nonsingular A, B and eigenvalues λ1, λ−1 +1 , . . . , λn, λ−1 +n . We set +ˆP(λ) := (A + BXCT ) − λB +and intend to determine C ∈ M2n×2(C) such that ˆP(λ) is again symplectic +and has the eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n +for a given value µ ∈ +C \ {0}. A direct calculation reveals that (39) is equivalent to B−1A being +a symplectic matrix, i.e. JT (B−1A)T J = (B−1A)−1. Thus, Theorem 3 can +be applied to the matrix S := B−1A that has the same eigenvalues as P(λ). +Now suppose +AX = BX +�λ1 +λ−1 +1 +� +, +X = +� +x1 +x2 +� +∈ M2n×2(C), +i.e. x1, x2 ∈ C2n are generalized eigenvectors for λ1 and λ−1 +1 , respectively. +Furthermore, assume xT +1 Jx2 ̸= 0. Then +ˆS := B−1A + XRXT JT B−1A +is symplectic with eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n +provided that R is +chosen according to the conditions in Theorem 3 for µ. Then, the matrix +pencil ˆP(λ) := B( ˆS − λI2n), i.e. +ˆP(λ) = +� +A + BXRXT JT B−1A +� +− λB, +(40) +has the same eigenvalues as ˆS [5, Sec. 3.1]. In fact, ˆP(λ) is again a symplectic +pencil. To show this, we have to check that +� +A + BXRXT JT B−1A +� +J +� +A + BXRXT JT B−1A +�T = BJBT +holds. To this end, it only remains to prove that +AJ +� +BXRXT JT B−1A +�T + +� +BXRXT JT B−1A +� +JAT ++ +� +BXRXT JT B−1A +� +J +� +BXRXT JT B−1A +�T = 0 +(41) +since AJAT = BJBT holds by assumption. Using this relation, (41) simplifies +to +−BXRT XT BT + BXRXT BT + BXRJ2RT XT BT = 0 +which can be rewritten as BX +� +R − RT + RJ2RT � +XT BT = 0. As in Section 2 +this relation holds if and only if R−RT +RJ2RT = 0. As for any R ∈ M2(C) +we have RJ2RT = RT J2R, it follows that the condition R −RT +RJ2RT = 0 +is equivalent to (6). Therefore, since R was constructed according to (13) +so that R − RT + RT J2R = 0 holds, this finally shows that (41) is true, so +ˆP(λ) is symplectic. As already mentioned above, ˆP(λ) and ˆS have the same +eigenvalues, so the eigenvalues of ˆP(λ) are µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n . +25 + +Notice that ˆP(λ) in (40) can be rewritten in various ways, e.g. +ˆP(λ) = +� +A + BXRXT BT A−T JT � +− λB +(42) += +� +A + BXRXT BT JT A +� +− λB +(43) +using the relation JT B−1A = BT A−T JT that follows from B−1A being sym- +plectic in (42) and A−T = JT AJ in (43). Using XT BT = Λ−1XT AT and +exchanging XT BT with Λ−1XT AT in (42) yields +ˆP(λ) = (A + BXRΛ−1XT AT A−T JT ) − λB += +� +A + BXR +� +λ−1 +1 +λ1 +� +XT JT +� +− λB. +(44) +which is an expression for ˆP(λ) similar to (14). We conclude our finding in +the following theorem. +Theorem 11. Let P(λ) = A − λB, A, B ∈ M2n(C) nonsingular, be a +symplectic pencil with eigenvalues λ1, λ−1 +1 , λ2, λ−1 +2 , . . . , λn, λ−1 +n +∈ C and let +µ ∈ C \ {0} be given. Let x1, x2 ∈ C2n be eigenvectors for λ1 and λ−1 +1 , respec- +tively, normalized such that XT J2nX = J2 for X = [ x1 x2 ] ∈ M2n(C) and +set d := (µ + µ−1) − (λ1 + λ−1 +1 ). Then the matrix pencil +ˆP(λ) := +� +A + BXR +� +λ−1 +1 +λ1 +� +XT J +� +− λB +(45) +is again symplectic and has the eigenvalues µ, µ−1, λ2, λ−1 +2 , . . . , λn, λ−1 +n +pro- +vided that R = [rij]ij ∈ M2(C) is chosen such that (12) and (13) hold. +Regarding ˆP(λ) in (40), let ˆP(λ) = ˆA−λ ˆB with ˆA = A+BXRXT JT B−1A +and ˆB = B. Then certainly ∥ ˆB − B∥/∥B∥ = 0 while +∥ ˆA − A∥ +∥A∥ +≤ κ(B)∥R∥∥X∥2, +where κ(B) = ∥B∥∥B−1∥ is the condition number of the matrix B. Thus, +using Theorem 5 we may immediately bound ∥ ˆA − A∥/∥A∥. Furthermore, +from the form of ˆP(λ) in (45), statements similar to those in Section 4.2 can +directly be derived. +7. Summary +In this work we showed how to modify a pair of eigenvalues λ, 1/λ of a sym- +plectic matrix S to desired target values µ, 1/µ for a symplecitc matrix ˆS in a +structure-preserving way. Universal bounds on the relative distance between +S and ˆS with modified spectrum were given. The eigenvalues Segre charac- +teristics of S were related to those of S and some statements on eigenvalue +condition numbers have been derived. The main results have been extended +to matrix pencils. +26 + +8. Acknowledgement +The author is grateful to Thomas Richter as this work was in parts developed +during the authors employment in Thomas Richter’s group at the Otto-von- +Guericke-Universit¨at Magdeburg. +27 + +References +[1] A. T. Alexandridis, H. E. Psillakis. The inverse optimal LQR problem +and its relation to passivity and eigenvalue assignment. +International +Journal of Tomography & Statistics, Vol. 5, 2007. +[2] T. Betcke, N. J. Higham, V. Mehrmann, C. Schr¨oder, F. Tisseur. +NLEVP: A Collection of Nonlinear Eigenvalue Problems. ACM Trans- +actions on Mathematical Software, Vol. 39(2), 2013. +[3] R. Bru, R. Canto, R. L. Soto and A. M. Urbano A Brauer’s theorem and +related Results. Cent. Eur. J. Math. 10(1), 2012. +[4] D. F. Delchamps. State Space and Input-Output Linear Systems. Springer +Verlag, New York, USA, 1988. +[5] F. De Ter´an, F. M. Dopico, D. S. Mackey. Spectral equivalence of matrix +polynomials and the Index Sum Theorem. Linear Algebra Appl., Vol. +459, 2014. +[6] H. Fassbender. +Symplectic Methods for the Symplectic Eigenproblem. +Kluwer Academic Publishers, New York, USA, 2002. +[7] R. A. Horn, C. R. Johnson. Matrix Analysis (Second Edition). Cambridge +University Press, New York, USA, 2013. +[8] D. P. Jagger. MATLAB Toolbox for Classical Matrix Groups. Masters +thesis, University of Manchester, 2003. MIMS EPrint 2007.99. +[9] C. R. Johnson, E. A. Schreiner. The Relationship between AB and BA. +The American Mathematical Monthly, Vol. 103(7), 1996. +[10] A. J.Laub, K. Meyer. Canonical Forms for Symplectic and Hamiltonian +Matrices. Celestial Mechanics, Vol. 9, 1974. +[11] D. G. Luenberger. Optimization by Vector Space Methods. John Wiley +& Sons, Inc., New York, USA, 1969. +[12] T. Lyche. Numerical Linear Algebra and Matrix Factorizations. Springer +Nature, Cham, Switzerland, 2020. +[13] H. Perfect. Methods of constructing certain stochastic matrices. II Duke +Math. J. 22(2), 1955. +[14] Y. Saad. +Numerical Methods for Large Eigenvalue Problems (Revised +Edition). Society for Industrial and Applied Mathematics, Philadelphia, +USA, 2011. +28 + +[15] H. Shapiro. Linear Algebra and Matrices. Topics for a Second Course. +American Mathematical Society, Providence, USA, 2015. +[16] R. L. Soto and O. Rojo. Applications of a Brauer theorem in the nonneg- +ative inverse eigenvalue problem. Linear Algebra Appl. 416(2-3), 2006. +[17] O. Taussky, H. Zassenhaus. On the similarity transformation between a +matirx and its transpose. Pacific J. Math. 9(3), 1959. +29 + diff --git a/ktFRT4oBgHgl3EQfYDeM/content/tmp_files/load_file.txt b/ktFRT4oBgHgl3EQfYDeM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91126c53d9c295c358c925bcc34fa9af0b40cf81 --- /dev/null +++ b/ktFRT4oBgHgl3EQfYDeM/content/tmp_files/load_file.txt @@ -0,0 +1,937 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf,len=936 +page_content='PHILIP SALTENBERGER Institute for Numerical Analysis, TU Braunschweig Braunschweig, Germany (E-Mail: philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='saltenberger@tu-bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='de) STRUCTURE-PRESERVING EIGENVALUE MODIFICATION OF SYMPLECTIC MATRICES AND MATRIX PENCILS Abstract A famous theorem by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Brauer shows how to modify a single eigen- value of a matrix A by a rank-one update without changing the remain- ing eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A generalization of this theorem (due to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Rado) is used to change a pair of eigenvalues λ, 1/λ of a symplectic matrix S in a structure-preserving way to desired target values µ, 1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Universal bounds on the relative distance between S and the newly constructed symplectic matrix ˆS with modified spectrum are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The eigenvalues Segre characteristics of ˆS are related to those of S and a statement on the eigenvalue condition numbers of ˆS is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The main results are extended to matrix pencils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Introduction In numerical linear algebra and matrix analysis one occasionally encounters the necessity of modifying special eigenvalues of a matrix without altering its remaining eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Techniques for changing certain eigenvalues of a matrix have, for instance, been applied to solve nonnegative inverse eigen- value problems [13, 16] or, in form of deflation methods, to remove dominant eigenvalues in eigenvalue computations [14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, the task of modifying eigenvalues of matrices is of interest in stability and feedback of linear systems [4, § 25], [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='3] or for passivity and eigenvalue assignment in control design [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' One basic result on how a single eigenvalue of a matrix may be changed without modifying any other eigenvalues is due to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Brauer and can be found in [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 1], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 1 (Brauer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let A ∈ Mn(C) have eigenvalues λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn ∈ C and let x1 ∈ Cn be an eigenvector for λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then, for any c ∈ Cn, the matrix ˆA = A + x1cT ∈ Mn(C) has the eigenvalues λ1 + cT x1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This work is concerned with the purposive change of certain eigenvalues of matrices with symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A complex 2n × 2n matrix S ∈ M2n(C) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='13548v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='NA] 31 Jan 2023 is called symplectic, if1 ST J2nS = J2n =: J, where J2n = �0n×n In −In 0n×n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (1) Defining S⋆ := JT ST J, we see that (1) is equivalent to S⋆S = I2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, a symplectic matrix S is always nonsingular and S⋆ = S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In consequence, as S⋆ is similar2 to S, the eigenvalues of a symplectic matrices arise in pairs λj, λ−1 j , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , n, where λj and λ−1 j have the same Segre characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Recall that for an eigenvalue λ of S, its Segre characteristic is the sequence of sizes of the Jordan blocks of S with eigenvalue λ in non-increasing order [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We denote the Segre characteristic of an eigenvalue by ((·, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' It is now im- mediate that Theorem 1 can in general not be used for a structure-preserving, symplectic change of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In fact, for a structure-preserving eigenvalue modification, the change of λj and λ−1 j must take place simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Without any structure-preservation in mind, changing two (or more) eigen- values simultaneously is possible with the following generalization of Theorem 1 attributed to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Rado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' It can be found in [13], see also [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 2 (Rado).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let A ∈ Mn(C) have eigenvalues λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn ∈ C and let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , xk ∈ Cn be linearly independent eigenvectors for λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Set X = [ x1 · · · xk ] ∈ Mn×k(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then, for any matrix C ∈ Mn×k(C), the matrix ˆA = A + XCT has the eigenvalues µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , µk, λk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, where µj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , k, are the eigenvalues of Ω = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn) + CT X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this work, we investigate how Theorem 2 can be utilized to change a pair of eigenvalues λj, λ−1 j of a symplectic matrix S ∈ M2n(C) (or a symplectic matrix pencil) to desired target values µ, µ−1 in a structure-preserving way without modifying any other eigenvalues of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Considering Theorem 2, the starting point of our discussion is thus the following question: Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n , linearly independent eigenvectors x1, x2 ∈ C2n for λ1 and λ−1 1 , respec- tively, and X = [ x1 x2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' How has C ∈ M2n×2(C) to be chosen, such that ˆS := S + XCT is symplectic with eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n for some given value µ ∈ C \\ {0}?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The above-mentioned problem will be discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In Section 3 we investigate whether we can find an upper bound b > 0 that only de- pends on λ1, µ, x1 and x2 that assures the existence of a symplectic matrix ˆS ∈ M2n(C) with eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n and relative dis- tance ∥ ˆS − S∥/∥S∥ ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We derive distinguished matrices ˆS1, ˆS2 for which such a bound b can be neatly expressed and related to the relative change 1Here and in the following, T denotes the transpose of a (maybe complex) matrix or vector, not its conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2By definition, S⋆ is similar to ST and by the Taussky-Zassenhaus Theorem [17], ST is similar to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2 in the eigenvalue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' |λ1 − µ|/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We discuss commutativity relations be- tween S and ˆS in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1 and characterize the Segre characteristics of the eigenvalues of ˆS in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1 will come in handy here to find a condition on the simultaneous diagonalizability of S and ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In Section 6 we partially extend our results from Section 2 to symplectic matrix pencils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Notation The set of all m × n matrices over K (where we use either K = C or K = R) is denoted by Mm×n(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever n = m we write Mn(K) instead of Mn×n(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For J2n ∈ M2n(R), see (1), we simply write J and add the index whenever it is necessary to specify the size of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The range of a matrix A ∈ Mm×n(C) is the vector space spanned by its columns and is denoted range(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For A ∈ Mm×n(C), we denote the Moore-Penrose pseudoinverse of A by A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In case m > n and rank(A) = n, we have A+ = (AHA)−1AH so that A+A = In, while for n > m and rank(A) = m, A+ = AH(AAH)−1 yields AA+ = Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The superscript H always denotes the conjugate transpose of a matrix or vector while T is used for the pure transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever λ ∈ C is some complex number, we denote by R(λ) and I(λ) its real and imaginary part, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Complex conjugation of a number x = a+ıb ∈ C is denoted by a bar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' x = a − ıb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Symplectic Eigenvalue Modification Let S ∈ M2n(C) be a symplectic matrix (see (1)) with eigenvalues λ1, λ−1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=', λn, λ−1 n and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, assume x1, x2 ∈ C2n are linearly independent3 eigenvectors of S for λ1 and λ−1 1 , respectively, and define X = [ x1 x2 ] ∈ M2n×2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this section, our goal is to determine all possible matrices C ∈ M2n×2(C) such that ˆS := S +XCT is again symplectic and has the eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, we will make use of Rado’s theorem and derive a structure-preserving version of Theorem 2 (see Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As it will become clear later, it seems appropriate to consider the situa- tions xT 1 Jx2 ̸= 0 and xT 1 Jx2 = 0 seperately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' First, we assume that for the eigenvalues λ1, λ−1 1 there exist eigenvectors x1, x2 ∈ C2n such that xT 1 Jx2 ̸= 0 (this immediately implies x1 and x2 to be linearly independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this case, we can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' xT 1 Jx2 = 1, which can be achieved by a scaling of x1 and/or x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' That is, we have XT JX = J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For the matrix ˆS := S + XCT to be symplectic, it has to hold that ˆST J ˆS = � S + XCT �T J � S + XCT � = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (2) 3If λ1 ̸= λ−1 1 , then x1 and x2 are necessarily linear independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, the linear independence is only a restrictive requirement if λ1 = λ−1 1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' λ1 = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3 Using ST JS = J, (2) is equivalent to the matrix equation CXT JS + ST JXCT + CXT JXCT = 0 (3) for the unknown matrix C ∈ M2n×2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Notice that (3) can be rewritten as C(XT JS + J2CT ) = −ST JXCT (4) using XT JX = J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Since ST JX ∈ M2n×2(C) is a matrix of full rank, (4) immediately implies range(C) ⊆ range(ST JX) for any solution C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, for every C satisfying (3), there is a matrix R = [rij]ij ∈ M2(C) such that C = ST JXRT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Plugging this ansatz into (3), we obtain a 2n × 2n equation for R, namely ST JXRT XT JS + ST JXRXT JT S + ST JXRT J2RXT JT S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Replacing XT JS by −XT JT S this can be rewritten as ST JX � R − RT + RT J2R � XT JT S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (5) Finally, we may multiply (5) with the pseudo inverses (ST JX)+ from the left and with (XT JT S)+ from the right to obtain R − RT + RT J2R = 0, (6) which is a matrix equation for R of size 2 × 2 that is equivalent to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As R − RT and RT J2R are both skew-symmetric, their diagonals are identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Comparing the entries of R − RT and RT J2R in the (1,2) position, we obtain the condition r12 − r21 + r11r22 − r12r21 = 0 (7) for (6) to hold (comparing the elements in the (2, 1) position certainly gives the same condition with a minus sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In summary, a matrix of the form ˆS = S + XCT is symplectic if and only if C = ST JXRT for some matrix R = [rij]ij ∈ M2(C) whose entries satisfy (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Next, to achieve the desired eigenvalue modification, according to Theorem 2 we need to assure that the eigenvalues of Ω := Λ + CT X = �λ1 0 0 λ−1 1 � + CT X = �λ1 0 0 λ−1 1 � + RXT JT SX (8) become equal to µ and µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, recall that SX = Xdiag(λ1, λ−1 1 ) (by construction of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus diag(λ1, λ−1 1 ) + RXT JT SX = diag(λ1, λ−1 1 ) − RXT JXdiag(λ1, λ−1 1 ) which, since XT JX = J2, yields Ω = �λ1 0 0 λ−1 1 � − RJ2 �λ1 0 0 λ−1 1 � = �λ1 + λ1r12 −λ−1 1 r11 λ1r22 λ−1 1 − λ−1 1 r21 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (9) 4 The characteristic polynomial of Ω is p(z) = z2 − � λ1(1 + r12) + λ−1 1 (1 − r21) � z + (1 + r12)(1 − r21) + r11r22, which should, by Theorem 2, be equal to q(z) = (z − µ)(z − µ−1) = z2 − (µ + µ−1) + 1 to achieve that ˆS will have the eigenvalues µ and µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This gives two more conditions: one the one hand λ1(1 + r12) + λ−1 1 (1 − r21) = µ + µ−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' λ1r12 − λ−1 1 r21 = (µ + µ−1) − (λ1 + λ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (10) One the other hand, (1 + r12)(1 − r21) + r11r22 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The latter condition, however, is equal to condition (7) obtained for the symplectic structure above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, additionally to (7), which is required for S + XCT to be symplectic, the equation (10) has to hold to achieve that µ, µ−1 become eigenvalues of S + XCT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In conclusion, we obtain the following version of Theorem 2 that answers the question stated in Section 1 on the eigenvalue modification for symplectic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 1 , λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let x1, x2 ∈ C2n be eigenvectors for λ1 and λ−1 1 , respectively, normalized such that XT JX = J2 for X = [ x1 x2 ] ∈ M2n×2(C) and set d := (µ + µ−1) − (λ1 + λ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the matrix ˆS := S + XCT ∈ M2n(C) (11) is symplectic and has the eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n if and only if CT = RXT JT S for some matrix R = [rij]ij ∈ M2(C) whose entries satisfy the conditions d = λ1r12 − λ−1 1 r21, and (12) 0 = r12 − r21 + r11r22 − r12r21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (13) Notice that the matrix ˆS = S + XRXT JT S in (11) can also be expressed as ˆS = (I2n + XRXT JT )S or as ˆS = S + XRΛ−1XT JT = S + XR � λ−1 1 0 0 λ1 � XT JT (14) according to the relation Λ−1XT JT = XT JT S (where Λ = diag(λ1, λ−1 1 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, we see from (12) and (13) that there exist infinitely many pos- sible choices for R that realize the desired eigenvalue modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Next, we discuss the case that the eigenvectors x1, x2 ∈ C2n of the sym- plectic matrix S for λ1 and λ−1 1 , respectively, satisfy xT 1 Jx2 = 0 and how this condition effects the result from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, first notice that a symplectic matrix S ∈ M2n(C) need in fact not have eigenvectors x1, x2 for 5 λ1 and λ−1 1 that satisfy xT 1 Jx2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A situation of this kind arises for the symplectic matrix S = � ��� λ1 1 0 0 0 λ1 0 0 0 0 λ−1 1 0 0 0 −λ−2 1 λ−1 1 � ��� and its eigenvalue λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The only eigenvectors for λ1 and λ−1 1 are e1 and e4, respectively, and we have eT 1 J4e4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, Theorem 3 cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A simple sufficient (but not necessary) criterion to assure that eigenvectors with xT 1 Jx2 ̸= 0 must exist, is that S is a diagonalizable matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' [10, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1] and Corollary 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever x1, x2 ∈ C2n are eigenvectors of S for λ1 and λ−1 1 with xT 1 Jx2 = 0, then XT JX = 0 follows for X = [ x1 x2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this case, it follows from (3) that (4) takes the form CXT JS = −ST JXCT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Again we obtain range(C) ⊆ range(ST JX), so there has to exist some matrix R = [rij]ij ∈ M2(C) such that C = ST JXRT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' However, despite the concrete form of R, analogously to (8) we obtain Ω = �λ 0 0 λ−1 � + CT X = �λ 0 0 λ−1 � − RXT JSX = �λ 0 0 λ−1 � since SX = Xdiag(λ, λ−1) and XT JX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, even if R is chosen ac- cording to (13) such that ˆS = S + XRXT JT S is symplectic, no change in the eigenvalues can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In consequence, a change of an eigenvalue pair λ1, λ−1 1 of a symplectic matrix by Rado’s theorem in a structure-preserving way is only possible if there exist eigenvectors x1 and x2 for λ1 and λ−1 1 , re- spectively, such that xT 1 Jx2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In the next section, we derive a universal criterion on the existence of such eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Applying Theorem 3: a criterion We will now characterize those symplectic matrices S ∈ M2n(C), for which an eigenvalue adjustment according to Theorem 3 is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The condition derived below involves the Segre characteristic of the eigenvalue λ1 ∈ σ(S) to be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' First, let λ1 ∈ σ(S) and Sx1 = λ1x1 and Sx2 = λ−1 1 x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now suppose at least one of both vectors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' x1, belongs to a nontrivial4 Jordan chain, that is, there is some z ∈ C2n such that (S − λ1I2n)z = x1 (and possibly more 4By nontrivial, we mean a Jordan chain of length ≥ 2 while a trivial Jordan chain refers to a chain of length one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 6 generalized eigenvectors beside z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then we have xT 1 Jx2 = � (S − λ1I2n)z �T Jx2 = zT ST Jx2 − λ1zT Jx2 = zT JJT ST Jx2 − λ1zT Jx2 = zT JS−1x2 − λ1zT Jx2 = λ1zT Jx2 − λ1zT Jx2 = 0 (15) as JT ST J = S⋆ = S−1 and S−1x2 = λ1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In consequence, xT 1 Jx2 = 0 whenever x1, x2 are eigenvectors of S for λ1 and λ−1 1 , respectively, and at least one of them belongs to a nontrivial Jordan chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In other words, we may have xT 1 Jx2 ̸= 0 only in case both x1 and x2 belong to trivial Jordan chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Next, we show that in case x1 belongs to a trivial Jordan chain there must exist x2 (also from a trivial Jordan chain) such that xT 1 Jx2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, assume that λ1 ∈ C is an eigenvalue of the symplectic ma- trix S ∈ M2n(C) with p ≥ 1 ones in its Segre characteristic (that is, there are p Jordan blocks of size 1 × 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' p trivial Jordan chains, and possibly other Jordan blocks of size ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then there exists a matrix F ∈ M2n(C) transforming S to the following Jordan form F −1SF =: G = � ���� λ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' λ1 0 0 ˆG � ���� (16) where the upper-left block is λ1Ip and ˆG contains all other Jordan blocks (note that there might also be other Jordan blocks for λ1 of size ≥ 2 contained in ˆG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now define x1 := Fe1 and ˜f H := eT 1 F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (17) Then x1 is a right eigenvector of S for λ1 (Sx1 = λ1x1) and ˜f is a left eigenvector of S for λ1 ( ˜f HS = λ1 ˜f H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Certainly, ˜f Hx1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now we define xT 2 := ˜f HJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then we have ˜f HS = λ1 ˜f H ⇔ xT 2 JT S = λ1xT 2 JT ⇔ xT 2 JT SJ = λ1xT 2 ⇔ xT 2 S−T = λ1xT 2 ⇔ Sx2 = λ−1 1 x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (18) It follows that x2 is an eigenvector of S for λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now we obtain xT 1 Jx2 = xT 2 JT x1 = ˜f Hx1 = 1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In conclusion, for any eigenvector x1 of S for λ1 belonging to a trivial Jordan chain, there always exists an eigenvector x2 of S for λ−1 1 such that xT 1 Jx2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Recall from our observation (15) above, that x2 must also be a vector from a trivial Jordan chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We conclude our findings in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 7 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C\\{0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then Theorem 3 is applicable to S for λ1 and µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' there exist eigenvectors x1, x2 ∈ C2n for λ1 and λ−1 1 , respectively, with xT 1 Jx2 ̸= 0, if and only if the Segre characteristic of S for λ1 contains a one, that is, it has the form ((⋆, ⋆, · · · , ⋆, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In particular, eigenvectors x1 for λ1 and x2 for λ−1 1 with xT 1 Jx2 ̸= 0 always belong to trivial Jordan chains of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Do not overlook that Theorem 4 applies also for λ1 = λ−1 1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' λ1 = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this case λ1 ∈ σ(S) necessarily has an even multiplicity and an even number of Jordan blocks of the same size, so a Segre characteristic of the form ((⋆, ⋆, · · · , ⋆, 1)) implies that there appears at least another one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' ((⋆, ⋆, · · · , ⋆, 1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the reasoning in (16), (17) and (18) applies in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If S is diagonalizable, the Segre characteristic of S for any eigen- value λj ∈ σ(S) consists only of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' So we immediately obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then Theorem 3 is applicable to S for λ1 and µ if S is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Bounding the relative change Let S ∈ M2n(C) be symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For the matrix ˆS = S + XRXT JT S in (11) we immediately obtain a bound on its (absolute or relative) change in norm with respect to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' That is, ∥S − ˆS∥ ≤ ∥R∥∥X∥∥XT ∥∥S∥ and ∥S − ˆS∥ ∥S∥ ≤ ∥R∥∥X∥∥XT ∥ (19) hold for any submultiplicative and unitarily invariant matrix norm ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this section, we intend to derive explicit bounds of the relative distance between S and ˆS for certain choices of R = [rij]ij ∈ M2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, we assume ∥ · ∥ = ∥ · ∥F so that ∥X∥F = ∥XT ∥F holds and the upper bound in (19) reduces to ∥R∥F ∥X∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To bound the relative change ∥ ˆS − S∥F /∥S∥F with respect to S consider again (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The solution set to (12) is an affine subspace of C2 and all solutions may be parameterized as r12 = ηλ−1 1 , r21 = −λ1d + ηλ1, η ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (20) Plugging these expressions for r12 and r21 into (13) yields a polynomial in η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' p(η) = −η2 + η � λ−1 1 + d − λ1 � + dλ1 + r11r22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (21) Thus, depending on r11 and r22 (which can both be arbitrary), in (21) there are always two solutions η1, η2 ∈ C of p(η) = 0 and, in consequence, two 8 matrices R(ηj, r11, r22) := � r11 ηjλ−1 1 λ1(ηj − d) r22 � , j = 1, 2, (22) so that their entries satisfy (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To find some R ∈ M2(C) that yields a small norm ∥R∥F and thus a small bound in (19), it seems natural to consider the case r11r22 = 0, in particular r11 = r22 = 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The two possible roots of p(η) for r11r22 = 0 are η1 = µ − λ1 and η2 = µ−1 − λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The matrices R1 := R(η1, 0, 0) and R2 := R(η2, 0, 0) that arise according to (22) are thus given by R1 = � 0 λ−1 1 (µ − λ1) µ−1(µ − λ1) 0 � , R2 = � 0 µ−1(λ−1 1 − µ) λ1(λ−1 1 − µ) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (23) Using R1 and R2 in (23), explicit bounds can be found on ∥ ˆS − S∥F /∥S∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' According to (19) and (23) such a bound b ≥ 0 only depends on λ1, µ and the eigenvectors of S for λ1 and λ−1 1 and guarantees the existence of a sym- plectic matrix ˆS ∈ M2n(C) that solves the problem from Section 1 with ∥ ˆS −S∥F /∥S∥F ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To formulate these bounds, we impose a condition on X to estimate ∥X∥F without computing the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In particular, we assume the eigenvectors x1, x2 ∈ C2n of S for λ1 and λ−1 1 , respectively, to be normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' ∥x1∥2 = ∥x2∥2 = 1 and X ∈ M2n×2(C) to be of the form X = 1 � xT 1 Jx2 � x1 x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (24) Then XT JX = J2 holds and it follows that ∥X∥2 F = ����� 1 � xT 1 Jx2 ����� 2 ��� x1 x2 ���2 F = 1 |xT 1 Jx2| ��� x1 x2 ���2 F = 2 |xT 1 Jx2| (25) The value 1/|xT 1 Jx2| has a nice interpretation whenever λ1 is a simple eigen- value of S and ∥x1∥2 = ∥x2∥2 = 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To see this, recall that, whenever A ∈ Mn(C) has a simple eigenvalue λ ∈ C (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' its algebraic multiplicity equals one), then κ(A, λ) := ∥u∥2∥v∥2 |vHu| is called its condition number, where u ∈ C2n and v ∈ C2n are right and left eigenvectors of A for λ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Au = λu and vHA = λvH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' It is a measure on how sensitive λ reacts to small changes in the matrix A, see [14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As we have seen in (18) above, (xT 2 JT )S = λ1(xT 2 JT ) whenever x2 satisfies 5Certainly, choosing r11 = 0 and r22 ̸= 0 gives the same roots of p(η) = 0 in (21), and thus the same values for r12 and r21, but a larger Frobenius norm of R than choosing r11 = r22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 9 Sx2 = λ−1 1 x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, for simple λ1 (which implies that λ−1 1 is simple as well) we can choose u = x1 and vH = xT 2 JT so that κ(S, λ1) = ∥u∥2∥v∥2 |vHu| = ∥x1∥2∥Jx2∥2 |xT 2 JT x1| = 1 |xT 1 Jx2| since ∥x1∥2 = 1 and ∥Jx2∥2 = ∥x2∥2 = ∥x2∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We can now formulate the following theorem which follows directly from the bound in (19), the observation in (25) and the Frobenius norms of the matrices R1, R2 in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1, λ−1 1 ∈ σ(S) and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let x1, x2 ∈ C2n be normalized eigenvectors for λ1 and λ−1 1 , respectively, and X ∈ M2n×2(C) as in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Define Φ := 2 |xT 1 Jx2| � = 2κ(S, λ1) if λ1 is simple � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (i) Let ˆS1 = S + XR1XT JT S be constructed according to Theorem 3 with R1 from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then ∥ ˆS1 − S∥F ∥S∥F ≤ |λ1 − µ| |λ1| � Φ � 1 + |λ1|2 |µ|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (26) (ii) Let ˆS2 = S + XR2XT JT S be constructed according to Theorem 3 with R2 from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then ∥ ˆS2 − S∥F ∥S∥F ≤ |λ−1 1 − µ| |λ−1 1 | � �Φ � 1 + |λ−1 1 |2 |µ|2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (27) As the following example shows, similar easy bounds can be found with the use of R1 and R2 when ∥ · ∥2 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In fact, in the 2-norm, such a bound can be sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For a symplectic matrix S ∈ M2n(C), the bound (19) for ˆS1 = S + XR1XT JT S with respect to ∥ · ∥2 can easily be determined as ∥ ˆS1 − S∥2 ∥S∥2 ≤ |λ1 − µ| |λ1| max � 1, |λ1| |µ| � ∥X∥2 2, (28) It can be seen for S = diag(Λ, Λ−1) with Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn) that the bound in (28) can be sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In particular, with eigenvectors e1, en+1 ∈ R2n for λ1, λ−1 1 , respectively, and X = [ e1 en+1 ] we have XR1XT JT S = diag � r12λ1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , 0, −r21λ−1 1 , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , 0 � 10 with nonzero entries in the first and (n+1)st position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As r12λ1 = µ−λ1 and −r21λ−1 1 = µ−1 − λ−1 1 we obtain under the assumption |λ1 − µ| ≥ |λ−1 1 − µ−1| ∥ ˆS − S∥2 = ∥XR1XT JT S∥2 = |λ1 − µ| and so ∥ ˆS − S∥2/∥S∥2 = |λ1 − µ|/|λ1| if λ1 is the largest eigenvalue of S in absolute value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' ∥S∥2 = |λ1|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' On the other hand, for X we certainly have ∥X∥2 2 = 1 and thus, whenever |µ| ≥ |λ1|, the bound on the right hand side in (28) also reduces to |λ1 − µ|/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Improved distance bounds Although the bound in (26) nicely relates ∥ ˆS1 − S∥F /∥S∥F to the relative value change |λ1 − µ|/|λ1| and the condition number κ(S, λ1), it can be quite bad6, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 1 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this section we derive sharper bounds under the additional assumption that ∥S∥F is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As before, let S ∈ M2n(C) be symplectic with eigenvectors x1, x2 ∈ C2n for λ1, λ−1 1 ∈ σ(S), respectively, such that xT 1 Jx2 ̸= 0 (and set X = [ x1 x2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As seen in (14), we have for Λ := diag(λ1, λ−1 1 ) and R ∈ M2(C) that satisfies (12) and (13) ˆS = S + XRXT JT S = S + XRΛ−1XT JT and therefore ∥S − ˆS∥F = ∥XRΛ−1XT JT ∥F = ∥XRΛ−1XT ∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever R = Rj (j = 1, 2) from (23), then XRjΛ−1XT = X � 0 ηj ηj − d 0 � XT =: X ˜RjXT , ˜Rj = RjΛ−1, according to (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Recall the solutions of p(η) = 0 in (21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' η1 = µ − λ1 (corresponding to R1) and η2 = µ−1−λ1 (corresponding to R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Instead of es- timating ∥X ˜RjXT ∥F by ∥ ˜Rj∥F ∥X∥2 F we now intend to estimate ∥X ˜RjXT ∥F directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, assume that X ∈ M2n×2(C) has the form (24) and ˆSj = S + XRjXT JT S, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then we have for Y = � xT 1 Jx2X = [ x1 x2 ] ∥S − ˆSj∥2 F = ∥X ˜RjXT ∥2 F = tr � Y ˜RjY T (Y ˜RjY T )H� |xT 1 Jx2|2 = tr � Y ˜RjY T Y ˜RH j Y H� |xT 1 Jx2|2 = tr � Y HY ˜Rj(Y HY ) ˜RH j � |xT 1 Jx2|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 6The same is true for the bound in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 11 Now we further obtain |xT 1 Jx2|2∥ ˆSj − S∥2 F = tr � Y HY ˜Rj(Y HY ) ˜RH j � = tr �� 1 xH 1 x2 xH 2 x1 1 � � 0 ηj ηj − d 0 � � 1 xH 2 x1 xH 1 x2 1 � � 0 ηj − d ηj 0 �� = |xH 1 x2|2ηj(ηj − d) + |ηj|2 + |ηj − d|2 + |xH 1 x2|2ηj(ηj − d) = |ηj|2 + |ηj − d|2 + 2|xH 1 x2|2 · R(ηj(ηj − d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (29) Note that |xH 1 x2| ≤ ∥x1∥2∥x2∥2 = 1 as x1 and x2 are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Fur- thermore, R(ηj(ηj − d)) = R(|ηj|2 − ηjd) = |ηj|2 − R(ηjd), and we may now derive upper (and lower) bounds for (29) depending on whether this term is positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (i) Suppose R(ηjd) < |ηj|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then R(ηj(ηj − d)) > 0 follows and, since |xH 1 x2|2 ≤ 1, we can estimate from (29), setting |xH 1 x2|2 = 1, |xT 1 Jx2|2∥S − ˆSj∥2 F ≤ |ηj|2 + |ηj − d|2 + 2 · R(ηj(ηj − d)) = � ηj + (ηj − d) �� ηj + (ηj − d) � = |2ηj − d|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' On the other hand, changing the sign of R(ηj(ηj −d)) we certainly have |xT 1 Jx2|2∥S − ˆSj∥2 F ≥ |ηj|2 + |ηj − d|2 − 2 · R(ηj(ηj − d)) = (ηj − (ηj − d))(ηj − (ηj − d)) = |d|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (ii) Suppose R(ηjd) ≥ |ηj|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then R(ηj(ηj − d)) ≤ 0 follows and we can estimate from (29), setting again |xH 1 x2|2 = 1, |xT 1 Jx2|2∥S − ˆSj∥2 F ≥ |ηj|2 + |ηj − d|2 + 2 · R(ηj(ηj − d)) = |2ηj − d|2 while on the other hand, with a change of sign, we obtain |xT 1 Jx2|2∥S − ˆSj∥2 F ≤ |ηj|2 + |ηj − d|2 − 2 · R(ηj(ηj − d)) = |d|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Before we state our findings in the next theorem, notice that there are neat expressions for the terms 2ηj − d, j = 1, 2, arising above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2η1 − d = µ − λ1 λ1 � λ1 + µ−1� , 2η2 − d = λ−1 1 − µ λ−1 1 � µ−1 + λ−1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As it turns out, also d can be rewritten in a similar fashion as d = µ − λ1 λ1 � λ1 − µ−1� = λ−1 1 − µ λ−1 1 � µ−1 − λ−1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (30) 12 Finally, the two conditions to be checked in (i) and (ii) above can be simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For η1 = µ − λ1 one finds, after some reformulations, R � η1d � − |η1|2 = 1 2 � (µ − λ1)d + (µ − λ1)d � − |µ − λ1|2 = 1 2 �|µ − λ1|2d µ − λ1 + |µ − λ1|2d µ − λ1 � − |µ − λ1|2 = −|µ − λ1|2 � 1 − 1 2 � d µ − λ1 + d µ − λ1 �� = −|µ − λ1|2 � 1 − R � d µ − λ1 �� = −|µ − λ1|2 � 1 − R �λ1 − µ−1 λ1 �� = −|µ − λ1|2R((µλ1)−1) where we used the first expression for d in (30) in the second-last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus R(η1d) ≥ |ηj|2 holds if and only if −|µ − λ1|2R((µλ1)−1) ≥ 0, which is the case if and only if R(λ1µ) ≤ 0 as (λ1µ)−1 and λ1µ are located in the same half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For η2 = µ−1 − λ1 we obtain analogously R � η2d � − |η2|2 = −|λ1µ − 1|2R � (λµ)−1� and so R(η2d) ≥ |ηj|2 holds if and only if R((λµ)−1) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This, in turn, holds if and only if R(λ1µ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In conclusion, we have proven the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1, λ−1 1 ∈ σ(S) and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let x1, x2 ∈ C2n be normalized eigenvectors for λ1 and λ−1 1 , respectively, with xT 1 Jx2 ̸= 0 and X ∈ M2n×2(C) as in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Define Φ := 1 |xT 1 Jx2| · ∥S∥F � = κ(S, λ1) ∥S∥F if λ1 is simple � (i) Let ˆS1 = S + XR1XT JT S be constructed according to Theorem 3 with R1 from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever R(λ1µ) ≤ 0, then |λ1 − µ| |λ1| � |λ1 + µ−1|Φ � ≤ ∥ ˆS1 − S∥F ∥S∥F ≤ |λ1 − µ| |λ1| � |λ1 − µ−1|Φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If R(λ1µ) > 0 the upper and lower bounds interchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (ii) Let ˆS2 = S + XR2XT JT S be constructed according to Theorem 3 with R2 from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever R(λ1µ) ≤ 0, then |λ−1 1 − µ| |λ−1 1 | � |λ−1 1 + µ−1|Φ � ≤ ∥ ˆS2 − S∥F ∥S∥F ≤ |λ−1 1 − µ| |λ−1 1 | � |λ−1 1 − µ−1|Φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If R(λ1µ) > 0 the upper and lower bounds interchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 13 It is shown in Section 5 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 1) that the bounds in Theorem 6 are significantly sharper compared to the bounds in (26) and (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For any matrix R = [rij] ∈ M2(C) that satisfies the conditions (12) and (13) the bounds in (19) can easily be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' However, there are several reasons for not considering other choices of R (beside R1 and R2 from (23)) in this section in detail: (a) If r11 ̸= 0, r22 ̸= 0, there are two possibilities for R whose entries r12 and r21 of R depend on c = r11r22 through (one of) the zeros of p(η) = 0, see (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, r12 and r21 involve the expression of a complex square root and there is no neat and compact expression for ∥R∥F compared to (26) and (27) or to the formulas in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, minimizing ∥S − ˆS∥F with respect to the entries of R under the side conditions (12) and (13) results in a difficult complex optimization problem for which the author is not aware of a closed form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (b) Among all matrices R that satisfy the conditions (12) and (13) the ma- trices R1 and R2 from (23) are the only possible choice when ˆS = S + XRXT JT S should inherit desirable properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' related to diagonaliz- ability) from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' These distinguishing features of R1 and R2 are discussed in the upcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (c) All numerical experiments that have been performed indicate that rarely a matrix R′ ∈ M2(C) different from R1 and R2 that satiesfies (12) and (13) was detected such that ∥ ˆS −S∥F /∥S∥F for ˆS = S +XR′XT JT S was smaller than the minimum of ∥ ˆS1 − S∥/∥S∥F and ∥ ˆS2 − S∥F /∥S∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This is visualized in Section 5, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Segre characteristics and commutativity relations In this section we discuss how the Segre characteristics of eigenvalues are effected by a change of a symplectic matrix S ∈ M2n(C) to ˆS ∈ M2n(C) according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In particular, we will show that the Segre character- istics of the eigenvalues of S and ˆS are either the same or connected in a direct way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, we make a statement on eigenvectors of S and ˆS that re- main unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Notice that, in the context of Theorem 2, the eigenvectors of A and ˆA = A + XCT are in general all different and not related in an immediate fashion [3] if no further restrictions are imposed on the form of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this section we show that, in the structure-preserving context of Theorem 3, the particular form of C allows for some explicit statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, we derive statements on the diagonalizability of ˆS and the simultaneous di- agonalizability of S and ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, we begin in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1 with a result on the commutativity of S and ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The Commutativity of S and ˆS Recall that the matrix ˆS in (11) can also be expressed as ˆS = � I2n + XRXT JT � S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (31) Since ˆS and S are both symplectic, the matrix ˆSS−1 = ˆSS⋆ = I2n+XRXT JT is symplectic, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As for any A, B ∈ Mn(C) the matrices AB and BA always have the same eigenvalues [9], beside ˆS, we may also define the symplectic matrix ˜S := S � I2n + XRXT JT � ∈ M2n(C) that solves the eigenvalue modi- fication problem stated in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A question naturally arising is whether there is a connection between ˆS from (31) and ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Such a connection is revealed in Theorem 7 which shows a distinguishing feature of the matrices from (23) among all matrices R that satisfy (12) and (13), see Remark 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The result from Theorem 7 will be used when the diagonalizability of ˆS is investigated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and assume ˆS = S + XRXT JT S has been constructed according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the following is true: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In case λ1 ̸= ±1, ˆS = � I2n + XRXT JT � S = S � I2n + XRXT JT � = ˜S (32) holds if and only if R = [rij]ij ∈ M2(C) is one of the matrices in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In case λ1 = ±1, (32) holds for any R = [rij]ij ∈ M2(C) satisfying (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' First, notice that ˆS = ˜S is equivalent to XRXT JT S = SXRXT JT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (33) Multiplying both equations with J (from the right) and using the relations SX = XΛ (where Λ = diag(λ1, λ−1 1 )) and JT SJ = S−T yields XRXT S−T = XΛRXT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Moreover, XT S−T = (S−1X)T = (XΛ−1)T = Λ−1XT and, equiv- alently to (33), it suffices to investigate the equation XRΛ−1XT = XΛRXT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (34) Now, as X ∈ M2n×2(C) has full rank, (34) is (by the multiplication with X+ from the left and (XT )+ from the right) equivalent to RΛ−1 = ΛR, that is, ΛRΛ = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For R = [rij]ij we obtain ΛRΛ = �λ1 0 0 λ−1 1 � �r11 r12 r21 r22 � �λ1 0 0 λ−1 1 � = �λ2 1r11 r12 r21 λ−2 1 r22 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This shows that ΛRΛ = R holds, in case λ1 ̸= ±1, if and only if r11 = r22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The two possibilities for R that satisfy the conditions (12) and (13) when r11 = r22 = 0 are the matrices in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, if λ1 = ±1, the equation always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 15 Theorem 7 shows that, in general (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' for λ1 ̸= ±1), only the two possible choices for R in (23) produce commutativity of S and I2n + XRXT JT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' to have ˆS = ˜S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In the special case λ1 = ±1, any matrix R determined from (12) and (13) will cause this commutativity relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Segre characteristics Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To analyse the consequences of the change S �→ ˆS = S + XRXT JT S on the Segre characteristics of the eigenvalues of S and ˆS, we discuss the cases of λ1, λ−1 1 (the eigenvalues that are changed), µ, µ−1 (the values λ1 and λ−1 1 are changed to) and all other eigenvalues (which are the same for S and ˆS) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As before, let x1, x2 ∈ C2n be eigenvectors for λ1 and λ−1 1 , respectively, normalized such that XT J2nX = J2 for X = [ x1 x2 ] ∈ M2n×2(C) and assume ˆS = S + XRXT JT S has been constructed as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We first consider eigenvalues different from λ1, λ−1 1 , µ and µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' These eigenvalues and their algebraic multiplicities are the same for S and ˆS and we show that their eigenspaces and Jordan chains (thus, in consequence, their Segre characteristics) remain completely unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To prove this, we need the following fact about the matrix S and its (generalized) eigenvectors (see also [10, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2]): assume that λ is some eigenvalue of S different from λ1 and λ−1 1 and let y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , yp ∈ C2n (p ≥ 1) be a Jordan chain for S and λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' it holds that (S − λI2n)y1 = 0 and (S − λI2n)yk+1 = yk for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then XT Jyk = 0 follows for any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To see this, first consider the eigenvector y1 of S for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We have λ1xT 1 Jy1 = xT 1 ST Jy1 = xT 1 JJT ST Jy1 = xT 1 JS−1y1 = λ−1xT 1 Jy1 This shows that xT 1 Jy1 = 0 if λ ̸= λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Similarly, xT 2 Jy1 = 0 follows for λ ̸= λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now assume that xT i Jyℓ = 0 holds for i = 1, 2 and ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For (S − λI2n)yk+1 = yk we thus obtain 0 = xT 1 Jyk = xT 1 J(S − λI2n)yk+1 = xT 1 JSyk+1 − λxT 1 Jyk+1 = xT 1 S−T Jyk+1 − λxT 1 Jyk+1 = λ−1 1 xT 1 Jyk+1 − λxT 1 Jyk+1 = (λ−1 1 − λ)xT 1 Jyk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, again xT 1 Jyk+1 = 0 follows whenever λ ̸= λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' With the same reasoning we obtain xT 2 Jyk+1 = 0 for λ ̸= λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In conclusion we have XT Jyk = 0 for any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , p whenever λ1 ̸= λ ̸= λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C\\{0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Suppose that ˆS ∈ M2n(C) has been constructed according to Theorem 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then for any λ ∈ σ( ˆS) which is neither equal to λ1 or λ−1 1 nor equal to µ or µ−1 the Segre characteristics of λ as an eigenvalue of S and ˆS and their corresponding Jordan chains, respectively, are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Assume λ ∈ σ( ˆS) is neither equal to λ1 or λ−1 1 nor equal to µ or µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' By construction of ˆS = S + XRXT JT S, λ is an eigenvalue of both S and ˆS with the same algebraic multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever y1 ∈ C2n is an eigenvector of S for λ it is also an eigenvector of ˆS for λ since XT Jy1 = 0, which implies ˆSy1 = Sy1 + XRXT JT Sy1 = Sy1 − λXRXT Jy1 = Sy1 = λy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (35) Next, let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , yp ∈ C2n be a Jordan chain for S and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then � ˆS − λI2n � yi+1 = � S + XRXT JT S � yi+1 − λyi+1 = � yi + λyi+1 � + XRXT JT Syi+1 − λyi+1 = yi − XRXT J(yi + λyi+1) = yi since XT Jyk = 0 for any yk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , p, from the Jordan chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Inductively, this shows that y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , yp remains to be a Jordan chain of ˆS for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, the Segre characteristic for λ of S and ˆS and the corresponding Jordan chains are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Next we consider λ1 and λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' When S ∈ M2n(C) is transformed to ˆS = S + XRXT JT S and (one instance of) λ1, λ−1 1 is replaced by µ and µ−1, the Segre characteristic of λ1 for ˆS is necessarily different from its Segre characteristic for S due to the eigenvalue modification that has taken place (if λ1 is a simple eigenvalue of S, then it is not even an eigenvalue of ˆS anymore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' However, if the algebraic multiplicity of λ1 as an eigenvalue of S is ≥ 2, then the Segre characteristics of λ1 as an eigenvalue of S and ˆS are connected in an easy fashion (see Theorem 8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This is obviously false in the general context of Rado’s Theorem, where nontrivial Jordan blocks may arise, as the following counterexample for A = I4 and ˆA = A + XCT shows: ˆA = � ��� 1 1 1 1 � ��� + � ��� 1 0 0 1 0 0 0 0 � ��� �1 0 0 0 0 0 1 0 � = � ��� 2 1 1 1 1 � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In this example, the Segre characteristic of 1 ∈ σ(A) is ((1, 1, 1, 1)) while it is ((2, 1)) for ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C \\ {0, λ1, λ−1 1 } be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Suppose that ˆS = S + XRXT JT S ∈ M2n(C) has been constructed according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the following hold: 17 (i) If the Segre characteristic of λ1 ̸= ±1 as an eigenvalue of S is ((sk, sk−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s2, s1)) (36) with7 sk ≥ sk−1 ≥ · · · ≥ s2 ≥ s1 = 1, then the Segre characteristic of λ1 as an eigenvalue of ˆS is ((sk, sk−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Moreover, if λ1 = ±1 and (36) is its Segre characteristic of S with7 s2 = s1 = 1, then the Segre characteristic of λ1 as an eigenvalue of ˆS is ((sk, sk−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (ii) Let µ /∈ σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the Segre characteristic of µ as an eigenvalue of ˆS is always ((1)) if µ ̸= µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If µ = µ−1 its Segre characteristic is ((1, 1)) if and only if R = R1 or R = R2 from (23), otherwise it is ((2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (i) We first assume λ1 ̸= λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' According to the Segre characteristic ((sk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s1)) of λ1 ∈ σ(S) there are k ≥ 1 Jordan blocks Lk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , L1 of sizes sk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As s1 = 1 let x1 be the corresponding eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We denote the generalized eigenvectors corresponding to the ℓ-th Jordan block Lℓ by xℓ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , xℓ sℓ and set Xℓ = [ xℓ 1 · · · xℓ sℓ ] and ˜ X := [ X2 · · · Xk ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Since λ−1 1 ∈ σ(S) has the same Segre characteristic as λ1, there are also k Jordan blocks Gk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , G1 of S for λ−1 1 of sizes sk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let y1, Yℓ = [ yℓ 1 · · · yℓ sℓ ] and ˜Y := [ Y2 · · · Yk ] be defined analogously from the (generalized) eigenvectors for λ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We now define the matrix U := [ x1 y1 ˜ X ˜Y ] ∈ M2n×2p(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the matrix ˆS = S + XRXT JT S can be written as ˆS = S + � x1 y1 ˜ X ˜Y � � ������ r11 r12 r21 r22 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 0 0 � ������ XT JT S =: S + U ˜RXT JT S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As SU = UP ′ for P ′ := diag(λ1, λ−1 1 , L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , Lk, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , Gk) we therefore obtain ˆSU = SU + U ˜RXT JT SU and thus ˆSU = U(P ′ + ˜RXT JT UP ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now set P = [pij]ij := P ′ + ˜RXT JT UP ′ and notice that the third to last row of ˜RXT JT UP ′ are identically zero (due to the form of ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, the form of P can be explicitly determined (with ⋆ indicating zero or nonzero entries 7Notice that for Theorem 3 to be applicable to λ1 ̸= ±1, s1 = 1 is a necessary condition according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If λ1 = ±1, then s1 = 1 implies s2 = 1 since then Jordan blocks of a particular size must appear an even number of times in the Jordan structure of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 18 that are not of further interest): P = � ���������������� λ1(1 + r12) −λ−1 1 r11 ⋆ · · ⋆ ⋆ · · ⋆ λ1r22 λ−1 1 (1 − r21) ⋆ · · ⋆ ⋆ · · ⋆ 0 0 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Lk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' G2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 0 0 Gk � ���������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (37) Now, its is easily seen that L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , Lk, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , Gk are part of the Jordan structure of P, hence they also arise in the Jordan structure of ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This shows that the Segre characterstic of λ1 ̸= ±1 ∈ σ( ˆS) and λ−1 1 ∈ σ( ˆS) is both ((sk, sk−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , s2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If λ = ±1 the proof follows the same lines without the use of ˜Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To prove (ii) we note that the upper-left 2 × 2 block of P is exactly Ω from (9), hence its eigenvalues are µ, µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If µ ̸= µ−1, then Ω is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, we obtain the Segre characteristic of µ and µ−1 as eigenvalues of ˆS both as ((1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' On the other hand, if µ = µ−1, then Ω is semisimple if and only if its minimal polynomial is p(z) = z − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now p(Ω′) = �λ1(1 + r12) − µ −λ−1 1 r11 λ1r22 λ−1 1 (1 − r21) − µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, for p(Ω) = 0 we must have r11 = r22 = 0 and the only possible choices for Ω to be semisimple are R1 and R2 from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' It is easy to check that in fact both choices result in p(Ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This proves that the Segre characteristic of µ as an eigenvalue of ˆS is ((1, 1)) if R = R1, R2 and that it has to be ((2)) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Finally, assume that µ was already an eigenvalue of S and Theorem 3 is applied for λ1 ∈ σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the arguments of the proof of Lemma 1 apply to µ (as an ’old’ eigenvalue of S) as well as the result from Theorem 8 (ii) (for µ as the ’new’ eigenvalue appearing in the spectrum of ˆS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, the Segre characteristic of µ ∈ σ( ˆS) is its old Segre characteristic from S extended by one of the cases described in Theorem 8 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As another consequence of Theorem 8 it follows that, if λ1 is semisimple for S, then λ1 remains semisimple for ˆS is case its multiplicity was ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Moreover, assume the symplectic matrix S ∈ M2n(C) is diagonalizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' all its eigenvalues are semisimple) and let ˆS be constructed according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As a consequence of Lemma 1 and Theorem 8, the diagonalizability of ˆS can then only be circumvented in case a 2×2 Jordan block arises for µ = µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As seen above, a 2 × 2 Jordan block for µ will arise if R is different from R1 19 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In other words, if R1 or R2 from (23) are chosen in Theorem 3, the matrix ˆS will be semisimple in case S was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In fact, this is the only situation in which S and ˆS are simultaneously diagonalizable since the simultaneous diagonalizability of S and ˆS is only possible if S and ˆS commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' According to Theorem 7 this is the case if and only if R = R1, R2 are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We summarize this result in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with λ1 ∈ σ(S) and let µ ∈ C \\ {0, λ1, λ−1 1 } be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Suppose that ˆS = S + XRXT JT S ∈ M2n(C) has been constructed according to Theorem 3 and that S is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then ˆS is semisimple if and only if R = R1 or R = R2 for one of the matrices in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Moreover, in this case, S and ˆS are simultaneously diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Example 2 below shows how the simultaneous diagonalization looks like if R1 or R2 are used to construct ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Whenever T −1ST = diag(Λ, Λ−1) with T = [ x1 Y x2 Z ] ∈ M2n(C) and Y, Z ∈ M2n×(n−1)(C) and Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn), then for Sj = S + XRjXT JT S with R1, R2 from (23) we have T −1 ˆS1T = �Λ1 Λ−1 1 � , and T −1 ˆS2T = �Λ2 Λ−1 2 � where Λ1 = diag(µ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn) and Λ2 = diag(µ−1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In fact, for S being diagonal, R1 and R2 are the only possible choice such that ˆS is diagonal, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A note on condition numbers We conclude this section with a result on eigenvalue condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' It is a surprising fact, that we can apply Theorem 3 to a symplectic matrix S ∈ M2n(C) without changing any eigenvalue condition number (for simple eigenvalues) at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For Rado’s theorem this is not true: an unstructured application of Theorem 2 typically changes all eigenvalue condition numbers, even those of eigenvalues that remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The main result on the behavior of condition numbers is stated in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For its proof, we need the following well-known fact that we state without proof in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let A ∈ Mn(C) and suppose x is a right eigenvector of A for λ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Ax = λx) and y is a left eigenvector of A for µ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' yHA = µyH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then yHx = 0 if λ ̸= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let S ∈ M2n(C) be symplectic with eigenvalues λ1, λ−1 1 , λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n and let µ ∈ C \\ {0} with µ /∈ σ(S) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Assume that λ1 is simple and let ˆS = S + XRXT JT S be constructed according to Theorem 3 so that µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n are the eigenvalues of ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the following holds: 20 (i) If ν is a simple eigenvalue of ˆS, µ ̸= ν ̸= µ−1, then κ( ˆS, ν) = κ(S, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (ii) For R = R1, where R1 is the matrix given in (23), it holds that κ( ˆS, µ) = κ(S, λ1) and κ( ˆS, µ−1) = κ(S, λ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (i) Under the assumption µ−1 ̸= ν ̸= µ and the simplicity of λ1, it follows that ν = λj or ν = λ−1 j for some j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' that ν = λ2 and let y1 ∈ C2n be some corresponding eigenvector of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' According to Lemma 1, ˆSy1 = λ2y1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Next suppose z1 ∈ C2n is some left eigenvector of S for λ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' zH 1 S = λ2zH 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then zH 1 ˆS = zH 1 � S + XRXT JT S � = zH 1 S = λ2zH 1 as zH 1 X = 0 according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus the left and right eigenvectors for λ2 of S and ˆS coincide, which directly implies κ( ˆS, λ2) = κ(S, λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (ii) Now assume R = R1 for R1 in (23) and suppose ν = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The condition µ /∈ σ(S) implies µ to be simple for ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If x1, x2 ∈ C2n are eigenvectors of S for λ1 and λ−1 1 , respectively, then one directly obtains ˆSx1 = Sx1 + XRXT JT Sx1 = λ1x1 − λ1XRXT Jx1 = λ1x1 − λ1XR � 0 −1 � = λ1(1 + r12)x1 = λ1 · (1 + λ−1 1 (µ − λ1))x1 = (λ1 + µ − λ1)x1 = µx1 (similarly, ˆSx2 = µ−1x2 follows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Analogously we have xT 2 JT ˆS = µxT 2 JT , which follows from xT 2 JT S = λ1xT 2 JT (see (18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, the left and right eigenvectors of S for λ1 and those of ˆS for µ coincide and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The proof is analogous for ν = µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' One can proceed as in the above proof to see that, if R = R2 is used, κ( ˆS, µ) = κ(S, λ−1 1 ) and κ( ˆS, µ−1) = κ(S, λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In fact, it is easy to show that now ˆSx1 = µ−1x1 and ˆSx2 = µx2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Experiments To perform numerical experiments in Matlab R2021b, we used the code available in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To obtain symplectic matrices S ∈ M200(C), a symplectic matrix S′ ∈ M200(C) constructed from [8], was modified as S = � D−1 1 D1 � S′ �D2 D−1 2 � , where D1 = diag(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , α100), D2 = diag(β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , β100) (38) and αi, βj are random numbers: rand(1) + 1i*rand(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Compared to S′, S has a more widespread spectrum ranging in magnitude from about 10−3 to 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 21 Figure 1: The plots show the effect of an eigenvalues modification when λ1 ∈ σ(S) undergoes a relative change of |λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Fifty experiments have been performed with S ∈ M200(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The plots show the use of ˆS1 = (I2n + XR1XT JT )S for R1 in (23), the coarse bound in (26) (left plot), the upper and lower bounds from Theorem 5 (right plot) and the relative change in the eigenvalue |λ1 − µ|/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In Figure 1, 50 examples are shown where a randomly selected eigenvalue λ1 of S has been changed to µ = λ1(1 + γz), where z ∈ C is a random complex number with |z| = 1 and γ = |λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, |µ − λ1|/|λ1| = |λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The plot shows the relative change ∥S − ˆS1∥F /∥S∥F , the coarse bound in (26), the upper and lower bounds from Theorem 5 and the relative change in the eigenvalue |λ1 − µ|/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The plots show clearly that the bounds from Theorem 5 are significantly sharper than the bound from (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' If R2 is used instead of R1 the plots do not alter essentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The most significant difference between using ˆS1 and ˆS2 can be seen when an eigenvalue is subject to a small or large relative change compared to its absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' This is shown in Figure 2 where the experimental set-up is the same as before but now γ = 10−3|λ1| (left plot) and γ = 103|λ1| (right plot) are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Both plots show the relative change ∥S − ˆSj∥F /∥S∥F for j = 1, 2 and the relative change in the eigenvalue |λ1 − µ|/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' It is seen that for a small change in the eigenvalue λ1, the relative change ∥S − ˆS1∥F /∥S∥F is significantly smaller than ∥S − ˆS2∥F /∥S∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' However, when λ1 undergoes a large change, then there is no big difference in using R1 or R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In Figure 3 we show the use of matrices R different from R1 and R2 in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, we fix a symplectic matrix S ∈ M200(C) and an eigenvalue λ1 of S that is to be modified by a relativ change of |λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We consider a mesh grid on [−1, 1]×[−1, 1] with 150 discretization points in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Each point (xj, yk) is associated with the complex number cjk := xj + ıyk from which we made up r11 = r22 = √cjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The values for r12 and r21 are found according to (20) and (21) so that we obtain two different matrices ˜R1 = R(η1, r11, r22) 22 IIS SiIle /ISI/e [入 μ/入, Upper Bound R 10-1 10-2 10°4 0 5 10 15 20 25 30 35 40 45 50100 0-II - Si/ /lI/e Upper Bound R Lower Bound Ri 10-1 10-2 10-3 10-4 0 5 10 15 20 25 30 35 40 45 50Figure 2: The plots show the effect of an eigenvalues modification when λ1 ∈ σ(S) undergoes a relative change of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='001 · |λ1| (left plot) and 1000 · |λ1| (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Fifty experiments have been performed with S ∈ M200(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The plots show the use of ˆS1 and ˆS2 and the relative change in the eigenvalue |λ1 − µ|/|λ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' and ˜R2 = R(η2, r11, r22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The matrices ˜Sj = (I2n + X ˜RjXT JT )S, j = 1, 2, where constructed and in Figure 3 the minimum of ∥S − ˜Sj∥F /∥S∥F is shown for each point cjk ∈ [−1, 1] × [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The plot indicates that the minimum numerically found among all values ∥S − ˜Sj∥F /∥S∥F is attained for cjk = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' r11 = r22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, the minimum is obtained for one of the matrices R in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In most examples that have been considered a plot similar to the one in Figure 3 arose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' However, seldom the numerical minimum was detected somewhere near cjk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Symplectic Matrix Pencils In this section we analyse the eigenvalue modification problem of Section 1 for symplectic matrix pencils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We call a matrix pencil P(λ) = A − λB, A, B ∈ M2n(C) symplectic (see [6, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1]) if it holds that AJAT = BJBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (39) From (39) it follows that for a symplectic pencil P(λ) = A − λB either A and B are both regular or singular, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' [6, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A scalar ν ∈ C is called an eigenvalue of P(λ) if there exists some nonzero x ∈ C2n with P(ν)x = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Ax = νBx [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Eigenvalues of symplectic pencils also arise in pairs (ν, ν−1) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In particular, if A and B are singular, it is also possible to have ν = 0 as an eigenvalue of P(λ) which implies that λ−1 = ∞ is also an eigenvalue of P(λ) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=', [5] for more information on the finite and infinite eigenstructure of matrix pencils).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In the following, we focus on symplectic 23 IIS Si/ /lI/e S S/S A /A, 10-2 10~4 10-5 10-6 10~ 0 5 10 15 20 25 30 35 40 45 50104 IS - S:l/e /lS/e IS S2[e/lSIIp 10 [入 /[入 102 10 7 100 10-1 10-2 10~3 0 5 10 15 20 25 30 35 40 45 50Figure 3: The plot shows the minimum of ∥S − ˜Sj∥F /∥S∥F for j = 1, 2 on the square [−1, 1] × [−1, 1] with 150 discretization points in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The minimum on the grid is attained for cjk = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' r11 = r22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' For random experiments the plots always look similar to the plot shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Notice that the surface has a sharp edge which arises due to the complex square root that has to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' pencils P(λ) = A − λB where both A and B are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, P(λ) has neither the eigenvalue zero nor the eigenvalue infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In general, for an arbitrary matrix pencil P(λ) = A − λB, A, B ∈ Mn(C), where B is nonsingular, we can easily derive an adapted version of Rado’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let P(λ) = A − λB, A, B ∈ Mn(C) and B nonsingular, be a matrix pencil with eigenvalues λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , xk ∈ Cn be eigenvectors for λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λk such that rank(X) = k for X = [ x1 x2 · · · xk ] ∈ Mn×k(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, let C ∈ Mn×k(C) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the matrix pencil ˜P(λ) = (A + BXCT ) − λB has the eigenvalues µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , µk, λk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, where µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , µk are the eigen- values of the matrix Λ + CT X with Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As B−1P(λ) = B−1A − λIn, the eigenvalues of P(λ) coincide with those of the matrix M := B−1A ∈ M2n(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In particular, P(λ) does not have λ = ∞ as an eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Assume that AX = BXΛ with Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λk) we obtain B−1AX = XΛ and Theorem 2 implies that ˆ M := B−1A+XCT has the eigenvalues µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , µk, λk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, where µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , µk are the eigenvalues of the matrix Λ+CT X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As ˆ M and the matrix pencil ˆP(λ) = (A+BXCT )−λB have the same eigenvalues, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 24 × 10°4 9 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 6 5 ~ A 3 > 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='8Now let P(λ) = A − λB be a symplectic matrix pencil according to (39) with nonsingular A, B and eigenvalues λ1, λ−1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We set ˆP(λ) := (A + BXCT ) − λB and intend to determine C ∈ M2n×2(C) such that ˆP(λ) is again symplectic and has the eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n for a given value µ ∈ C \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' A direct calculation reveals that (39) is equivalent to B−1A being a symplectic matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' JT (B−1A)T J = (B−1A)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, Theorem 3 can be applied to the matrix S := B−1A that has the same eigenvalues as P(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Now suppose AX = BX �λ1 λ−1 1 � , X = � x1 x2 � ∈ M2n×2(C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' x1, x2 ∈ C2n are generalized eigenvectors for λ1 and λ−1 1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, assume xT 1 Jx2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then ˆS := B−1A + XRXT JT B−1A is symplectic with eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n provided that R is chosen according to the conditions in Theorem 3 for µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then, the matrix pencil ˆP(λ) := B( ˆS − λI2n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' ˆP(λ) = � A + BXRXT JT B−1A � − λB, (40) has the same eigenvalues as ˆS [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' In fact, ˆP(λ) is again a symplectic pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To show this, we have to check that � A + BXRXT JT B−1A � J � A + BXRXT JT B−1A �T = BJBT holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' To this end, it only remains to prove that AJ � BXRXT JT B−1A �T + � BXRXT JT B−1A � JAT + � BXRXT JT B−1A � J � BXRXT JT B−1A �T = 0 (41) since AJAT = BJBT holds by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Using this relation, (41) simplifies to −BXRT XT BT + BXRXT BT + BXRJ2RT XT BT = 0 which can be rewritten as BX � R − RT + RJ2RT � XT BT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As in Section 2 this relation holds if and only if R−RT +RJ2RT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As for any R ∈ M2(C) we have RJ2RT = RT J2R, it follows that the condition R −RT +RJ2RT = 0 is equivalent to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Therefore, since R was constructed according to (13) so that R − RT + RT J2R = 0 holds, this finally shows that (41) is true, so ˆP(λ) is symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' As already mentioned above, ˆP(λ) and ˆS have the same eigenvalues, so the eigenvalues of ˆP(λ) are µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 25 Notice that ˆP(λ) in (40) can be rewritten in various ways, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' ˆP(λ) = � A + BXRXT BT A−T JT � − λB (42) = � A + BXRXT BT JT A � − λB (43) using the relation JT B−1A = BT A−T JT that follows from B−1A being sym- plectic in (42) and A−T = JT AJ in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Using XT BT = Λ−1XT AT and exchanging XT BT with Λ−1XT AT in (42) yields ˆP(λ) = (A + BXRΛ−1XT AT A−T JT ) − λB = � A + BXR � λ−1 1 λ1 � XT JT � − λB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' (44) which is an expression for ˆP(λ) similar to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' We conclude our finding in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let P(λ) = A − λB, A, B ∈ M2n(C) nonsingular, be a symplectic pencil with eigenvalues λ1, λ−1 1 , λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n ∈ C and let µ ∈ C \\ {0} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Let x1, x2 ∈ C2n be eigenvectors for λ1 and λ−1 1 , respec- tively, normalized such that XT J2nX = J2 for X = [ x1 x2 ] ∈ M2n(C) and set d := (µ + µ−1) − (λ1 + λ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then the matrix pencil ˆP(λ) := � A + BXR � λ−1 1 λ1 � XT J � − λB (45) is again symplectic and has the eigenvalues µ, µ−1, λ2, λ−1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' , λn, λ−1 n pro- vided that R = [rij]ij ∈ M2(C) is chosen such that (12) and (13) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Regarding ˆP(λ) in (40), let ˆP(λ) = ˆA−λ ˆB with ˆA = A+BXRXT JT B−1A and ˆB = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Then certainly ∥ ˆB − B∥/∥B∥ = 0 while ∥ ˆA − A∥ ∥A∥ ≤ κ(B)∥R∥∥X∥2, where κ(B) = ∥B∥∥B−1∥ is the condition number of the matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Thus, using Theorem 5 we may immediately bound ∥ ˆA − A∥/∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Furthermore, from the form of ˆP(λ) in (45), statements similar to those in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content='2 can directly be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Summary In this work we showed how to modify a pair of eigenvalues λ, 1/λ of a sym- plectic matrix S to desired target values µ, 1/µ for a symplecitc matrix ˆS in a structure-preserving way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Universal bounds on the relative distance between S and ˆS with modified spectrum were given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The eigenvalues Segre charac- teristics of S were related to those of S and some statements on eigenvalue condition numbers have been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' The main results have been extended to matrix pencils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 26 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Soto and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Rojo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Applications of a Brauer theorem in the nonneg- ative inverse eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 416(2-3), 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Taussky, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Zassenhaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' On the similarity transformation between a matirx and its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 9(3), 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFRT4oBgHgl3EQfYDeM/content/2301.13548v1.pdf'} diff --git a/ldFIT4oBgHgl3EQfsCtp/vector_store/index.faiss b/ldFIT4oBgHgl3EQfsCtp/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8f682097eac6d0fb8223291282aa80c859a09e33 --- /dev/null +++ b/ldFIT4oBgHgl3EQfsCtp/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d577d778a8c77b55cb4a490a3456a0016a11258ed5678f2e6ea48f71971b5648 +size 1769517 diff --git a/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf b/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf new file mode 100644 index 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its magnetic field using spectropolarimetry +M. Nelissen1, P. McGinnis1, C. P. Folsom2, 3, T. Ray1, A. A. Vidotto4, E. Alecian5, J. Bouvier5, J. Morin6, J.-F. Donati7, +and R. Devaraj1 +1 Dublin Institute for Advanced Studies, Astronomy & Astrophysics Section, 31 Fitzwilliam Place, Dublin 2, Ireland +e-mail: nelissen@cp.dias.ie +2 Tartu Observatory, University of Tartu, Observatooriumi 1, Tõravere, 61602 Tartumaa, Estonia +3 University of Western Ontario, Department of Physics & Astronomy, London, Ontario, N6A 3K7, Canada +4 Leiden Observatory, Leiden University, PO Box 9513, 2300RA, Leiden, The Netherlands +5 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +6 LUPM, Université de Montpellier & CNRS, Montpellier, Cedex 05, France +7 Univ. de Toulouse, CNRS, IRAP, 14 avenue Belin, 31400 Toulouse, France +Accepted 22 December 2022 +ABSTRACT +Context. Misalignments between a forming star’s rotation axis and its outer disk axis, although not predicted by standard theories of +stellar formation, have been observed in several classical T Tauri stars (cTTs). The low-mass cTTs DK Tau is suspected of being among +them. In addition, it is an excellent subject to investigate the interaction between stellar magnetic fields and material accreting from +the circumstellar disk, as it presents clear signatures of accretion. +Aims. The goal of this paper is to study DK Tau’s average line-of-sight magnetic field in both photospheric absorption lines and +emission lines linked to accretion, using spectropolarimetric observations, as well as to examine inconsistencies regarding its rotation +axis. +Methods. We used data collected with the ESPaDOnS spectropolarimeter, at the Canada-France-Hawaii Telescope, and the NARVAL +spectropolarimeter, at the Télescope Bernard Lyot, probing two distinct epochs (December 2010 to January 2011 and November to +December 2012), each set spanning a few stellar rotation cycles. We first determined the stellar parameters of DK Tau, such as effective +temperature and 𝑣 sin𝑖. Next, we removed the effect of veiling from the spectra, then obtained least-squares deconvolution (LSD) +profiles of the photospheric absorption lines for each observation, before determining the average line-of-sight magnetic field from +them. We also investigated accretion-powered emission lines, namely the 587.6 nm Hei line and the Caii infrared triplet (at 849.8 nm, +854.2 nm and 866.2 nm), as tracers of the magnetic fields present in the accretion shocks. +Results. We find that DK Tau experiences accretion onto a magnetic pole at an angle of ∼ 30° from the pole of its rotation axis, with +a positive field at the base of the accretion funnels. In 2010 we find a magnetic field of up to 0.95kG (from the Caii infrared triplet) +and 1.77kG (from the Hei line) and in 2012 we find up to 1.15kG (from the Caii infrared triplet) and 1.99kG (from the Hei line). +Additionally, using our derived values of period, 𝑣 sin𝑖 and stellar radius, we find a value of 58° (+18)(-11) for the inclination of the +stellar rotation axis, which is significantly different from the outer disk axis inclination of 21° given in the literature. +Conclusion. We find that DK Tau’s outer disk axis is likely misaligned compared to its rotation axis by 37°. +Key words. Stars: individual: DK Tau - Stars: variables: T Tauri - Stars: magnetic field - Accretion, accretion disks - Techniques: +polarimetric - Techniques: spectroscopic +1. Introduction +Stellar magnetic fields are omnipresent and play an essential role +in the formation of stars and planets. Understanding their impact +is therefore crucial to the study of stellar birth. We do not, how- +ever, yet possess a complete picture of how stellar magnetic fields +originate, how they evolve over time, or the extent of their impact +on circumstellar disks and the accretion process in the early stages +of a star’s life (see e.g., Bouvier et al. 2007; Gregory et al. 2012; +Folsom et al. 2016; Hartmann et al. 2016; Villebrun et al. 2019). +Stellar magnetic fields of accreting T Tauri stars play an essential +role in driving accretion and strongly impact the geometry of the +accretion flow (see Hartmann et al. 2016). By analyzing the mag- +netic field along the line-of-sight and integrated over the visible +stellar hemisphere measured in the acc retion-powered emission +lines, one can recreate a picture of the component of the stellar +magnetic field that dominates the accretion process. This is an +integral quantity that relates Stokes I and Stokes V, making use +of spectropolarimetric data. The interested reader is referred to, +for example, Rees & Semel (1979); Donati & Landstreet (2009); +Morin (2012). The Stokes parameters and the magnetic field are +connected through the Zeeman effect. This effect describes the +impact of a magnetic field on a spectrum: its atomic (and molecu- +lar) lines are broadened or split, depending on the strength of the +field and the sensitivity of the line in question (see e.g., Tennyson +2011; Hussain & Alecian 2014). +In this work, we analyze the spectropolarimetry of the clas- +sical T Tauri star (cTTs) DK Tau. DK Tau is a young low-mass +star surrounded by a circumstellar disk which is actively accret- +ing from its inner regions. It is a wide binary (separation 2′′.38, +Article number, page 1 of 18 +arXiv:2301.01175v1 [astro-ph.SR] 3 Jan 2023 + +A&A proofs: manuscript no. main +equivalent to 307 au - see e.g., Manara et al. 2019), which allows +DK Tau A (hereafter "DK Tau") to be spatially resolved with +spectropolarimetry and studied on its own. It is located in the +Taurus Molecular Cloud at a distance of 132.6 pc (Gaia Collab- +oration et al. 2016, 2018, 2022). Its spectral type is K7 (see e.g., +Johns-Krull 2007; Fischer et al. 2011) with a heliocentric radial +velocity of +16.2 km s−1 (Kounkel et al. 2019). Rota et al. (2022) +measured the inclination, with respect to the line of sight, of its +outer (>20 au) gaseous disk axis via projection to be 21°, using +the 12CO (J = 2–1) line with ALMA. The star’s rotation period 𝑃, +as determined using photometry, has been measured as 8.4 days +(Bouvier et al. 1993) and 8.18 days (Percy et al. 2010; Artemenko +et al. 2012). +DK Tau presents with significant and variable veiling (a +strong signature of accretion - see e.g., Hartigan et al. 1995; +Fischer et al. 2011). In addition to accretion, it also shows ev- +idence of ejection (see e.g., Hartigan et al. 1995), in particular +inner disk winds and a jet as revealed by the detection of both +low and high velocity forbidden [Oi] emission by Hartigan et al. +(1995) and Banzatti et al. (2019). +The effect of veiling on spectra complicates magnetic field +measurements, yet the study of active accretors is very valuable +for understanding the interaction between stellar magnetic fields +and accreting material from the circumstellar disk. Indeed, our +current understanding of magnetospheric accretion (see e.g., Shu +et al. 1994; Romanova et al. 2002; Bessolaz et al. 2008; Hartmann +et al. 2016) involves the truncation of the inner circumstellar disk +at a few stellar radii and the channeling of accretion in funnel +flows by the stellar magnetic field. When the accreted matter falls +onto the star at near free-fall velocities, it produces an accretion +shock close to the stellar surface, which gives rise to contin- +uum and line veiling (Calvet & Gullbring 1998) and generates a +localized bright/hot spot at the level of the chromosphere. +Simple models of stellar formation do not account for a mis- +alignment between the star’s rotation axis and its outer disk axis. +However, misalignments may be common, as several dippers dis- +play a low inclination of their outer disk axis (see e.g., Ansdell +et al. 2020; Sicilia-Aguilar et al. 2020). Dippers are stars that +show flux dips in their light curves. The standard explanation in- +volves circumstellar material from the inner disk passing in front +of the star, occulting it periodically or aperiodically (see e.g., +McGinnis et al. 2015; Roggero et al. 2021). Dippers therefore re- +quire a relatively high inclination for the inner disk axis (which is +here assumed to point in the same direction as the stellar rotation +axis), that is inconsistent with their measured outer disk axis in- +clination. Such a misalignment has been directly measured in few +cTTs (see e.g., Alencar et al. 2018; Bouvier et al. 2020). These +observations suggest a more complex formation mechanism than +normally considered, though it is not clear what gives rise to such +a misalignment. As examples of potential causes, Sicilia-Aguilar +et al. (2020) suggest the possibility of two different protostellar +collapses, whereas Alencar et al. (2018) invoke the presence of a +massive planet inside the disk gap, and Benisty et al. (2018) the +effects of a low-mass stellar companion. DK Tau shows signs of +being one of these systems with a misaligned outer disk. +In this paper, we describe our observations in Sect. 2. We de- +tail our analysis and results regarding stellar parameters, veiling, +and magnetic characterizations in Sect. 3. In Section 4 we discuss +the implications of the inclination we measure for DK Tau and +its magnetic field. Finally, Sect. 5 contains our conclusions. +2. Observations +Our data set is comprised of two sets of circularly polarized +spectra of DK Tau, collected from December 2010 to January +2011, and from the end of November to the end of December +2012, with the spectropolarimeters ESPaDOnS (Echelle Spec- +troPolarimetric Device for the Observation of Stars), mounted +at the CFHT (Canada-France-Hawaii Telescope) 3.6 meter tele- +scope in Hawaii (Donati et al. 2006), and NARVAL, mounted at +the 2 meter TBL (Télescope Bernard Lyot) on the Pic du Midi in +France (Aurière 2003). These echelle spectropolarimeters cover +the visible domain, from 370 to 1 050 nm, in a single exposure, +and have a resolving power of 65 000. ESPaDOnS has a fiber +aperture of 1′′.66, while NARVAL has one of 2′′.80. +Table 1 lists the dates of the middle of the observations for the +two ESPaDOnS and NARVAL data sets. The total exposure time +was 4 996.0 s for each ESPaDOnS observation and 4 800.0 s for +each NARVAL observation. In 2010 a total of 15 observations +were taken over 39 days, and in 2012 a total of 12 observations +were taken over 35 days, with the intention of capturing a few +rotation cycles of the target in each set. +The data are public and were downloaded from the archive +of the PolarBase website1 (see e.g., Petit et al. 2014). These +observations were made as a result of proposals 10BP12 and +12BP12, with J.-F. Donati as P.I in both cases and obtained as +part of the MaPP (Magnetic Protostars and Planets) large pro- +gram at the CFHT. We also downloaded the corresponding im- +age files for the ESPaDOnS data from the Canadian Astronomy +Data Centre (CADC) website2. The data had been previously re- +duced at the CFHT and TBL. The reduction was carried out with +the LibreESpRIT (for "Echelle Spectra Reduction: an Interactive +Tool") reduction package specifically built for extracting polar- +ization echelle spectra from raw data. This includes subtracting +the bias and the dark frames, and correcting for the variations in +sensitivity using flat field frames (Donati et al. 1997). The spec- +tra were continuum normalized in addition to the LibreESpRIT +automatic continuum normalization, as the automatic procedure +is not tailored for stars presenting with emission lines and it did +not manage to properly adjust the continuum. +3. Analysis & results +3.1. Veiling +Accretion shocks are at a higher temperature than the photo- +sphere. This adds an extra continuum to the stellar continuum, +artificially decreasing the depth of the photospheric absorption +lines. This is known as veiling and it varies with the wavelength, +and can also vary from night to night. Its effect needs to be re- +moved from the spectra in order to analyze the stellar magnetic +field from the photospheric absorption lines. +Veiling (𝑅) is defined as the ratio between the flux of the ac- +cretion shock and the photospheric flux. For a normalized spec- +trum, it can be expressed using the following equation: +𝐼v(𝜆) = [𝐼ph(𝜆) + 𝑅(𝜆)]𝑁(𝜆) +(1) +where 𝐼v(𝜆) refers to the veiled intensity at wavelength 𝜆, 𝐼ph(𝜆) +to the intensity of the photosphere at wavelength 𝜆, 𝑅(𝜆) to the +veiling at wavelength 𝜆 and 𝑁(𝜆) is a normalization constant at +wavelength 𝜆. +In order to measure the veiling in our spectra, we used a +technique based on the fitting of a rotationally broadened and +1 http://polarbase.irap.omp.eu +2 http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en +Article number, page 2 of 18 + +M. Nelissen et al.: Misalignment of the outer disk of DK Tau +Table 1: Dates for the 2010 and 2012 ESPaDOnS and NARVAL data sets. +Date +Heliocentric Julian +Rotation cycle +S/N of the continuum +Airmass +Instrument +(yyyy-mm-dd) +date (UTC) +(8.2 day period) +at the central wavelength +2010-11-26 +2 455 527.436 03 +0.00 +70 +1.1 +NARVAL +2010-12-09 +2 455 540.393 30 +1.58 +77 +1.2 +NARVAL +2010-12-10 +2 455 541.397 11 +1.70 +53 +1.1 +NARVAL +2010-12-13 +2 455 544.418 52 +2.07 +42 +1.1 +NARVAL +2010-12-14 +2 455 544.979 74 +2.14 +72 +1.1 +ESPaDOnS +2010-12-15 +2 455 545.853 82 +2.25 +97 +1.0 +ESPaDOnS +2010-12-16 +2 455 546.854 61 +2.37 +100 +1.0 +ESPaDOnS +2010-12-17 +2 455 547.826 04 +2.49 +108 +1.1 +ESPaDOnS +2010-12-18 +2 455 548.819 03 +2.61 +113 +1.1 +ESPaDOnS +2010-12-19 +2 455 549.786 41 +2.73 +81 +1.2 +ESPaDOnS +2010-12-19 +2 455 550.390 86 +2.80 +54 +1.1 +NARVAL +2010-12-24 +2 455 554.883 63 +3.35 +83 +1.0 +ESPaDOnS +2010-12-26 +2 455 557.034 94 +3.61 +101 +1.8 +ESPaDOnS +2010-12-30 +2 455 560.977 48 +4.09 +96 +1.3 +ESPaDOnS +2011-01-03 +2 455 565.451 64 +4.64 +68 +1.1 +NARVAL +2012-11-19 +2 456 250.509 43 +0.00 +68 +1.1 +NARVAL +2012-11-25 +2 456 256.919 80 +0.78 +91 +1.0 +ESPaDOnS +2012-11-28 +2 456 259.895 31 +1.15 +121 +1.1 +ESPaDOnS +2012-11-29 +2 456 260.991 38 +1.28 +127 +1.1 +ESPaDOnS +2012-12-01 +2 456 262.947 48 +1.52 +105 +1.0 +ESPaDOnS +2012-12-02 +2 456 263.865 70 +1.63 +84 +1.1 +ESPaDOnS +2012-12-04 +2 456 265.965 15 +1.89 +120 +1.0 +ESPaDOnS +2012-12-07 +2 456 268.846 50 +2.24 +94 +1.1 +ESPaDOnS +2012-12-09 +2 456 271.387 97 +2.55 +70 +1.2 +NARVAL +2012-12-10 +2 456 271.825 04 +2.60 +107 +1.2 +ESPaDOnS +2012-12-12 +2 456 273.557 58 +2.81 +63 +1.2 +NARVAL +2012-12-23 +2 456 284.762 49 +4.18 +95 +1.3 +ESPaDOnS +artificially veiled weak-lined T Tauri star (wTTs) spectrum to the +spectrum of DK Tau. By choosing a wTTs with the same spectral +type as our cTTs and coming from the same star forming region3, +this wTTs can be seen as the purely photospheric version of our +star because it experiences no accretion. We tested several wTTs +with a K7 spectral type and a line-of-sight-projected equatorial +rotational velocity 𝑣 sin𝑖 lower than the 𝑣 sin𝑖 of DK Tau (see +Sect. 3.2), in order to find the one that provided the best fit. We +ultimately used the spectrum of TAP45 (which has a 𝑣 sin𝑖 of +11.5 km s−1 - see Feigelson et al. 1987; Bouvier et al. 1993) as +a template to estimate the veiling across DK Tau’s spectra as the +extra continuum that has to be added to the wTTs spectrum in +order to reproduce the veiled cTTs spectrum (at the lowest 𝜒2 +level). +We obtained values of the veiling as a function of the wave- +length (in bins of ∼20 nm), for each spectrum. When the value is +less than ∼0.4 throughout the spectrum, we see that it is approxi- +mately constant in wavelength, therefore we took the mean value +as 𝑅 for the whole spectrum. When it is larger than this, we fit +a linear relation through the points and considered this function +as our 𝑅(𝜆). We then inverted Eq. 1 to recover the normalized +photospheric spectrum of each observation. Figure 1 shows the +night with the least veiling and the one with the most veiling on +the top and bottom panel respectively. +In 2010, the peak values of veiling (at ∼550 nm) for each +observation range from 0.2 to 1.8. Three observations out of +fifteen have nearly constant veiling values across their spectrum. +3 This implies that both T Tauri stars would have the same chemical +composition and very similar age and log 𝑔. We also assume that the +microturbulence and macroturbulence velocities should be very similar. +In 2012, the peak values of veiling for each observation range +from 0.2 to 1.3 (lower than the value from two years before). +Only one observation out of twelve has a nearly constant veiling +value across its spectrum. +For the two sets of observations, we see that the slope of +veiling as a function of wavelength steepens when the veiling is +higher as well. We also find that when the veiling is high, there is +more scatter. We believe this scatter is due to the correlation of +stronger line emission with stronger mass accretion rate (see e.g., +Dodin & Lamzin 2012; Rei et al. 2018). In that case, individual +line emission would contribute more to the veiling by filling +in absorption lines; whereas for nights when the veiling is low, +continuum veiling would remain the dominant form of veiling. +The interpretation of these trends goes beyond the scope of this +work, and will be investigated in a subsequent paper. +3.2. Stellar parameters +Given the importance of accurate stellar parameters, we took +advantage of ESPaDOnS and NARVAL’s continuum normalized +high resolution spectra to derive precise values for DK Tau, in +particular of the line-of-sight-projected equatorial rotational ve- +locity 𝑣 sin𝑖, needed to investigate the potential misalignment +of DK Tau’s outer disk, and for the effective temperature 𝑇eff, +which is needed for our analysis. For this, we first obtained the +mean spectrum of the four nights with the least amount of veiling +(i.e., 𝑅 = 0.2 for all four nights, see Sect. 3.1) in order to get a +spectrum with a higher signal to noise ratio of 136, whereas the +individual spectra had a signal to noise ratio of 103 on average. +We then used the ZEEMAN spectrum synthesis program, devel- +Article number, page 3 of 18 + +A&A proofs: manuscript no. main +550 +600 +650 +700 +750 +800 +850 +Wavelength (nm) +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Veiling +DK Tau 2010-12-17 +Mean: 0.21 +Standard deviation +550 +600 +650 +700 +750 +800 +850 +Wavelength (nm) +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Veiling +DK Tau 2012-12-07 +Fit: -2.62*10 +3x + 2.76 +Uncertainty of fit +Fig. 1: Veiling (gray dots), the best linear fit (blue line) and the +standard deviation (light blue shaded region) as a function of +wavelength for the fourth night of the 2010 ESPaDOnS observa- +tions (top panel) and for the seventh night of the 2012 ESPaDOnS +observations (bottom panel). +oped by Landstreet (1988) and Wade et al. (2001) and modified +by Folsom et al. (2016) to derive the stellar parameters. Under +the assumption of local thermodynamic equilibrium, this code +solves radiative transfer equations. It iteratively compares a syn- +thetic spectrum (calculated using a grid of model atmospheres +and a list of atomic data) with the observation, by 𝜒2 minimiza- +tion, to determine free model parameters. Veiling was included +in the model. +We used the Vienna Atomic Line Database (VALD) website4 +(see e.g., Ryabchikova et al. 2015) to acquire a continuous atomic +line list ranging from 400 nm to 1 000 nm for a star of 𝑇eff = +4 000 K, logarithmic surface gravity log 𝑔 = 4 (in cgs units) and +a solar metallicity. We used the MARCS model atmosphere grid +of Gustafsson et al. (2008). In the ZEEMAN code, we specified +the following initial values for the stellar parameters: for 𝑇eff and +𝑣 sin𝑖, we used the values provided in the literature, of 𝑇eff = +4 000 K (Herczeg & Hillenbrand 2014), and 𝑣 sin𝑖 = 12.7 km s−1 +(McGinnis et al. 2020). We used log 𝑔 = 4 (in cgs units), as +this is typical of T Tauri stars (TTs), microturbulence velocity +𝑣mic = 1 km s−1, macroturbulence velocity 𝑣mac = 0 km s−1, solar +metallicity and a veiling of 0.1. We started with a model with +initial values of 𝑇eff, 𝑣 sin𝑖, log 𝑔 and veiling for a fixed value +of 𝑣mic, 𝑣mac and metallicity. We then ran fits with only 𝑇eff and +𝑣 sin𝑖 as free parameters. When fitting the observation, we used +a wavelength range from 400 nm to 1 000 nm. We ran the code +4 http://vald.astro.uu.se +on several windows throughout the spectrum. We calculated the +average of the values obtained for the different spectral windows +and took the standard deviation of the spread as the error bars. +We derive a 𝑇eff of 4 150 ± 110 K and a 𝑣 sin𝑖 of 13.0 ± 1.3 +km s−1. We note that our values are in good agreement with the +ones found in the literature (see Herczeg & Hillenbrand 2014; +McGinnis et al. 2020). +We also determined DK Tau’s radius. We first calculated the +stellar luminosity 𝐿★ using the J-band magnitudes of Eisner et al. +(2007). As DK Tau A and B are not spatially resolved in their +observations, we corrected the J-band magnitude to remove the +contribution of DK Tau B. This was done by extrapolating the +brightness ratio given by Eisner et al. (2007) for the K-band (of +3.3) to the J-band, taking into consideration the shape of the +continua of the two stars based on their respective spectral types, +as well as the individual extinction each star suffers (i.e., AV += 0.7 mag for DK Tau A, and AV = 1.80 mag for DK Tau B - +Herczeg & Hillenbrand 2014). We find a flux ratio of 4.4 for the +J-band. We then corrected the magnitude for the extinction in +DK Tau A (i.e., 0.7 mag) (Herczeg & Hillenbrand 2014)5. It can +be noted that the value for the extinction quoted by Fischer et al. +(2011) is a factor 2 higher. The variability of the extinction is +probably connected to the dipper behavior of the star (Roggero +et al. 2021). We also corrected for a veiling of 𝑅𝐽 = 0.3 ± 0.1, +based on an interpolation of our measurements of veiling as a +function of wavelength (see Sect. 3.1). This is very similar to +the values observed by Fischer et al. (2011). We then used the +bolometric correction in the J band from Pecaut & Mamajek +(2013). We find a value of 𝐿★ = (1.65 ± 0.25) 𝐿 ⊙. Using the +relation 𝑅2 = 𝐿★/(4𝜋𝜎𝑇4 +eff) and our measured value of 𝑇eff, we +derive a stellar radius of 𝑅★ = (2.48 ± 0.25) 𝑅⊙. +Table 2 summarizes the measured properties of DK Tau. The +inclination of the outer disk axis was measured by Rota et al. +(2022) using ALMA observations. We derive a different value +for the inclination 𝑖 of the stellar rotation axis (see Sect. 4.1). The +stellar rotation period 𝑃 mentioned in the literature is based on +photometry (see Bouvier et al. 1993; Percy et al. 2010; Arte- +menko et al. 2012). The mass 𝑀★ was derived by Johns-Krull +(2007) from pre-main sequence evolutionary tracks. Using the +Siess et al. (2000) models6, we find that the values of 𝑇eff and 𝐿★ +that we obtain give a slightly higher 𝑀★ than the one of 0.7 𝑀⊙ +derived by Johns-Krull (2007), but it agrees with 0.7 𝑀⊙ within +2𝜎 (see Appendix A). +3.3. Least-squares deconvolution +Least-squares deconvolution (LSD) is a cross-correlation tech- +nique used to add the signatures from hundreds of photospheric +absorption lines and obtain an average line profile of very high +signal to noise ratio (see e.g., Donati et al. 1997; Kochukhov et al. +2010). It makes use of a line mask, a list describing the position +and depth of the chosen absorption lines. We first created a line +mask using the VALD line list (with a detection threshold of 0.01, +𝑇eff = 4 000 K, log 𝑔 = 4 and 𝑣mic = 1 km s−1) and assuming an +ATLAS9 stellar atmosphere model (Kurucz 1993). Next, using +our line mask, we applied LSD to all observations in the range +from 500 nm to 1000 nm (i.e., excluding the blue edge of the spec- +tra because of excess noise), excluding regions with telluric and +5 We chose to use the value quoted by Herczeg & Hillenbrand (2014) +as they give a value for both components of the binary. +6 http://www.astro.ulb.ac.be/~siess/pmwiki/pmwiki.php? +n=WWWTools.PMS +Article number, page 4 of 18 + +M. Nelissen et al.: Misalignment of the outer disk of DK Tau +Table 2: Summary of the measured properties of DK Tau. +Stellar parameter +Value +Reference +𝑖 (°) +58 (+18)(-11) +This work +Outer disk axis (°) +21 ± 3 +Rota et al. (2022) +𝑃 (days) +8.4 +Bouvier et al. (1993) +8.18 +Percy et al. (2010) +8.18 +Artemenko et al. (2012) +8.20 ± 0.13 +This work +𝑇eff (K) +4 150 ± 110 +This work +𝑣 sin𝑖 (km s−1) +13.0 ± 1.3 +This work +𝑅★ (𝑅⊙) +2.48 ± 0.25 +This work +𝑀★ (𝑀⊙) +0.68 +Johns-Krull (2007) +𝐿★ (𝐿 ⊙) +1.65 ± 0.25 +This work +The inclination is derived from 𝑣 sin𝑖 = 2𝜋𝑅★𝑃−1 sin𝑖 (see Sect. +4.1). +emission lines, as well as the lines most affected by accretion7, +in order to obtain a mean line profile for each veiling-corrected +spectrum. We set the effective Landé factor 𝑔0 = 1.4, the central +intensity 𝑑0 = 0.47 and the equivalent wavelength 𝜆0 = 650.0 nm +for all the nights (see Kochukhov et al. 2010). The LSD profiles +were then normalized to be at the same equivalent width. The lat- +ter was done in order to correct for the residual effects of veiling. +For all nights, we obtained a definite Zeeman signal detection +in the LSD profiles, with a false alarm probability smaller than +0.001 % (see the LSD profiles in Appendix B). +The Stokes I LSD profiles of the ESPaDOnS and NARVAL +observations made on 19 December 2010 presented a small ab- +sorption feature blueward of the main absorption line (see e.g., +Fig. 2). The other nights all showed a single photospheric ab- +sorption line. This small absorption feature is due to scattered +moonlight, as the full Moon was close to DK Tau on that night +(with an angular separation of about 8°), and as the contamina- +tion is at the expected lunar radial velocity in the heliocentric rest +frame8. This type of contamination has been seen before (see +e.g., Donati et al. 2011). +We fit the wing of the main absorption feature with that of a +Voigt profile and manually removed the contamination. +3.4. Average line-of-sight magnetic field +We measured the magnetic field along the line-of-sight and inte- +grated over the visible stellar hemisphere, 𝐵los9, by using equation +3.3 from Morin (2012): +𝐵los(𝐺) = −2.14 × 1011 +∫ +𝑣 𝑉(𝑣) d𝑣 +𝜆0 𝑔eff 𝑐 +∫ +[𝐼𝑐 − 𝐼𝑣] d𝑣 +(2) +7 We identified them by comparing with the spectrum of RU Lup, a +cTTs of the same spectral type as DK Tau but presenting with more +accretion, and determining the lines in emission. +8 The online applet at https://astroutils.astronomy.osu.edu/ +exofast/barycorr.html, based on Wright & Eastman (2014), allows +to calculate the correction applied to geocentric observations in order to +transpose them in the heliocentric rest frame. In our case, this correction +is -8.8 km s−1. Applied to the Moon’s radial velocity of 0 km s−1 in the +geocentric rest frame, this translates into a radial velocity of -8.8 km s−1 +in the heliocentric rest frame. +9 𝐵los is also referred to as the longitudinal field 𝐵ℓ. We chose not to +use this term to avoid confusion with its homonym 𝐵𝜙, the field along +the east-west direction or azimuthal field (see e.g., Vidotto 2016). +Fig. 2: LSD profile (in the heliocentric velocity frame) of +Stokes V (top) and Stokes I (bottom) parameters normalized to +the continuum for the sixth night of the 2010 ESPaDOnS obser- +vations. Note on this night scattered light from the Moon caused +the small blue-ward absorption feature. For comparison, we also +show the seventh night of the 2010 ESPaDOnS observations +(dashed line), which did not suffer from moonlight contamina- +tion. +on the LSD profiles of the photospheric absorption lines (see also +Rees & Semel 1979; Donati et al. 1997; Wade et al. 2000). In this +equation, 𝑣 is the radial velocity in the rest frame of the star, 𝑉 +refers to Stokes V, 𝜆0 is the wavelength of the line center in nm, +𝑔eff is the effective Landé factor of the line, 𝐼 refers to Stokes I +and 𝐼𝑐 to the unpolarized continuum. We find values ranging +from -0.19 ± 0.05 kG to 0.20 ± 0.03 kG in 2010 and from -0.13 +± 0.02 kG to 0.08 ± 0.02 kG in 2012 (see Fig. 3, and see the +list of values in Appendix B). It should be noted that, since 𝐵los +represents a signed average over the visible stellar hemisphere, +regions of opposite polarities partly cancel out. +We applied a phase dispersion minimization (PDM; Stelling- +werf 1978) technique on the 2010 𝐵los values and found a period +of 8.20 ± 0.13 days. Since this is the period of the modulation +of the stellar magnetic field, it accurately represents the stellar +rotation period. This period is consistent with values found in the +literature from photometry (8.4 days from Bouvier et al. 1993, +and 8.18 days from Percy et al. 2010 and Artemenko et al. 2012). +Small discrepancies could be explained by differential rotation: +different measurements tracking features at various latitudes. +3.5. Emission lines +After analyzing the LSD profiles of photospheric absorption lines +to derive the associated 𝐵los linked to non-accreting regions, we +also investigated emission lines associated with accretion shocks, +in particular the narrow component of the 587.6 nm Hei line +and of the Caii infrared triplet (IRT - at 849.8 nm, 854.2 nm +and 866.2 nm). They can be used to get more information on +DK Tau’s magnetic fields, as they are tracers of the fields present +at the footpoints of accretion funnels (see e.g., Donati et al. +Article number, page 5 of 18 + +0.4 +Vllc +0.2 +100× +MAMW.M +0.0 +-0.2 +1.00 +0.95 +I/lc +0.90 +0.85 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)A&A proofs: manuscript no. main +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +200 +0 +200 +Blos (G) +Blos (2010) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Phase (P = 8.2 days) +0.5 +1.0 +1.5 +Veiling +Veiling at 617.50 nm (2010) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 3 +Cycle 4 +0.0 +0.2 +0.4 +0.6 +0.8 +100 +0 +100 +Blos (G) +Blos (2012) +0.0 +0.2 +0.4 +0.6 +0.8 +Phase (P = 8.2 days) +0.5 +1.0 +Veiling +Veiling at 617.50 nm (2012) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 4 +Fig. 3: Average line-of-sight magnetic field 𝐵los as derived from the photospheric absorption lines and veiling over time, shown +folded in phase with the derived 8.2 day period, for the 2010 (left panel) and 2012 (right panel) datasets. Different colors and symbols +represent different rotation cycles. +2007). These lines are known to have multiple components (see +e.g., Beristain et al. 2001; McGinnis et al. 2020), often containing +a narrow component (NC), originating from the accretion shock, +and a much broader component (or components), which likely +forms farther out in the accretion columns or in hot winds. For +this reason, we decomposed the profiles using a fit of 2 or more +Gaussians in order to isolate the NC (examples of these fits can +be seen in Appendix C). For the Caii lines, we then averaged the +three lines into a single LSD-like profile in order to increase the +signal to noise ratio, as it has been done in other studies (see e.g., +Donati et al. 2007, 2008, 2012). Since this is a triplet, the shape +of all three lines should be the same (see e.g., Azevedo et al. +2006). In addition, in our data, we see an intensity ratio close to +1:1:1 and the NCs are not contaminated by the nearby Paschen +emission lines at 850.2 nm, 854.5 nm and 866.5 nm. The NC of +the Hei line is believed to be generated in the post-shock region +at the base of the magnetic accretion funnels that connect the +surface of the star to its inner disk. The NC of the Caii IRT is +thought to probe the accretion regions and the chromosphere. +The Stokes V and Stokes I profiles of the Hei emission line +and the Caii IRT, for both epochs, can be seen in Fig. 4. The +Stokes V profiles of the emission lines show similar signatures +with phase. This indicates that the accretion spot is mostly likely +located at a high enough latitude to be visible with the same +polarity at all times. They also indicate that the field is positive +(i.e., pointing toward the observer) at the base of the accretion +funnels that connect the star to its circumstellar disk. +We measured 𝐵los intheemission linesusingthesamemethod +as described in Sect. 3.4. We find values ranging from 0.20 ± 0.51 +kG to 1.77 ± 0.08 kG for Hei in 2010, from 0.48 ± 0.09 kG to 1.99 +± 0.09 kG for Hei in 2012, from 0.34 ± 0.04 kG to 0.95 ± 0.02 +kG for Caii in 2010, from 0.26 ± 0.02 kG to 1.41 ± 0.05 kG for +Caii in 2012. Figure 5 shows the obtained values folded in phase +(and the list of values can be found in Appendix C). The values of +𝐵los are higher in 2012. We find that the values measured through +the Caii IRT are lower than the ones measured through the Hei +line. It has been hypothesized that the lower intensity of 𝐵los +found using the Caii IRT stems from the dilution of the emission +from the accretion shock by chromospheric emission (see e.g., +Donati et al. 2019), which results in the Stokes V/I profile being +shallower for the Caii IRT than for the Hei line. +We find extreme values of 𝐵los in the emission lines that are +one order of magnitude larger than the extreme values of 𝐵los +derived from the LSD profiles of the photospheric absorption +lines (see Fig. 3), showing strong fields in the accretion shocks. +This is consistent with the current understanding of accretion +shocks as compact regions with some of the strongest magnetic +field concentrating in dark polar regions at the surface of the star +and magnetic field lines reaching to the circumstellar disk. For +both epochs, we only see the positive pole and never the negative +one. +The plots of the 𝐵los in the emission lines in 2012 do not fold +well in phase. We believe this may stem from the location of the +accretion shocks being more dynamic and/or the accretion being +more complex, with more than one accretion shock. This is also +observed in the variability of veiling in 2012 (see Fig. 3), which +does not fold well with the rotation phase, indicating that there is +considerable intrinsic variability in the mass accretion rate. +We derived the equivalent width (EW) of the Hei emission +line. Figure 6 shows the obtained values folded in phase (and +the list of values can be found in Appendix C). When the EW +is larger, we are seeing more of the accretion shock in our line- +of-sight. This is also when we measure stronger magnetic fields +using emission lines tracing the accretion shock (see Fig. 5), as +expected following the paradigm of magnetospheric accretion. +3.6. Magnetic Obliquity +We used the third equation of Preston (1967), which assumes +a pure dipole, a simplification of the magnetic field present in +the accretion shocks, to calculate an estimate of the magnetic +obliquity (i.e., the angle between the stellar rotation axis and the +magnetic field axis) derived from the emission lines. We used +the extreme values found for 𝐵los in the emission lines and an +inclination 𝑖 of 58°. For the Hei emission line, we find a magnetic +obliquity10 of 26° for the 2010 epoch, and of 21° for the 2012 +10 This is not the magnetic obliquity of the entirety of DK Tau’s magnetic +field. It is based solely on the average line-of-sight magnetic field present +Article number, page 6 of 18 + +M. Nelissen et al.: Misalignment of the outer disk of DK Tau +Fig. 4: Stokes V and Stokes I profiles (in gray) and average (in red) of the Hei emission line (top panels) and the Caii IRT (bottom +panels), for the 2010 (left panels) and 2012 (right panels) observations. +epoch. For the Caii IRT, we find a magnetic obliquity of 16° +for the 2010 epoch, and of 23° for the 2012 epoch. These esti- +mates are consistent with the Stokes V signatures of the emission +lines. They are also consistent with the magnetic obliquity of 18° +(+8)(-7) in 2011 derived by McGinnis et al. (2020), using the +radial velocity variability of the Hei emission line and assuming +one accretion spot. We therefore have an agreement between the +in the accretion shocks and probed through emission lines. Furthermore, +the calculation uses the extreme values for 𝐵los and does not account for +variability between nights. +values derived from the magnetic field that drives the accretion +and the value derived from a result of accretion. +The estimates of the magnetic obliquity are consistent with +only seeing the positive pole in the plots of the 𝐵los in the emis- +sion lines, confirming that DK Tau experiences nearly poleward +accretion, with a positive field at the base of the accretion funnels. +The Hei emission lines show a radial velocity variability with +a small amplitude, which is another indication that the accretion +spot is most likely close to the pole. As the star rotates, if the +accretion spot were located at the equator, the velocity variation +would be large as the spot gets red and blueshifted. +Article number, page 7 of 18 + +Hel line (2010) +10 +0 +100 × Vllc +-10 +-20 +-30 +3.0 +2.5 +2.0 +1.5 +1.0 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)Hel line (2012) +10 +100 × Vllc +-10 +-20 +-30 +3.0 +2.5 +2.0 +1.5 +1.0 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)Call IRT (2010) +15 +10 +100 × Vllc +5 +0 +-5 +-10 +-15 +2.4 +2.2 +2.0 +1.8 +1.6 +1.4 +1.2 +1.0 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)Call IRT (2012) +20 +15 +10 +100 × Vllc +5 +0 +-5 +-10 +-15 +2.39 +2.50 +2.25 +2.00 += 1.75 +1.50 +1.25 +1.00 +0.75 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)A&A proofs: manuscript no. main +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Phase (P = 8.2 days) +500 +0 +500 +1000 +1500 +2000 +Blos (G) +HeI line (2010) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 3 +Cycle 4 +0.0 +0.2 +0.4 +0.6 +0.8 +Phase (P = 8.2 days) +500 +750 +1000 +1250 +1500 +1750 +2000 +Blos (G) +HeI line (2012) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Phase (P = 8.2 days) +300 +400 +500 +600 +700 +800 +900 +1000 +Blos (G) +CaII IRT (2010) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 3 +Cycle 4 +0.0 +0.2 +0.4 +0.6 +0.8 +Phase (P = 8.2 days) +200 +400 +600 +800 +1000 +1200 +1400 +Blos (G) +CaII IRT (2012) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 4 +Fig. 5: Average line-of-sight magnetic field 𝐵los over time, shown folded in phase with an 8.2 day period, for the Hei emission line +(top panels) and the Caii IRT (bottom panels), for the 2010 (left panel) and 2012 (right panel) datasets. Different colors and symbols +represent different rotation cycles. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Phase (P = 8.2 days) +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +EW (Å) +HeI line (2010) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 3 +Cycle 4 +0.0 +0.2 +0.4 +0.6 +0.8 +Phase (P = 8.2 days) +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +EW (Å) +HeI line (2012) +Cycle 0 +Cycle 1 +Cycle 2 +Cycle 4 +Fig. 6: Equivalent width (in ˚𝐴) of the Hei emission line over time, shown folded in phase with an 8.2 day period, for the 2010 (left +panel) and 2012 (right panel) datasets. Different colors and symbols represent different rotation cycles. +Article number, page 8 of 18 + +M. Nelissen et al.: Misalignment of the outer disk of DK Tau +3.7. Truncation & co-rotation radii +In order to calculate the truncation radius, we need to know the +star’s mass accretion rate. For this, we measured the equivalent +width (EW) of several emission lines (i.e., H𝛼, H𝛽, H𝛾, the Hei +lines at 447.1 nm, 667.8 nm and 706.5 nm, as well as the Caii +IRT at 849.8 nm, 854.2 nm and 866.2 nm). Since ESPaDOnS and +NARVAL’s spectra are not flux calibrated, we created a template +for DK Tau based on SO879, a weak-lined T Tauri star with a +K7 spectral type (described in Stelzer et al. 2013). The template +was corrected for extinction, then scaled to have the same lumi- +nosity (i.e., 1.65 𝐿 ⊙) and be at the same distance (i.e., 132.6 pc) +as DK Tau. After correcting the EW for veiling, we used this +template to flux calibrate them through the following formula: +𝐹line = 𝐸𝑊line · 𝐹cont +(3) +where 𝐹line is the flux of the line, 𝐸𝑊line is the veiling corrected +EW of the line and 𝐹cont is the flux of the continuum of the +template at the wavelength of the line in question. Then we ob- +tained the luminosity in each line. Next, we used the relations in +Table B.1. from Alcalá et al. (2017) to calculate the accretion lu- +minosity from each line and averaged these values for each night. +We then took the average over all nights in each epoch as the +accretion luminosity, and the standard deviation of the spread in +the values found from different nights as the error bars. We find +𝐿acc = 0.26 ± 0.18 𝐿 ⊙ in 2010 and 𝐿acc = 0.49 ± 0.42 𝐿 ⊙ in 2012. +These values are similar to the ones found e.g., by Fischer et al. +(2011) (i.e., 0.17 𝐿 ⊙) or by Fang et al. (2018) (i.e., 0.16 𝐿 ⊙). +We then converted the accretion luminosity into mass ac- +cretion rate using Eq. 8 from Gullbring et al. (1998) with the +values of 𝑅★ and 𝑀★ from Table 2 and 𝑅in = 5 𝑅★ (as is typ- +ically used). We find log ( �𝑀acc[𝑀⊙ yr−1]) = -7.43 in 2010, and +log ( �𝑀acc[𝑀⊙ yr−1]) = -7.15 in 2012. These values are consistent +with the one of -7.42 quoted by Gullbring et al. (1998). Finally, +we used Eq. 6 from Bessolaz et al. (2008) to estimate the trun- +cation radius. This equation assumes an axisymmetric dipole, +which is a simplification of DK Tau’s magnetic topology. It also +uses the dipolar field calculated at the equator as 𝐵★. Consid- +ering the equatorial value is half of the value at the pole, we +estimated the latter (see Preston 1967) using the values of 𝐵los in +the emission lines 11. This is an approximation, as part of the 𝐵los +could come from higher order multipoles, in particular from the +octupole, rather than the dipole. We find 𝑟trunc ∼ (5.2 ± 1.1) 𝑅★ +for the Hei emission line in 2010, 𝑟trunc ∼ (3.9 ± 0.8) 𝑅★ for the +Caii IRT12 in 2010, 𝑟trunc ∼ (4.7 ± 1.2) 𝑅★ for the Hei emission +line in 2012, and 𝑟trunc ∼ (3.9 ± 1.0) 𝑅★ for the Caii IRT in 2012. +We calculated the co-rotation radius as well, using Kepler’s third +law: 𝑟co-rot = 6.1 𝑅★. We find that the truncation radius values are +consistent with the co-rotation radius within the error bars. This +implies that DK Tau is unlikely to be in the propeller regime, an +unstable accretion regime, as the truncation radius is not farther +than the co-rotation radius (Romanova et al. 2018). For the 2010 +epoch, DK Tau may be in the stable accretion accretion regime, +since the 𝐵los and accretion tracers seem fairly periodic. The trun- +cation radius being slightly smaller than the co-rotation radius is +consistent with this as well (Blinova et al. 2016). +11 We used the magnetic field derived from the accretion-powered emis- +sion lines, considering it will dominate over the magnetic field derived +from the photospheric absorption lines at the distance of the truncation +radius. +12 We find lower values for the truncation radius estimates when using +the Caii IRT. This stems from the lower values found for 𝐵los in those +lines (see Sect. 3.5). +4. Discussion +4.1. Inconsistencies regarding the inclination +Naively one might assume that the inclination angle 𝑖 of the stel- +lar rotation axis with respect to the line-of-sight is 21° based +on the inclination of the outer gaseous disk axis (Rota et al. +2022). DK Tau’s lightcurve however classifies the star as a dipper +(Roggero et al. 2021). The traditional explanation invokes cir- +cumstellar material passing in front of the star and occulting it. If +the disk is seen close to edge on, matter lifted above the disk plane +could cause these occultations. However, DK Tau’s outer disk is +seen nearly pole on (Rota et al. 2022), which is inconsistent with +this scenario, unless the stellar rotation axis is at a very different +angle than that of the outer disk axis. +Furthermore, based on the star’s rotational properties (see +Table 2) and using the following relation +𝑣 sin𝑖 = 2𝜋𝑅★ +𝑃 +sin𝑖 +(4) +we derive a much higher inclination of 58° (+18)(-11). Therefore, +if DK Tau is in fact seen nearly pole on, then 𝑃, 𝑣 sin𝑖 and 𝑅★ are +not consistent with each other. As the stellar radius is the most +uncertain of these parameters13, it is possible that it may have +been underestimated. However, if we consider that 𝑃, 𝑣 sin𝑖 and +𝑖 are accurately determined, then we would need 𝑅★ = (6 ±1) 𝑅⊙ +for this formula to agree, which is unrealistically large for a TTs. +Another possibility is that the period or 𝑣 sin𝑖 may be inac- +curate. Regarding 𝑣 sin𝑖, the value we derive agrees within error +bars with the one measured by McGinnis et al. (2020), despite +using two different assessment methods. It is therefore a value +that can be trusted. This leaves the stellar rotation period. +In the literature, the stellar rotation period of DK Tau has been +measured using photometry with values ranging from 8.18 days +(Percy et al. 2010; Artemenko et al. 2012) to 8.4 days (Bouvier +et al. 1993). However, since the photometry is dominated by +flux dips that might be due to extinction events (Roggero et al. +2021), it is possible that these dips are caused by circumstellar +material that is not located at the co-rotation radius. In that case, +the measured period would not be the same as the stellar rotation +period. +In Sect. 3.4 we derived a period from the rotational mod- +ulation of the line-of-sight magnetic field 𝐵los, which should +accurately represent the stellar rotation period. In the context of +exoplanet search programs, 𝐵los is indeed often considered as the +most reliable indicator of stellar rotation period (see e.g., Hébrard +et al. 2016). Additionally, the value we find of 8.20 ± 0.13 days +is consistent with those found in the literature from photometry. +We therefore find that the value for the period can be trusted. +Moreover, this rotation period can be seen in a number of +datasets at our disposal. For example, we computed bidimen- +sional periodograms of the intensity of the Hei (at 587.6 nm) +emission line, and the Stokes I and Stokes V LSD profiles of the +photospheric absorption lines (see Appendix D). The Hei line +comes from the accretion shock and should therefore vary with +the stellar rotation period. In 2010 we find a period around 8 days +for the entire red-shifted part of the Hei line (from 0 to 50 km s−1), +however this period is very uncertain. The Stokes V profile also +shows a period near 8.5 days between ∼-20 and -7 km s−1 and +13 This is because it depends on evolutionary models as well as an ac- +curate determination of the effective temperature and stellar luminosity. +Both can be subject to fairly large uncertainties, particularly the luminos- +ity for a star with a dipper light curve which likely suffers from variable +extinction. +Article number, page 9 of 18 + +A&A proofs: manuscript no. main +between ∼0 and 8 km s−1, but again the uncertainty is large. The +Stokes I profile does not show a clear period. In 2012 there is +no clear period found from the Hei line, likely because accretion +is more intrinsically variable in this epoch than 2 years prior. +Again no clear period is observed from the Stokes I profile, but +the Stokes V profile shows a possible periodicity at around 8 days +(albeit with a large uncertainty, same as in 2010). +Furthermore, the variation of the 2010 veiling as a function +of time is also consistent with an 8.2 day period (see Fig. 3). The +variation of the 2012 veiling as a function of time, however, does +not seem to follow any trend with the period. This is consistent +with the intensity of the HeI line not showing a clear correlation +with period in this epoch, since both are tracing accretion. This +is another indication that there must be some intrinsic variability +in the mass accretion rate in 2012, which masks any rotational +modulation of the accretion spot(s). +We looked at the possibility of the rotation period or 𝑣 sin𝑖 be- +ing inaccurate and found evidence to the contrary. Because their +values appear to be accurate, we deduce that it is the value for the +inclination that is problematic. We conclude that the inclination +measured for the outer circumstellar disk axis must not represent +the inclination of the rotation axis of the star. This suggests that +there is a considerable misalignment between the rotation axis of +DK Tau and its outer disk. When we calculate DK Tau’s inclina- +tion based on its rotational properties using Eq. 4, we find 𝑖 = 58° +(+18)(-11). This value is based on 𝑣 sin𝑖 = (13.0 ± 1.3) km s−1, +𝑃 = (8.2 ± 0.2) days and 𝑅★ = (2.48 ± 0.25) 𝑅⊙. It follows that +the outer disk axis of DK Tau is likely misaligned by 37° with its +rotation axis (see Fig. 7). +Fig. 7: Sketch (not to scale) showing DK Tau in the center, sur- +rounded first by its inner disk, then by its outer disk which is +considerably misaligned. The rotation axis at 58° is in red, the +outer disk axis at 21° is in blue, and the line-of-sight axis is in +gray (by C. Delvaux). +Misalignments between the inner and outer circumstellar disk +axes of T Tauri stars are starting to be observed, when combining +near infrared interferometric VLTI/GRAVITY data and millime- +ter interferometric ALMA data (see e.g., Ansdell et al. 2020; Bou- +vier et al. 2020), or with shadows observed with VLT/SPHERE +(see e.g., Benisty et al. 2018; Sicilia-Aguilar et al. 2020), or with +VLTI/GRAVITY (see e.g., Bohn et al. 2022). We find a misalign- +ment between the outer disk axis and the rotation axis. What of +the inner disk axis? In young dippers like DK Tau, the material +that causes the dips is believed to be located in the inner accretion +disk (Bouvier et al. 2007; McGinnis et al. 2015), which need to +be observed at high inclinations in order for material to cross +our line-of-sight to produce the dips. Therefore an inclination of +21° of the inner disk of DK Tau is difficult to reconcile with its +dipper light curve. In addition, the inner disk axis is normally +expected to be aligned with the stellar rotation axis. We thus find +it very likely that the inner disk axis of DK Tau has the same +inclination of 𝑖 = 58° as was calculated for its rotation axis. This +inclination is sufficiently high to support the dipper behavior and +there are other cases of dippers with similar inclinations (see e.g., +McGinnis et al. 2015; Roggero et al. 2021). DK Tau is one more +example of a TTS with a misalignment between its inner and +outer circumstellar disk axes. +It is also a wide binary system, and the misalignment could +stem from the binary formation mechanism: turbulent fragmen- +tation might generate disk axis that are more randomly oriented. +It is however interesting to note as well that the outer disk axes +of both components of the binary are misaligned by 43° (see +Rota et al. 2022), which is close to the value (of 37°) of the mis- +alignment between the inner and outer disk axes of DK Tau A. +This could suggest a quasi-alignment of the inner disk axis of +DK Tau A with the outer disk axis of DK Tau B, assuming that +they are not only aligned compared to our line-of-sight, but that +the orientation of their nodes are aligned as well, which is un- +known. +4.2. Magnetic field in the accretion-powered emission lines +The 𝐵los derived from the photospheric absorption lines (see +Sect. 3.4), gives a partial view of DK Tau’s magnetic fields that +exclude the accreting regions, as photospheric absorption lines +and accretion-powered emission lines form in different regions of +the stellar surface. It is the field present in these accreting regions +that is understood to best probe the global stellar magnetic field +that reaches to the circumstellar disk. The magnetic obliquity +derived from the accretion-powered emission lines is therefore +likely to be close to the actual value. We find that the low magnetic +obliquity that we derive (see Sect. 3.6), the positive polarity of +the 𝐵los in the emission lines, as well as their Stokes V signatures +and the range of radial velocity of the Hei line are consistent with +the presence of an accretion spot always visible and close to the +pole. This is where the accretion funnels connecting DK Tau to +its disk would be anchored. This is similar to what has been found +for several other cTTs (see e.g., Johnstone et al. 2014; McGinnis +et al. 2020). +For the 2010 epoch, we find that the magnetic field in the +Caii IRT (and in the Hei line - see Fig. 5) is at a maximum close +to the same phase (i.e., around phase 0.3) as the maximum in +the veiling (see Fig. 3), which is when the accretion shock is in +our line-of-sight. Around phase 0.5, we see a small redshifted +absorption in the H𝛼 line (see Appendix E), indicating that the +accretion column is in our line-of-sight, which is directly after +the maximum of the magnetic field in the emission lines and the +increase in veiling, therefore probably directly after the accretion +shock was in our line-of-sight. Because this small redshifted +absorption is not perfectly simultaneous with the increase in +Article number, page 10 of 18 + +i= 58° +i=21°M. Nelissen et al.: Misalignment of the outer disk of DK Tau +veiling and in the emission lines, it might be an indication of +differential rotation. +5. Conclusions +In this paper, we have studied DK Tau, a low-mass classical +T Tauri star (cTTs) with significant veiling (defined as the ra- +tio between the accretion shock flux and the photospheric flux), +using dual-epoch spectropolarimetric observations (collected in +2010 and 2012). We derive an effective temperature 𝑇eff of 4 150 +± 110 K and a line-of-sight-projected equatorial rotational veloc- +ity 𝑣 sin𝑖 of 13.0 ± 1.3 km s−1, in agreement with the literature. +We find peak values of veiling in the optical (∼550 nm) ranging +from 0.2 to 1.8 in 2010, and from 0.2 to 1.3 in 2012. +We derive the line-of-sight magnetic field integrated over the +visible hemisphere 𝐵los from the photospheric absorption lines +(linked to non-accreting regions). We find values ranging from +-0.19 ± 0.05 kG to 0.20 ± 0.03 kG in 2010 and from -0.13 ± 0.02 +kG to 0.08 ± 0.02 kG in 2012. +We recover a rotation period of 8.2 days using the values of +𝐵los for the 2010 dataset. We confirmed the period by analyzing +the intensity of the Hei line in 2010, the intensity of the Stokes V +profiles in both 2010 and 2012, and the variation of veiling as a +function of time in 2010. They are all consistent with an 8.2 day +period. This also agrees with the values of period given in the +literature from photometry. +We find several inconsistencies related to the inclination of +the stellar rotation axis with respect to the line-of-sight 𝑖. The +measurement of the inclination of the outer circumstellar disk +axis gives a value of 21° (Rota et al. 2022). DK Tau’s lightcurve, +however, classifies it as a dipper (Roggero et al. 2021), for which +the simplest explanation involves a star seen close to edge on. +Furthermore, using Eq. 4, we find that the inclination of 21°, the +period, 𝑣 sin𝑖 and the radius are not consistent with each other. +When using the values of period, 𝑣 sin𝑖 and stellar radius that +we derive to estimate the inclination of the stellar rotation axis, +we find a value of 𝑖 = 58° (+18)(-11). We thus find a substantial +misalignment between DK Tau’s rotation axis (at 58°) and its +outer disk axis (at 21°) to be likely. +To complement the partial picture of the 𝐵los derived from the +photospheric absorption lines, we analyzed emission lines that +are tracers of the magnetic fields present in the accretion shocks. +We examined the narrow component of the 587.67 nm Hei emis- +sion line and of the Caii infrared triplet (IRT - at at 849.8 nm, +854.2 nm and 866.2 nm). We found that their Stokes V profiles +show similar signatures with phase, indicating that DK Tau ex- +periences poleward accretion, with a positive field at the base of +the accretion funnels connecting the star to its circumstellar disk. +We measured 𝐵los within the accretion shocks. We find values +ranging from 0.92 ± 0.09 kG to 1.77 ± 0.08 kG for Hei in 2010, +from 0.48 ± 0.09 kG to 1.99 ± 0.09 kG for Hei in 2012, from 0.42 +± 0.02 kG to 0.95 ± 0.02 kG for Caii in 2010, from 0.30 ± 0.01 +kG to 1.15 ± 0.02 kG for Caii in 2012. The positive polarity of the +𝐵los in the emission lines is again consistent with the presence +of an accretion spot always visible and close to the pole. This +geometry is similar to what has been found for other cTTs (see +e.g., Johnstone et al. 2014; McGinnis et al. 2020). +We derived an estimate of the magnetic obliquity from the +emission lines using the third equation of Preston (1967). This +equation assumes a pure dipole, which is a simplification of the +magnetic field present in the accretion shocks. It is the field +present in these accretion shocks that is understood to best probe +the global stellar magnetic field that reaches the circumstel- +lar disk. The magnetic obliquities derived from the accretion- +powered emission lines are therefore a reasonable reflection of +the real dipole present above the surface of the star. For the Hei +emission line, we find a magnetic obliquity of 26° for the 2010 +epoch, and of 21° for the 2012 epoch. For the Caii IRT, we find a +magnetic obliquity of 16° for the 2010 epoch, and of 23° for the +2012 epoch. These estimates are consistent with the magnetic +obliquity of 18° (+8)(-7) in 2011 derived by McGinnis et al. +(2020), using the Hei emission line and assuming one accretion +spot. +We also estimated the truncation radius using the values of +𝐵los in the emission lines, and find 𝑟trunc ∼ (5.2 ± 1.1) 𝑅★ for the +Hei emission line in 2010, 𝑟trunc ∼ (3.9 ± 0.8) 𝑅★ for the Caii +IRT in 2010, 𝑟trunc ∼ (4.7 ± 1.2) 𝑅★ for the Hei emission line in +2012, and 𝑟trunc ∼ (3.9 ± 1.0) 𝑅★ for the Caii IRT in 2012. We +calculated the co-rotation radius as well, and find 𝑟co-rot = 6.1 𝑅★. +We find that the truncation radius values are consistent with the +co-rotation radius within the error bars. +In conclusion, we find that DK Tau, presenting with signif- +icant veiling, has similar magnetic properties to the more mod- +erately accreting cTTs studied so far. In addition, we find that +DK Tau’s outer disk axis is likely to be misaligned compared to +its rotation axis by 38°. This poses questions with regards to stan- +dard models of circumstellar disk formation. More observations +of cTTs are needed to better understand the prevalence of such +misalignments, while the geometry of DK Tau’s system requires +additional studies to characterize it further. +Acknowledgements +The authors thank A. Natta for helpful discussions and C. Delvaux +for the sketch of DK Tau. We also thank the referee for valuable +comments that improved the paper. +Based on observations obtained at the Canada-France-Hawaii +Telescope (CFHT) which is operated by the National Research +Council of Canada, the Institut National des Sciences de l’Univers +of the Centre National de la Recherche Scientifique of France, +and the University of Hawaii. 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D. 2014, PASP, 126, 838 +Article number, page 12 of 18 + +M. Nelissen et al.: Misalignment of the outer disk of DK Tau +Appendix A: Stellar parameters +Using the Siess et al. (2000) models14, we checked the compatibility of the values of 𝑀★, 𝑇eff and 𝐿★ and obtained Fig. A.1. The +range of 𝑇eff and 𝐿★ (accounting for their error bars) that we derive correspond to masses that are close within 2𝜎 to the value of 𝑀★ += 0.7 𝑀⊙ quoted by Johns-Krull (2007). +Fig. A.1: Hertzsprung-Russell diagram with PMS evolutionary tracks from Siess et al. (2000). The colored lines correspond to +different masses (in 𝑀⊙). The gray rectangle highlights our values of 𝑇eff and 𝐿★ with their error bars. +Appendix B: Photospheric absorption lines +Figure B.1 shows the Stokes V and Stokes I profiles of the photospheric absorption lines for both epochs. Table B.1 lists the values +of the 𝐵los, the line-of-sight magnetic field integrated over the visible hemisphere, for the photospheric absorption lines. +14 http://www.astro.ulb.ac.be/~siess/pmwiki/pmwiki.php?n=WWWTools.PMS +Article number, page 13 of 18 + +HR diagram +1.5 +0.5 +0.6 +0.7 +0.8 +0.9 +0.5 +(Lo) +L +0 +-0.5 +.1 +-1.5 +3.74 +3.72 +3.7 +3.68 +3.66 +3.64 +3.62 +3.6 +3.58 +3.56 +log Teff(K)A&A proofs: manuscript no. main +Fig. B.1: Stokes V and Stokes I profiles (in gray) and average (in red) of the absorption lines, for the 2010 (left panels) and 2012 +(right panels) observations. +Table B.1: 𝐵los for the photospheric absorption lines. +Date +Rotation cycle +𝐵los +(yyyy-mm-dd) +(8.2 day period) +(G) +2010-11-26 +0.00 +-148.49 +2010-12-09 +1.58 +131.92 +2010-12-10 +1.70 +16.08 +2010-12-13 +2.07 +-185.01 +2010-12-14 +2.14 +-186.52 +2010-12-15 +2.25 +-90.83 +2010-12-16 +2.37 +30.36 +2010-12-17 +2.49 +205.23 +2010-12-18 +2.61 +87.10 +2010-12-19 +2.73 +31.08 +2010-12-19 +2.80 +6.72 +2010-12-24 +3.35 +10.11 +2010-12-26 +3.61 +100.33 +2010-12-30 +4.09 +-148.73 +2011-01-03 +4.64 +93.22 +2012-11-19 +0.00 +-29.55 +2012-11-25 +0.78 +-124.88 +2012-11-28 +1.15 +-17.76 +2012-11-29 +1.28 +81.99 +2012-12-01 +1.52 +-15.75 +2012-12-02 +1.63 +-78.84 +2012-12-04 +1.89 +-74.67 +2012-12-07 +2.24 +-5.17 +2012-12-09 +2.55 +-87.57 +2012-12-10 +2.60 +-102.34 +2012-12-12 +2.81 +-35.25 +2012-12-23 +4.18 +26.34 +Article number, page 14 of 18 + +LSD profiles (2010) +0.4 +0.2 +1 × 00 +0.0 +-0.2 +L +-0.4 +-0.6 +1.00 +0.95 +0.90 +0.85 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)LSD profiles (2012) +0.6 +0.4 +0.2 +X +0.0 +00 +-0.2 +-0.4 +1.00 +0.95 +0.85 +0.80 +-100 +-50 +0 +50 +100 +Velocity (km. s-1)M. Nelissen et al.: Misalignment of the outer disk of DK Tau +Appendix C: Emission lines +The emission lines associated with accretion shocks have multiple components (usually a broad and a narrow component). The +narrow component is believed to come from the accretion shock, and that is the component we wish to isolate to probe the magnetic +field in the shock region. We fit several components of each emission line that we studied and then subtracted all the components +except the narrow one, to get a residual profile. Figure C.1 shows two examples, for the Hei line (at 587.6 nm) and for the average of +the Caii IRT (at 849.8 nm, 854.2 nm and 866.2 nm). +Fig. C.1: Fit of the Hei line for the first night of the 2010 ESPaDOnS observations (left panel) and of the average of the Caii IRT for +the second night of the 2012 ESPaDOnS observations (right panel). The observed line is in black. The different components are in +red and their sum is in blue. +Table C.1 lists the values of the 𝐵los, the line-of-sight magnetic field integrated over the visible hemisphere, for the Hei line and +the Caii IRT. +Table C.2 lists the values of the equivalent width (EW) of the Hei line. +Article number, page 15 of 18 + +Hel line (2010) +3.0 +2.5 +Normalized intensity +2.0 +1.5 +1.0 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +Velocity (km/s)Call IRT (2012) +3.00 +2.75 +2.50 +Normalized intensity +2.25 +2.00 +1.75 +1.50 +1.25 +1.00 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +Velocity (km/s)A&A proofs: manuscript no. main +Table C.1: 𝐵los for the emission lines. +Date +Rotation cycle +𝐵los +𝐵los +(yyyy-mm-dd) +(8.2 day period) +for Hei (G) +for Caii (G) +2010-11-26 +0.00 +688.77 +341.07 +2010-12-09 +1.58 +912.45 +778.92 +2010-12-10 +1.70 +1601.15 +781.50 +2010-12-13 +2.07 +-201.64 +450.74 +2010-12-14 +2.14 +970.06 +610.67 +2010-12-15 +2.25 +1392.66 +621.40 +2010-12-16 +2.37 +1473.35 +949.06 +2010-12-17 +2.49 +1579.56 +826.53 +2010-12-18 +2.61 +1321.08 +699.57 +2010-12-19 +2.73 +1759.99 +663.43 +2010-12-19 +2.80 +1023.12 +760.05 +2010-12-24 +3.35 +1767.61 +944.87 +2010-12-26 +3.61 +923.56 +671.05 +2010-12-30 +4.09 +1198.59 +426.48 +2011-01-03 +4.64 +803.45 +771.46 +2012-11-19 +0.00 +1957.62 +1411.28 +2012-11-25 +0.78 +1988.06 +876.93 +2012-11-28 +1.15 +1130.63 +838.51 +2012-11-29 +1.28 +939.35 +782.20 +2012-12-01 +1.52 +479.01 +300.70 +2012-12-02 +1.63 +606.14 +397.38 +2012-12-04 +1.89 +1670.70 +1052.93 +2012-12-07 +2.24 +1872.10 +1149.95 +2012-12-09 +2.55 +1178.70 +828.18 +2012-12-10 +2.60 +1349.05 +561.12 +2012-12-12 +2.81 +1137.52 +263.62 +2012-12-23 +4.18 +1435.56 +850.50 +Table C.2: EW of the Hei line. +Date +Rotation cycle +EW for +(yyyy-mm-dd) +(8.2 day period) +Hei ( ˚𝐴) +2010-11-26 +0.00 +1.37 +2010-12-09 +1.58 +1.49 +2010-12-10 +1.70 +1.54 +2010-12-13 +2.07 +1.32 +2010-12-14 +2.14 +2.44 +2010-12-15 +2.25 +2.04 +2010-12-16 +2.37 +2.05 +2010-12-17 +2.49 +1.46 +2010-12-18 +2.61 +1.74 +2010-12-19 +2.73 +1.50 +2010-12-19 +2.80 +0.86 +2010-12-24 +3.35 +2.20 +2010-12-26 +3.61 +1.50 +2010-12-30 +4.09 +1.95 +2011-01-03 +4.64 +1.43 +2012-11-19 +0.00 +2.61 +2012-11-25 +0.78 +1.74 +2012-11-28 +1.15 +1.61 +2012-11-29 +1.28 +1.44 +2012-12-01 +1.52 +1.43 +2012-12-02 +1.63 +1.78 +2012-12-04 +1.89 +2.57 +2012-12-07 +2.24 +2.12 +2012-12-09 +2.55 +1.91 +2012-12-10 +2.60 +1.67 +2012-12-12 +2.81 +1.19 +2012-12-23 +4.18 +1.91 +Article number, page 16 of 18 + +M. Nelissen et al.: Misalignment of the outer disk of DK Tau +Appendix D: Bidimensional periodograms +Fig. D.1 shows the bidimensional periodograms of the intensity of the Hei (at 587.6 nm) emission line, the Stokes I and Stokes V +LSD profiles of the photospheric absorption lines. Bidimensional periodograms analyze the intensity of the line in several bins over +the velocity range. This allows us to see if different parts of the lines have different periods and therefore distinct origins. A dark color +on the plots indicates a peak in the periodogram, meaning that a period was found, but a large spot indicates a large uncertainty.We +would expect to find the same period in all the bins if the entire line has a single origin. This is seen for instance in the plot of the +Hei line in 2010, where the same period is found for the whole redshifted part of the line. However, the width of the peak of the +periodogram is very broad, indicating that there is a large uncertainty in this period. +−100 +−50 +0 +50 +100 +2 +4 +6 +8 +10 +−100 +−50 +0 +50 +100 +v (km/s) +2 +4 +6 +8 +10 +Period (days) +DK Tau HeI 2010 +−100 +−50 +0 +50 +100 +2 +4 +6 +8 +10 +0.0 +1.2 +2.4 +3.6 +4.8 +−100 +−50 +0 +50 +100 +2 +4 +6 +8 +10 +−100 +−50 +0 +50 +100 +v (km/s) +2 +4 +6 +8 +10 +Period (days) +DK Tau HeI 2012 +−100 +−50 +0 +50 +100 +2 +4 +6 +8 +10 +0.0 +1.1 +2.1 +3.2 +4.2 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +−40 +−20 +0 +20 +40 +v (km/s) +2 +4 +6 +8 +10 +Period (days) +DK Tau LSD Profile 2010 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +0.0 +1.3 +2.5 +3.8 +5.0 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +−40 +−20 +0 +20 +40 +v (km/s) +2 +4 +6 +8 +10 +Period (days) +DK Tau LSD Profile 2012 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +0.0 +1.2 +2.4 +3.7 +4.9 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +−40 +−20 +0 +20 +40 +v (km/s) +2 +4 +6 +8 +10 +Period (days) +DK Tau Stokes V 2010 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +0.0 +1.3 +2.6 +3.9 +5.2 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +−40 +−20 +0 +20 +40 +v (km/s) +2 +4 +6 +8 +10 +Period (days) +DK Tau Stokes V 2012 +−40 +−20 +0 +20 +40 +2 +4 +6 +8 +10 +0.0 +1.2 +2.5 +3.7 +4.9 +Fig. D.1: Bidimensional periodograms of the intensity of the Hei (at 587.6 nm) emission line (top panel), the Stokes I (middle panel) +and Stokes V LSD profiles (bottom panel) of the photospheric absorption lines, for the 2010 (left panels) and 2012 epoch (right +panels). The power of the periodogram is showed using the color code. A light color represents a zero power intensity, while a dark +color represents the maximum power intensity. +Article number, page 17 of 18 + +A&A proofs: manuscript no. main +Appendix E: H𝜶 lines +Fig. E.1 shows the H𝛼 line for the fourth and fifth night of the 2010 ESPaDOnS observations, corresponding to phase ∼0.5. For both +nights, we see a small redshifted absorption, indicating that the accretion column is in our line-of-sight. +400 +200 +0 +200 +400 +Velocity (km.s +1) +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Intensity +DK Tau 2010-12-17 +400 +200 +0 +200 +400 +Velocity (km.s +1) +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Intensity +DK Tau 2010-12-18 +Fig. E.1: H𝛼 line for the fourth night (right panel) and the fifth night (left panel) of the 2010 ESPaDOnS observations. The dotted +line highlights the continuum. +Article number, page 18 of 18 + diff --git a/q9AzT4oBgHgl3EQfO_td/content/tmp_files/load_file.txt b/q9AzT4oBgHgl3EQfO_td/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8477d97716f9a500138370e395d776ffeba3027e --- /dev/null +++ b/q9AzT4oBgHgl3EQfO_td/content/tmp_files/load_file.txt @@ -0,0 +1,1909 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf,len=1908 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main ©ESO 2023 January 4, 2023 Misalignment of the outer disk of DK Tau and a first look at its magnetic field using spectropolarimetry M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' McGinnis1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Folsom2, 3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Ray1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Vidotto4, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Alecian5, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Bouvier5, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Morin6, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Donati7, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Devaraj1 1 Dublin Institute for Advanced Studies, Astronomy & Astrophysics Section, 31 Fitzwilliam Place, Dublin 2, Ireland e-mail: nelissen@cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='dias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='ie 2 Tartu Observatory, University of Tartu, Observatooriumi 1, Tõravere, 61602 Tartumaa, Estonia 3 University of Western Ontario, Department of Physics & Astronomy, London, Ontario, N6A 3K7, Canada 4 Leiden Observatory, Leiden University, PO Box 9513, 2300RA, Leiden, The Netherlands 5 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 6 LUPM, Université de Montpellier & CNRS, Montpellier, Cedex 05, France 7 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' de Toulouse, CNRS, IRAP, 14 avenue Belin, 31400 Toulouse, France Accepted 22 December 2022 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Misalignments between a forming star’s rotation axis and its outer disk axis, although not predicted by standard theories of stellar formation, have been observed in several classical T Tauri stars (cTTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The low-mass cTTs DK Tau is suspected of being among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In addition, it is an excellent subject to investigate the interaction between stellar magnetic fields and material accreting from the circumstellar disk, as it presents clear signatures of accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The goal of this paper is to study DK Tau’s average line-of-sight magnetic field in both photospheric absorption lines and emission lines linked to accretion, using spectropolarimetric observations, as well as to examine inconsistencies regarding its rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We used data collected with the ESPaDOnS spectropolarimeter, at the Canada-France-Hawaii Telescope, and the NARVAL spectropolarimeter, at the Télescope Bernard Lyot, probing two distinct epochs (December 2010 to January 2011 and November to December 2012), each set spanning a few stellar rotation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We first determined the stellar parameters of DK Tau, such as effective temperature and 𝑣 sin𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Next, we removed the effect of veiling from the spectra, then obtained least-squares deconvolution (LSD) profiles of the photospheric absorption lines for each observation, before determining the average line-of-sight magnetic field from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also investigated accretion-powered emission lines, namely the 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 nm Hei line and the Caii infrared triplet (at 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 nm, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm), as tracers of the magnetic fields present in the accretion shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find that DK Tau experiences accretion onto a magnetic pole at an angle of ∼ 30° from the pole of its rotation axis, with a positive field at the base of the accretion funnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In 2010 we find a magnetic field of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95kG (from the Caii infrared triplet) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='77kG (from the Hei line) and in 2012 we find up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15kG (from the Caii infrared triplet) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='99kG (from the Hei line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Additionally, using our derived values of period, 𝑣 sin𝑖 and stellar radius, we find a value of 58° (+18)(-11) for the inclination of the stellar rotation axis, which is significantly different from the outer disk axis inclination of 21° given in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find that DK Tau’s outer disk axis is likely misaligned compared to its rotation axis by 37°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Stars: individual: DK Tau - Stars: variables: T Tauri - Stars: magnetic field - Accretion, accretion disks - Techniques: polarimetric - Techniques: spectroscopic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Introduction Stellar magnetic fields are omnipresent and play an essential role in the formation of stars and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Understanding their impact is therefore crucial to the study of stellar birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We do not, how- ever, yet possess a complete picture of how stellar magnetic fields originate, how they evolve over time, or the extent of their impact on circumstellar disks and the accretion process in the early stages of a star’s life (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Gregory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Folsom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Villebrun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Stellar magnetic fields of accreting T Tauri stars play an essential role in driving accretion and strongly impact the geometry of the accretion flow (see Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' By analyzing the mag- netic field along the line-of-sight and integrated over the visible stellar hemisphere measured in the acc retion-powered emission lines, one can recreate a picture of the component of the stellar magnetic field that dominates the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is an integral quantity that relates Stokes I and Stokes V, making use of spectropolarimetric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The interested reader is referred to, for example, Rees & Semel (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Donati & Landstreet (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Morin (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Stokes parameters and the magnetic field are connected through the Zeeman effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This effect describes the impact of a magnetic field on a spectrum: its atomic (and molecu- lar) lines are broadened or split, depending on the strength of the field and the sensitivity of the line in question (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Tennyson 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Hussain & Alecian 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In this work, we analyze the spectropolarimetry of the clas- sical T Tauri star (cTTs) DK Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' DK Tau is a young low-mass star surrounded by a circumstellar disk which is actively accret- ing from its inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is a wide binary (separation 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='38, Article number, page 1 of 18 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='01175v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='SR] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main equivalent to 307 au - see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2019), which allows DK Tau A (hereafter "DK Tau") to be spatially resolved with spectropolarimetry and studied on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is located in the Taurus Molecular Cloud at a distance of 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 pc (Gaia Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016, 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Its spectral type is K7 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Johns-Krull 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2011) with a heliocentric radial velocity of +16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 km s−1 (Kounkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2022) measured the inclination, with respect to the line of sight, of its outer (>20 au) gaseous disk axis via projection to be 21°, using the 12CO (J = 2–1) line with ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The star’s rotation period 𝑃, as determined using photometry, has been measured as 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 days (Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1993) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 days (Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Artemenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' DK Tau presents with significant and variable veiling (a strong signature of accretion - see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In addition to accretion, it also shows ev- idence of ejection (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1995), in particular inner disk winds and a jet as revealed by the detection of both low and high velocity forbidden [Oi] emission by Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (1995) and Banzatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The effect of veiling on spectra complicates magnetic field measurements, yet the study of active accretors is very valuable for understanding the interaction between stellar magnetic fields and accreting material from the circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Indeed, our current understanding of magnetospheric accretion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Bessolaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016) involves the truncation of the inner circumstellar disk at a few stellar radii and the channeling of accretion in funnel flows by the stellar magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When the accreted matter falls onto the star at near free-fall velocities, it produces an accretion shock close to the stellar surface, which gives rise to contin- uum and line veiling (Calvet & Gullbring 1998) and generates a localized bright/hot spot at the level of the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Simple models of stellar formation do not account for a mis- alignment between the star’s rotation axis and its outer disk axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' However, misalignments may be common, as several dippers dis- play a low inclination of their outer disk axis (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Dippers are stars that show flux dips in their light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The standard explanation in- volves circumstellar material from the inner disk passing in front of the star, occulting it periodically or aperiodically (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Dippers therefore re- quire a relatively high inclination for the inner disk axis (which is here assumed to point in the same direction as the stellar rotation axis), that is inconsistent with their measured outer disk axis in- clination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Such a misalignment has been directly measured in few cTTs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These observations suggest a more complex formation mechanism than normally considered, though it is not clear what gives rise to such a misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' As examples of potential causes, Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2020) suggest the possibility of two different protostellar collapses, whereas Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2018) invoke the presence of a massive planet inside the disk gap, and Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2018) the effects of a low-mass stellar companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' DK Tau shows signs of being one of these systems with a misaligned outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In this paper, we describe our observations in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We de- tail our analysis and results regarding stellar parameters, veiling, and magnetic characterizations in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In Section 4 we discuss the implications of the inclination we measure for DK Tau and its magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Finally, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 5 contains our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Observations Our data set is comprised of two sets of circularly polarized spectra of DK Tau, collected from December 2010 to January 2011, and from the end of November to the end of December 2012, with the spectropolarimeters ESPaDOnS (Echelle Spec- troPolarimetric Device for the Observation of Stars), mounted at the CFHT (Canada-France-Hawaii Telescope) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 meter tele- scope in Hawaii (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2006), and NARVAL, mounted at the 2 meter TBL (Télescope Bernard Lyot) on the Pic du Midi in France (Aurière 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These echelle spectropolarimeters cover the visible domain, from 370 to 1 050 nm, in a single exposure, and have a resolving power of 65 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' ESPaDOnS has a fiber aperture of 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='66, while NARVAL has one of 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Table 1 lists the dates of the middle of the observations for the two ESPaDOnS and NARVAL data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The total exposure time was 4 996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 s for each ESPaDOnS observation and 4 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 s for each NARVAL observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In 2010 a total of 15 observations were taken over 39 days, and in 2012 a total of 12 observations were taken over 35 days, with the intention of capturing a few rotation cycles of the target in each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The data are public and were downloaded from the archive of the PolarBase website1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Petit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These observations were made as a result of proposals 10BP12 and 12BP12, with J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Donati as P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='I in both cases and obtained as part of the MaPP (Magnetic Protostars and Planets) large pro- gram at the CFHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also downloaded the corresponding im- age files for the ESPaDOnS data from the Canadian Astronomy Data Centre (CADC) website2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The data had been previously re- duced at the CFHT and TBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The reduction was carried out with the LibreESpRIT (for "Echelle Spectra Reduction: an Interactive Tool") reduction package specifically built for extracting polar- ization echelle spectra from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This includes subtracting the bias and the dark frames, and correcting for the variations in sensitivity using flat field frames (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The spec- tra were continuum normalized in addition to the LibreESpRIT automatic continuum normalization, as the automatic procedure is not tailored for stars presenting with emission lines and it did not manage to properly adjust the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Analysis & results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Veiling Accretion shocks are at a higher temperature than the photo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This adds an extra continuum to the stellar continuum, artificially decreasing the depth of the photospheric absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is known as veiling and it varies with the wavelength, and can also vary from night to night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Its effect needs to be re- moved from the spectra in order to analyze the stellar magnetic field from the photospheric absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Veiling (𝑅) is defined as the ratio between the flux of the ac- cretion shock and the photospheric flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For a normalized spec- trum, it can be expressed using the following equation: 𝐼v(𝜆) = [𝐼ph(𝜆) + 𝑅(𝜆)]𝑁(𝜆) (1) where 𝐼v(𝜆) refers to the veiled intensity at wavelength 𝜆, 𝐼ph(𝜆) to the intensity of the photosphere at wavelength 𝜆, 𝑅(𝜆) to the veiling at wavelength 𝜆 and 𝑁(𝜆) is a normalization constant at wavelength 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In order to measure the veiling in our spectra, we used a technique based on the fitting of a rotationally broadened and 1 http://polarbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='eu 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='cadc-ccda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='hia-iha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='nrc-cnrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='ca/en Article number, page 2 of 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau Table 1: Dates for the 2010 and 2012 ESPaDOnS and NARVAL data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Date Heliocentric Julian Rotation cycle S/N of the continuum Airmass Instrument (yyyy-mm-dd) date (UTC) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period) at the central wavelength 2010-11-26 2 455 527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='436 03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 NARVAL 2010-12-09 2 455 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='393 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='58 77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 NARVAL 2010-12-10 2 455 541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='397 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 NARVAL 2010-12-13 2 455 544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='418 52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='07 42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 NARVAL 2010-12-14 2 455 544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='979 74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='14 72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2010-12-15 2 455 545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='853 82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ESPaDOnS 2010-12-16 2 455 546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='854 61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='37 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ESPaDOnS 2010-12-17 2 455 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='826 04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2010-12-18 2 455 548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='819 03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 113 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2010-12-19 2 455 549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='786 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='73 81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 ESPaDOnS 2010-12-19 2 455 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='390 86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80 54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 NARVAL 2010-12-24 2 455 554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='883 63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='35 83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ESPaDOnS 2010-12-26 2 455 557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='034 94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 ESPaDOnS 2010-12-30 2 455 560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='977 48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 ESPaDOnS 2011-01-03 2 455 565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='451 64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='64 68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 NARVAL 2012-11-19 2 456 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='509 43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 NARVAL 2012-11-25 2 456 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='919 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='78 91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ESPaDOnS 2012-11-28 2 456 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='895 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15 121 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2012-11-29 2 456 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='991 38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='28 127 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2012-12-01 2 456 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='947 48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='52 105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ESPaDOnS 2012-12-02 2 456 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='865 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='63 84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2012-12-04 2 456 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='965 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='89 120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ESPaDOnS 2012-12-07 2 456 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='846 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='24 94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 ESPaDOnS 2012-12-09 2 456 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='387 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='55 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 NARVAL 2012-12-10 2 456 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='825 04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='60 107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 ESPaDOnS 2012-12-12 2 456 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='557 58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='81 63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 NARVAL 2012-12-23 2 456 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='762 49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 ESPaDOnS artificially veiled weak-lined T Tauri star (wTTs) spectrum to the spectrum of DK Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' By choosing a wTTs with the same spectral type as our cTTs and coming from the same star forming region3, this wTTs can be seen as the purely photospheric version of our star because it experiences no accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We tested several wTTs with a K7 spectral type and a line-of-sight-projected equatorial rotational velocity 𝑣 sin𝑖 lower than the 𝑣 sin𝑖 of DK Tau (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2), in order to find the one that provided the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We ultimately used the spectrum of TAP45 (which has a 𝑣 sin𝑖 of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 km s−1 - see Feigelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1993) as a template to estimate the veiling across DK Tau’s spectra as the extra continuum that has to be added to the wTTs spectrum in order to reproduce the veiled cTTs spectrum (at the lowest 𝜒2 level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We obtained values of the veiling as a function of the wave- length (in bins of ∼20 nm), for each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When the value is less than ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 throughout the spectrum, we see that it is approxi- mately constant in wavelength, therefore we took the mean value as 𝑅 for the whole spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When it is larger than this, we fit a linear relation through the points and considered this function as our 𝑅(𝜆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then inverted Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1 to recover the normalized photospheric spectrum of each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Figure 1 shows the night with the least veiling and the one with the most veiling on the top and bottom panel respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In 2010, the peak values of veiling (at ∼550 nm) for each observation range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Three observations out of fifteen have nearly constant veiling values across their spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3 This implies that both T Tauri stars would have the same chemical composition and very similar age and log 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also assume that the microturbulence and macroturbulence velocities should be very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In 2012, the peak values of veiling for each observation range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 (lower than the value from two years before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Only one observation out of twelve has a nearly constant veiling value across its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the two sets of observations, we see that the slope of veiling as a function of wavelength steepens when the veiling is higher as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also find that when the veiling is high, there is more scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We believe this scatter is due to the correlation of stronger line emission with stronger mass accretion rate (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Dodin & Lamzin 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Rei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In that case, individual line emission would contribute more to the veiling by filling in absorption lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' whereas for nights when the veiling is low, continuum veiling would remain the dominant form of veiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The interpretation of these trends goes beyond the scope of this work, and will be investigated in a subsequent paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Stellar parameters Given the importance of accurate stellar parameters, we took advantage of ESPaDOnS and NARVAL’s continuum normalized high resolution spectra to derive precise values for DK Tau, in particular of the line-of-sight-projected equatorial rotational ve- locity 𝑣 sin𝑖, needed to investigate the potential misalignment of DK Tau’s outer disk, and for the effective temperature 𝑇eff, which is needed for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For this, we first obtained the mean spectrum of the four nights with the least amount of veiling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 for all four nights, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1) in order to get a spectrum with a higher signal to noise ratio of 136, whereas the individual spectra had a signal to noise ratio of 103 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then used the ZEEMAN spectrum synthesis program, devel- Article number, page 3 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main 550 600 650 700 750 800 850 Wavelength (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 Veiling DK Tau 2010-12-17 Mean: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='21 Standard deviation 550 600 650 700 750 800 850 Wavelength (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 Veiling DK Tau 2012-12-07 Fit: -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='62*10 3x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='76 Uncertainty of fit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1: Veiling (gray dots), the best linear fit (blue line) and the standard deviation (light blue shaded region) as a function of wavelength for the fourth night of the 2010 ESPaDOnS observa- tions (top panel) and for the seventh night of the 2012 ESPaDOnS observations (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' oped by Landstreet (1988) and Wade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2001) and modified by Folsom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2016) to derive the stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Under the assumption of local thermodynamic equilibrium, this code solves radiative transfer equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It iteratively compares a syn- thetic spectrum (calculated using a grid of model atmospheres and a list of atomic data) with the observation, by 𝜒2 minimiza- tion, to determine free model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Veiling was included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We used the Vienna Atomic Line Database (VALD) website4 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2015) to acquire a continuous atomic line list ranging from 400 nm to 1 000 nm for a star of 𝑇eff = 4 000 K, logarithmic surface gravity log 𝑔 = 4 (in cgs units) and a solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We used the MARCS model atmosphere grid of Gustafsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In the ZEEMAN code, we specified the following initial values for the stellar parameters: for 𝑇eff and 𝑣 sin𝑖, we used the values provided in the literature, of 𝑇eff = 4 000 K (Herczeg & Hillenbrand 2014), and 𝑣 sin𝑖 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 km s−1 (McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We used log 𝑔 = 4 (in cgs units), as this is typical of T Tauri stars (TTs), microturbulence velocity 𝑣mic = 1 km s−1, macroturbulence velocity 𝑣mac = 0 km s−1, solar metallicity and a veiling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We started with a model with initial values of 𝑇eff, 𝑣 sin𝑖, log 𝑔 and veiling for a fixed value of 𝑣mic, 𝑣mac and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then ran fits with only 𝑇eff and 𝑣 sin𝑖 as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When fitting the observation, we used a wavelength range from 400 nm to 1 000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We ran the code 4 http://vald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='se on several windows throughout the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We calculated the average of the values obtained for the different spectral windows and took the standard deviation of the spread as the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We derive a 𝑇eff of 4 150 ± 110 K and a 𝑣 sin𝑖 of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We note that our values are in good agreement with the ones found in the literature (see Herczeg & Hillenbrand 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also determined DK Tau’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We first calculated the stellar luminosity 𝐿★ using the J-band magnitudes of Eisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' As DK Tau A and B are not spatially resolved in their observations, we corrected the J-band magnitude to remove the contribution of DK Tau B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This was done by extrapolating the brightness ratio given by Eisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2007) for the K-band (of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3) to the J-band, taking into consideration the shape of the continua of the two stars based on their respective spectral types, as well as the individual extinction each star suffers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 mag for DK Tau A, and AV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80 mag for DK Tau B - Herczeg & Hillenbrand 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find a flux ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 for the J-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then corrected the magnitude for the extinction in DK Tau A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 mag) (Herczeg & Hillenbrand 2014)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It can be noted that the value for the extinction quoted by Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2011) is a factor 2 higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The variability of the extinction is probably connected to the dipper behavior of the star (Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also corrected for a veiling of 𝑅𝐽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1, based on an interpolation of our measurements of veiling as a function of wavelength (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is very similar to the values observed by Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then used the bolometric correction in the J band from Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find a value of 𝐿★ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25) 𝐿 ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Using the relation 𝑅2 = 𝐿★/(4𝜋𝜎𝑇4 eff) and our measured value of 𝑇eff, we derive a stellar radius of 𝑅★ = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25) 𝑅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Table 2 summarizes the measured properties of DK Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The inclination of the outer disk axis was measured by Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2022) using ALMA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We derive a different value for the inclination 𝑖 of the stellar rotation axis (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The stellar rotation period 𝑃 mentioned in the literature is based on photometry (see Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Arte- menko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The mass 𝑀★ was derived by Johns-Krull (2007) from pre-main sequence evolutionary tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Using the Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2000) models6, we find that the values of 𝑇eff and 𝐿★ that we obtain give a slightly higher 𝑀★ than the one of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 𝑀⊙ derived by Johns-Krull (2007), but it agrees with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 𝑀⊙ within 2𝜎 (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Least-squares deconvolution Least-squares deconvolution (LSD) is a cross-correlation tech- nique used to add the signatures from hundreds of photospheric absorption lines and obtain an average line profile of very high signal to noise ratio (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Kochukhov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It makes use of a line mask, a list describing the position and depth of the chosen absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We first created a line mask using the VALD line list (with a detection threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='01, 𝑇eff = 4 000 K, log 𝑔 = 4 and 𝑣mic = 1 km s−1) and assuming an ATLAS9 stellar atmosphere model (Kurucz 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Next, using our line mask, we applied LSD to all observations in the range from 500 nm to 1000 nm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', excluding the blue edge of the spec- tra because of excess noise), excluding regions with telluric and 5 We chose to use the value quoted by Herczeg & Hillenbrand (2014) as they give a value for both components of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 6 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='ulb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='be/~siess/pmwiki/pmwiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' n=WWWTools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='PMS Article number, page 4 of 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau Table 2: Summary of the measured properties of DK Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Stellar parameter Value Reference 𝑖 (°) 58 (+18)(-11) This work Outer disk axis (°) 21 ± 3 Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2022) 𝑃 (days) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (1993) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2010) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 Artemenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2012) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='13 This work 𝑇eff (K) 4 150 ± 110 This work 𝑣 sin𝑖 (km s−1) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 This work 𝑅★ (𝑅⊙) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 This work 𝑀★ (𝑀⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='68 Johns-Krull (2007) 𝐿★ (𝐿 ⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 This work The inclination is derived from 𝑣 sin𝑖 = 2𝜋𝑅★𝑃−1 sin𝑖 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' emission lines, as well as the lines most affected by accretion7, in order to obtain a mean line profile for each veiling-corrected spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We set the effective Landé factor 𝑔0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4, the central intensity 𝑑0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='47 and the equivalent wavelength 𝜆0 = 650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 nm for all the nights (see Kochukhov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The LSD profiles were then normalized to be at the same equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The lat- ter was done in order to correct for the residual effects of veiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For all nights, we obtained a definite Zeeman signal detection in the LSD profiles, with a false alarm probability smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='001 % (see the LSD profiles in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Stokes I LSD profiles of the ESPaDOnS and NARVAL observations made on 19 December 2010 presented a small ab- sorption feature blueward of the main absorption line (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The other nights all showed a single photospheric ab- sorption line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This small absorption feature is due to scattered moonlight, as the full Moon was close to DK Tau on that night (with an angular separation of about 8°), and as the contamina- tion is at the expected lunar radial velocity in the heliocentric rest frame8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This type of contamination has been seen before (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We fit the wing of the main absorption feature with that of a Voigt profile and manually removed the contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Average line-of-sight magnetic field We measured the magnetic field along the line-of-sight and inte- grated over the visible stellar hemisphere, 𝐵los9, by using equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 from Morin (2012): 𝐵los(𝐺) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='14 × 1011 ∫ 𝑣 𝑉(𝑣) d𝑣 𝜆0 𝑔eff 𝑐 ∫ [𝐼𝑐 − 𝐼𝑣] d𝑣 (2) 7 We identified them by comparing with the spectrum of RU Lup, a cTTs of the same spectral type as DK Tau but presenting with more accretion, and determining the lines in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 8 The online applet at https://astroutils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='edu/ exofast/barycorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='html, based on Wright & Eastman (2014), allows to calculate the correction applied to geocentric observations in order to transpose them in the heliocentric rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In our case, this correction is -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Applied to the Moon’s radial velocity of 0 km s−1 in the geocentric rest frame, this translates into a radial velocity of -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 km s−1 in the heliocentric rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 9 𝐵los is also referred to as the longitudinal field 𝐵ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We chose not to use this term to avoid confusion with its homonym 𝐵𝜙, the field along the east-west direction or azimuthal field (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Vidotto 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2: LSD profile (in the heliocentric velocity frame) of Stokes V (top) and Stokes I (bottom) parameters normalized to the continuum for the sixth night of the 2010 ESPaDOnS obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Note on this night scattered light from the Moon caused the small blue-ward absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For comparison, we also show the seventh night of the 2010 ESPaDOnS observations (dashed line), which did not suffer from moonlight contamina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' on the LSD profiles of the photospheric absorption lines (see also Rees & Semel 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Wade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In this equation, 𝑣 is the radial velocity in the rest frame of the star, 𝑉 refers to Stokes V, 𝜆0 is the wavelength of the line center in nm, 𝑔eff is the effective Landé factor of the line, 𝐼 refers to Stokes I and 𝐼𝑐 to the unpolarized continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find values ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 kG to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='03 kG in 2010 and from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG in 2012 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3, and see the list of values in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It should be noted that, since 𝐵los represents a signed average over the visible stellar hemisphere, regions of opposite polarities partly cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We applied a phase dispersion minimization (PDM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Stelling- werf 1978) technique on the 2010 𝐵los values and found a period of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='13 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Since this is the period of the modulation of the stellar magnetic field, it accurately represents the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This period is consistent with values found in the literature from photometry (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 days from Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1993, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 days from Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2010 and Artemenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Small discrepancies could be explained by differential rotation: different measurements tracking features at various latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Emission lines After analyzing the LSD profiles of photospheric absorption lines to derive the associated 𝐵los linked to non-accreting regions, we also investigated emission lines associated with accretion shocks, in particular the narrow component of the 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 nm Hei line and of the Caii infrared triplet (IRT - at 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 nm, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' They can be used to get more information on DK Tau’s magnetic fields, as they are tracers of the fields present at the footpoints of accretion funnels (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 5 of 18 0.' metadata={'source': 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+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 200 0 200 Blos (G) Blos (2010) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 Veiling Veiling at 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 nm (2010) Cycle 0 Cycle 1 Cycle 2 Cycle 3 Cycle 4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 Veiling Veiling at 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 nm (2012) Cycle 0 Cycle 1 Cycle 2 Cycle 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3: Average line-of-sight magnetic field 𝐵los as derived from the photospheric absorption lines and veiling over time, shown folded in phase with the derived 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period, for the 2010 (left panel) and 2012 (right panel) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Different colors and symbols represent different rotation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These lines are known to have multiple components (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Beristain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020), often containing a narrow component (NC), originating from the accretion shock, and a much broader component (or components), which likely forms farther out in the accretion columns or in hot winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For this reason, we decomposed the profiles using a fit of 2 or more Gaussians in order to isolate the NC (examples of these fits can be seen in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the Caii lines, we then averaged the three lines into a single LSD-like profile in order to increase the signal to noise ratio, as it has been done in other studies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2007, 2008, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Since this is a triplet, the shape of all three lines should be the same (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Azevedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In addition, in our data, we see an intensity ratio close to 1:1:1 and the NCs are not contaminated by the nearby Paschen emission lines at 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 nm and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The NC of the Hei line is believed to be generated in the post-shock region at the base of the magnetic accretion funnels that connect the surface of the star to its inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The NC of the Caii IRT is thought to probe the accretion regions and the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Stokes V and Stokes I profiles of the Hei emission line and the Caii IRT, for both epochs, can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Stokes V profiles of the emission lines show similar signatures with phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This indicates that the accretion spot is mostly likely located at a high enough latitude to be visible with the same polarity at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' They also indicate that the field is positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', pointing toward the observer) at the base of the accretion funnels that connect the star to its circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We measured 𝐵los intheemission linesusingthesamemethod as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='51 kG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 kG for Hei in 2010, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 kG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 kG for Hei in 2012, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='04 kG to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG for Caii in 2010, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 kG for Caii in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Figure 5 shows the obtained values folded in phase (and the list of values can be found in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The values of 𝐵los are higher in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find that the values measured through the Caii IRT are lower than the ones measured through the Hei line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It has been hypothesized that the lower intensity of 𝐵los found using the Caii IRT stems from the dilution of the emission from the accretion shock by chromospheric emission (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2019), which results in the Stokes V/I profile being shallower for the Caii IRT than for the Hei line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find extreme values of 𝐵los in the emission lines that are one order of magnitude larger than the extreme values of 𝐵los derived from the LSD profiles of the photospheric absorption lines (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3), showing strong fields in the accretion shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is consistent with the current understanding of accretion shocks as compact regions with some of the strongest magnetic field concentrating in dark polar regions at the surface of the star and magnetic field lines reaching to the circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For both epochs, we only see the positive pole and never the negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The plots of the 𝐵los in the emission lines in 2012 do not fold well in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We believe this may stem from the location of the accretion shocks being more dynamic and/or the accretion being more complex, with more than one accretion shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is also observed in the variability of veiling in 2012 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3), which does not fold well with the rotation phase, indicating that there is considerable intrinsic variability in the mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We derived the equivalent width (EW) of the Hei emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Figure 6 shows the obtained values folded in phase (and the list of values can be found in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When the EW is larger, we are seeing more of the accretion shock in our line- of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is also when we measure stronger magnetic fields using emission lines tracing the accretion shock (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 5), as expected following the paradigm of magnetospheric accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Magnetic Obliquity We used the third equation of Preston (1967), which assumes a pure dipole, a simplification of the magnetic field present in the accretion shocks, to calculate an estimate of the magnetic obliquity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', the angle between the stellar rotation axis and the magnetic field axis) derived from the emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We used the extreme values found for 𝐵los in the emission lines and an inclination 𝑖 of 58°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the Hei emission line, we find a magnetic obliquity10 of 26° for the 2010 epoch, and of 21° for the 2012 10 This is not the magnetic obliquity of the entirety of DK Tau’s magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is based solely on the average line-of-sight magnetic field present Article number, page 6 of 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4: Stokes V and Stokes I profiles (in gray) and average (in red) of the Hei emission line (top panels) and the Caii IRT (bottom panels), for the 2010 (left panels) and 2012 (right panels) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the Caii IRT, we find a magnetic obliquity of 16° for the 2010 epoch, and of 23° for the 2012 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These esti- mates are consistent with the Stokes V signatures of the emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' They are also consistent with the magnetic obliquity of 18° (+8)(-7) in 2011 derived by McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2020), using the radial velocity variability of the Hei emission line and assuming one accretion spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We therefore have an agreement between the in the accretion shocks and probed through emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Furthermore, the calculation uses the extreme values for 𝐵los and does not account for variability between nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' values derived from the magnetic field that drives the accretion and the value derived from a result of accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The estimates of the magnetic obliquity are consistent with only seeing the positive pole in the plots of the 𝐵los in the emis- sion lines, confirming that DK Tau experiences nearly poleward accretion, with a positive field at the base of the accretion funnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Hei emission lines show a radial velocity variability with a small amplitude, which is another indication that the accretion spot is most likely close to the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' As the star rotates, if the accretion spot were located at the equator, the velocity variation would be large as the spot gets red and blueshifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 7 of 18 Hel line (2010) 10 0 100 × Vllc 10 20 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 100 50 0 50 100 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' s-1)Hel line (2012) 10 100 × Vllc 10 20 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 100 50 0 50 100 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' s-1)Call IRT (2010) 15 10 100 × Vllc 5 0 5 10 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 100 50 0 50 100 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' s-1)Call IRT (2012) 20 15 10 100 × Vllc 5 0 5 10 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 100 50 0 50 100 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' s-1)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 500 0 500 1000 1500 2000 Blos (G) HeI line (2010) Cycle 0 Cycle 1 Cycle 2 Cycle 3 Cycle 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 500 750 1000 1250 1500 1750 2000 Blos (G) HeI line (2012) Cycle 0 Cycle 1 Cycle 2 Cycle 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 300 400 500 600 700 800 900 1000 Blos (G) CaII IRT (2010) Cycle 0 Cycle 1 Cycle 2 Cycle 3 Cycle 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 200 400 600 800 1000 1200 1400 Blos (G) CaII IRT (2012) Cycle 0 Cycle 1 Cycle 2 Cycle 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 5: Average line-of-sight magnetic field 𝐵los over time, shown folded in phase with an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period, for the Hei emission line (top panels) and the Caii IRT (bottom panels), for the 2010 (left panel) and 2012 (right panel) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Different colors and symbols represent different rotation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 EW (Å) HeI line (2010) Cycle 0 Cycle 1 Cycle 2 Cycle 3 Cycle 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 Phase (P = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 EW (Å) HeI line (2012) Cycle 0 Cycle 1 Cycle 2 Cycle 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 6: Equivalent width (in ˚𝐴) of the Hei emission line over time, shown folded in phase with an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period, for the 2010 (left panel) and 2012 (right panel) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Different colors and symbols represent different rotation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 8 of 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Truncation & co-rotation radii In order to calculate the truncation radius, we need to know the star’s mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For this, we measured the equivalent width (EW) of several emission lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', H𝛼, H𝛽, H𝛾, the Hei lines at 447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 nm, 667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 nm and 706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 nm, as well as the Caii IRT at 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 nm, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Since ESPaDOnS and NARVAL’s spectra are not flux calibrated, we created a template for DK Tau based on SO879, a weak-lined T Tauri star with a K7 spectral type (described in Stelzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The template was corrected for extinction, then scaled to have the same lumi- nosity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='65 𝐿 ⊙) and be at the same distance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 pc) as DK Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' After correcting the EW for veiling, we used this template to flux calibrate them through the following formula: 𝐹line = 𝐸𝑊line · 𝐹cont (3) where 𝐹line is the flux of the line, 𝐸𝑊line is the veiling corrected EW of the line and 𝐹cont is the flux of the continuum of the template at the wavelength of the line in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Then we ob- tained the luminosity in each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Next, we used the relations in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' from Alcalá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2017) to calculate the accretion lu- minosity from each line and averaged these values for each night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then took the average over all nights in each epoch as the accretion luminosity, and the standard deviation of the spread in the values found from different nights as the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find 𝐿acc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 𝐿 ⊙ in 2010 and 𝐿acc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='42 𝐿 ⊙ in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These values are similar to the ones found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', by Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2011) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='17 𝐿 ⊙) or by Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2018) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='16 𝐿 ⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We then converted the accretion luminosity into mass ac- cretion rate using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 8 from Gullbring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (1998) with the values of 𝑅★ and 𝑀★ from Table 2 and 𝑅in = 5 𝑅★ (as is typ- ically used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find log ( �𝑀acc[𝑀⊙ yr−1]) = -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='43 in 2010, and log ( �𝑀acc[𝑀⊙ yr−1]) = -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15 in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These values are consistent with the one of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='42 quoted by Gullbring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Finally, we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 6 from Bessolaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2008) to estimate the trun- cation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This equation assumes an axisymmetric dipole, which is a simplification of DK Tau’s magnetic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It also uses the dipolar field calculated at the equator as 𝐵★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Consid- ering the equatorial value is half of the value at the pole, we estimated the latter (see Preston 1967) using the values of 𝐵los in the emission lines 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is an approximation, as part of the 𝐵los could come from higher order multipoles, in particular from the octupole, rather than the dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find 𝑟trunc ∼ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1) 𝑅★ for the Hei emission line in 2010, 𝑟trunc ∼ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8) 𝑅★ for the Caii IRT12 in 2010, 𝑟trunc ∼ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2) 𝑅★ for the Hei emission line in 2012, and 𝑟trunc ∼ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0) 𝑅★ for the Caii IRT in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We calculated the co-rotation radius as well, using Kepler’s third law: 𝑟co-rot = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 𝑅★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find that the truncation radius values are consistent with the co-rotation radius within the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This implies that DK Tau is unlikely to be in the propeller regime, an unstable accretion regime, as the truncation radius is not farther than the co-rotation radius (Romanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the 2010 epoch, DK Tau may be in the stable accretion accretion regime, since the 𝐵los and accretion tracers seem fairly periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The trun- cation radius being slightly smaller than the co-rotation radius is consistent with this as well (Blinova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 11 We used the magnetic field derived from the accretion-powered emis- sion lines, considering it will dominate over the magnetic field derived from the photospheric absorption lines at the distance of the truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 12 We find lower values for the truncation radius estimates when using the Caii IRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This stems from the lower values found for 𝐵los in those lines (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Inconsistencies regarding the inclination Naively one might assume that the inclination angle 𝑖 of the stel- lar rotation axis with respect to the line-of-sight is 21° based on the inclination of the outer gaseous disk axis (Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' DK Tau’s lightcurve however classifies the star as a dipper (Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The traditional explanation invokes cir- cumstellar material passing in front of the star and occulting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' If the disk is seen close to edge on, matter lifted above the disk plane could cause these occultations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' However, DK Tau’s outer disk is seen nearly pole on (Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2022), which is inconsistent with this scenario, unless the stellar rotation axis is at a very different angle than that of the outer disk axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Furthermore, based on the star’s rotational properties (see Table 2) and using the following relation 𝑣 sin𝑖 = 2𝜋𝑅★ 𝑃 sin𝑖 (4) we derive a much higher inclination of 58° (+18)(-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Therefore, if DK Tau is in fact seen nearly pole on, then 𝑃, 𝑣 sin𝑖 and 𝑅★ are not consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' As the stellar radius is the most uncertain of these parameters13, it is possible that it may have been underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' However, if we consider that 𝑃, 𝑣 sin𝑖 and 𝑖 are accurately determined, then we would need 𝑅★ = (6 ±1) 𝑅⊙ for this formula to agree, which is unrealistically large for a TTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Another possibility is that the period or 𝑣 sin𝑖 may be inac- curate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Regarding 𝑣 sin𝑖, the value we derive agrees within error bars with the one measured by McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2020), despite using two different assessment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is therefore a value that can be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This leaves the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In the literature, the stellar rotation period of DK Tau has been measured using photometry with values ranging from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 days (Percy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Artemenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2012) to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 days (Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' However, since the photometry is dominated by flux dips that might be due to extinction events (Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2021), it is possible that these dips are caused by circumstellar material that is not located at the co-rotation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In that case, the measured period would not be the same as the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 we derived a period from the rotational mod- ulation of the line-of-sight magnetic field 𝐵los, which should accurately represent the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In the context of exoplanet search programs, 𝐵los is indeed often considered as the most reliable indicator of stellar rotation period (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Hébrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Additionally, the value we find of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='13 days is consistent with those found in the literature from photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We therefore find that the value for the period can be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Moreover, this rotation period can be seen in a number of datasets at our disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For example, we computed bidimen- sional periodograms of the intensity of the Hei (at 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 nm) emission line, and the Stokes I and Stokes V LSD profiles of the photospheric absorption lines (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Hei line comes from the accretion shock and should therefore vary with the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In 2010 we find a period around 8 days for the entire red-shifted part of the Hei line (from 0 to 50 km s−1), however this period is very uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Stokes V profile also shows a period near 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 days between ∼-20 and -7 km s−1 and 13 This is because it depends on evolutionary models as well as an ac- curate determination of the effective temperature and stellar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Both can be subject to fairly large uncertainties, particularly the luminos- ity for a star with a dipper light curve which likely suffers from variable extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 9 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main between ∼0 and 8 km s−1, but again the uncertainty is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The Stokes I profile does not show a clear period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In 2012 there is no clear period found from the Hei line, likely because accretion is more intrinsically variable in this epoch than 2 years prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Again no clear period is observed from the Stokes I profile, but the Stokes V profile shows a possible periodicity at around 8 days (albeit with a large uncertainty, same as in 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Furthermore, the variation of the 2010 veiling as a function of time is also consistent with an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The variation of the 2012 veiling as a function of time, however, does not seem to follow any trend with the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is consistent with the intensity of the HeI line not showing a clear correlation with period in this epoch, since both are tracing accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is another indication that there must be some intrinsic variability in the mass accretion rate in 2012, which masks any rotational modulation of the accretion spot(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We looked at the possibility of the rotation period or 𝑣 sin𝑖 be- ing inaccurate and found evidence to the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Because their values appear to be accurate, we deduce that it is the value for the inclination that is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We conclude that the inclination measured for the outer circumstellar disk axis must not represent the inclination of the rotation axis of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This suggests that there is a considerable misalignment between the rotation axis of DK Tau and its outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When we calculate DK Tau’s inclina- tion based on its rotational properties using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4, we find 𝑖 = 58° (+18)(-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This value is based on 𝑣 sin𝑖 = (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3) km s−1, 𝑃 = (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2) days and 𝑅★ = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25) 𝑅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It follows that the outer disk axis of DK Tau is likely misaligned by 37° with its rotation axis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 7: Sketch (not to scale) showing DK Tau in the center, sur- rounded first by its inner disk, then by its outer disk which is considerably misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The rotation axis at 58° is in red, the outer disk axis at 21° is in blue, and the line-of-sight axis is in gray (by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Delvaux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Misalignments between the inner and outer circumstellar disk axes of T Tauri stars are starting to be observed, when combining near infrared interferometric VLTI/GRAVITY data and millime- ter interferometric ALMA data (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Bou- vier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020), or with shadows observed with VLT/SPHERE (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020), or with VLTI/GRAVITY (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Bohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find a misalign- ment between the outer disk axis and the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' What of the inner disk axis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In young dippers like DK Tau, the material that causes the dips is believed to be located in the inner accretion disk (Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2015), which need to be observed at high inclinations in order for material to cross our line-of-sight to produce the dips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Therefore an inclination of 21° of the inner disk of DK Tau is difficult to reconcile with its dipper light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In addition, the inner disk axis is normally expected to be aligned with the stellar rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We thus find it very likely that the inner disk axis of DK Tau has the same inclination of 𝑖 = 58° as was calculated for its rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This inclination is sufficiently high to support the dipper behavior and there are other cases of dippers with similar inclinations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' DK Tau is one more example of a TTS with a misalignment between its inner and outer circumstellar disk axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is also a wide binary system, and the misalignment could stem from the binary formation mechanism: turbulent fragmen- tation might generate disk axis that are more randomly oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is however interesting to note as well that the outer disk axes of both components of the binary are misaligned by 43° (see Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2022), which is close to the value (of 37°) of the mis- alignment between the inner and outer disk axes of DK Tau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This could suggest a quasi-alignment of the inner disk axis of DK Tau A with the outer disk axis of DK Tau B, assuming that they are not only aligned compared to our line-of-sight, but that the orientation of their nodes are aligned as well, which is un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Magnetic field in the accretion-powered emission lines The 𝐵los derived from the photospheric absorption lines (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4), gives a partial view of DK Tau’s magnetic fields that exclude the accreting regions, as photospheric absorption lines and accretion-powered emission lines form in different regions of the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is the field present in these accreting regions that is understood to best probe the global stellar magnetic field that reaches to the circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The magnetic obliquity derived from the accretion-powered emission lines is therefore likely to be close to the actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find that the low magnetic obliquity that we derive (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6), the positive polarity of the 𝐵los in the emission lines, as well as their Stokes V signatures and the range of radial velocity of the Hei line are consistent with the presence of an accretion spot always visible and close to the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is where the accretion funnels connecting DK Tau to its disk would be anchored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is similar to what has been found for several other cTTs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the 2010 epoch, we find that the magnetic field in the Caii IRT (and in the Hei line - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 5) is at a maximum close to the same phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', around phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3) as the maximum in the veiling (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 3), which is when the accretion shock is in our line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Around phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5, we see a small redshifted absorption in the H𝛼 line (see Appendix E), indicating that the accretion column is in our line-of-sight, which is directly after the maximum of the magnetic field in the emission lines and the increase in veiling, therefore probably directly after the accretion shock was in our line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Because this small redshifted absorption is not perfectly simultaneous with the increase in Article number, page 10 of 18 i= 58° i=21°M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau veiling and in the emission lines, it might be an indication of differential rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Conclusions In this paper, we have studied DK Tau, a low-mass classical T Tauri star (cTTs) with significant veiling (defined as the ra- tio between the accretion shock flux and the photospheric flux), using dual-epoch spectropolarimetric observations (collected in 2010 and 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We derive an effective temperature 𝑇eff of 4 150 ± 110 K and a line-of-sight-projected equatorial rotational veloc- ity 𝑣 sin𝑖 of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 km s−1, in agreement with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find peak values of veiling in the optical (∼550 nm) ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 in 2010, and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We derive the line-of-sight magnetic field integrated over the visible hemisphere 𝐵los from the photospheric absorption lines (linked to non-accreting regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 kG to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='03 kG in 2010 and from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We recover a rotation period of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 days using the values of 𝐵los for the 2010 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We confirmed the period by analyzing the intensity of the Hei line in 2010, the intensity of the Stokes V profiles in both 2010 and 2012, and the variation of veiling as a function of time in 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' They are all consistent with an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This also agrees with the values of period given in the literature from photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find several inconsistencies related to the inclination of the stellar rotation axis with respect to the line-of-sight 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The measurement of the inclination of the outer circumstellar disk axis gives a value of 21° (Rota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' DK Tau’s lightcurve, however, classifies it as a dipper (Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2021), for which the simplest explanation involves a star seen close to edge on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Furthermore, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 4, we find that the inclination of 21°, the period, 𝑣 sin𝑖 and the radius are not consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' When using the values of period, 𝑣 sin𝑖 and stellar radius that we derive to estimate the inclination of the stellar rotation axis, we find a value of 𝑖 = 58° (+18)(-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We thus find a substantial misalignment between DK Tau’s rotation axis (at 58°) and its outer disk axis (at 21°) to be likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' To complement the partial picture of the 𝐵los derived from the photospheric absorption lines, we analyzed emission lines that are tracers of the magnetic fields present in the accretion shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We examined the narrow component of the 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='67 nm Hei emis- sion line and of the Caii infrared triplet (IRT - at at 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 nm, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We found that their Stokes V profiles show similar signatures with phase, indicating that DK Tau ex- periences poleward accretion, with a positive field at the base of the accretion funnels connecting the star to its circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We measured 𝐵los within the accretion shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 kG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 kG for Hei in 2010, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 kG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 kG for Hei in 2012, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG for Caii in 2010, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='01 kG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='02 kG for Caii in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The positive polarity of the 𝐵los in the emission lines is again consistent with the presence of an accretion spot always visible and close to the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This geometry is similar to what has been found for other cTTs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=', Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We derived an estimate of the magnetic obliquity from the emission lines using the third equation of Preston (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This equation assumes a pure dipole, which is a simplification of the magnetic field present in the accretion shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' It is the field present in these accretion shocks that is understood to best probe the global stellar magnetic field that reaches the circumstel- lar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The magnetic obliquities derived from the accretion- powered emission lines are therefore a reasonable reflection of the real dipole present above the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the Hei emission line, we find a magnetic obliquity of 26° for the 2010 epoch, and of 21° for the 2012 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For the Caii IRT, we find a magnetic obliquity of 16° for the 2010 epoch, and of 23° for the 2012 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' These estimates are consistent with the magnetic obliquity of 18° (+8)(-7) in 2011 derived by McGinnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2020), using the Hei emission line and assuming one accretion spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also estimated the truncation radius using the values of 𝐵los in the emission lines, and find 𝑟trunc ∼ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1) 𝑅★ for the Hei emission line in 2010, 𝑟trunc ∼ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8) 𝑅★ for the Caii IRT in 2010, 𝑟trunc ∼ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2) 𝑅★ for the Hei emission line in 2012, and 𝑟trunc ∼ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0) 𝑅★ for the Caii IRT in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We calculated the co-rotation radius as well, and find 𝑟co-rot = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 𝑅★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We find that the truncation radius values are consistent with the co-rotation radius within the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In conclusion, we find that DK Tau, presenting with signif- icant veiling, has similar magnetic properties to the more mod- erately accreting cTTs studied so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' In addition, we find that DK Tau’s outer disk axis is likely to be misaligned compared to its rotation axis by 38°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This poses questions with regards to stan- dard models of circumstellar disk formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' More observations of cTTs are needed to better understand the prevalence of such misalignments, while the geometry of DK Tau’s system requires additional studies to characterize it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Acknowledgements The authors thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Natta for helpful discussions and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Delvaux for the sketch of DK Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We also thank the referee for valuable comments that improved the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Based on observations obtained at the Télescope Bernard Lyot (TBL) which is operated by the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This work has made use of the VALD database, operated at Uppsala University, 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851 Wright, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' & Eastman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 2014, PASP, 126, 838 Article number, page 12 of 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau Appendix A: Stellar parameters Using the Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2000) models14, we checked the compatibility of the values of 𝑀★, 𝑇eff and 𝐿★ and obtained Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The range of 𝑇eff and 𝐿★ (accounting for their error bars) that we derive correspond to masses that are close within 2𝜎 to the value of 𝑀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 𝑀⊙ quoted by Johns-Krull (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: Hertzsprung-Russell diagram with PMS evolutionary tracks from Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The colored lines correspond to different masses (in 𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The gray rectangle highlights our values of 𝑇eff and 𝐿★ with their error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Appendix B: Photospheric absorption lines Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 shows the Stokes V and Stokes I profiles of the photospheric absorption lines for both epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 lists the values of the 𝐵los, the line-of-sight magnetic field integrated over the visible hemisphere, for the photospheric absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 14 http://www.' 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number, page 13 of 18 HR diagram 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 (Lo) L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='56 log Teff(K)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: Stokes V and Stokes I profiles (in gray) and average (in red) of the absorption lines, for the 2010 (left panels) and 2012 (right panels) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: 𝐵los for the photospheric absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Date Rotation cycle 𝐵los (yyyy-mm-dd) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period) (G) 2010-11-26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 2010-12-09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='58 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='92 2010-12-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 2010-12-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='07 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='01 2010-12-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='14 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='52 2010-12-15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='83 2010-12-16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='37 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='36 2010-12-17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='23 2010-12-18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='10 2010-12-19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='73 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 2010-12-19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='72 2010-12-24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='35 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='11 2010-12-26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='33 2010-12-30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='73 2011-01-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='64 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='22 2012-11-19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='55 2012-11-25 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 2012-12-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='63 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='84 2012-12-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='89 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='67 2012-12-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='17 2012-12-09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='55 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='57 2012-12-10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='60 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='34 2012-12-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='81 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 2012-12-23 4.' metadata={'source': 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+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='85 100 50 0 50 100 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' s-1)LSD profiles (2012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80 100 50 0 50 100 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' s-1)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau Appendix C: Emission lines The emission lines associated with accretion shocks have multiple components (usually a broad and a narrow component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The narrow component is believed to come from the accretion shock, and that is the component we wish to isolate to probe the magnetic field in the shock region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' We fit several components of each emission line that we studied and then subtracted all the components except the narrow one, to get a residual profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 shows two examples, for the Hei line (at 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 nm) and for the average of the Caii IRT (at 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 nm, 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm and 866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: Fit of the Hei line for the first night of the 2010 ESPaDOnS observations (left panel) and of the average of the Caii IRT for the second night of the 2012 ESPaDOnS observations (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The observed line is in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The different components are in red and their sum is in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 lists the values of the 𝐵los, the line-of-sight magnetic field integrated over the visible hemisphere, for the Hei line and the Caii IRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 lists the values of the equivalent width (EW) of the Hei line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 15 of 18 Hel line (2010) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 Normalized intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 200 150 100 50 0 50 100 150 200 Velocity (km/s)Call IRT (2012) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 Normalized intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 200 150 100 50 0 50 100 150 200 Velocity (km/s)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: 𝐵los for the emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Date Rotation cycle 𝐵los 𝐵los (yyyy-mm-dd) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period) for Hei (G) for Caii (G) 2010-11-26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='77 341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='07 2010-12-09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='58 912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='45 778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='92 2010-12-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15 781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 2010-12-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='07 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='64 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='74 2010-12-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='14 970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='06 610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='67 2010-12-15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 1392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='66 621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='40 2010-12-16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='37 1473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='35 949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='06 2010-12-17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 1579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='56 826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='53 2010-12-18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 1321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='08 699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='57 2010-12-19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='73 1759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='99 663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='43 2010-12-19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80 1023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='12 760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 2010-12-24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='35 1767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='87 2010-12-26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='56 671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 2010-12-30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 1198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='59 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='48 2011-01-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='64 803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='45 771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='46 2012-11-19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='62 1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='28 2012-11-25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='78 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='06 876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='93 2012-11-28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15 1130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='63 838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='51 2012-11-29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='28 939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='35 782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 2012-12-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='52 479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='01 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 2012-12-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='63 606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='14 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='38 2012-12-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='89 1670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 1052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='93 2012-12-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='24 1872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='10 1149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95 2012-12-09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='55 1178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 2012-12-10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='60 1349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='12 2012-12-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='81 1137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='52 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='62 2012-12-23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 1435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='56 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2: EW of the Hei line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Date Rotation cycle EW for (yyyy-mm-dd) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 day period) Hei ( ˚𝐴) 2010-11-26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='37 2010-12-09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 2010-12-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='54 2010-12-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='32 2010-12-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='44 2010-12-15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='04 2010-12-16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='05 2010-12-17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='46 2010-12-18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='74 2010-12-19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 2010-12-19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='86 2010-12-24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='20 2010-12-26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='50 2010-12-30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='95 2011-01-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='43 2012-11-19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 2012-11-25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='74 2012-11-28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='61 2012-11-29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='44 2012-12-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='43 2012-12-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='78 2012-12-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='57 2012-12-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='12 2012-12-09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='91 2012-12-10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='67 2012-12-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='19 2012-12-23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='91 Article number, page 16 of 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Nelissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' : Misalignment of the outer disk of DK Tau Appendix D: Bidimensional periodograms Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 shows the bidimensional periodograms of the intensity of the Hei (at 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 nm) emission line, the Stokes I and Stokes V LSD profiles of the photospheric absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Bidimensional periodograms analyze the intensity of the line in several bins over the velocity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This allows us to see if different parts of the lines have different periods and therefore distinct origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' A dark color on the plots indicates a peak in the periodogram, meaning that a period was found, but a large spot indicates a large uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='We would expect to find the same period in all the bins if the entire line has a single origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' This is seen for instance in the plot of the Hei line in 2010, where the same period is found for the whole redshifted part of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' However, the width of the peak of the periodogram is very broad, indicating that there is a large uncertainty in this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' −100 −50 0 50 100 2 4 6 8 10 −100 −50 0 50 100 v (km/s) 2 4 6 8 10 Period (days) DK Tau HeI 2010 −100 −50 0 50 100 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 −100 −50 0 50 100 2 4 6 8 10 −100 −50 0 50 100 v (km/s) 2 4 6 8 10 Period (days) DK Tau HeI 2012 −100 −50 0 50 100 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 −40 −20 0 20 40 2 4 6 8 10 −40 −20 0 20 40 v (km/s) 2 4 6 8 10 Period (days) DK Tau LSD Profile 2010 −40 −20 0 20 40 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 −40 −20 0 20 40 2 4 6 8 10 −40 −20 0 20 40 v (km/s) 2 4 6 8 10 Period (days) DK Tau LSD Profile 2012 −40 −20 0 20 40 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 −40 −20 0 20 40 2 4 6 8 10 −40 −20 0 20 40 v (km/s) 2 4 6 8 10 Period (days) DK Tau Stokes V 2010 −40 −20 0 20 40 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 −40 −20 0 20 40 2 4 6 8 10 −40 −20 0 20 40 v (km/s) 2 4 6 8 10 Period (days) DK Tau Stokes V 2012 −40 −20 0 20 40 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: Bidimensional periodograms of the intensity of the Hei (at 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='6 nm) emission line (top panel), the Stokes I (middle panel) and Stokes V LSD profiles (bottom panel) of the photospheric absorption lines, for the 2010 (left panels) and 2012 epoch (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The power of the periodogram is showed using the color code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' A light color represents a zero power intensity, while a dark color represents the maximum power intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 17 of 18 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' main Appendix E: H𝜶 lines Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1 shows the H𝛼 line for the fourth and fifth night of the 2010 ESPaDOnS observations, corresponding to phase ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' For both nights, we see a small redshifted absorption, indicating that the accretion column is in our line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' 400 200 0 200 400 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='s 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 Intensity DK Tau 2010-12-17 400 200 0 200 400 Velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='s 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='0 Intensity DK Tau 2010-12-18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content='1: H𝛼 line for the fourth night (right panel) and the fifth night (left panel) of the 2010 ESPaDOnS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' The dotted line highlights the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} +page_content=' Article number, page 18 of 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfO_td/content/2301.01175v1.pdf'} diff --git a/qNA0T4oBgHgl3EQfKf_Y/vector_store/index.faiss b/qNA0T4oBgHgl3EQfKf_Y/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a5323c6b6876479dc2da4c0c9c47af32e6c66ae0 --- /dev/null +++ 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sha256:17257f8be1dc5d18a9851f95a2f0291bb6259e15b94e7a09cc77f08657d0055c +size 5963821 diff --git a/qNFKT4oBgHgl3EQfHi2k/content/tmp_files/2301.11730v1.pdf.txt b/qNFKT4oBgHgl3EQfHi2k/content/tmp_files/2301.11730v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a90578f38d7769082dd167e40f61ebe7fd4fa192 --- /dev/null +++ b/qNFKT4oBgHgl3EQfHi2k/content/tmp_files/2301.11730v1.pdf.txt @@ -0,0 +1,1142 @@ +arXiv:2301.11730v1 [cs.IT] 27 Jan 2023 +Two-Server Private Information Retrieval with +Optimized Download Rate and Result Verification +Stanislav Kruglik∗, Son Hoang Dau†, Han Mao Kiah∗, Huaxiong Wang∗ +∗ School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore +† School of Computing Technologies, STEM College, RMIT University, Melbourne, Australia +stanislav.kruglik@ntu.edu.sg, sonhoang.dau@rmit.edu.au, hmkiah@ntu.edu.sg, hxwang@ntu.edu.sg +Abstract—Private Information Retrieval (PIR) schemes allow a +client to retrieve any file of interest, while hiding the file identity +from the database servers. In contrast to most existing PIR schemes +that assume honest-but-curious servers, we study the case of +dishonest servers. The latter provide incorrect answers and try +to persuade the client to output the wrong result. We introduce +several PIR schemes with information-theoretic privacy and result +verification for the case of two servers. Security guarantees can +be information-theoretical or computational, and the verification +keys can be public or private. In this work, our main performance +metric is the download rate. +I. INTRODUCTION +A private information retrieval (PIR) scheme allows a user +to retrieve a given file xi from a database xT = x1 · · · xm, +while keeping its identity or index i ∈ [m] private from +the database servers [1]. The problem is motivated by the +necessity to preserve the privacy of not only the sensitive +content downloaded from public databases, but also the identity +of the queried record [2], [3]. Examples include the price of +a specific stock or a specific blockchain transaction. A trivial +solution is to simply download the entire database and this +clearly incurs tremendous communication costs. Unfortunately, +in the case of a single server, Chor et al. [1] showed that this was +the best information-theoretically secure solution. Nevertheless, +in the same seminal paper, Chor et al. [1] showed that when the +content are replicated among several servers, the communication +cost can be significantly reduced. Following [1], several authors +have introduced PIR schemes that progressively reduced the +communications cost [4]–[6]. Formally, in this model, the client +queries each of the k servers (each stores x1 · · · xm) once and +retrieves xi, while keeping the index i private from up to t +honest-but-curious servers. In PIR literature, such a scheme is +called t-private k-server PIR scheme and such a property is +known as t-privacy. Motivated by the large size of stored files, +we ignore the upload cost. In other words, we assume queries +are small compared to the downloaded file. Hence, our main +performance metric is the download rate, defined as the ratio of +the retrieved file size to the amount of information downloaded +by the user [7]–[9]. The PIR capacity is defined as the maximum +achievable download rate and the capacity is shown in [10] to +be equal to +Cm(t, k) = +1 − t/k +1 − (t/k)m . +(1) +Since the number of files is also large, we are interested in +asymptotic capacity values. So, we let m → ∞, and we have +that +C(t, k) ≜ lim +m→∞ Cm(t, k) = 1 − t +k . +(2) +Most of the current schemes assume that servers are honest- +but-curious and that they provide correct responses. However, +such an assumption cannot be guaranteed within a cloud en- +vironment. This poses an interesting question: what can the +user do if servers provide wrong responses? Here, we provide +three different interpretations of this question and their formal +definitions. +• s-verifiablility [referred as s-security in [11]]. The client +can detect the presence of up to s servers that persuade the +client to output a wrong result. +• a-accountability. The client can identify each of up to a +servers that persuade the client to output a wrong result. +• b-byzantine resistance/b-byzantine robustness. The client +can retrieve the correct result in presence of up to b servers +that persuade the client to output a wrong result. +It is clear that a-accountability implies a-verifiablility, while +b-byzantine resistance implies both b-accountability and b- +verifiablility. We also note that existing verifiable PIR schemes +[12], [13] provide accountability - but rely on computational +assumptions and require a trusted setup. Other previously stud- +ied schemes are b-byzantine-resistant PIR schemes [14]–[17]. +Typically, they rely on error-correcting techniques and require +at least three servers, while protocols considered in this paper +are deployed in the two-server scenario. +Both verifiable a-accountable and b-byzantine PIR schemes +identify malicious servers. However, in certain low-latency +applications, such as private media browsing, this may be +excessive [18]. By simply requiring s-verifiability, we can then +have some savings of communication cost. s-verifiable PIR +schemes was considered in papers [11], [19]. In these papers, +authors measure the communication cost as the sum of upload +and download costs, while in our case, we ignore the upload +cost. +In this paper, we consider the notion of s-verifiability and +propose a two-server verifiable PIR scheme with an optimized +download rate by modifying a linear secret-sharing-based PIR +scheme that achieves PIR asymptotic capacity (2) from [20]. We +also propose a generalization that allows the PIR scheme to be +publicly verifiable. After which, to reduce the communication +cost, we introduce the use of homomorphic hashing in the +verification step [21]. +II. PRELIMINARIES +A. Notation +For any integer n > 0 we denote [n] = {1, . . ., n}. For any +prime number p, we denote an extended finite field with pt +elements as Fpt. The base field with p elements is denoted as +Fp. By superscript T , we denote the transpose of a vector. + +B. PIR Based Secret-Sharing Schemes +In what follows, we focus on the two-server scenario and +add a result verification to the linear secret-sharing-based PIR +scheme that achieves PIR capacity [20, Proposition 2]. Let us +formally define it and denote as Π0. Each server S1 and S2 +stores m files x1, . . . , xm ∈ Fpt that form the data vector xT = +(x1, . . . , xm). Queries from client to servers to retrieve the file +i are formed by Shamir secret sharing scheme as f(u1) and +f(u2), where f(u) = ei + ru ∈ Fm +p , ei is all zero vector of +length m with a ‘1’ in the position i, r is a random vector over +Fm +p , u1 and u2 are pre-selected points over Fp that correspond +to servers one and two. As a response, server Sj computes the +scalar product of data vector x and query f(uj) that can be +written as f(u)xT = eT +i x + rT xu = xi + rT xu. By collecting +responses from two servers, the client can retrieve xi from +� xi +rT x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +f(u1)T x +f(u2)T x +� +(3) +We do note that such a scheme has the download rate 1 +2 and its +1-privacy is ensured by the properties of underlined Shamir’s +secret sharing scheme. +C. PIR Model +Before we proceed further, let us formally define two server +verifiable PIR. Our model consist of two servers S1 and S2. +Each server stores m files x1, . . . , xm ∈ Fpt that form the data +vector xT = (x1, . . . , xm). The client wants to privately retrieve +the correct value of xi, while one server can be malicious and +provide a wrong response. +Definition 1 (PIR with results verification). A two-server PIR +scheme Π with results verification consists of three algorithms +that can be described as follows: +• (vk, σ1, σ2, auxA, auxV ) ← QueriesGen(m, i) is a ran- +domized query-generating algorithm for the client. As +input, it takes the database size m and retrieval index i +and outputs two queries σ1 and σ2 that will be given +to servers S1 and S2, value vk further employed by the +client for verification purposes and, if necessary, auxiliary +information auxA used in answer-generation and auxV +used in verification. +• πj +← +AnswerGen(j, σj, x, auxA) is a deterministic +answer-generation algorithm for server Sj. As input it takes +server number j, query σj, database x, and, if necessary, +auxiliary information auxA, and outputs the query response +πj . +• {xi, ⊥} ← Verify(i, vk, π1, π2, auxV ) is a deterministic +verification algorithm for the client. As input, it takes the +retrieval index i, verification key vk, servers answers π1, +π2, if necessary, auxiliary information auxV , and uses them +to determine if it can reconstruct the correct value of xi. If +it cannot do so, it outputs the special symbol ⊥ indicating +that at least one of the answers is incorrect, otherwise, it +reconstructs xi and outputs it. +A scheme is called publicly verifiable if vk is public. Oth- +erwise, the scheme is privately verifiable. Public verification is +generally preferred, but usually relies on computational assump- +tions, while privately verifiable schemes can be information- +theoretically secure. Considered PIR scheme should satisfy +correctness, privacy, and security properties. Let us formally +define them. +Definition +2 +(Correctness). +The +scheme +Π +is +correct +if +for +any +m, +i +∈ +[m] +and +x +∈ +Fm +pt +and +any +(vk, σ1, σ2, auxA, auxV ) ← QueriesGen(m, i) it holds that +Verify(i, vk, AnswerGen(1, σ1, x, auxA), AnswerGen(2, σ2, +x, auxV ) = xi. +Definition 3 (Privacy). The scheme Π is private if for +any m, any i, i′ +∈ +[m] and (vk, σ1, σ2, auxA, auxV ) +← +QueriesGen(m, i), +(vk′, σ′ +1, σ′ +2, auxA, auxV ) +← +QueriesGen(m, i′) and any j ∈ [2], queries σj and σ′ +j +are identically distributed. The latter means that a given server +j cannot distinguish between them. +Following [22] and recent papers on multi-server verifiable +computations [11], [19], the verifiablity property can be defined +through the notion of security experiment: an adversary A +controls the dishonest server Sj, knows the database x, retrieval +index i, and, if necessary, the auxiliary information auxA and +crafts an answer ˆπj after receiving the query σj. We consider +this setup for the privately verifiable case and we denote +the corresponding experiment as EXPP riV +A,Π (m, x, i, j). For the +publicly verifiable case, we borrow the same ideas from [11] +and generalize the security experiment of [23] in a single-server +setting to our two-server setup. The resulting security experi- +ment EXPP ubV +A,Π (m, x, i, j) is identical to EXPP riV +A,Π (m, x, i, j) +except that adversary also knows verification key vk. Let us +formally define them. +Definition 4 (Security experiment). An interactive security +experiment EXPP riV +A,Π (m, x, i, j)/EXPP ubV +A,Π (m, x, i, j) between +adversary A that controls dishonest server j ∈ [2] and its +challenger in privately/publicly verifiable case can be described +as follows: +• Challenger +generates +(vk, σ1, σ2, auxA, auxV ) +← +QueriesGen(m, i) and +sends σj +to +the adversary +A. +• The adversary A generates a crafted answer ˆπi +← +A(j, σj, x, i, auxA)/ˆπi +← +A(j, σj, x, i, vk, auxA) and +sends it to the challenger. +• The challenger computes π3−j +← +AnswerGen(3 − +j, σ3−j, x, auxA) +• The challenger runs the verification algorithm Verify +with inputs i, vk, ˆπj and π3−j, if necessary, auxV and +computes an output y. +• If +y +/∈ +{xi, ⊥} +set +the +outcome +EXPP riV +A,Π (m, x, i, j)/EXPP ubV +A,Π (m, x, i, j) +of +the +experiment to be 1, otherwise set it to be 0. +In what follows, we define the following two notions of +verifiablity – information-theoretical and computational. +Definition 5 (Negligible function). A function from N to R+ is +negligible and denoted as negl(.) if for all c > 0 there exists +a natural number λ0 such that negl(λ) < +1 +λc for all λ > λ0. +Definition 6 (Information-theoretical virifiability). The protocol +Π is (1, ǫ)-verifiable if for any adversary A, any j ∈ [2], m, +x ∈ Fm +pt and any i ∈ [m], we have Pr[EXPP riV +A,Π (m, x, i, j) = +1] ≤ ǫ. Here, the probability is taken over the randomness of +A and the experiment. +Definition 7 (Computational verifiability). The scheme Π +is 1-verifiable if for any probabilistic poly-time (PPT) ad- +versary A, any j +∈ +[2], m, x +∈ +Fm +qt +and any i +∈ + +[m], Pr[EXPP riV +A,Π (m, x, i, j)/EXPP ubV +A,Π (m, x, i, j) += +1] +≤ +negl(λ), where the probability is taken over the randomness +of A and the experiment. +The notion of computational verifiability relies on certain +cryptographic assumptions. Schemes proposed in this paper rely +on the following Discrete logarithm (DLog) assumption. +Definition 8 (DLog assumption). Let G be a cyclic group of +order p > 2λ. For generator g and α ∈ Fp\{0} we define the ad- +vantage AdvDLog +A +of adversary A in solving discrete logarithm +problem in group G as the probability that he can find α from +values of g and gα. We say that discrete logarithm assumption +holds if for any PPT adversary A, AdvDLog +A +(λ) ≤ negl(λ). +III. OUR CONSTRUCTIONS +In this section, we introduce two-server PIR schemes in +which the client can verify the result in the presence of one dis- +honest server. The basic idea is to provide one more independent +set of queries to the servers, so that the user can compute two +versions of xi or xi and vxi for some secret parameter v and +verify that both servers are truthful. If we keep the verification +key in secret and download whole responses to queries, we +obtain an information-theoretical (1, ǫ)-verifiable scheme with +private verification, or shortly (1, ǫ) privately verifiable scheme +(see section III-A). Later on, we generalize this scheme to +a publicly verifiable setup (see section III-B) and reduce the +communication cost by introducing homomorphic hashes (see +section III-C). +A. Information-Theoretical Privately Verifiable Scheme +Let us modify the linear secret-sharing-based PIR scheme +Π0 to the case of results verification by creating one more set +of independent queries. The resulting two server verifiable PIR +scheme Π1 can be described as follows. +• QueriesGen(m, i): Choose v ← Fp \ {0}, r ← Fm +p , +rv ← Fm +p . Let f(u) = ei + ru ∈ Fm +p and fv(u) = vei + +rvu ∈ Fm +p , ei is all zero vector of length m with a ‘1’ +in the position i. Compute σj = (f(uj), fv(uj)) for all +j ∈ [2]. Output vk = v, σ1, σ2, and auxV = {u1, u2}. +• AnswerGen(j, σj, x): Parse σj as (f(uj), fv(uj)), com- +pute zj = f(uj)T x, wj = fv(uj)T x, output πj = (zj, wj). +• Verify(i, vk, π1, π2, auxV ): Parse auxV as {u1, u2}. Re- +trieve xi and vxi as +� xi +rT x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +z1 +z2 +� += +� +a +b +c +d +� +· +� +z1 +z2 +� +(4) +and � +vxi +rT +v x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +w1 +w2 +� += +� +a +b +c +d +� +· +� +w1 +w2 +� +(5) +If the equation below holds, output xi; otherwise, output +⊥. +v (az1 + bz2) = aw1 + bw2 +(6) +The proofs of correctness of our scheme and proof of privacy are +almost identical to that of [20] and omitted here. The download +rate is equal to 1 +4 and below we show that Π1 is (1, ǫ)-verifiable. +Theorem 1. The scheme Π1 is (1, ǫ)-verifiable where ǫ = +1 +p−1. +Proof. Without loss of generality, assume that Adversary A +controls the server S1 and set j = 1. Let π1 = (z1, w1) and +π2 = (z2, w2) be the answers obtained by correctly executing +algorithm AnswerGen by each server. Let ˆπ1 = (ˆz1, ˆw1) be +the answer chosen by A for S1. +From the description of Π1 it is clear that +xi = A = az1 + bz2 +and +vxi = V = aw1 + bw2 +while +ˆxi = ˆA = aˆz1 + bz2 +and +ˆvˆxi = ˆV = a ˆw1 + bw2. +A wins the security experiment EXPP riV +A,Π1 (m, x, i, j) if the +ˆA ̸= A and ˆV = v ˆA. It is clear that V = vA. Hence, in this +case, +ˆV − V = v( ˆA − A). +From equations (4), (5) and the fact that server S2 is honest, it +is clear that +ˆA − A = a (ˆz1 − z1) = a∆0 ̸= 0 +(7) +ˆV − V = a ( ˆw1 − w1) = a∆1 +(8) +Hence A wins the security experiment EXPP riV +A,Π1 (m, x, i, j) iff +it finds the solution for the equation +G(v) = a∆1 − av∆0 = a(∆1 − v∆0) = 0. +(9) +From the description of the security experiment, it follows that +∆0, ∆1 are known to A and independent from v. Hence +Pr[EXPP riV +A,Π1 (m, x, i, j) = 1] ≤ Pr[G(v) = 0], +(10) +for v chosen independently and uniformly at random from +Fp \ {0}. By Schwartz-Zippel Lemma [24], [25] this value can +be estimated from above as +1 +p−1, and the theorem statement +follows. +B. Computational Publicly Verifiable Scheme +Next, we modify information-theoretical privately verifiable +scheme Π1 to a publicly verifiable setup. To do so, we choose +a cyclic group G with generator g of prime order p ≥ 2λ and +made vk = gv public. The resulting PIR scheme Π2 can be +described as follows. +• QueriesGen(m, i): Choose x ← Fp \ {0}, r ← Fm +p , +rv ← Fm +p . Let f(u) = ei + ru ∈ Fm +p and fv(u) = vei + +rvu ∈ Fm +p . Compute σj = (f(uj), fv(uj)) for all j ∈ [2]. +Choose a cyclic group G of prime order p with generator +g. Output vk = gv, σ1, σ2 and auxV = {u1, u2}. +• AnswerGen(j, σj, x): Parse σj as (f(uj), fv(uj)), com- +pute zj = f(uj)T x, wj = fv(uj)T x, output πj = (zj, wj). +• Verify(i, vk, π1, π2, auxV ): Parse auxV as {u1, u2}. Re- +trieve xi and vxi as +� xi +rT x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +z1 +z2 +� += +� +a +b +c +d +� +· +� +z1 +z2 +� +(11) +and � vxi +rT +v x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +w1 +w2 +� += +� +a +b +c +d +� +· +� +w1 +w2 +� +(12) +If the equation below holds, output xi; otherwise, output +⊥: +(gv)(az1+bz2) = gaw1+bw2. +(13) + +The proofs of correctness of our scheme and proof of privacy are +almost identical to that of [20] and omitted here. The download +rate is equal to 1 +4. +Theorem 2. The scheme Π2 is 1-verifiable under DLog as- +sumption in G. +Proof. Without +loss +of +generality, +assume +that +Adver- +sary +A +controls the +server S1 +and +set +j += +1. +Let +π1 += +(f(u1)T x, fv(u1)T x) += +(z1, w1) +and +π2 += +(f(u2)T x, fv(u2)T x) = (z2, w2) be the answers obtained by +correctly executing algorithm AnswerGen by each server. Let +ˆπ1 = (ˆz1, ˆw1) is answers chosen by A for S1. +From the description of Π1 it is clear that +xi = A = az1 + bz2 +and +vxi = V = aw1 + bw2 +while +ˆxi = ˆA = aˆz1 + bz2 +and +ˆvˆxi = ˆV = a ˆw1 + bw2. +A wins the security experiment EXPP ubV +A,Π2 (m, x, i, j) if the +ˆA ̸= A and g ˆV = gv ˆ +A. From equations (11),(12) and the fact +that server 2 is honest it is clear that +ˆA − A = a (ˆz1 − z1) = a∆0 ̸= 0 +(14) +ˆV − V = a ( ˆw1 − w1) = a∆1. +(15) +As +gV += +(gv)A, +A +wins +the +security +experiment +EXPP ubV +A,Π2 (m, x, i, j) iff +ga∆0 = (gv)a∆1. +(16) +From the description of security experiment it follows that +∆0, ∆1 are known to A and independent from v and gv. Hence +Pr[EXPP riV +A,Π1 (m, x, i, j) = 1] can be estimated from above as +probability of learning the value v = a∆0 +a∆1 = ∆0 +∆1 where ∆0 ̸= 0 +from the discrete-logarithm relationship in the group G. As a +result, the theorem statement follows. +C. Computational Privately Verifiable Scheme +The main drawback of scheme Π1 is that it doubles the +download cost in comparison to capacity-achieving scheme Π0. +The possible solution to a problem of this kind is to apply +homomorphic hashes to the second part of responses π1 and +π2. The construction of homomorphic hashes is based on DLog +assumption in the multiplicative group of order p ≥ 2λ in the +finite field and was introduced for the first time for verification +of digital content distributed by rateless erasure codes in [26] +and further applied for network coding in [21]. Resulted PIR +scheme Π3 can be described as follows. +• QueriesGen(m, i): Choose x ← Fp \ {0}, r ← Fm +p , +rv ← Fm +p . Let f(u) = ei + ru ∈ Fm +p and fv(u) = vei + +rvu ∈ Fm +p . Compute σj = (f(uj), fv(uj)) for all j ∈ [2]. +Choose a prime number r so that r − 1 is divisible by p +and random elements g1, . . . , gt of Zr of order p. Choose +a basis B of Fpt over Fp. Output vk = v, σ1, σ2, auxA = +{g1, . . . , gt, B} and auxV = {g1, . . . , gt, B, u1, u2}. +• AnswerGen(j, σj, x, auxA): Parse σj as (f(uj), fv(uj)), +auxA +as +{g1, . . . , gt, B}, +compute zj += +f(uj)T x, +wj += +fv(uj)T x. Represent wj +in basis B of Fpt +over Fp as (wj,1, . . . , wj,t)T +and compute h(wj) += +�t +l=1 gwj,l +l +mod r. Output πj = (zj, h(wj)). +• Verify(i, vk, π1, π2, auxV ): +Parse +auxV +as +{g1, . . . , gt, B, u1, u2}. Retrieve xi as +� +xi +rT x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +z1 +z2 +� += +� +a +b +c +d +� +· +� +z1 +z2 +� +(17) +Represent xi in basis B of Fpt over Fp as (xi,1, . . . , xi,t)T +and compute h(xi) = �t +l=1 gxi,l +l +mod r. +If the equation below holds, output xi; otherwise, output +⊥. +h(xi)v = h(w1)ah(w2)b mod r. +(18) +The proofs of correctness of our scheme and proof of privacy +are almost identical to that of [20] and omitted here. The +download rate is equal to +t log p +2(t log p+log r) that can be arbitrarily +close to 1 +2. +Theorem 3. The scheme Π3 is 1-verifiable under DLog as- +sumption in a multiplicative group of order p in a finite field. +Proof. Without loss of generality, assume that Adversary +A controls the server S1 and set j += +1. Let π1 += +(f(u1)T x, h(fv(u1)T x)) = (z1, h(w1)) and π2 = (z2, h(w2)) +be the answers obtained by correctly executing algorithm +AnswerGen by each server. Let ˆπ1 = (ˆz1, h( ˆw1) be the answer +chosen by A for S1. +Let HC and HCc denote the event of hash collision and its +complement, respectively. From the description of Π3 and the +law of total probability, it is clear that +Pr[EXPP riV +A,Π3 (m, x, i, j) = 1] += Pr[EXPP riV +A,Π3 (m, x, i, j) = 1|HC]Pr[HC] ++ Pr[EXPP riV +A,Π3 (m, x, i, j) = 1|HCc]Pr[HCc] +≤ Pr[HC] + Pr[EXPP riV +A,Π3 (m, x, i, j) = 1|HCc]Pr[HCc] +≤ Pr[HC] + Pr[EXPP riV +A,Π1 (m, x, i, j) = 1] ≤ negl(λ). (19) +The final inequality (19) follows from the collision resistance +of homomorphic hash functions under the DLog assumption in +a multiplicative group of order p ≥ 2λ. +Remark 9. We can replace homomorphic hashes construction +based on DLog assumption with homomorphic hashes construc- +tion based on t-poly DH assumption in the group of points of an +elliptic curve [27], [28]. Despite the latter requiring a smaller +field size to deploy, it requires the existence of a trusted setup +that can be impossible in some scenarios. +IV. COMPARISONS +Let us consider setup when we have m files over a finite field +Fpt and possible solutions to the problem of verifiable PIR with +optimized download cost. By Pr we denote the probability that +an adversary with access to one server wins in the security +experiment. We measure the file size, upload cost (UP), and +download cost (DC) in bits. Also, we differentiate schemes by +the notion of 1-verifiablity (1-verif.), 1-privacy (1-priv.), and +presence of verifiability (verif.). The comparison is presented +in Table 1. For comparison we take schemes Π0, Π1, Π2 +and Π3 decribed in this paper. Also, we apply a two-server +verifiable computation scheme of low-degree polynomials from +[11, Scheme 3] at the top of scheme Π0 to construct the +two-server verifiable scheme. We note that this is the only +scheme from [11] that can be deployed in two server scenarios. + +Π0 +[11, Scheme 3] +A +Π1 +Π2 +Π3 +File size +t · log(p) +t · log(p) +t · log(p) +t · log(p) +t · log(p) +t · log(p) +UC +2m · log(p) +4m · log(p) +4m · log(p) +4m · log(p) +4m · log(p) +4m · log(p) +DC +2t · log(p) +4t · log(p) +4t · log(p) +4t · log(p) +4t · log(p) +2t · log(p) + 2 · log(r) +Pr +1 +p−1 +p2−3 +2(p−1) +(p−2)2 +1 +p−1 +negl(λ) +negl(λ) +1-verif. +no +IT +IT +IT +DLog +DLog +1-priv. +IT +IT +IT +IT +IT +IT +verif. +no +private +private +private +public +private +Table 1: Comparison of two-server PIR schemes that optimize the download rate +It cannot be generalized to publicly verifiable cases, and we +cannot apply homomorphic hashing schemes to it as we cannot +separate server responses into two parts (one is responsible for +the retrieval, and one is for the verification). The parameters of +the resulting scheme are presented in column 3. +Two server-verifiable schemes can also be created from +scheme Π0 by forming two independent queries for each server. +The important thing here is that in comparison to previously +described schemes, points of evaluation must be kept secret. +Such an approach offers a higher probability that an adversary +with access to one server wins in the security experiment. Let +us formally describe it below and prove its (1, ǫ)-verifiability. +In that follows, we denote this scheme as A. +• QueriesGen(m, i): Choose u1, u2 ← Fp, u1 ̸= u2, +˜u1, ˜u2 ← Fp, ˜u1 ̸= ˜u2, r ← Fm +p , rv ← Fm +p . Let +f(u) = ei + ru ∈ Fm +p +and fv(u) = ei + rvu ∈ Fm +p . +Compute σj = (f(uj), fv(˜uj)) for all j ∈ [2]. Output +vk = auxV = {u1, u2, ˜u1, ˜u2}, σ1 and σ2. +• AnswerGen(j, σj, x): Parse σj as (f(uj), fv(uj)), com- +pute zj = f(uj)T x, wj = fv(uj)T x, output πj = (zj, wj). +• Verify(i, vk, π1, π2, auxV ): +Parse +vk += +auxV += +{u1, u2, ˜u1, ˜u2}. Retrieve xi as +� xi +rT x +� += +� +1 +u1 +1 +u2 +�−1 +· +� +z1 +z2 +� += +� +a +b +c +d +� +· +� +z1 +z2 +� +(20) +and � xi +rT +v x +� += +� +1 +˜u1 +1 +˜u2 +�−1 +· +� +w1 +w2 +� += +� +˜a +˜b +˜c +˜d +� +· +� +w1 +w2 +� +(21) +If the equation below holds, output xi; otherwise, output +⊥. +az1 + bz2 = ˜aw1 + ˜bw2 +(22) +The proofs of correctness of this scheme and proof of privacy +are almost identical to that of [20] and omitted here. The +download rate is equal to 1 +4. +Theorem 4. The scheme A is (1, ǫ)-verifiable where ǫ = +2(p−1) +(p−2)2 . +Proof. Without loss of generality, assume that Adversary A +controls the server S1 and set j = 1. Let π1 = (z1, w1) and +π2 = (z2, w2) be the answers obtained by correctly executing +algorithm AnswerGen by each server. Let ˆπ1 = (ˆz1, ˆw1) be +the answer chosen by A for S1. +From the description of A it is clear that +xi = A = az1 + bz2 +and +xi = V = ˜aw1 + ˜bw2 +while +ˆxi = ˆA = a · ˆz1 + bz2 +and +˜xi = ˆV = ˜a ˆw1 + ˜bw2. +A wins the security experiment EXPP riV +A,A (m, x, i, j) if the +ˆA ̸= A and ˆV = ˆA. It is clear that A = V . Hence, +ˆV − V = ˆA − A. +From equations (20), (21) and the fact that server 2 is honest it +is clear that +ˆA − A = a (ˆz1 − z2) = a∆0 ̸= 0 +(23) +ˆV − V = ˜a ( ˆw1 − w2) = ˜a∆1, +(24) +where a = +u2 +u2−u1 and ˜a = +˜u2 +˜u2−˜u1 . Hence A wins the security +experiment EXPP riV +A,Π1 (m, x, i, j) iff it finds the solution for the +equation +G(u1, u2, ˜u1, ˜u2) = +˜u2 +˜u2 − ˜u1 +∆1 − +u2 +u2 − u1 +∆0 += ˜u2(u2 − u1)∆1 − u2(˜u2 − ˜u1)∆0 +(˜u2 − ˜u1)(u2 − u1) += 0. +(25) +From the description of security experiment, it follows that +∆0, ∆1 are known to A and independent from u1, u2, ˜u1, ˜u2. +Hence, +Pr[EXPP riV +A,A (m, x, i, j) = 1] +≤ Pr(G(u1, u2, ˜u1, ˜u2) = 0|∆1 ̸= ∆0 ̸= 0) += +� +(c1,c2,c3,c4)∈L +Pr(G(c1, c2, c3, c4) = 0|∆1 ̸= ∆0 ̸= 0) +· Pr[(u1, u2, ˜u1, ˜u2) = (c1, c2, c3, c4)] = +g +(q − 1)2(q − 2)2 , +(26) +where L = {(c1, c2, c3, c4)|c1, c2, c3, c4 ∈ Fq \ {0}, c1 ̸= +c2, c3 +̸= +c4} and g is the number of (c1, c2, c3, c4) +∈ +L so that G(c1, c2, c3, c4) = 0. Since the denominator of +G is non-zero, we can estimate g as number of zeros of +H(u1, u2, ˜u1, ˜u2) = ˜u2(u2 − u1)∆1 − u2(˜u2 − ˜u1)∆0. Let +L′ = {(c1, c2, c3, c4)|c1, c2, c3, c4 ∈ Fq \ {0}}. It is clear that +L ⊆ L′ and we can estimate g from above as number of zeros +of H(τ1, τ2, τ3, τ4) where τ1, τ2, τ3, τ4 are chosen uniformly +from L′. By Schwartz-Zippel Lemma [24], [25] this value can +be estimated from above as +2 +q−1. As a result, we have that +g ≤ 2(q − 1)3, and the theorem statement follows. +V. CONCLUSION +We considered the problem of reflecting malicious server +behavior in private information retrieval schemes with optimized +download rates. We focused on the extreme two-server case + +and propose generalizations of linear secret-sharing-based PIR +schemes to verifiable setups. Considered schemes can detect +the presence of one cheating server and offers information- +theoretical private verifiability, computational public verifiabil- +ity, and computational private verifiability with download rate +close to those of non-verifiable scheme. +ACKNOWLEDGMENT +Authors thank L. F. 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Mu, “Practical anonymous +divisible e-cash from bounded accumulators,” in Finan- +cial Cryptography and Data Security, 2008, pp. 287–301. + diff --git a/qNFKT4oBgHgl3EQfHi2k/content/tmp_files/load_file.txt b/qNFKT4oBgHgl3EQfHi2k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed3a4dde422f519c71f3c253b5534f8e3c7620c9 --- /dev/null +++ b/qNFKT4oBgHgl3EQfHi2k/content/tmp_files/load_file.txt @@ -0,0 +1,477 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf,len=476 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='11730v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='IT] 27 Jan 2023 Two-Server Private Information Retrieval with Optimized Download Rate and Result Verification Stanislav Kruglik∗, Son Hoang Dau†, Han Mao Kiah∗, Huaxiong Wang∗ ∗ School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore † School of Computing Technologies, STEM College, RMIT University, Melbourne, Australia stanislav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='kruglik@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='sg, sonhoang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='dau@rmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='au, hmkiah@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='sg, hxwang@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content='sg Abstract—Private Information Retrieval (PIR) schemes allow a client to retrieve any file of interest, while hiding the file identity from the database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In contrast to most existing PIR schemes that assume honest-but-curious servers, we study the case of dishonest servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The latter provide incorrect answers and try to persuade the client to output the wrong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We introduce several PIR schemes with information-theoretic privacy and result verification for the case of two servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Security guarantees can be information-theoretical or computational, and the verification keys can be public or private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In this work, our main performance metric is the download rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' INTRODUCTION A private information retrieval (PIR) scheme allows a user to retrieve a given file xi from a database xT = x1 · · · xm, while keeping its identity or index i ∈ [m] private from the database servers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The problem is motivated by the necessity to preserve the privacy of not only the sensitive content downloaded from public databases, but also the identity of the queried record [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Examples include the price of a specific stock or a specific blockchain transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A trivial solution is to simply download the entire database and this clearly incurs tremendous communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Unfortunately, in the case of a single server, Chor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' [1] showed that this was the best information-theoretically secure solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Nevertheless, in the same seminal paper, Chor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' [1] showed that when the content are replicated among several servers, the communication cost can be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Following [1], several authors have introduced PIR schemes that progressively reduced the communications cost [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Formally, in this model, the client queries each of the k servers (each stores x1 · · · xm) once and retrieves xi, while keeping the index i private from up to t honest-but-curious servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In PIR literature, such a scheme is called t-private k-server PIR scheme and such a property is known as t-privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Motivated by the large size of stored files, we ignore the upload cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In other words, we assume queries are small compared to the downloaded file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence, our main performance metric is the download rate, defined as the ratio of the retrieved file size to the amount of information downloaded by the user [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The PIR capacity is defined as the maximum achievable download rate and the capacity is shown in [10] to be equal to Cm(t, k) = 1 − t/k 1 − (t/k)m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (1) Since the number of files is also large, we are interested in asymptotic capacity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' So, we let m → ∞, and we have that C(t, k) ≜ lim m→∞ Cm(t, k) = 1 − t k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (2) Most of the current schemes assume that servers are honest- but-curious and that they provide correct responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' However, such an assumption cannot be guaranteed within a cloud en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' This poses an interesting question: what can the user do if servers provide wrong responses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Here, we provide three different interpretations of this question and their formal definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' s-verifiablility [referred as s-security in [11]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The client can detect the presence of up to s servers that persuade the client to output a wrong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' a-accountability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The client can identify each of up to a servers that persuade the client to output a wrong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' b-byzantine resistance/b-byzantine robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The client can retrieve the correct result in presence of up to b servers that persuade the client to output a wrong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' It is clear that a-accountability implies a-verifiablility, while b-byzantine resistance implies both b-accountability and b- verifiablility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We also note that existing verifiable PIR schemes [12], [13] provide accountability - but rely on computational assumptions and require a trusted setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Other previously stud- ied schemes are b-byzantine-resistant PIR schemes [14]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Typically, they rely on error-correcting techniques and require at least three servers, while protocols considered in this paper are deployed in the two-server scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Both verifiable a-accountable and b-byzantine PIR schemes identify malicious servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' However, in certain low-latency applications, such as private media browsing, this may be excessive [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' By simply requiring s-verifiability, we can then have some savings of communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' s-verifiable PIR schemes was considered in papers [11], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In these papers, authors measure the communication cost as the sum of upload and download costs, while in our case, we ignore the upload cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In this paper, we consider the notion of s-verifiability and propose a two-server verifiable PIR scheme with an optimized download rate by modifying a linear secret-sharing-based PIR scheme that achieves PIR asymptotic capacity (2) from [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We also propose a generalization that allows the PIR scheme to be publicly verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' After which, to reduce the communication cost, we introduce the use of homomorphic hashing in the verification step [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' PRELIMINARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Notation For any integer n > 0 we denote [n] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' For any prime number p, we denote an extended finite field with pt elements as Fpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The base field with p elements is denoted as Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' By superscript T , we denote the transpose of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' PIR Based Secret-Sharing Schemes In what follows, we focus on the two-server scenario and add a result verification to the linear secret-sharing-based PIR scheme that achieves PIR capacity [20, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let us formally define it and denote as Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Each server S1 and S2 stores m files x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , xm ∈ Fpt that form the data vector xT = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Queries from client to servers to retrieve the file i are formed by Shamir secret sharing scheme as f(u1) and f(u2), where f(u) = ei + ru ∈ Fm p , ei is all zero vector of length m with a ‘1’ in the position i, r is a random vector over Fm p , u1 and u2 are pre-selected points over Fp that correspond to servers one and two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' As a response, server Sj computes the scalar product of data vector x and query f(uj) that can be written as f(u)xT = eT i x + rT xu = xi + rT xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' By collecting responses from two servers, the client can retrieve xi from � xi rT x � = � 1 u1 1 u2 �−1 � f(u1)T x f(u2)T x � (3) We do note that such a scheme has the download rate 1 2 and its 1-privacy is ensured by the properties of underlined Shamir’s secret sharing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' PIR Model Before we proceed further, let us formally define two server verifiable PIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Our model consist of two servers S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Each server stores m files x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , xm ∈ Fpt that form the data vector xT = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The client wants to privately retrieve the correct value of xi, while one server can be malicious and provide a wrong response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 1 (PIR with results verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A two-server PIR scheme Π with results verification consists of three algorithms that can be described as follows: (vk, σ1, σ2, auxA, auxV ) ← QueriesGen(m, i) is a ran- domized query-generating algorithm for the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' As input, it takes the database size m and retrieval index i and outputs two queries σ1 and σ2 that will be given to servers S1 and S2, value vk further employed by the client for verification purposes and, if necessary, auxiliary information auxA used in answer-generation and auxV used in verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' πj ← AnswerGen(j, σj, x, auxA) is a deterministic answer-generation algorithm for server Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' As input it takes server number j, query σj, database x, and, if necessary, auxiliary information auxA, and outputs the query response πj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' {xi, ⊥} ← Verify(i, vk, π1, π2, auxV ) is a deterministic verification algorithm for the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' As input, it takes the retrieval index i, verification key vk, servers answers π1, π2, if necessary, auxiliary information auxV , and uses them to determine if it can reconstruct the correct value of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' If it cannot do so, it outputs the special symbol ⊥ indicating that at least one of the answers is incorrect, otherwise, it reconstructs xi and outputs it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A scheme is called publicly verifiable if vk is public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Oth- erwise, the scheme is privately verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Public verification is generally preferred, but usually relies on computational assump- tions, while privately verifiable schemes can be information- theoretically secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Considered PIR scheme should satisfy correctness, privacy, and security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let us formally define them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 2 (Correctness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme Π is correct if for any m, i ∈ [m] and x ∈ Fm pt and any (vk, σ1, σ2, auxA, auxV ) ← QueriesGen(m, i) it holds that Verify(i, vk, AnswerGen(1, σ1, x, auxA), AnswerGen(2, σ2, x, auxV ) = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 3 (Privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme Π is private if for any m, any i, i′ ∈ [m] and (vk, σ1, σ2, auxA, auxV ) ← QueriesGen(m, i), (vk′, σ′ 1, σ′ 2, auxA, auxV ) ← QueriesGen(m, i′) and any j ∈ [2], queries σj and σ′ j are identically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The latter means that a given server j cannot distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Following [22] and recent papers on multi-server verifiable computations [11], [19], the verifiablity property can be defined through the notion of security experiment: an adversary A controls the dishonest server Sj, knows the database x, retrieval index i, and, if necessary, the auxiliary information auxA and crafts an answer ˆπj after receiving the query σj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We consider this setup for the privately verifiable case and we denote the corresponding experiment as EXPP riV A,Π (m, x, i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' For the publicly verifiable case, we borrow the same ideas from [11] and generalize the security experiment of [23] in a single-server setting to our two-server setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The resulting security experi- ment EXPP ubV A,Π (m, x, i, j) is identical to EXPP riV A,Π (m, x, i, j) except that adversary also knows verification key vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let us formally define them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 4 (Security experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' An interactive security experiment EXPP riV A,Π (m, x, i, j)/EXPP ubV A,Π (m, x, i, j) between adversary A that controls dishonest server j ∈ [2] and its challenger in privately/publicly verifiable case can be described as follows: Challenger generates (vk, σ1, σ2, auxA, auxV ) ← QueriesGen(m, i) and sends σj to the adversary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The adversary A generates a crafted answer ˆπi ← A(j, σj, x, i, auxA)/ˆπi ← A(j, σj, x, i, vk, auxA) and sends it to the challenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The challenger computes π3−j ← AnswerGen(3 − j, σ3−j, x, auxA) The challenger runs the verification algorithm Verify with inputs i, vk, ˆπj and π3−j, if necessary, auxV and computes an output y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' If y /∈ {xi, ⊥} set the outcome EXPP riV A,Π (m, x, i, j)/EXPP ubV A,Π (m, x, i, j) of the experiment to be 1, otherwise set it to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In what follows, we define the following two notions of verifiablity – information-theoretical and computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 5 (Negligible function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A function from N to R+ is negligible and denoted as negl(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=') if for all c > 0 there exists a natural number λ0 such that negl(λ) < 1 λc for all λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 6 (Information-theoretical virifiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The protocol Π is (1, ǫ)-verifiable if for any adversary A, any j ∈ [2], m, x ∈ Fm pt and any i ∈ [m], we have Pr[EXPP riV A,Π (m, x, i, j) = 1] ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Here, the probability is taken over the randomness of A and the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 7 (Computational verifiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme Π is 1-verifiable if for any probabilistic poly-time (PPT) ad- versary A, any j ∈ [2], m, x ∈ Fm qt and any i ∈ [m], Pr[EXPP riV A,Π (m, x, i, j)/EXPP ubV A,Π (m, x, i, j) = 1] ≤ negl(λ), where the probability is taken over the randomness of A and the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The notion of computational verifiability relies on certain cryptographic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Schemes proposed in this paper rely on the following Discrete logarithm (DLog) assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Definition 8 (DLog assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let G be a cyclic group of order p > 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' For generator g and α ∈ Fp\\{0} we define the ad- vantage AdvDLog A of adversary A in solving discrete logarithm problem in group G as the probability that he can find α from values of g and gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We say that discrete logarithm assumption holds if for any PPT adversary A, AdvDLog A (λ) ≤ negl(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' OUR CONSTRUCTIONS In this section, we introduce two-server PIR schemes in which the client can verify the result in the presence of one dis- honest server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The basic idea is to provide one more independent set of queries to the servers, so that the user can compute two versions of xi or xi and vxi for some secret parameter v and verify that both servers are truthful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' If we keep the verification key in secret and download whole responses to queries, we obtain an information-theoretical (1, ǫ)-verifiable scheme with private verification, or shortly (1, ǫ) privately verifiable scheme (see section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Later on, we generalize this scheme to a publicly verifiable setup (see section III-B) and reduce the communication cost by introducing homomorphic hashes (see section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Information-Theoretical Privately Verifiable Scheme Let us modify the linear secret-sharing-based PIR scheme Π0 to the case of results verification by creating one more set of independent queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The resulting two server verifiable PIR scheme Π1 can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' QueriesGen(m, i): Choose v ← Fp \\ {0}, r ← Fm p , rv ← Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let f(u) = ei + ru ∈ Fm p and fv(u) = vei + rvu ∈ Fm p , ei is all zero vector of length m with a ‘1’ in the position i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Compute σj = (f(uj), fv(uj)) for all j ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Output vk = v, σ1, σ2, and auxV = {u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' AnswerGen(j, σj, x): Parse σj as (f(uj), fv(uj)), com- pute zj = f(uj)T x, wj = fv(uj)T x, output πj = (zj, wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Verify(i, vk, π1, π2, auxV ): Parse auxV as {u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Re- trieve xi and vxi as � xi rT x � = � 1 u1 1 u2 �−1 � z1 z2 � = � a b c d � � z1 z2 � (4) and � vxi rT v x � = � 1 u1 1 u2 �−1 � w1 w2 � = � a b c d � � w1 w2 � (5) If the equation below holds, output xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' otherwise, output ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' v (az1 + bz2) = aw1 + bw2 (6) The proofs of correctness of our scheme and proof of privacy are almost identical to that of [20] and omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The download rate is equal to 1 4 and below we show that Π1 is (1, ǫ)-verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme Π1 is (1, ǫ)-verifiable where ǫ = 1 p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Without loss of generality, assume that Adversary A controls the server S1 and set j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let π1 = (z1, w1) and π2 = (z2, w2) be the answers obtained by correctly executing algorithm AnswerGen by each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let ˆπ1 = (ˆz1, ˆw1) be the answer chosen by A for S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From the description of Π1 it is clear that xi = A = az1 + bz2 and vxi = V = aw1 + bw2 while ˆxi = ˆA = aˆz1 + bz2 and ˆvˆxi = ˆV = a ˆw1 + bw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A wins the security experiment EXPP riV A,Π1 (m, x, i, j) if the ˆA ̸= A and ˆV = v ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' It is clear that V = vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence, in this case, ˆV − V = v( ˆA − A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From equations (4), (5) and the fact that server S2 is honest, it is clear that ˆA − A = a (ˆz1 − z1) = a∆0 ̸= 0 (7) ˆV − V = a ( ˆw1 − w1) = a∆1 (8) Hence A wins the security experiment EXPP riV A,Π1 (m, x, i, j) iff it finds the solution for the equation G(v) = a∆1 − av∆0 = a(∆1 − v∆0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (9) From the description of the security experiment, it follows that ∆0, ∆1 are known to A and independent from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence Pr[EXPP riV A,Π1 (m, x, i, j) = 1] ≤ Pr[G(v) = 0], (10) for v chosen independently and uniformly at random from Fp \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' By Schwartz-Zippel Lemma [24], [25] this value can be estimated from above as 1 p−1, and the theorem statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Computational Publicly Verifiable Scheme Next, we modify information-theoretical privately verifiable scheme Π1 to a publicly verifiable setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' To do so, we choose a cyclic group G with generator g of prime order p ≥ 2λ and made vk = gv public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The resulting PIR scheme Π2 can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' QueriesGen(m, i): Choose x ← Fp \\ {0}, r ← Fm p , rv ← Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let f(u) = ei + ru ∈ Fm p and fv(u) = vei + rvu ∈ Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Compute σj = (f(uj), fv(uj)) for all j ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Choose a cyclic group G of prime order p with generator g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Output vk = gv, σ1, σ2 and auxV = {u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' AnswerGen(j, σj, x): Parse σj as (f(uj), fv(uj)), com- pute zj = f(uj)T x, wj = fv(uj)T x, output πj = (zj, wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Verify(i, vk, π1, π2, auxV ): Parse auxV as {u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Re- trieve xi and vxi as � xi rT x � = � 1 u1 1 u2 �−1 � z1 z2 � = � a b c d � � z1 z2 � (11) and � vxi rT v x � = � 1 u1 1 u2 �−1 � w1 w2 � = � a b c d � � w1 w2 � (12) If the equation below holds, output xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' otherwise, output ⊥: (gv)(az1+bz2) = gaw1+bw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (13) The proofs of correctness of our scheme and proof of privacy are almost identical to that of [20] and omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The download rate is equal to 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme Π2 is 1-verifiable under DLog as- sumption in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Without loss of generality, assume that Adver- sary A controls the server S1 and set j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let π1 = (f(u1)T x, fv(u1)T x) = (z1, w1) and π2 = (f(u2)T x, fv(u2)T x) = (z2, w2) be the answers obtained by correctly executing algorithm AnswerGen by each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let ˆπ1 = (ˆz1, ˆw1) is answers chosen by A for S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From the description of Π1 it is clear that xi = A = az1 + bz2 and vxi = V = aw1 + bw2 while ˆxi = ˆA = aˆz1 + bz2 and ˆvˆxi = ˆV = a ˆw1 + bw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A wins the security experiment EXPP ubV A,Π2 (m, x, i, j) if the ˆA ̸= A and g ˆV = gv ˆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From equations (11),(12) and the fact that server 2 is honest it is clear that ˆA − A = a (ˆz1 − z1) = a∆0 ̸= 0 (14) ˆV − V = a ( ˆw1 − w1) = a∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (15) As gV = (gv)A, A wins the security experiment EXPP ubV A,Π2 (m, x, i, j) iff ga∆0 = (gv)a∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (16) From the description of security experiment it follows that ∆0, ∆1 are known to A and independent from v and gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence Pr[EXPP riV A,Π1 (m, x, i, j) = 1] can be estimated from above as probability of learning the value v = a∆0 a∆1 = ∆0 ∆1 where ∆0 ̸= 0 from the discrete-logarithm relationship in the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' As a result, the theorem statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Computational Privately Verifiable Scheme The main drawback of scheme Π1 is that it doubles the download cost in comparison to capacity-achieving scheme Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The possible solution to a problem of this kind is to apply homomorphic hashes to the second part of responses π1 and π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The construction of homomorphic hashes is based on DLog assumption in the multiplicative group of order p ≥ 2λ in the finite field and was introduced for the first time for verification of digital content distributed by rateless erasure codes in [26] and further applied for network coding in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Resulted PIR scheme Π3 can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' QueriesGen(m, i): Choose x ← Fp \\ {0}, r ← Fm p , rv ← Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let f(u) = ei + ru ∈ Fm p and fv(u) = vei + rvu ∈ Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Compute σj = (f(uj), fv(uj)) for all j ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Choose a prime number r so that r − 1 is divisible by p and random elements g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , gt of Zr of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Choose a basis B of Fpt over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Output vk = v, σ1, σ2, auxA = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , gt, B} and auxV = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , gt, B, u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' AnswerGen(j, σj, x, auxA): Parse σj as (f(uj), fv(uj)), auxA as {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , gt, B}, compute zj = f(uj)T x, wj = fv(uj)T x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Represent wj in basis B of Fpt over Fp as (wj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , wj,t)T and compute h(wj) = �t l=1 gwj,l l mod r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Output πj = (zj, h(wj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Verify(i, vk, π1, π2, auxV ): Parse auxV as {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , gt, B, u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Retrieve xi as � xi rT x � = � 1 u1 1 u2 �−1 � z1 z2 � = � a b c d � � z1 z2 � (17) Represent xi in basis B of Fpt over Fp as (xi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' , xi,t)T and compute h(xi) = �t l=1 gxi,l l mod r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' If the equation below holds, output xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' otherwise, output ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' h(xi)v = h(w1)ah(w2)b mod r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (18) The proofs of correctness of our scheme and proof of privacy are almost identical to that of [20] and omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The download rate is equal to t log p 2(t log p+log r) that can be arbitrarily close to 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme Π3 is 1-verifiable under DLog as- sumption in a multiplicative group of order p in a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Without loss of generality, assume that Adversary A controls the server S1 and set j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let π1 = (f(u1)T x, h(fv(u1)T x)) = (z1, h(w1)) and π2 = (z2, h(w2)) be the answers obtained by correctly executing algorithm AnswerGen by each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let ˆπ1 = (ˆz1, h( ˆw1) be the answer chosen by A for S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let HC and HCc denote the event of hash collision and its complement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From the description of Π3 and the law of total probability, it is clear that Pr[EXPP riV A,Π3 (m, x, i, j) = 1] = Pr[EXPP riV A,Π3 (m, x, i, j) = 1|HC]Pr[HC] + Pr[EXPP riV A,Π3 (m, x, i, j) = 1|HCc]Pr[HCc] ≤ Pr[HC] + Pr[EXPP riV A,Π3 (m, x, i, j) = 1|HCc]Pr[HCc] ≤ Pr[HC] + Pr[EXPP riV A,Π1 (m, x, i, j) = 1] ≤ negl(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (19) The final inequality (19) follows from the collision resistance of homomorphic hash functions under the DLog assumption in a multiplicative group of order p ≥ 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We can replace homomorphic hashes construction based on DLog assumption with homomorphic hashes construc- tion based on t-poly DH assumption in the group of points of an elliptic curve [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Despite the latter requiring a smaller field size to deploy, it requires the existence of a trusted setup that can be impossible in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' COMPARISONS Let us consider setup when we have m files over a finite field Fpt and possible solutions to the problem of verifiable PIR with optimized download cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' By Pr we denote the probability that an adversary with access to one server wins in the security experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We measure the file size, upload cost (UP), and download cost (DC) in bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Also, we differentiate schemes by the notion of 1-verifiablity (1-verif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' ), 1-privacy (1-priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' ), and presence of verifiability (verif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The comparison is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' For comparison we take schemes Π0, Π1, Π2 and Π3 decribed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Also, we apply a two-server verifiable computation scheme of low-degree polynomials from [11, Scheme 3] at the top of scheme Π0 to construct the two-server verifiable scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We note that this is the only scheme from [11] that can be deployed in two server scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Π0 [11, Scheme 3] A Π1 Π2 Π3 File size t · log(p) t · log(p) t · log(p) t · log(p) t · log(p) t · log(p) UC 2m · log(p) 4m · log(p) 4m · log(p) 4m · log(p) 4m · log(p) 4m · log(p) DC 2t · log(p) 4t · log(p) 4t · log(p) 4t · log(p) 4t · log(p) 2t · log(p) + 2 · log(r) Pr 1 p−1 p2−3 2(p−1) (p−2)2 1 p−1 negl(λ) negl(λ) 1-verif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' no IT IT IT DLog DLog 1-priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' IT IT IT IT IT IT verif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' no private private private public private Table 1: Comparison of two-server PIR schemes that optimize the download rate It cannot be generalized to publicly verifiable cases, and we cannot apply homomorphic hashing schemes to it as we cannot separate server responses into two parts (one is responsible for the retrieval, and one is for the verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The parameters of the resulting scheme are presented in column 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Two server-verifiable schemes can also be created from scheme Π0 by forming two independent queries for each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The important thing here is that in comparison to previously described schemes, points of evaluation must be kept secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Such an approach offers a higher probability that an adversary with access to one server wins in the security experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let us formally describe it below and prove its (1, ǫ)-verifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' In that follows, we denote this scheme as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' QueriesGen(m, i): Choose u1, u2 ← Fp, u1 ̸= u2, ˜u1, ˜u2 ← Fp, ˜u1 ̸= ˜u2, r ← Fm p , rv ← Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let f(u) = ei + ru ∈ Fm p and fv(u) = ei + rvu ∈ Fm p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Compute σj = (f(uj), fv(˜uj)) for all j ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Output vk = auxV = {u1, u2, ˜u1, ˜u2}, σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' AnswerGen(j, σj, x): Parse σj as (f(uj), fv(uj)), com- pute zj = f(uj)T x, wj = fv(uj)T x, output πj = (zj, wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Verify(i, vk, π1, π2, auxV ): Parse vk = auxV = {u1, u2, ˜u1, ˜u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Retrieve xi as � xi rT x � = � 1 u1 1 u2 �−1 � z1 z2 � = � a b c d � � z1 z2 � (20) and � xi rT v x � = � 1 ˜u1 1 ˜u2 �−1 � w1 w2 � = � ˜a ˜b ˜c ˜d � � w1 w2 � (21) If the equation below holds, output xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' otherwise, output ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' az1 + bz2 = ˜aw1 + ˜bw2 (22) The proofs of correctness of this scheme and proof of privacy are almost identical to that of [20] and omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The download rate is equal to 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' The scheme A is (1, ǫ)-verifiable where ǫ = 2(p−1) (p−2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Without loss of generality, assume that Adversary A controls the server S1 and set j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let π1 = (z1, w1) and π2 = (z2, w2) be the answers obtained by correctly executing algorithm AnswerGen by each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let ˆπ1 = (ˆz1, ˆw1) be the answer chosen by A for S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From the description of A it is clear that xi = A = az1 + bz2 and xi = V = ˜aw1 + ˜bw2 while ˆxi = ˆA = a · ˆz1 + bz2 and ˜xi = ˆV = ˜a ˆw1 + ˜bw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' A wins the security experiment EXPP riV A,A (m, x, i, j) if the ˆA ̸= A and ˆV = ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' It is clear that A = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence, ˆV − V = ˆA − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' From equations (20), (21) and the fact that server 2 is honest it is clear that ˆA − A = a (ˆz1 − z2) = a∆0 ̸= 0 (23) ˆV − V = ˜a ( ˆw1 − w2) = ˜a∆1, (24) where a = u2 u2−u1 and ˜a = ˜u2 ˜u2−˜u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence A wins the security experiment EXPP riV A,Π1 (m, x, i, j) iff it finds the solution for the equation G(u1, u2, ˜u1, ˜u2) = ˜u2 ˜u2 − ˜u1 ∆1 − u2 u2 − u1 ∆0 = ˜u2(u2 − u1)∆1 − u2(˜u2 − ˜u1)∆0 (˜u2 − ˜u1)(u2 − u1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' (25) From the description of security experiment, it follows that ∆0, ∆1 are known to A and independent from u1, u2, ˜u1, ˜u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Hence, Pr[EXPP riV A,A (m, x, i, j) = 1] ≤ Pr(G(u1, u2, ˜u1, ˜u2) = 0|∆1 ̸= ∆0 ̸= 0) = � (c1,c2,c3,c4)∈L Pr(G(c1, c2, c3, c4) = 0|∆1 ̸= ∆0 ̸= 0) Pr[(u1, u2, ˜u1, ˜u2) = (c1, c2, c3, c4)] = g (q − 1)2(q − 2)2 , (26) where L = {(c1, c2, c3, c4)|c1, c2, c3, c4 ∈ Fq \\ {0}, c1 ̸= c2, c3 ̸= c4} and g is the number of (c1, c2, c3, c4) ∈ L so that G(c1, c2, c3, c4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Since the denominator of G is non-zero, we can estimate g as number of zeros of H(u1, u2, ˜u1, ˜u2) = ˜u2(u2 − u1)∆1 − u2(˜u2 − ˜u1)∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Let L′ = {(c1, c2, c3, c4)|c1, c2, c3, c4 ∈ Fq \\ {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' It is clear that L ⊆ L′ and we can estimate g from above as number of zeros of H(τ1, τ2, τ3, τ4) where τ1, τ2, τ3, τ4 are chosen uniformly from L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' By Schwartz-Zippel Lemma [24], [25] this value can be estimated from above as 2 q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' As a result, we have that g ≤ 2(q − 1)3, and the theorem statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' CONCLUSION We considered the problem of reflecting malicious server behavior in private information retrieval schemes with optimized download rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' We focused on the extreme two-server case and propose generalizations of linear secret-sharing-based PIR schemes to verifiable setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Considered schemes can detect the presence of one cheating server and offers information- theoretical private verifiability, computational public verifiabil- ity, and computational private verifiability with download rate close to those of non-verifiable scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' ACKNOWLEDGMENT Authors thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Zhang from ShanghaiTech University for numerous fruitful discussions during the preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Chor, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfHi2k/content/2301.11730v1.pdf'} +page_content=' Goldreich, E.' metadata={'source': 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b/qNFLT4oBgHgl3EQfii8S/content/tmp_files/2301.12107v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c709b433e662d57a1096ad269c9bdd409e03e495 --- /dev/null +++ b/qNFLT4oBgHgl3EQfii8S/content/tmp_files/2301.12107v1.pdf.txt @@ -0,0 +1,1828 @@ +Constraints on the Parameterized Deceleration Parameter in FRW Universe +Amine Bouali,1, ∗ Himanshu Chaudhary,2, † Ujjal Debnath,3, ‡ Tanusree Roy,3, § and G.Mustafa4, ¶ +1Laboratory of Physics of Matter and Radiation, +Mohammed I University, BP 717, Oujda, Morocco, +2Department of Applied Mathematics, Delhi Technological University, Delhi-110042, India, +3Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711 103, India, +4Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China, +Confirmation of accelerated expansion of the universe probed the concept of dark energy theory, +and since then, numerous models have been introduced to explain its origin and nature. The present +work is based on reconstructing dark energy by parametrization of the deceleration parameter +in the FRW universe filled with radiation, dark matter and dark energy. We have chosen some +well-motivated parametrized models 1-3 in an attempt to investigate the energy density in terms +of deceleration parameters by estimating the cosmological parameters with the help of different +observational datasets. Also, we have introduced a new model 4 for the parametrization of the +deceleration parameter. Then we analyzed the cosmography parameters using the best-fit values of +the parameters. Using the information criteria, we have examined the viability of the models. +I. +INTRODUCTION +The concept of cosmic acceleration was probably one +of the most promising discoveries in the modern cos- +mology paradigm. +Recently, two independent research +works involving distant supernovae suggested that in the +present epoch, the universe is undergoing an accelerated +expansion [1, 2]. This phenomenon has been favorably +explained later by the existence of an energy compo- +nent with massive negative pressure and comprising +nearly 70% of the universe. +This is known as “dark +energy” (DE). The nature of this is still unidentified. +Synchronizing with the observed data, many DE models +have been proposed so far. +Among them, the ΛCDM +model is widely accepted as supposedly it ‘best accom- +modates’ the observations but also it comes with some +disadvantages like a fine-tuning problem, coincidence +problems and so on [3–5]. To overcome these drawbacks, +alternative DE models have been explored like a quite +favorable phantom, k-essence, Chaplygin gas, etc., [6] +for possible explanations of the origin and nature of the +dark energy. +However, prior to the accelerated phase, +the universe had gone through a decelerated phase in +the early epoch where the effects of dark energy were +absent or subdominant. +It is believed that density +perturbations occurred in this epoch which played a key +role in the structural formation of the universe. So, to +cover the entire evolution, we would have to employ a +cosmological model which would simultaneously describe +the accelerated and decelerated phases. +Earlier, the +definition of cosmology was addressed by Sandage[7] as +a search for two simple but fundamental cosmographic +parameters: Hubble parameter (H0), which determined +∗ a1.bouali@ump.ac.ma +† himanshuch1729@gmail.com +‡ ujjaldebnath@gmail.com +§ tanusreeroy1995@gmail.com +¶ gmustafa3828@gmail.com +the expansion rate and a small correction q0 due to grav- +ity, known as deceleration parameter (DP), responsible +for slowing down the expansion. Though the inclusion +of ‘dark energy’ has completely changed the scenario, +but still, any practical aspect of cosmological evolution +is tightly bound to DP. It is defined as q = − a¨a +˙a2 where +a(t) is the scale factor of the universe. q > 0 (¨a < 0) +indicates a decelerating universe and vice versa. While +HP describes the linear part of the time dependence of +the scale factor, the non-linear correction term q0 opens +up possibilities like the presence of local instabilities +or the existence of chaotic regimes [8]. +Moreover, the +dynamics of observable galaxy number variation can +be determined through DP. Like tenergyr cosmological +models, DE has been subjected to numerous modifica- +tions to be better fitted with observational data. +Parametrization of DP as a function of scale factor a +or redshift z can be accounted as a suitable approach +to it. +Limitations to such parametric assumptions +are: +(i) Most of the parametrizations diverge in the +distant future, and some of them are only valid at low +redshift limit (z << 1) [9, 10]; (ii) For prior parametric +assumptions true nature of the dark energy can be +misleading; (iii) In non-parametric models, evolution +can be directly deduced from observational data avoiding +parametric assumptions [11–15]. +However, it can help +to improve the efficiency of future cosmological surveys. +So in pursuit of understanding the transition from +decelerated to accelerated phase, the parametrization +approach can be proved fruitful. +Recently Capozziello +reconstructed a divergence-free form of DP starting +with Pade polynomials and analyzed the corresponding +observational data [16]. A logarithmic parametrization +of DP was proposed in ref. +[17], and the constraints +were obtained by using type Ia supernova, BAO and +CMB data sets. +Motivated by these ideas, we have +adopted some well-motivated parametrizations of DP +to reconstruct dark energy and, consequently, Hubble +parameter H(z) in terms of redshift z. +Mainly, well- +arXiv:2301.12107v1 [gr-qc] 28 Jan 2023 + +2 +established parametrized models have been introduced +for dark energy equation of state and constrain the +model parameters by observational data analysis. +In +the study of the generalized holographic dark energy +model, some well-known parametrization type models +have been considered [18, 19]. Till now, some authors +have assumed some possible forms of parametrization +of deceleration parameter [20–25]. The main advantage +of the parameterization deceleration parameter is that +the final outcome may not depend on any particular +gravitational theory. +In the ref. +[26, 27], the authors +have introduced the analogous parametrized deceleration +parameter instead of parametrized dark energy equation +of state for the well-established models. Here we have +considered +the +analogous +parametrized +deceleration +parameter for the well-established models and constrains +the model parameters by Markov Chain Monte Carlo +(MCMC) data analysis. We compare all the models with +the ΛCDM model (which is the base model) to examine +which model is more viable than others. +The main focus of the work is to constrain the model +parameters using recently released data. Here, in par- +ticular, we have chosen to use the updated astronomical +datasets: the measurements of Hubble parameter from +the differential evolution of cosmic chronometers (CC); +Pantheon datasets from Type Ia Supernovae sample +comprising 1048 measurements; 17 uncorrelated baryons +acoustic oscillation (BAO) data. +New constraints on +DP have been provided by jointly analyzing the above +datasets and implementing Monte Carlo Markov Chain +(MCMC) method. +The era of modern cosmology pro- +motes the study of kinematic quantities, vastly known as +”Cosmography” or ”Cosmo-kinetics”. The very idea of +it is observationally driven and completely independent +of any prior assumption of the gravity theory or elected +cosmological model. Cosmography presents itself with a +compelling advantage as it simply follows the symmetry +principles and direct observation- without involving Ein- +stein equations (Friedmann equations). +Consequently, +we can steadfastly avoid some arguable speculations +regarding ’dark energy’, ’dark matter’ and others. While +pure cosmography does not envision the scale factor a(t) +itself but the history of its evolution can be inferred to +some extent. +Dunajski and Gibbons [28] have studied +the constraints on the cosmographic parameters like the +deceleration, jerk, and snap parameters for different dark +energy models. The parametrization of these quantities +is discussed in [29–32]. Shafieloo, Kim and Linder [22] +have discussed the non-parametric reconstruction of +these quantities. +The organization of the paper is as follows: In section +II, we consider the basic equations of the FRW model. +The Hubble parameter is written in terms of the decel- +eration parameter. Section III deals with the data de- +scriptions like cosmic chronometric datasets, Pantheonic +datasets and BAO datasets. In section IV, we consider +parametrized deceleration parameter models like models +1, 2, 3 and 4. In section V, we fit the models with H(z) +and SNe-IA datasets. In section VI, we discuss the cos- +mography parameters. +In section VII, we analyze the +detailed description of the model parameters. In section +VIII, we present the information criteria for our models. +Finally, the results are presented in section IX. +II. +BASIC EQUATIONS OF FRW MODEL +We have considered a spatially flat, homogeneous, +isotropic FRW universe with line element +ds2 = −dt2 + a2(t) +� +dr2 + r2 � +dθ2 + sin2θdφ2�� +(1) +a(t) being the scale factor. +The energy-momentum tensor of the fluid reads as +Tµν = (ρ + p)uµuν + pgµν +(2) +where ρ and p are the energy density and pressure +density of the fluid respectively. +The fluid 4-velocity +uµ = dxµ +ds satisfies the relation uµuµ = −1. +For the FRW Univere, the Friedmann equations in Ein- +stein’s gravity are given by +H2 = 8πG +3 +ρ +(3) +and +˙H = −4πG(ρ + p) +(4) +where, H = ˙a/a is the Hubble parameter and overhead +dot denotes derivative with cosmic time t. Considering +that the universe is filled with fluid matter of total en- +ergy density ρ and total pressure p, it obeys the energy +conservation equation +˙ρ + 3H(ρ + p) = 0 +(5) +We start with the prediction that the universe is com- +posed of matter content comprising radiation, dark mat- +ter (DM) and dark energy (DE). So, ρ and p consist +of densities and pressures of radiation, DM and DE. So +ρ = ρr +ρm +ρd and p = pr +pm +pd. Now assume that +radiation, DM and DE follows the conservation equation +separately so that +˙ρr + 3H(ρr + pr) = 0, +(6) +˙ρm + 3H(ρm + pm) = 0 +(7) +and + +3 +˙ρd + 3H(ρd + pd) = 0 +(8) +For radiation, pr = 1 +3ρr, so from equation (6) we obtain +ρr = ρr0a−4. Since the DM follows negligible pressure +(i.e., pm = 0), so from equation (7) we obtain ρm = +ρm0a−3. +Let us consider the deceleration parameter +q = −1 − +˙H +H2 +(9) +The corresponding deceleration parameter for DE is +given by [? ] +qd = −1 − +˙Hd +H2 +d +(10) +where Hd is the Hubble expansion rate of the dark energy +term. So from equations (3) and (4), we can write +H2 +d = 8πG +3 +ρd +(11) +and +˙Hd = −4πG(ρd + pd) +(12) +Using the field equations (11) and (12) and the energy- +conservation equation (8), the fluid energy density be- +comes +ρd = ρd0 e +� +2(1+qd) +1+z +dz +(13) +where ρd0 represents the present value of the density +parameter and z is the redshift parameter described as +1 + z = 1 +a (presently, a0 = 1). Defining Ωr0 = 8πGρr0 +3H2 +0 +, +Ωm0 = 8πGρm0 +3H2 +0 +and Ωd0 = 8πGρd0 +3H2 +0 +, from equation (3), we +obtain the Hubble parameter as +H2(z) = H2 +0 +� +Ωr0(1 + z)4 + Ωm0(1 + z)3 + Ωd0 e +� +2(1+qd) +1+z +dz� +(14) +where Ωd0 = 1 − Ωr0 − Ωm0. +III. +DATA DESCRIPTION +In this section, we will constrain our model param- +eters by using three types of dataset. The CC datasets +consist 31 measurements, Pantheon dataset consists 1048 +measurements and 17 uncorrelated BAO measurements +to obtain the best fit value of our model parameters. +We have implemented the Markov Chain Monte Carlo +(MCMC) [33] and implemented with the open source +package Polychord [34] and GetDist [35]. The total χ2 +function of the combination CC + BAO + Pantheon and +define as +χ2 = χ2 +CC + χ2 +SN + χ2 +BAO. +A. +Cosmic Chronometric (CC) datasets +We consider the compilation of 31 measurements of +CC lying between the redshift range 0.07 ≤ z ≤ 1.965. +The underlying principle for these measurements was +proposed in [36], by relating the Hubble parameter with +redshift z, and cosmic time t +H(z) = − +1 +1 + z +dz +dt +The χ2 function for these measurements, denoted by +χ2 +CC, is +χ2 +CC = +31 +� +i=1 +� +Hth (zi) − Hobs (zi) +�2 +σ2 +Hobs(zi) +, +(15) +where Hth (zi, k, α, h) represent the theoretical value +obtained from our cosmological model, Hobs (zi) and +represent the observed value of hubble parameter with +standard deviation σ2 +Hobs(zi). (to see more and rundown +all measurements see [37]) +B. +Pantheon datasets +The Pantheon datasets comprises 1048 measurements +of type Ia supernovae from five different sub-samples +SNLS, SDSS, PSI, low- z, and HST in the redshift range +of 0.01 < z < 2.3 [38] . The χ2 function of the Pantheon +data is given as +χ2 +Pan = ∆µC−1 +Pan∆µT , +(16) +where ∆µ = µobs +i +− µth . Where +� +µobs +i +� +represented +as observed distance modulus and evaluated as +µobs +i += µB,i + M, +(17) +µB,i represents the observed peak magnitude at max- +imum in the rest frame of the B band for redshift zi, +while M represents nuisance parameter. The theoretical +distance modulus evaluated as +µth = 5 log10 DL + M, +(18) +where +DL = (1 + zhel) +� zcmb +0 +H0dz +H(z) , +(19) +with zhel is heliocentric and zcmb is CMB rest frame +redshifts. The covariance matrix is measured as CPan = + +4 +Csys + Dstat . Where Csys is systematic covariance ma- +trix and Dstat stands for diagonal of covariance matrix +of the statistical uncertainty and calculated as +Dstat ,ii = σ2 +µB,i. +(20) +The description and the systematic covariance matrix +together with µB,i, σ2 +µB,i, zcmb, and zhel for the i th SnIa +are mentioned in [39]. +C. +Uncorrelated Baryon Acoustic Oscillations +(unCor BAO) +We picked 17 uncorrelated BAO measurements from +the greatest collection of BAO dataset of (333) measure- +ments since adopting the entire catalogue of BAO might +lead in a very significant error owing to data correlations, +therefore we opted a small dataset to minimise inaccura- +cies. Transverse BAO studies contribute measurements +of DH(z)/rd = c/H(z)rd with comoving angular diame- +ter distance.[40] [41]. +DM = +c +H0 +Sk +�� z +0 +dz′ +E (z′) +� +, +(21) +where +Sk(x) = +� +� +� +� +� +1 +√Ωk sinh +�√Ωkx +� +if +Ωk > 0 +x +if +Ωk = 0 +1 +√−Ωk sin +�√−Ωkx +� +if +Ωk < 0. +(22) +We also consider the angular diameter distance DA = +DM/(1 + z) and the DV (z)/rd. which is combination of +the BAO peak coordinates and rd is the sound horizon +at the drag epoch. Finally we can obtain ”line-of-sight” +(or ”radial”) observations directly the Hubble parameter +DV (z) ≡ +� +zDH(z)D2 +M(z) +�1/3 . +(23) +IV. +PARAMETERIZED DECELERATION +PARAMETER +Most simplest parametrization of q which contains two +parameters can be taken as +q(z) = q0 + q1X(z) +(24) +where q0 and q1 are constants and X(z) is a function +of redshift z. In search of satisfactory solutions to the +cosmological puzzles, many forms of X(z) has been sug- +gested. As mentioned earlier, most of them were inad- +equate in explaining future evolution scenarios. So the +persuasion of an ideal divergence-free parametrization of +DP is still relevant. The well-known parametrized equa- +tion of state parameter models has been introduced by +several authors, and the corresponding analogous of these +models for parametrized deceleration parameters have +been introduced in [26, 27]. Here, we have adopted the +analogous of some well-known parametrized models for +the deceleration parameter, which contains two unknown +parameters and calculated the corresponding Hubble pa- +rameter in terms of redshift z. +A. +Model 1 (Wetterich type) +The Wetterich model for parametrized equation of +state parameter has been studied in [42, 43]. The analo- +gous Wetterich type parametrization of deceleration pa- +rameter has been introduced in [26, 27] and is given by +qd(z) = +q0 +1 + q11og(1 + z) +(25) +where q0 and q1 are constants. Then the energy density +will be given by +ρd = ρd0 (1 + z)2 {1 + q1 log(1 + z)} +2q0 +q1 +(26) +From equation (14), we obtain +H2(z) = H2 +0 +� +Ωr0(1 + z)4 + Ωm0(1 + z)3 ++ (1 − Ωr0 − Ωm0) (1 + z)2 {1 + q1 log(1 + z)} +2q0 +q1 +� +(27) +B. +Model 2 (Barboza-Alcaniz type) +The Barboza-Alcaniz model for parametrized equation +of state parameter has been studied in [44]. The anal- +ogous Barboza-Alcaniz type parametrization of deceler- +ation parameter has been introduced in [26, 27] and is +given by +qd(z) = q0 + q1 +z(1 + z) +1 + z2 +(28) +where q0 and q1 are constants. Then the energy density +will be +ρd = ρd0 (1 + z)2(1+q0) � +1 + z2�q1 +(29) +From equation (14), we obtain +H2(z) = H2 +0 +� +Ωr0(1 + z)4 + Ωm0(1 + z)3 ++ (1 − Ωr0 − Ωm0) (1 + z)2(1+q0) � +1 + z2�q1� +(30) + +5 +C. +Model 3 (CPL type) +The famous Chevallier-Polarski-Linder (CPL) model +for parametrized equation of state parameter has been +studied in [29, 30]. The analogous CPL type parametriza- +tion of deceleration parameter has been introduced in +[26, 27] and is given by +qd(z) = q0 + q1 +z +1 + z +(31) +where q0 and q1 are constants. Subsequently, the en- +ergy density (13) becomes +ρd = ρd0 (1 + z)2(1+q0+q1) e +2q1 +1+z +(32) +From equation (14), we obtain +H2(z) = H2 +0 +� +Ωr0(1 + z)4 + Ωm0(1 + z)3 ++ (1 − Ωr0 − Ωm0) (1 + z)2(1+q0+q1) e +2q1 +1+z +� +(33) +D. +Model 4 +Here we propose a new parametrized model for decel- +eration parameter and is given as in the form: +qd(z) = q0 + q1 +1 + z +2 + z +(34) +where q0 and q1 are constants. Subsequently, the en- +ergy density (13) becomes +ρd = ρd0 (1 + z)2(1+q0)(2 + z)2q1 +(35) +From equation (14), we obtain +H2(z) = H2 +0 +� +Ωr0(1 + z)4 + Ωm0(1 + z)3 ++ (1 − Ωr0 − Ωm0) (1 + z)2(1+q0)(2 + z)2q1� (36) +68 +69 +70 +H0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q1 +1.0 +0.9 +0.8 +q0 +0.00 +0.01 +0.02 +0.03 +r0 +0.15 +0.20 +0.25 +m0 +0.15 +0.20 +0.25 +m0 +0.00 +0.01 +0.02 +0.03 +r0 +1.0 +0.9 +0.8 +q0 +0.0 +0.5 +1.0 +q1 +CC + SN + BAO +FIG. 1. The above figure shows the MCMC confidence con- +tours at 1σ and 2σ obtained from CC+SNIa+BAO dataset for +Model 1 (Wetterich type). +68 +69 +70 +H0 +0.0 +0.1 +0.2 +0.3 +q1 +1.0 +0.9 +0.8 +0.7 +q0 +0.00 +0.01 +0.02 +0.03 +r0 +0.15 +0.20 +0.25 +m0 +0.15 +0.20 +0.25 +m0 +0.00 +0.01 +0.02 +0.03 +r0 +1.0 +0.9 +0.8 +q0 +0.0 +0.1 +0.2 +0.3 +q1 +CC + SN + BAO +FIG. 2. The above figure shows the MCMC confidence con- +tours at 1σ and 2σ obtained from CC+SNIa+BAO dataset for +Model 2 (Barboza-Alcaniz type). + +6 +59 +60 +61 +62 +H0 +0.18 +0.19 +0.20 +q1 +1.1 +1.0 +0.9 +q0 +0.000 +0.005 +0.010 +0.015 +r0 +0.30 +0.32 +0.34 +0.36 +0.38 +m0 +0.30 +0.33 +0.36 +m0 +0.000 0.005 0.010 0.015 +r0 +1.1 +1.0 +0.9 +q0 +0.18 +0.19 +0.20 +q1 +CC + SN + BAO +FIG. 3. The above figure shows the MCMC confidence con- +tours at 1σ and 2σ obtained from CC+SNIa+BAO dataset for +Model 3 (CPL type). +70 +75 +80 +H0 +0.2 +0.1 +0.0 +q1 +1.0 +0.9 +0.8 +q0 +0.000 +0.005 +0.010 +0.015 +r0 +0.20 +0.22 +0.24 +0.26 +0.28 +m0 +0.20 +0.24 +0.28 +m0 +0.004 +0.010 +0.016 +r0 +1.0 +0.9 +0.8 +q0 +0.2 +0.1 +0.0 +q1 +CC + SN + BAO +FIG. 4. The above figure shows the MCMC confidence con- +tours at 1σ and 2σ obtained from CC+SNIa+BAO dataset for +Model 4. +MCMC Results +Model +Parameters +Best fit Value +ΛCDM Model +H0 +69.854848+1.259100 +−1.259100 +Ωm0 +0.268654+0.012822 +−0.012822 +ΩΛ +0.724585+0.009373 +−0.009373 +Model 1 +H0 +68.990853+0.502428 +−0.502428 +Ωm0 +0.221900+0.022982 +−0.022982 +Ωr0 +0.007835+0.005651 +−0.005651 +q0 +−0.941550+0.043352 +−0.043352 +q1 +0.740557+0.220903 +−0.220903 +Model 2 +H0 +69.061167+0.498457 +−0.498457 +Ωm0 +0.231719+0.025169 +−0.025169 +Ωr0 +0.006670+0.005154 +−0.005154 +q0 +−0.924723+0.052163 +−0.052163 +q1 +0.251461+0.084597 +−0.084597 +Model 3 +H0 +60.450021+0.511464 +−0.511464 +Ωm0 +0.338454+0.015533 +−0.015533 +Ωr0 +0.004485+0.003527 +−0.003527 +q0 +−0.968857+0.046034 +−0.046034 +q1 +0.190554+0.006576 +−0.006576 +Model 4 +H0 +71.392060+2.608372 +−2.608372 +Ωm0 +0.240129+0.021352 +−0.021352 +Ωr0 +0.003402+0.002740 +−0.002740 +q0 +−0.897355+0.056271 +−0.056271 +q1 +-0.091362+0.073319 +−0.073319 +TABLE I. Summary of the MCMC results using CC + Pan ++ BAO dataset. +V. +OBSERVATIONAL, AND THEORETICAL +COMPARISONS OF THE HUBBLE AND +DISTANCE MODULUS FUNCTIONS +Once we have the free parameters of the Model (1- +4), we can compare the model predictions to the ob- +servational data and the LambdaCDM model with error +bands, respectively. +A. +Comparison with the Hubble data points. +First, We consider the comparison of the Models (1 - +4) with the 31 (CC) data points and the ΛCDM model +with 1 σ and 2 σ error bands . The comparison findings +are shown in Figure 5, 6, 7 and 8. The Figure shows that +all model fit with (CC) dataset quite well. + +7 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +0 +50 +100 +150 +200 +250 +300 +H(z) +mean +CDM +Model 1 +1 +2 +H(z) Dataset +FIG. 5. The figure show that the theoretical curve of Hubble +function H(z) of Model 1 and ΛCDM models against the CC +datasets. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +0 +50 +100 +150 +200 +250 +300 +H(z) +mean +CDM +Model 2 +1 +2 +H(z) Dataset +FIG. 6. The figure show that the theoretical curve of Hubble +function H(z) of Model 2 and ΛCDM models against the CC +datasets. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +0 +50 +100 +150 +200 +250 +300 +H(z) +mean +CDM +Model 3 +1 +2 +H(z) Dataset +FIG. 7. The figure show that the theoretical curve of Hubble +function H(z) of Model 2 and ΛCDM models against the CC +datasets. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +0 +50 +100 +150 +200 +250 +300 +H(z) +mean +CDM +Model 4 +1 +2 +H(z) Dataset +FIG. 8. The figure show that the theoretical curve of Hubble +function H(z) of Model 4 and ΛCDM models against the CC +datasets. +1. +Comparison with the Pantheon data. +We now compare the µ(z) distance modulus function of +Models (1-4) with the Pantheon data. From Fig. 9,10,11 +and 12 one can see that all Models fit with the Pantheon +dataset, very well. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +14 +16 +18 +20 +22 +24 +26 +28 +30 +CDM +Model 1 +mean +1 +2 +Pantheon +FIG. 9. The figure show that the theoretical curve of distance +modulus µ(z) of Model 1 and ΛCDM models against the Pan- +theon data. + +8 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +14 +16 +18 +20 +22 +24 +26 +28 +30 +CDM +Model 2 +mean +1 +2 +Pantheon +FIG. 10. +The figure show that the theoretical curve of dis- +tance modulus µ(z) of Model 2 and ΛCDM models against the +Pantheon data. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +14 +16 +18 +20 +22 +24 +26 +28 +30 +CDM +Model 3 +mean +1 +2 +Pantheon +FIG. 11. +The figure show that the theoretical curve of dis- +tance modulus µ(z) of Model 3 and ΛCDM models against the +Pantheon data. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +z +14 +16 +18 +20 +22 +24 +26 +28 +30 +CDM +Model 4 +mean +1 +2 +Pantheon +FIG. 12. +The figure show that the theoretical curve of dis- +tance modulus µ(z) of Model 4 and ΛCDM models against the +Pantheon data. +VI. +COSMOGRAPHY PARAMETERS +To study the early evolution and late evolution of the +universe, some other parameters named cosmographical +parameters can be analyzed. The cosmographical param- +eter like jerk (j), snap (s) parameters are [45–48] +j = +...a +aH3 = (1 + z)dq +dz + q(1 + 2q), +(37) +s = a(4) +aH4 = −(1 + z) dj +dz − j(2 + 3q) +(38) +(39) +So +the +cosmographical +parameters +contains +the +higher-order derivatives of the deceleration parameter +q. +The ’jerk’ parameter is considered to have a very +useful feature that is for standard ΛCDM model, j +always takes the value unity which helps us assess the +deviation regarding different dark energy models. Sahni +et al. +and Alam et al. +analyzed the importance of +the jerk parameter j for discriminating various dark +energy models. We have explored the evolution of such +kinematical quantities with respect to the redshift for +the involved parametric models. +0 +2 +4 +6 +8 +10 +-1.0 +-0.5 +0 +0.5 +Model 1 +FIG. 13. Evolution of deceleration parameter of Model 1 with +respect to redshift. + +9 +0 +2 +4 +6 +8 +10 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Model 1 +FIG. 14. Evolution of jerk parameter of Model 2 with respect +to redshift. +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +1 +2 +3 +Model 1 +FIG. 15. Evolution of snap parameter with of Model 3 with +respect to redshift. +0 +2 +4 +6 +8 +10 +-1.0 +-0.5 +0 +0.5 +Model 2 +FIG. 16. Evolution of deceleration parameter of Model 1 with +respect to redshift. +0 +2 +4 +6 +8 +10 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Model 2 +FIG. 17. Evolution of jerk parameter of Model 2 with respect +to redshift. +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +1 +2 +3 +Model 2 +FIG. 18. Evolution of snap parameter with of Model 2 with +respect to redshift. +0 +2 +4 +6 +8 +10 +-1.0 +-0.5 +0 +0.5 +Model 3 +FIG. 19. Evolution of decceleration parameter with of Model +3 with respect to redshift. + +10 +0 +2 +4 +6 +8 +10 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Model 3 +FIG. 20. Evolution of jerk parameter of Model 2 with respect +to redshift. +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +1 +2 +3 +Model 3 +FIG. 21. Evolution of snap parameter with of Model 3 with +respect to redshift. +-1 +0 +1 +2 +3 +-1.0 +-0.5 +0 +0.5 +New Model +FIG. 22. Evolution of decceleration parameter with of Model +4 with respect to redshift. +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +New Model +FIG. 23. Evolution of jerk parameter with of Model 4 with +respect to redshift. +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-1 +0 +1 +2 +3 +New Model +FIG. 24. Evolution of snap parameter with of Model 4 respect +to redshift. + +11 +VII. +DETAILED DESCRIPTION +A. +Cosmological Parameters +1. +Hubble parameter (H0) +This Hubble parameter is the normalized rate of ex- +pansion, although this parameter is not constant. The +current estimates of this parameter are “65-75”. Model +3 has the “H0 < 65 ”, but on the other hand, Model 1 +and Model 2 have this parameter in the Hubble range +of ”70 < H0 < 75”, and finally, in Model 4, “H0 > 70 +”(Table: I). +2. +The matter density parameter (Ωm0) +This parameter is the ratio of the actual (or observed) +density ρm to the critical density ρc. The Model 1, Model +2 and Model 4 have a matter density in the range of +“0.22 < Ωm0 < 0.26”, but in Model 3, this parameter +has the range “Ωm0 ¡ 30” (Table: I). +3. +The radiation density parameter (Ωr0) +This parameter is the ratio of the actual (or observed) +radiation density ρrad to the critical density ρc. +This +parameter for Model 1, Model 2, Model 3 and Model 4 +lies in the range “0.003 < Ωr0 < 0.008” (Table: I). +B. +Cosmographic Analysis +The cosmographic analysis provides a universal and +effective way to compare the solutions of the theoretical +models with cosmological observations. From the obser- +vational data, we obtain a set of cosmological parameters, +which must be compared with the predicted values of the +same parameters obtained from a given model. The re- +sult of the comparison allows us to conclude the accept- +ability of the considered model. +Thus, for a complete +comparison of all models with the observations and the +ΛCDM model, we will consider an extended set of param- +eters constructed from the higher-order time derivatives +of the scale factor. More exactly, we will concentrate on +the comparative behaviour of the deceleration, jerk, and +snap parameters of all models and ΛCDM models. +1. +Deceleration parameter +While analyzing Model 1’s trajectory, the behavior of +the model’s deceleration parameter is nearly comparable +to the ΛCDM model in the redshift range of q ∈ [-0,2], +but Model 1 endures a super acceleration in the future +( Fig: 13). Model 2 appears to have the same behavior +as ΛCDM in the q ∈ [-0.8,4], but Model 2 is slower since +it achieves the value −0.83565 as z → 0. (Fig: 16). In +Model 3, this parameter appears to behave similarly to +the ΛCDM model, in q ∈ [-0.5,6], before experiencing a +super acceleration in the near future as ”q = -1.25464” +(Fig: 16). Model 4 behaves differently from the ΛCDM. +but it aquire the same value as ΛCDM in near future 22). +2. +The jerk parameter +The jerk parameter of Model 1 basically different from +ΛCDM at high as well as low redshift. However, impor- +tant Model 1 predicts the higher value of j = 1.357595 +at z = 0 (Fig: 15), Meanwhile, on the other hand, Model +2 also shows different behaviour at both high and low +redshift, and seems to coincide with ΛCDM at redshift +value of z = 2 and z = 0.342566, and this model also pre- +dicts the higher value of j = 1.134545 (Fig: 18). Model 3 +shows different behavior than the ΛCDM as at high red- +shift z > 2, it having a higher value than the ΛCDM of +j = 1.2657, but it cuts the trajectory of ΛCDM, twice at +z = 1.5 and z = 0.3, finally, at lower redshift, it attains +the j value of 1.05417, which is a higher than ΛCDM +(Fig: 21). Although the jerk of Model 4 is inconsistent +with ΛCDM, since it shows different evolution at both +high and low redshifts and predicts the lower value of +j = 0.608251 (Fig: 24). +3. +The snap parameter +This parameter of Model 1 and Model 2 is significantly +systematic the difference with ΛCDM within the whole +redshift range, but this parameter of Model 1 monoton- +ically decreases in the redshift range of z > 0.3 and ac- +quires a sudden increase in ”s” value and predicts the +higher value of j = 1.608251 then ΛCDM, but Model 2 +predicts the same value of j as ΛCDM,(Fig: 15, 18). +Model 3 trajectory shows a proper systematic differ- +ence with ΛCDM and predicts a higher value of ”s” as +0.734546 21 Finally, the snap trajectory of Model 4 is +notably non-identical with the ΛCDM, in the given red- +shift range and accommodate the “s” value of −0.222743 +as z → 0. which is lower than ΛCDM (Fig: 24). +VIII. +INFORMATION CRITERIA +To discuss the viable model analysis, we need to +know the study of information criteria (IC). The Akaike +Information Criteria (AIC) [49] is merely used among +all ICs. +The AIC is an asymptotically unbiased esti- +mator of Kullback-Leibler information as the AIC is +an approximate minimization of the Kullback-Leibler +information. +The Gaussian estimator for the AIC can +be written as [50–53] AIC = −2 ln(Lmax) + 2κ + 2κ(κ+1) +N−κ−1 +where Lmax is the maximum likelihood function, κ is + +12 +the number of parameters of the models, and N is the +number of data points used in the data fit of the models. +Since for the models, N ≫ 1, so for this assumption, +the above expression converts to the original AIC like +AIC = −2 ln(Lmax) + 2κ. +If the set of models are +given, the deviations of the IC values are reduced to +△AIC = AICmodel − AICmin = △χ2 +min + 2△κ In the +study of data analysis, the more favorable range of +△AIC is (0, 2). +The low favorable range of △AIC is +(4, 7), while △AIC > 10 provides less support model. +Model +χ2 +min +χ2 +red +AIC +∆AIC +ΛCDM Model 1102.67 0.981 1106.67 +0 +Model 1 +1103.21 0.961 1109.69 +0.54 +Model 2 +1103.05 0.963 1107.05 +0.38 +Model 3 +1103.85 0.965 1107.85 +1.18 +Model 4 +1103.76 0.972 1109.76 +3.09 +TABLE II. Summary of the χ2 +min, χ2 +red, AIC and ∆AIC. +IX. +DISCUSSIONS AND CONCLUSIONS +We have assumed the FRW model of the universe in +the presence of radiation, dark matter and dark energy. +Instead of considering the well-known parametrized dark +energy equation of state, we have considered the analo- +gous form of parametrized deceleration parameter for the +dark energy component and found the Hubble parameter +in terms of redshift with other model parameters. Here +we have assumed Model 1 (Wetterich type), Model +2 (Barboza-Alcaniz type) and Model 3 (CPL type), +which contains two unknown parameters. Also, we have +introduced a new Model 4 for parametrized deceleration +parameters, which also contains two unknown parame- +ters. The model parameters have been constrained for +H(z) datasets, Pantheon datasets, and BAO datasets +by MCMC method. 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Cuesta, et al., The clustering of galax- +ies in the sdss-iii baryon oscillation spectroscopic survey: +baryon acoustic oscillations in the data release 9 spec- +troscopic galaxy sample, Monthly Notices of the Royal +Astronomical Society 427 (4) (2012) 3435–3467. +[60] J. E. Bautista, M. Vargas-Maga˜na, K. S. Dawson, W. J. +Percival, J. Brinkmann, J. Brownstein, B. Camacho, +J. Comparat, H. Gil-Mar´ın, E.-M. Mueller, et al., The +sdss-iv extended baryon oscillation spectroscopic survey: +baryon acoustic oscillations at redshift of 0.72 with the +dr14 luminous red galaxy sample, The Astrophysical +Journal 863 (1) (2018) 110. +[61] T. +Abbott, +F. +Abdalla, +A. +Alarcon, +S. +Allam, +F. Andrade-Oliveira, J. Annis, S. Avila, M. Banerji, +N. Banik, K. Bechtol, et al., Dark energy survey year 1 +results: Measurement of the baryon acoustic oscillation +scale in the distribution of galaxies to redshift 1, Monthly +Notices of the Royal Astronomical Society 483 (4) (2019) +4866–4883. +[62] R. Neveux, E. Burtin, A. de Mattia, A. Smith, A. J. Ross, +J. Hou, J. Bautista, J. Brinkmann, C.-H. Chuang, K. S. +Dawson, et al., The completed sdss-iv extended baryon +oscillation spectroscopic survey: Bao and rsd measure- +ments from the anisotropic power spectrum of the quasar +sample between redshift 0.8 and 2.2, Monthly Notices of +the Royal Astronomical Society 499 (1) (2020) 210–229. +[63] M. Ata, +F. Baumgarten, +J. Bautista, +F. Beutler, +D. Bizyaev, M. R. Blanton, J. A. Blazek, A. S. Bolton, +J. Brinkmann, J. R. Brownstein, et al., The cluster- +ing of the sdss-iv extended baryon oscillation spectro- +scopic survey dr14 quasar sample: +first measurement +of baryon acoustic oscillations between redshift 0.8 and +2.2, Monthly Notices of the Royal Astronomical Society +473 (4) (2018) 4773–4794. +[64] V. de Sainte Agathe, C. Balland, H. D. M. Des Bour- +boux, M. Blomqvist, J. Guy, J. Rich, A. Font-Ribera, +M. M. Pieri, J. E. Bautista, K. Dawson, et al., Baryon +acoustic oscillations at z= 2.34 from the correlations of +lyα absorption in eboss dr14, Astronomy & Astrophysics +629 (2019) A85. +[65] C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch, +S. Croom, D. Croton, T. M. Davis, M. J. Drinkwater, +K. Forster, et al., The wigglez dark energy survey: Joint +measurements of the expansion and growth history at z¡ +1, Monthly Notices of the Royal Astronomical Society +425 (1) (2012) 405–414. +X. +APPENDIX + +15 +BAO name +redshift z Experiment Measurement Standarddeviation Ref. +6dFGS +0.106 +rs/DV +0.336 +0.015 +[54] +SDSS DR7 +0.15 +DV (rs,fidd/rs) +664 +25.0 +[55] +SDSS-DR7 + 2dFGRS +0.275 +rs/DV +0.1390 +0.0037 +[56] +SDSS-DR11 LOWZ +0.32 +DV (rd,fidd/rs) +1264 +25 +[57] +SDSS-III DR8 +0.54 +DA/rs +9.212 +0.41 +[58] +SDSSIII/ DR9 +0.57 +DV /rs +13.67 +0.22 +[59] +SDSS-IV DR14 +0.72 +DV (rs,fidd/rs) +2353 +63 +[60] +DES Year 1 +0.81 +DA/rs +10.75 +0.43 +[61] +DECals DR8 +0.874 +DA(rs,fidd/rs) +1680 +109 +[60] +0.72 +DV (rs,fidd/rs) +2353 +63 +eBoss DR16 BAO+RSD +1.48 +DH.rs +13.23 +0.47 +[62] +SDSS-IV/DR14 +1.52 +DV (rs,fidd/rs) +3843 +147.0 +[? ] +Boss Lya quasars DR9 +2.3 +H.rs +34188 +1188 +[63] +BOSS DR14 Lya in LyBeta +2.34 +DH.rs +8.86 +0.29 +[64] +WiggleZ +0.44 +0.0870 +0.0042 +0.6 +rs/DV +0.0672 +0.0031 +[65] +0.73 +0.0593 +0.0020 +TABLE III. Summary of the Baryon Acoustic Oscillations measurements used in this paper. + diff --git a/qNFLT4oBgHgl3EQfii8S/content/tmp_files/load_file.txt b/qNFLT4oBgHgl3EQfii8S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c21dbd2ed1a4dd98088d31ee90cfcc2572531705 --- /dev/null +++ b/qNFLT4oBgHgl3EQfii8S/content/tmp_files/load_file.txt @@ -0,0 +1,1160 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf,len=1159 +page_content='Constraints on the Parameterized Deceleration Parameter in FRW Universe Amine Bouali,1, ∗ Himanshu Chaudhary,2, † Ujjal Debnath,3, ‡ Tanusree Roy,3, § and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='Mustafa4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' ¶ 1Laboratory of Physics of Matter and Radiation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Mohammed I University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' BP 717,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Oujda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Morocco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 2Department of Applied Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Delhi Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Delhi-110042,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 3Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Indian Institute of Engineering Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Shibpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Howrah-711 103,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' India,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Zhejiang Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Jinhua 321004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' People’s Republic of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Confirmation of accelerated expansion of the universe probed the concept of dark energy theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' and since then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' numerous models have been introduced to explain its origin and nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The present work is based on reconstructing dark energy by parametrization of the deceleration parameter in the FRW universe filled with radiation, dark matter and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' We have chosen some well-motivated parametrized models 1-3 in an attempt to investigate the energy density in terms of deceleration parameters by estimating the cosmological parameters with the help of different observational datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Also, we have introduced a new model 4 for the parametrization of the deceleration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Then we analyzed the cosmography parameters using the best-fit values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Using the information criteria, we have examined the viability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' INTRODUCTION The concept of cosmic acceleration was probably one of the most promising discoveries in the modern cos- mology paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Recently, two independent research works involving distant supernovae suggested that in the present epoch, the universe is undergoing an accelerated expansion [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' This phenomenon has been favorably explained later by the existence of an energy compo- nent with massive negative pressure and comprising nearly 70% of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' This is known as “dark energy” (DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The nature of this is still unidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Synchronizing with the observed data, many DE models have been proposed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Among them, the ΛCDM model is widely accepted as supposedly it ‘best accom- modates’ the observations but also it comes with some disadvantages like a fine-tuning problem, coincidence problems and so on [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' To overcome these drawbacks, alternative DE models have been explored like a quite favorable phantom, k-essence, Chaplygin gas, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=', [6] for possible explanations of the origin and nature of the dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' However, prior to the accelerated phase, the universe had gone through a decelerated phase in the early epoch where the effects of dark energy were absent or subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' It is believed that density perturbations occurred in this epoch which played a key role in the structural formation of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' So, to cover the entire evolution, we would have to employ a cosmological model which would simultaneously describe the accelerated and decelerated phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Earlier, the definition of cosmology was addressed by Sandage[7] as a search for two simple but fundamental cosmographic parameters: Hubble parameter (H0), which determined ∗ a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='bouali@ump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='ma † himanshuch1729@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='com ‡ ujjaldebnath@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='com § tanusreeroy1995@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='com ¶ gmustafa3828@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='com the expansion rate and a small correction q0 due to grav- ity, known as deceleration parameter (DP), responsible for slowing down the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Though the inclusion of ‘dark energy’ has completely changed the scenario, but still, any practical aspect of cosmological evolution is tightly bound to DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' It is defined as q = − a¨a ˙a2 where a(t) is the scale factor of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' q > 0 (¨a < 0) indicates a decelerating universe and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' While HP describes the linear part of the time dependence of the scale factor, the non-linear correction term q0 opens up possibilities like the presence of local instabilities or the existence of chaotic regimes [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Moreover, the dynamics of observable galaxy number variation can be determined through DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Like tenergyr cosmological models, DE has been subjected to numerous modifica- tions to be better fitted with observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Parametrization of DP as a function of scale factor a or redshift z can be accounted as a suitable approach to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Limitations to such parametric assumptions are: (i) Most of the parametrizations diverge in the distant future, and some of them are only valid at low redshift limit (z << 1) [9, 10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (ii) For prior parametric assumptions true nature of the dark energy can be misleading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (iii) In non-parametric models, evolution can be directly deduced from observational data avoiding parametric assumptions [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' However, it can help to improve the efficiency of future cosmological surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' So in pursuit of understanding the transition from decelerated to accelerated phase, the parametrization approach can be proved fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Recently Capozziello reconstructed a divergence-free form of DP starting with Pade polynomials and analyzed the corresponding observational data [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' A logarithmic parametrization of DP was proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' [17], and the constraints were obtained by using type Ia supernova, BAO and CMB data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Motivated by these ideas, we have adopted some well-motivated parametrizations of DP to reconstruct dark energy and, consequently, Hubble parameter H(z) in terms of redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Mainly, well- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='12107v1 [gr-qc] 28 Jan 2023 2 established parametrized models have been introduced for dark energy equation of state and constrain the model parameters by observational data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In the study of the generalized holographic dark energy model, some well-known parametrization type models have been considered [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Till now, some authors have assumed some possible forms of parametrization of deceleration parameter [20–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The main advantage of the parameterization deceleration parameter is that the final outcome may not depend on any particular gravitational theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In the ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' [26, 27], the authors have introduced the analogous parametrized deceleration parameter instead of parametrized dark energy equation of state for the well-established models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Here we have considered the analogous parametrized deceleration parameter for the well-established models and constrains the model parameters by Markov Chain Monte Carlo (MCMC) data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' We compare all the models with the ΛCDM model (which is the base model) to examine which model is more viable than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The main focus of the work is to constrain the model parameters using recently released data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Here, in par- ticular, we have chosen to use the updated astronomical datasets: the measurements of Hubble parameter from the differential evolution of cosmic chronometers (CC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Pantheon datasets from Type Ia Supernovae sample comprising 1048 measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 17 uncorrelated baryons acoustic oscillation (BAO) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' New constraints on DP have been provided by jointly analyzing the above datasets and implementing Monte Carlo Markov Chain (MCMC) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The era of modern cosmology pro- motes the study of kinematic quantities, vastly known as ”Cosmography” or ”Cosmo-kinetics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The very idea of it is observationally driven and completely independent of any prior assumption of the gravity theory or elected cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Cosmography presents itself with a compelling advantage as it simply follows the symmetry principles and direct observation- without involving Ein- stein equations (Friedmann equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Consequently, we can steadfastly avoid some arguable speculations regarding ’dark energy’, ’dark matter’ and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' While pure cosmography does not envision the scale factor a(t) itself but the history of its evolution can be inferred to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Dunajski and Gibbons [28] have studied the constraints on the cosmographic parameters like the deceleration, jerk, and snap parameters for different dark energy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The parametrization of these quantities is discussed in [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Shafieloo, Kim and Linder [22] have discussed the non-parametric reconstruction of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The organization of the paper is as follows: In section II, we consider the basic equations of the FRW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The Hubble parameter is written in terms of the decel- eration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Section III deals with the data de- scriptions like cosmic chronometric datasets, Pantheonic datasets and BAO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In section IV, we consider parametrized deceleration parameter models like models 1, 2, 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In section V, we fit the models with H(z) and SNe-IA datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In section VI, we discuss the cos- mography parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In section VII, we analyze the detailed description of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In section VIII, we present the information criteria for our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Finally, the results are presented in section IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' BASIC EQUATIONS OF FRW MODEL We have considered a spatially flat, homogeneous, isotropic FRW universe with line element ds2 = −dt2 + a2(t) � dr2 + r2 � dθ2 + sin2θdφ2�� (1) a(t) being the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The energy-momentum tensor of the fluid reads as Tµν = (ρ + p)uµuν + pgµν (2) where ρ and p are the energy density and pressure density of the fluid respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The fluid 4-velocity uµ = dxµ ds satisfies the relation uµuµ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' For the FRW Univere, the Friedmann equations in Ein- stein’s gravity are given by H2 = 8πG 3 ρ (3) and ˙H = −4πG(ρ + p) (4) where, H = ˙a/a is the Hubble parameter and overhead dot denotes derivative with cosmic time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Considering that the universe is filled with fluid matter of total en- ergy density ρ and total pressure p, it obeys the energy conservation equation ˙ρ + 3H(ρ + p) = 0 (5) We start with the prediction that the universe is com- posed of matter content comprising radiation, dark mat- ter (DM) and dark energy (DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' So, ρ and p consist of densities and pressures of radiation, DM and DE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' So ρ = ρr +ρm +ρd and p = pr +pm +pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Now assume that radiation, DM and DE follows the conservation equation separately so that ˙ρr + 3H(ρr + pr) = 0, (6) ˙ρm + 3H(ρm + pm) = 0 (7) and 3 ˙ρd + 3H(ρd + pd) = 0 (8) For radiation, pr = 1 3ρr, so from equation (6) we obtain ρr = ρr0a−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Since the DM follows negligible pressure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=', pm = 0), so from equation (7) we obtain ρm = ρm0a−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Let us consider the deceleration parameter q = −1 − ˙H H2 (9) The corresponding deceleration parameter for DE is given by [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' ] qd = −1 − ˙Hd H2 d (10) where Hd is the Hubble expansion rate of the dark energy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' So from equations (3) and (4), we can write H2 d = 8πG 3 ρd (11) and ˙Hd = −4πG(ρd + pd) (12) Using the field equations (11) and (12) and the energy- conservation equation (8), the fluid energy density be- comes ρd = ρd0 e � 2(1+qd) 1+z dz (13) where ρd0 represents the present value of the density parameter and z is the redshift parameter described as 1 + z = 1 a (presently, a0 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Defining Ωr0 = 8πGρr0 3H2 0 , Ωm0 = 8πGρm0 3H2 0 and Ωd0 = 8πGρd0 3H2 0 , from equation (3), we obtain the Hubble parameter as H2(z) = H2 0 � Ωr0(1 + z)4 + Ωm0(1 + z)3 + Ωd0 e � 2(1+qd) 1+z dz� (14) where Ωd0 = 1 − Ωr0 − Ωm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' DATA DESCRIPTION In this section, we will constrain our model param- eters by using three types of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The CC datasets consist 31 measurements, Pantheon dataset consists 1048 measurements and 17 uncorrelated BAO measurements to obtain the best fit value of our model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' We have implemented the Markov Chain Monte Carlo (MCMC) [33] and implemented with the open source package Polychord [34] and GetDist [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The total χ2 function of the combination CC + BAO + Pantheon and define as χ2 = χ2 CC + χ2 SN + χ2 BAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Cosmic Chronometric (CC) datasets We consider the compilation of 31 measurements of CC lying between the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='07 ≤ z ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The underlying principle for these measurements was proposed in [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' by relating the Hubble parameter with redshift z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' and cosmic time t H(z) = − 1 1 + z dz dt The χ2 function for these measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' denoted by χ2 CC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' is χ2 CC = 31 � i=1 � Hth (zi) − Hobs (zi) �2 σ2 Hobs(zi) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (15) where Hth (zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' h) represent the theoretical value obtained from our cosmological model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Hobs (zi) and represent the observed value of hubble parameter with standard deviation σ2 Hobs(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (to see more and rundown all measurements see [37]) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Pantheon datasets The Pantheon datasets comprises 1048 measurements of type Ia supernovae from five different sub-samples SNLS, SDSS, PSI, low- z, and HST in the redshift range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='01 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='3 [38] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The χ2 function of the Pantheon data is given as χ2 Pan = ∆µC−1 Pan∆µT , (16) where ∆µ = µobs i − µth .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Where � µobs i � represented as observed distance modulus and evaluated as µobs i = µB,i + M, (17) µB,i represents the observed peak magnitude at max- imum in the rest frame of the B band for redshift zi, while M represents nuisance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The theoretical distance modulus evaluated as µth = 5 log10 DL + M, (18) where DL = (1 + zhel) � zcmb 0 H0dz H(z) , (19) with zhel is heliocentric and zcmb is CMB rest frame redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The covariance matrix is measured as CPan = 4 Csys + Dstat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Where Csys is systematic covariance ma- trix and Dstat stands for diagonal of covariance matrix of the statistical uncertainty and calculated as Dstat ,ii = σ2 µB,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (20) The description and the systematic covariance matrix together with µB,i, σ2 µB,i, zcmb, and zhel for the i th SnIa are mentioned in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Uncorrelated Baryon Acoustic Oscillations (unCor BAO) We picked 17 uncorrelated BAO measurements from the greatest collection of BAO dataset of (333) measure- ments since adopting the entire catalogue of BAO might lead in a very significant error owing to data correlations, therefore we opted a small dataset to minimise inaccura- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Transverse BAO studies contribute measurements of DH(z)/rd = c/H(z)rd with comoving angular diame- ter distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' [40] [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' DM = c H0 Sk �� z 0 dz′ E (z′) � , (21) where Sk(x) = � � � � � 1 √Ωk sinh �√Ωkx � if Ωk > 0 x if Ωk = 0 1 √−Ωk sin �√−Ωkx � if Ωk < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (22) We also consider the angular diameter distance DA = DM/(1 + z) and the DV (z)/rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' which is combination of the BAO peak coordinates and rd is the sound horizon at the drag epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Finally we can obtain ”line-of-sight” (or ”radial”) observations directly the Hubble parameter DV (z) ≡ � zDH(z)D2 M(z) �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (23) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' PARAMETERIZED DECELERATION PARAMETER Most simplest parametrization of q which contains two parameters can be taken as q(z) = q0 + q1X(z) (24) where q0 and q1 are constants and X(z) is a function of redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In search of satisfactory solutions to the cosmological puzzles, many forms of X(z) has been sug- gested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' As mentioned earlier, most of them were inad- equate in explaining future evolution scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' So the persuasion of an ideal divergence-free parametrization of DP is still relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The well-known parametrized equa- tion of state parameter models has been introduced by several authors, and the corresponding analogous of these models for parametrized deceleration parameters have been introduced in [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Here, we have adopted the analogous of some well-known parametrized models for the deceleration parameter, which contains two unknown parameters and calculated the corresponding Hubble pa- rameter in terms of redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 1 (Wetterich type) The Wetterich model for parametrized equation of state parameter has been studied in [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The analo- gous Wetterich type parametrization of deceleration pa- rameter has been introduced in [26, 27] and is given by qd(z) = q0 1 + q11og(1 + z) (25) where q0 and q1 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Then the energy density will be given by ρd = ρd0 (1 + z)2 {1 + q1 log(1 + z)} 2q0 q1 (26) From equation (14), we obtain H2(z) = H2 0 � Ωr0(1 + z)4 + Ωm0(1 + z)3 + (1 − Ωr0 − Ωm0) (1 + z)2 {1 + q1 log(1 + z)} 2q0 q1 � (27) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 2 (Barboza-Alcaniz type) The Barboza-Alcaniz model for parametrized equation of state parameter has been studied in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The anal- ogous Barboza-Alcaniz type parametrization of deceler- ation parameter has been introduced in [26, 27] and is given by qd(z) = q0 + q1 z(1 + z) 1 + z2 (28) where q0 and q1 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Then the energy density will be ρd = ρd0 (1 + z)2(1+q0) � 1 + z2�q1 (29) From equation (14), we obtain H2(z) = H2 0 � Ωr0(1 + z)4 + Ωm0(1 + z)3 + (1 − Ωr0 − Ωm0) (1 + z)2(1+q0) � 1 + z2�q1� (30) 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 3 (CPL type) The famous Chevallier-Polarski-Linder (CPL) model for parametrized equation of state parameter has been studied in [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The analogous CPL type parametriza- tion of deceleration parameter has been introduced in [26, 27] and is given by qd(z) = q0 + q1 z 1 + z (31) where q0 and q1 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Subsequently, the en- ergy density (13) becomes ρd = ρd0 (1 + z)2(1+q0+q1) e 2q1 1+z (32) From equation (14), we obtain H2(z) = H2 0 � Ωr0(1 + z)4 + Ωm0(1 + z)3 + (1 − Ωr0 − Ωm0) (1 + z)2(1+q0+q1) e 2q1 1+z � (33) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 4 Here we propose a new parametrized model for decel- eration parameter and is given as in the form: qd(z) = q0 + q1 1 + z 2 + z (34) where q0 and q1 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Subsequently, the en- ergy density (13) becomes ρd = ρd0 (1 + z)2(1+q0)(2 + z)2q1 (35) From equation (14), we obtain H2(z) = H2 0 � Ωr0(1 + z)4 + Ωm0(1 + z)3 + (1 − Ωr0 − Ωm0) (1 + z)2(1+q0)(2 + z)2q1� (36) 68 69 70 H0 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 q1 CC + SN + BAO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The above figure shows the MCMC confidence con- tours at 1σ and 2σ obtained from CC+SNIa+BAO dataset for Model 1 (Wetterich type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 68 69 70 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The above figure shows the MCMC confidence con- tours at 1σ and 2σ obtained from CC+SNIa+BAO dataset for Model 2 (Barboza-Alcaniz type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 6 59 60 61 62 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='20 q1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='1 1.' metadata={'source': 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+page_content='897355+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='056271 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='056271 q1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='091362+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='073319 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='073319 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Summary of the MCMC results using CC + Pan + BAO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' OBSERVATIONAL, AND THEORETICAL COMPARISONS OF THE HUBBLE AND DISTANCE MODULUS FUNCTIONS Once we have the free parameters of the Model (1- 4), we can compare the model predictions to the ob- servational data and the LambdaCDM model with error bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Comparison with the Hubble data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' First, We consider the comparison of the Models (1 - 4) with the 31 (CC) data points and the ΛCDM model with 1 σ and 2 σ error bands .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The comparison findings are shown in Figure 5, 6, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The Figure shows that all model fit with (CC) dataset quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 0 50 100 150 200 250 300 H(z) mean CDM Model 1 1 2 H(z) Dataset FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of Hubble function H(z) of Model 1 and ΛCDM models against the CC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 0 50 100 150 200 250 300 H(z) mean CDM Model 2 1 2 H(z) Dataset FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of Hubble function H(z) of Model 2 and ΛCDM models against the CC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 0 50 100 150 200 250 300 H(z) mean CDM Model 3 1 2 H(z) Dataset FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of Hubble function H(z) of Model 2 and ΛCDM models against the CC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 0 50 100 150 200 250 300 H(z) mean CDM Model 4 1 2 H(z) Dataset FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of Hubble function H(z) of Model 4 and ΛCDM models against the CC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Comparison with the Pantheon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' We now compare the µ(z) distance modulus function of Models (1-4) with the Pantheon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 9,10,11 and 12 one can see that all Models fit with the Pantheon dataset, very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 14 16 18 20 22 24 26 28 30 CDM Model 1 mean 1 2 Pantheon FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of distance modulus µ(z) of Model 1 and ΛCDM models against the Pan- theon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 14 16 18 20 22 24 26 28 30 CDM Model 2 mean 1 2 Pantheon FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of dis- tance modulus µ(z) of Model 2 and ΛCDM models against the Pantheon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 14 16 18 20 22 24 26 28 30 CDM Model 3 mean 1 2 Pantheon FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of dis- tance modulus µ(z) of Model 3 and ΛCDM models against the Pantheon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 z 14 16 18 20 22 24 26 28 30 CDM Model 4 mean 1 2 Pantheon FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The figure show that the theoretical curve of dis- tance modulus µ(z) of Model 4 and ΛCDM models against the Pantheon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' COSMOGRAPHY PARAMETERS To study the early evolution and late evolution of the universe, some other parameters named cosmographical parameters can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The cosmographical param- eter like jerk (j), snap (s) parameters are [45–48] j = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='a aH3 = (1 + z)dq dz + q(1 + 2q), (37) s = a(4) aH4 = −(1 + z) dj dz − j(2 + 3q) (38) (39) So the cosmographical parameters contains the higher-order derivatives of the deceleration parameter q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The ’jerk’ parameter is considered to have a very useful feature that is for standard ΛCDM model, j always takes the value unity which helps us assess the deviation regarding different dark energy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Sahni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' and Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' analyzed the importance of the jerk parameter j for discriminating various dark energy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' We have explored the evolution of such kinematical quantities with respect to the redshift for the involved parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 Model 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of deceleration parameter of Model 1 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 9 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 Model 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of jerk parameter of Model 2 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0 1 2 3 Model 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of snap parameter with of Model 3 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 Model 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of deceleration parameter of Model 1 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 Model 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of jerk parameter of Model 2 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0 1 2 3 Model 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of snap parameter with of Model 2 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 Model 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of decceleration parameter with of Model 3 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 10 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 Model 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of jerk parameter of Model 2 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0 1 2 3 Model 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of snap parameter with of Model 3 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 1 0 1 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 New Model FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of decceleration parameter with of Model 4 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 New Model FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of jerk parameter with of Model 4 with respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 1 0 1 2 3 New Model FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Evolution of snap parameter with of Model 4 respect to redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 11 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' DETAILED DESCRIPTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Cosmological Parameters 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Hubble parameter (H0) This Hubble parameter is the normalized rate of ex- pansion, although this parameter is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The current estimates of this parameter are “65-75”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 3 has the “H0 < 65 ”, but on the other hand, Model 1 and Model 2 have this parameter in the Hubble range of ”70 < H0 < 75”, and finally, in Model 4, “H0 > 70 ”(Table: I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The matter density parameter (Ωm0) This parameter is the ratio of the actual (or observed) density ρm to the critical density ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The Model 1, Model 2 and Model 4 have a matter density in the range of “0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='22 < Ωm0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='26”, but in Model 3, this parameter has the range “Ωm0 ¡ 30” (Table: I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The radiation density parameter (Ωr0) This parameter is the ratio of the actual (or observed) radiation density ρrad to the critical density ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' This parameter for Model 1, Model 2, Model 3 and Model 4 lies in the range “0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='003 < Ωr0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='008” (Table: I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Cosmographic Analysis The cosmographic analysis provides a universal and effective way to compare the solutions of the theoretical models with cosmological observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' From the obser- vational data, we obtain a set of cosmological parameters, which must be compared with the predicted values of the same parameters obtained from a given model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The re- sult of the comparison allows us to conclude the accept- ability of the considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Thus, for a complete comparison of all models with the observations and the ΛCDM model, we will consider an extended set of param- eters constructed from the higher-order time derivatives of the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' More exactly, we will concentrate on the comparative behaviour of the deceleration, jerk, and snap parameters of all models and ΛCDM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Deceleration parameter While analyzing Model 1’s trajectory, the behavior of the model’s deceleration parameter is nearly comparable to the ΛCDM model in the redshift range of q ∈ [-0,2], but Model 1 endures a super acceleration in the future ( Fig: 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 2 appears to have the same behavior as ΛCDM in the q ∈ [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='8,4], but Model 2 is slower since it achieves the value −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='83565 as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' (Fig: 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' In Model 3, this parameter appears to behave similarly to the ΛCDM model, in q ∈ [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5,6], before experiencing a super acceleration in the near future as ”q = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='25464” (Fig: 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 4 behaves differently from the ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' but it aquire the same value as ΛCDM in near future 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The jerk parameter The jerk parameter of Model 1 basically different from ΛCDM at high as well as low redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' However, impor- tant Model 1 predicts the higher value of j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='357595 at z = 0 (Fig: 15), Meanwhile, on the other hand, Model 2 also shows different behaviour at both high and low redshift, and seems to coincide with ΛCDM at redshift value of z = 2 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='342566, and this model also pre- dicts the higher value of j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='134545 (Fig: 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 3 shows different behavior than the ΛCDM as at high red- shift z > 2, it having a higher value than the ΛCDM of j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='2657, but it cuts the trajectory of ΛCDM, twice at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='5 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='3, finally, at lower redshift, it attains the j value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='05417, which is a higher than ΛCDM (Fig: 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Although the jerk of Model 4 is inconsistent with ΛCDM, since it shows different evolution at both high and low redshifts and predicts the lower value of j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='608251 (Fig: 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The snap parameter This parameter of Model 1 and Model 2 is significantly systematic the difference with ΛCDM within the whole redshift range, but this parameter of Model 1 monoton- ically decreases in the redshift range of z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='3 and ac- quires a sudden increase in ”s” value and predicts the higher value of j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='608251 then ΛCDM, but Model 2 predicts the same value of j as ΛCDM,(Fig: 15, 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model 3 trajectory shows a proper systematic differ- ence with ΛCDM and predicts a higher value of ”s” as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='734546 21 Finally, the snap trajectory of Model 4 is notably non-identical with the ΛCDM, in the given red- shift range and accommodate the “s” value of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='222743 as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' which is lower than ΛCDM (Fig: 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' INFORMATION CRITERIA To discuss the viable model analysis, we need to know the study of information criteria (IC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The Akaike Information Criteria (AIC) [49] is merely used among all ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The AIC is an asymptotically unbiased esti- mator of Kullback-Leibler information as the AIC is an approximate minimization of the Kullback-Leibler information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The Gaussian estimator for the AIC can be written as [50–53] AIC = −2 ln(Lmax) + 2κ + 2κ(κ+1) N−κ−1 where Lmax is the maximum likelihood function, κ is 12 the number of parameters of the models, and N is the number of data points used in the data fit of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Since for the models, N ≫ 1, so for this assumption, the above expression converts to the original AIC like AIC = −2 ln(Lmax) + 2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' If the set of models are given, the deviations of the IC values are reduced to △AIC = AICmodel − AICmin = △χ2 min + 2△κ In the study of data analysis, the more favorable range of △AIC is (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The low favorable range of △AIC is (4, 7), while △AIC > 10 provides less support model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Model χ2 min χ2 red AIC ∆AIC ΛCDM Model 1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='981 1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='67 0 Model 1 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='961 1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='54 Model 2 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='963 1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='38 Model 3 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='965 1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='18 Model 4 1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='972 1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='09 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Summary of the χ2 min, χ2 red, AIC and ∆AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' DISCUSSIONS AND CONCLUSIONS We have assumed the FRW model of the universe in the presence of radiation, dark matter and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Instead of considering the well-known parametrized dark energy equation of state, we have considered the analo- gous form of parametrized deceleration parameter for the dark energy component and found the Hubble parameter in terms of redshift with other model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Here we have assumed Model 1 (Wetterich type), Model 2 (Barboza-Alcaniz type) and Model 3 (CPL type), which contains two unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Also, we have introduced a new Model 4 for parametrized deceleration parameters, which also contains two unknown parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The model parameters have been constrained for H(z) datasets, Pantheon datasets, and BAO datasets by MCMC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Using the best-fit parameters, we have shown the nature of the deceleration parameter, jerk parameter and snap parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' The viability of the models has been studied by the information criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' We have compared all the models as well as compared with the ΛCDM model (which is the base model) to get which model is more viable than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' From Table: II, we observe that Models 1 - 4 are all viable models, but (i) Model 3 is more viable than Model 4, (ii) Model 1 is more viable than Model 3 and (iii) Model 2 is more viable than the Model 1 compared to the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Acknowledgement: TR is thankful to IIEST, Shibpur, India for providing Institute fellowship (SRF).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Forster, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=', The wigglez dark energy survey: Joint measurements of the expansion and growth history at z¡ 1, Monthly Notices of the Royal Astronomical Society 425 (1) (2012) 405–414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' APPENDIX 15 BAO name redshift z Experiment Measurement Standarddeviation Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' 6dFGS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='106 rs/DV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='015 [54] SDSS DR7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='15 DV (rs,fidd/rs) 664 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 [55] SDSS-DR7 + 2dFGRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='275 rs/DV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='1390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0037 [56] SDSS-DR11 LOWZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='32 DV (rd,fidd/rs) 1264 25 [57] SDSS-III DR8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='54 DA/rs 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='41 [58] SDSSIII/ DR9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='57 DV /rs 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='22 [59] SDSS-IV DR14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='72 DV (rs,fidd/rs) 2353 63 [60] DES Year 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='81 DA/rs 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='43 [61] DECals DR8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='874 DA(rs,fidd/rs) 1680 109 [60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='72 DV (rs,fidd/rs) 2353 63 eBoss DR16 BAO+RSD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='48 DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='rs 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='47 [62] SDSS-IV/DR14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='52 DV (rs,fidd/rs) 3843 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0 [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' ] Boss Lya quasars DR9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='rs 34188 1188 [63] BOSS DR14 Lya in LyBeta 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='34 DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='rs 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='29 [64] WiggleZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='6 rs/DV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0031 [65] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content='0020 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} +page_content=' Summary of the Baryon Acoustic Oscillations measurements used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFLT4oBgHgl3EQfii8S/content/2301.12107v1.pdf'} diff --git a/qtE0T4oBgHgl3EQfaQD2/content/tmp_files/2301.02334v1.pdf.txt b/qtE0T4oBgHgl3EQfaQD2/content/tmp_files/2301.02334v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..230c613ac7cef79a7421ff608e6991903b701abe --- /dev/null +++ b/qtE0T4oBgHgl3EQfaQD2/content/tmp_files/2301.02334v1.pdf.txt @@ -0,0 +1,1164 @@ +arXiv:2301.02334v1 [cs.IT] 5 Jan 2023 +1 +Capacity Region of Asynchronous Multiple Access +Channels with FTN +Zichao Zhang, Student Member, IEEE, Melda Yuksel, Senior Member, IEEE, +Gokhan M. Guvensen, Member, IEEE, and Halim Yanikomeroglu, Fellow, IEEE +Abstract—This paper studies the capacity region of asyn- +chronous multiple access +channel (MAC) with faster-than- +Nyquist (FTN) signaling. We first express the capacity region +in the frequency domain. Next, we calculate an achievable rate +region in time domain and prove that it is identical to the capacity +region calculated in the frequency domain. Our analysis confirms +that asynchronous transmission and FTN bring in significant +gains. +Index Terms—Capacity, faster-than-Nyquist (FTN), multiple +access channel (MAC), asynchronous transmission. +I. INTRODUCTION +The rapid growth of need in rate and number of devices pro- +poses a challenge to modern communication systems. Multiple +access communications is considered to be one of the potential +solutions for 5G and beyond [1]. Compared to orthogonal mul- +tiple access (OMA), multiple access performs non-orthogonal +resource allocation. For instance, one frequency band can be +shared by more than one users. Besides increased connectivity, +multiple access achieves rate pairs that OMA is not able to +achieve. +Faster-than-Nyquist signaling is another promising physical +layer technology for future communication systems [2]. It +improves spectral efficiency by increasing signaling rate while +maintaining power consumption [3]. Since the groundbreaking +work of Mazo in 1975 [2], there has been a substantial amount +of research on FTN [4]. The information-theoretical study +show that applying FTN to communication systems improves +capacity [3] and this improvement becomes more favorable +when FTN is applied to multi-antenna communication systems +[5]. +In order to support multiple devices sharing the same re- +sources as well as satisfying rate requirements, it is beneficial +to exploit the multiple access channel (MAC) with FTN. A +realistic problem follows, in practice, that each device will +experience a random time delay. However, instead of being +a hazard to the system, this asynchronism is analyzed in +[6], [7] and [8] and is shown to be beneficial to multiple +access transmission. In [6], the author explored the capacity +This work was supported in part by the Natural Sciences and Engineering +Research Council of Canada, NSERC, under a Discovery Grant and in part +by the Scientific and Technological Research Council of Turkey, TUBITAK, +under Grant 122E248. +Z. Zhang and H. Yanikomeroglu are with the Department of Systems and +Computer Engineering at Carleton University, Ottawa, ON, K1S 5B6, Canada +e-mail: zichaozhang@cmail.carleton.ca, halim@sce.carleton.ca. +M. Yuksel and G. M. Guvensen are with the Department of Electrical and +Electronics Engineering, Middle East Technical University, Ankara, 06800, +Turkey, e-mail: ymelda@metu.edu.tr, guvensen@metu.edu.tr. +region of asynchronous MAC with fixed or random time +delay differences and showed that these differences bring in +additional gains. However, the analysis in [6] is constrained to +rectangular pulse shapes in time. The authors of [7] extended +this limitation and derived the capacity region for band-limited +pulse shapes. In [8], the authors studied FTN in asynchronous +MAC and obtained an achievable rate region with fixed power +allocation. In this paper we derive the capacity region of the +asynchronous MAC with FTN. +The organization of the paper is as follows. In Section II +we establish the system model. In Section III we derive the +capacity region. In Section IV we show that the capacity region +for discrete MAC with finite memory defined in [6] actually +leads to the same region as in Section III. In Section V we plot +the rate regions for finite number of symbols and in Section +VI we conclude the paper. +II. SYSTEM MODEL +The MAC is composed of K transmitters and one receiver. +Due to imperfect clock generation or different propagation +delays, signals coming from each transmitter have differ- +ent time delays. We denote them as τ1, τ2, . . . , τK, τk +∈ +[0, T ], k = 1, . . . K. Without loss of generality, we assume +τ1 ≤ τ2 · · · ≤ τK. +All the transmitters use the same pulse shaping filter p(t) +and the same acceleration factor δ for FTN. The signal +transmitted from the lth user, xl(t) then has the form +xl(t) = +N−1 +� +m=0 +al[m]p(t − mδT − τl), +(1) +where al[m] are the symbols transmitted from the lth user and +N is the number of symbols transmitted. At the receiver, the +matched filter p∗(−t) is applied. +An additive white Gaussian noise ξ(t) with power spectral +density σ2 +0 is added at the receiver. After passing through +the matched filter this white noise becomes correlated. We +denote this noise as η(t) = ξ(t)⋆p∗(−t), where ⋆ denotes the +convolution operation. The signal at the output of the matched +filter is y(t) = +��K +k=1 xl(t) + ξ(t) +� +⋆ p∗(−t). +In order to obtain the sufficient statistics in this asyn- +chronous MAC with FTN, we need to sample according to +the time delay of each user. Thus, we sample at all t = +nδT +τk, n = 0, 1, . . . , N−1, k = 1, . . . , K and obtain K sets +of samples instead of a single set [7]. Then, the samples yl[n] +corresponding to user l are written by sampling the output of + +2 +the matched filter, y(t), at time nδT + τl, n = 0, . . . , N − 1, +and we write +yl[n] = +K +� +k=1 +N−1 +� +m=0 +ak[m]g +� +(n − m)δT + (τl − τk) +� ++ ηl[n]. +(2) +Here g(t) = p(t) ⋆ p∗(−t). Furthermore, +ηl[n] = η(nδT + τl) = ξ(t) ⋆ p∗(−t)|t=nδT +τl. +(3) +By defining the N × 1 vectors yl, al and ηl to represent +respectively the output samples, data symbols and noise, the +input-output relationship in (2) can be written in a compact +matrix product form as + + +y1 +y2 +... +yK + + = + + +G11 +G12 +. . . +G1K +G21 +G22 +. . . +G2K +... +... +... +... +GK1 +GK2 +. . . +GKK + + + + +a1 +a2 +... +aK + ++ + + +η1 +η2 +... +ηK + + . (4) +This expression can further be simplified as +y = ˜Ga + η, +(5) +where y = [y⊤ +1 , . . . , y⊤ +K]⊤, a = [a⊤ +1 , . . . , a⊤ +K]⊤ and η = +[η⊤ +1 , . . . , η⊤ +K]⊤. The matrix ˜G in (5) is KN×KN. The matrix +Glk in (4) is the N × N interference matrix. It represents +user k’s effect on the samples of user l and its (n, m)th entry, +n, m = 1, . . . , N, is (Glk)n,m = g +� +(n−m)δT +(τl−τk) +� +. In +this paper, we focus on the special case of K = 2. Note that, +the matrix Glk is a Toeplitz matrix. An N ×N Toeplitz matrix +TN has the structure (TN)i,j = ti−j, i, j = 0, . . . , N − 1. Its +generating function is defined as +G(TN) = +∞ +� +k=−∞ +tkejkλ, λ ∈ +� +−1 +2, 1 +2 +� +(6) +where we denote the operation of generating function compu- +tation by G. +III. THE CAPACITY REGION ANALYSIS +In this section we derive the capacity region in the frequency +domain. The capacity region C of the two-user multiple access +channel with memory is defined as [7] +C = +� +� 1 +2 +− 1 +2 +Sk(λ)dλ≤Pk +Sk(λ)≥0,λ∈[− 1 +2 , 1 +2] +k=1,2 +� +(R1, R2) : +0 ≤ R1 ≤ lim +N→∞ +1 +N IN(a1; y|a2) +0 ≤ R2 ≤ lim +N→∞ +1 +N IN(a2; y|a1) +0 ≤ R1 + R2 ≤ lim +N→∞ +1 +N IN(a1, a2; y) +� +, +(7) +where S1(fn) and S2(fn) are the power spectral densities of +user 1 and user 2, while P1 and P2 are the power constraints. +In (7), IN is the mutual information between two random +vectors with length N. +In FTN signaling, the input power spectrum to the physical +channel contains the effect of both data symbols as well as +FTN [5], [7]. This can be written as +Sk(λ) = 1 +δT Gδ(λ)Sak(λ), +(8) +where Gδ(λ) is the folded spectrum defined as +Gδ(λ) = 1 +δT +∞ +� +n=−∞ +����P +�λ − n +δT +����� +2 += 1 +δT +∞ +� +n=−∞ +G +�λ − n +δT +� +(9) +and P(·) and G(·) are respectively the continuous time +Fourier transforms of p(t) and g(t). The data power spectrum +Sak(λ), k = 1, 2, is obtained by the discrete-time Fourier +transform of the autocorrelation function of input symbols, +Rak[n] = E[ak[m + n]a∗ +k[m]]; i.e., +Sak(λ) = +∞ +� +n=−∞ +Rak[n]e−jλn, k = 1, 2. +(10) +Therefore the power constraint of user k is +1 +δT +� +1 +2 +− 1 +2 +Gδ(fn)Sk(fn)dfn ≤ Pk. +(11) +In order to obtain a closed-form expression for (7), we +need to calculate the mutual information expressions. The +differential entropy of a Gaussian vector y is +h(y) = 1 +2 log2((2π)2N det(Σy)), +(12) +where Σy = E[yy†], with † denoting the Hermitian conjuga- +tion. Define matrix G = G11 = G22, the (n, m)th entry of +which is g((n−m)δT ), it is easy to see that G is a Hermitian +matrix. Notice that G† +12 = G21, thus ˜G is a Hermitian matrix. +According to [6], for any non-zero vector a, a† ˜Ga is the +energy of x1(t) + x2(t); therefore, the quadratic form a† ˜Ga +is guaranteed to be greater than zero, and ˜G is positive definite. +The colored Gaussian noise vector η has the correla- +tion E[ηi[n]ηj[m]] = σ2 +0(Gij)n,m, +i, j ∈ {1, 2}, n, m ∈ +{0, 1, . . ., N − 1}. As this noise process is a stationary, zero +mean, colored Gaussian process; therefore, the optimal input +is also a stationary Gaussian process [9]. It is also reasonable +to assume that data symbols from the two users a1 and a2 +are independent. Then the covariance matrix of each user is +E[aka† +k] = Rk, k = 1, 2, and the covariance matrix Σy can +be written as +Σy = ˜G +�R1 +0 +0 +R2 +� +˜G† + σ2 +0 ˜G, +(13) +where 0 is an all-zero matrix of size N × N. +Then, mutual information expressions for the single-user +rate constraints in (7) can be calculated as +IN(a1; y|a2) = h(y1|a2) − h(y1|a1, a2) +(14) +≤ +1 +2N log2 det +� +E +� +(Ga1 + η1)(Ga1 + η1)†�� +− +1 +2N log2 det +� +E +� +η1η† +1 +�� +(15) += +1 +2N log2 det +� +GR1G + σ2 +0G +� +− 1 +2N log2 det +� +σ2 +0G +� +(16) += +1 +2N log2 det +� +IN + σ−2 +0 GR1 +� +, +(17) + +3 +and +IN(a2; y|a1) = +1 +2N log2 det +� +IN + σ−2 +0 GR2 +� +. +(18) +Remark 1: In order to calculate (16), we need the matrix +G to be invertible. Theoretically, a matrix is invertible as +long as it is positive definite. However, this inversion may +not be numerically stable. For root raised cosine pulses p(t), +numerical stability is achieved if δ(1+β) ≥ 1, where β is the +roll-off factor. [5]. +Note that the matrices G, G12, G21, R1 and R2 are all +Toeplitz matrices. In addition, comparing (6) and (10), we +observe that Sak(−λ) is the generating function of the matrix +Rk. Since Gδ(λ) in (9) is an even function, Sak(−λ) = +Sak(λ). Then, applying Szeg¨o’s theorem [10] and [11, Theo- +rem 2] on the single-user rate constraints of (7), we have +Ri ≤ 1 +2 +� +1 +2 +− 1 +2 +log2(1 + σ−2 +0 Sai(λ)Gδ(λ))dλ, i = 1, 2. +(19) +To find the sum-rate constraint, we first observe that ˜G and +˜R = +� +R1 +0 +0 +R2 +� +are block Toeplitz matrices [11]. Then, we +derive the sum-rate constraint in (7) as +R1 + R2 +≤ lim +N→∞ +� 1 +2N log2 det +� +E +� +yy†�� +− +1 +2N log2 det +� +E +� +η1η† +1 +�� � +(20) += lim +N→∞ +1 +2N log2 det +� +I2N + σ−2 +0 +˜G ˜R +� +. +(21) +In (21), ˜G ˜R is a block Toeplitz matrix, because the product +of block Toeplitz matrices is also block Toeplitz [11, Theorem +2]. Then applying [11, Theorem 6] on the sum-rate constraint +(21) we write +lim +N→∞IN(a1, a2; y) += 1 +2 +� +1 +2 +− 1 +2 +log2 σ−2 +0 +���� +1 + Sa1(λ)Gδ(λ) +Sa2(λ)G12,δ(−λ) +Sa1(λ)G21,δ(−λ) +1 + Sa2(λ)Gδ(λ) +���� dλ +(22) += 1 +2 +� +1 +2 +− 1 +2 +log2 +� +1 + σ−2 +0 Sa1(λ)Gδ(λ) + σ−2 +0 Sa2(λ)Gδ(λ) ++ σ−4 +0 Sa1(λ)Sa2(λ) +� +|Gδ(λ)|2 − |G12,δ(λ)|2� � +dλ, +(23) +where G12,δ(λ) is the generating function of the matrix G12 +obtained via (6) and written as +G12,δ(λ) = +∞ +� +n=∞ +g(nδT + (τ1 − τ2))ejλn +(24) += 1 +δT +∞ +� +n=∞ +G(λ − n +δT +)ej(τ1−τ2) λ−n +δT . +(25) +Similarly G21,δ(λ) is the generating function of the matrix +G21. It is easy to see that G12,δ(λ) = (G21,δ(λ))∗ and +|G12,δ(λ)|2 = |G12,δ(λ)|2. +Theorem 1: The capacity region of the two-user asyn- +chronous MAC with FTN is given as +C = +� +� 1 +2 +− 1 +2 +Sk(λ)dλ≤Pi +Sk(λ)≥0, ∀λ∈[− 1 +2 , 1 +2 ],k=1,2 +� +(R1, R2) : +R1 ≤ 1 +2 +� +1 +2 +− 1 +2 +log2(1 + σ−2 +0 S1(λ))dλ +R2 ≤ 1 +2 +� +1 +2 +− 1 +2 +log2(1 + σ−2 +0 S2(λ))dλ +(26) +R1 + R2 ≤ 1 +2 +� +1 +2 +− 1 +2 +log2 +� +1 + σ−2 +0 S1(λ) + σ−2 +0 S2(λ) ++ σ−4 +0 S1(λ)S2(λ) +� +1 − +���� +G12,δ(λ) +Gδ(λ) +���� +2� � +dλ +� +. +It is worth mentioning that this expression is more general than +[7, Theorem 1] in the sense that when δ = 1, (26) degrades +to [7, Theorem 1] for orthogonal signaling. +IV. AN ALTERNATIVE CAPACITY CALCULATION +The capacity region of the asynchronous MAC with finite +memory is defined as [6] +C = closure +� +lim inf +N→∞ CN +� +. +(27) +Here CN is the achievable region for N symbols defined as +CN = +� +p(a1)p(a2) +� +(R1, R2) :0 ≤ R1 ≤ I(a1; y|a2) +0 ≤ R2 ≤ I(a2; y|a1) +0 ≤ R1 + R2 ≤ I(a1, a2; y) +� +, +(28) +where p(a1) and p(a2) means the distribution of a1 and +a2. Capacity for an arbitrary MAC with infinite memory +cannot be defined in general. However, we believe that this +same expression is valid as long as the limit exists [9], [12]. +Therefore, in this section we calculate (27) and prove that it +is equal to the capacity region in (26). +By combining (17), (18) and (21), CN for the asynchronous +MAC with FTN can be written as +CN = +� +1 +N tr(GRk)≤Pk +Rk⪰0,k=1,2 +� +(R1, R2) : R1 ≤ +1 +2N log2 +��IN + σ−2 +0 GR1 +�� +R2 ≤ +1 +2N log2 +��IN + σ−2 +0 GR2 +�� +R1 + R2 ≤ +1 +2N log2 +����I2N + σ−2 +0 +� G +G12 +G21 +G +� �R1 +0 +0 +R2 +����� +� +. +(29) +In order to further push the sum-rate upper bound, we +suggest the novel derivation in (30)-(35). In order for (30) +to be computable, we need Remark 1 to be valid. In step +(a), we define Φ ≜ G− 1 +2 G12G− 1 +2 , Ψ1 ≜ G +1 +2 R1G +1 +2 and +Ψ2 ≜ G +1 +2 R2G +1 +2 , where Ψ1 and Ψ2 are Hermitian matrices. +In (b), we perform singular value decomposition on the matrix +Φ = UΦΛΦV † +Φ, where ΛΦ = diag{λ1, λ2, . . . , λN} is a +diagonal matrix and λi, i = 1, . . . , N are the singular values of + +4 +I(a1, a2; y) = +1 +2N log2 det +� +I + σ−2 +0 +� +G +1 +2 +0 +0 +G +1 +2 +� � +I +G− 1 +2 G12G− 1 +2 +G− 1 +2 G21G− 1 +2 +I +� � +G +1 +2 +0 +0 +G +1 +2 +� � +R1 +0 +0 +R2 +�� +(30) += +1 +2N log2 det +� +I + σ−2 +0 +� +I +G− 1 +2 G12G− 1 +2 +G− 1 +2 G21G− 1 +2 +I +� � +G +1 +2 R1G +1 +2 +0 +0 +G +1 +2 R2G +1 +2 +�� +(31) +(a) += +1 +2N log2 det +� +I + σ−2 +0 +� +I +Φ +Φ† +I +� � +Ψ1 +0 +0 +Ψ2 +�� +(32) +(b) += +1 +2N log2 det +� +I + σ−2 +0 +�UΦ +0 +0 +VΦ +� � I +ΛΦ +Λ† +Φ +I +� �U† +Φ +0 +0 +V † +Φ +� �Ψ1 +0 +0 +Ψ2 +�� +(33) += +1 +2N log2 det +� +I + σ−2 +0 +� I +ΛΦ +Λ† +Φ +I +� �U† +ΦΨ1UΦ +0 +0 +V † +ΦΨ2VΦ +�� +(34) +(c) += +1 +2N log2 det +� +I + σ−2 +0 +� I +ΛΦ +Λ† +Φ +I +� � ˜Ψ1 +0 +0 +˜Ψ2 +�� +. +(35) +Φ. In (c) we define ˜Ψ1 ≜ U† +ΦΨ1UΦ and ˜Ψ2 ≜ V † +ΦΨ2VΦ, +where ψ1i and ψ2i are the diagonal entries of ˜Ψ1 and ˜Ψ2. +Then we apply [6, Lemma 2] on (35) to upper bound the +mutual information as +I(a1, a2; y) = +1 +2N log2 +� +I + σ−2 +0 +� I +ΛΦ +Λ† +Φ +I +� � ˜Ψ1 +0 +0 +˜Ψ2 +�� +≤ +1 +2N +N−1 +� +i=0 +log2 +� +1 + ψ1i +σ2 +0 ++ ψ2i +σ2 +0 ++ ψ1iψ2i +σ4 +0 +(1 − |λi|2) +� +. +(36) +The equality in (36) is achieved when ˜Ψ1 and ˜Ψ2 are diagonal +matrices. Moreover, in order for [6, Lemma 2] to be valid or +the upper bound to be achieved, we need the complex scalars +λi to satisfy |λi| ≤ 1, i = 0, 1, ..., N − 1. As the matrix ˜G is +positive definite, for any non-zero vector v, we have +v† ˜Gv +(a) += ˜v† +�G +1 +2 +0 +0 +G +1 +2 +�† +˜G +�G +1 +2 +0 +0 +G +1 +2 +� +˜v += ˜v† +� I +Φ +Φ† +I +� +˜v > 0, +where (a) is because v ≜ +� +G +1 +2 +0 +0 +G +1 +2 +� +˜v. Thus, the matrix +� I +Φ +Φ† +I +� +is positive definite as well. Then according to [13], +|λi| < 1, i = 0, 1, ..., N − 1. +Next, the upper bound for I(a1; y|a2) and I(a2; y|a1) can +be obtained as in [6] as +I(a1; y|a2) ≤ +1 +2N +N−1 +� +i=0 +log2 +� +1 + ψ1i +σ2 +0 +� +(37) +I(a2; y|a1) ≤ +1 +2N +N−1 +� +i=0 +log2 +� +1 + ψ2i +σ2 +0 +� +. +(38) +The transmit power constraint for each user in this N-block +asynchronous multiple access channel with FTN is calculated +as +1 +NδT tr(GRk) = +1 +NδT tr(G +1 +2 RkG +1 +2 ) = +1 +NδT +N−1 +� +i=0 +ψki +≤ Pk, +k = 1, 2 (39) +where ψki ≥ 0, i = 1, . . . , N. Therefore, the region CN +in (29) is obtained using (36)-(39). As in the asynchronous +MAC with rectangular pulses and without FTN [6], the input +distribution achieving the upper bound in (37) has a covari- +ance matrix G−1 scaled according to the power constraint. +However, this distribution does not achieve the upper bound +in (36). +Lemma 1: If we have �∞ +n=−∞ |n|tn +< ∞, then the +discrete Fourier transform (DFT) vectors are asymptotically +the eigenvectors of Toeplitz matrix TN. +Proof 1: Although this result is discussed in [14], it is not +proved. We omit the full proof due to page limitations. +Corollary 1: The region defined in (27) with CN defined +using (36), (37), and (38) with the power constraint in (39) is +the same as the capacity region in (26). +Proof 2: Since the raised cosine filter g[n] satisfies +�∞ +n=−∞ |n|g[n] < ∞, by Lemma 1, G− 1 +2 is an asymp- +totically Toeplitz matrix. Its generating function is G +1 +2 +δ (λ). +We know that |λi|2’s are the eigenvalues of the Hermitian +matrix Φ†Φ. As G− 1 +2 is asymptotically Toeplitz and the +product of asymptotically Toeplitz matrices is also asymp- +totically Toeplitz [10, Theorem 5.3], the product Φ†Φ = +G− 1 +2 G† +12G−1G12G− 1 +2 is also asymptotically Toeplitz. The +generating function of Φ†Φ is +��� +G12,δ(λ) +Gδ(λ) +��� +2 +, which is the +product of the generating functions of the individual matrices +in the above Φ†Φ expansion. The eigenvalues of a Toeplitz +matrix asymptotically approximate the samples of its gener- +ating function [15]. Thus we have |λi|2 ≈ +��� +G12,δ(i/N) +Gδ(i/N) +��� +2 +, i = +0, 1, . . ., N − 1. Moreover, the values |λi|2 are the samples +from the constant spectrum +��� +G12,δ(λ) +Gδ(λ) +��� +2 +. Hence the discussion +in [6] about time and frequency domain capacity region +comparison applies, and we conclude that the two regions are +the same. +V. NUMERICAL RESULTS +In this section, we plot the capacity region for N = 10 for +root raised cosine pulses p(t) with roll-off factor β. We set +the SNR for both users to be 20 dB, and the signaling period +T = 1. + +5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +R1(bit/s/Hz) +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +R2(bit/s/Hz) +aMAC,( , )=(1,0) +aMAC,( , )=(0.8,0.25) +aMAC,( , )=(0.8,0.25),iid +aMAC,( , )=(0.9,0.25) +aMAC,( , )=(1,0.25) +MAC,( , )=(1,0.25) +Fig. 1. Capacity regions for asynchronous MAC with different (δ, β) pairs, +asynchronous MAC with FTN with independent and identically distributed +(iid) inputs, and synchronous MAC. +In Fig. 1, we compare the capacity region (aMAC, (δ, β) = +(0.8, 0.25)) with the performance of synchronous MAC +(MAC, (δ, β) = 1, 0.25)) [9], asynchronous MAC without +FTN (aMAC, (δ, β) = 1, 0.25)) [7], and asynchronous MAC +with FTN without power optimization (aMAC, (δ, β) += +0.8, 0.25), iid) [8]. For all the asynchronous simulations in +this figure, we set the time difference τ to be δT +2 . We can see +that FTN brings a significant gain for both single-user rates as +well as the sum-rate. The region without power optimization +(aMAC, (δ, β) = 0.8, 0.25), iid) is obtained by setting ψki’s +to be PkδT, ∀i = 0, 1, . . ., N − 1, k = 1, 2. We observe +that both asynchronous transmission and FTN significantly +improve the rate region and optimal power allocation is +necessary to establish the capacity region. In this figure, we +also compare the results with asynchronous MAC with FTN +(aMAC, (δ, β) = 0.9, 0.25)) and with asynchronous MAC +with Nyquist signaling (aMAC, (δ, β) = 1, 0)) as an upper +bound. We observe that for a given β value, we should +let δ(1 + β) as close to 1 as possible. This also supports +the discussion of choices of (δ, β) pairs in [5]. Although +impractical, the best choice is Nyquist transmission with ideal +sinc pulses; i.e. (δ, β) = (1, 0). +We then study the influence of time difference τ between +two users on the performance of the system. In this figure +we set (δ, β) = (0.8, 0.25). We can see that when the time +difference between two users is half the sampling period δT +2 , +the performance is better than the performance with other +values of τ. This suggests that [7, Proposition 2] also holds +in the presence of FTN. +VI. CONCLUSION +In this paper we derive the capacity region of asynchronous +multiple access channels with FTN signaling. We calculate the +capacity region both in frequency and time domains, and show +that the two regions are equal to each other. As a side result, we +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +R1(bit/s/Hz) +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +R2(bit/s/Hz) +=0.5 T +=0.3 T +=0.8 T +=0.1 T +Fig. 2. +Capacity regions for asynchronous MAC with FTN with different +time differences τ between users. +prove that the DFT vectors are asymptotically the eigenvectors +of the Toeplitz matrix TN as long as �∞ +n=−∞ |n|tn < ∞. +REFERENCES +[1] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo, +“Non-orthogonal multiple access for 5G and beyond,” Proceedings of +the IEEE, vol. 105, no. 12, pp. 2347–2381, 2017. +[2] J. E. Mazo, “Faster-than-Nyquist signaling,” The Bell System Technical +Journal, vol. 54, no. 8, pp. 1451–1462, 1975. +[3] F. Rusek and J. B. Anderson, “Constrained capacities for faster-than- +Nyquist signaling,” IEEE Transactions on Information Theory, vol. 55, +no. 2, pp. 764–775, 2009. +[4] J. B. Anderson, F. Rusek, and V. ¨Owall, “Faster-than-Nyquist signaling,” +Proceedings of the IEEE, vol. 101, no. 8, pp. 1817–1830, 2013. +[5] Z. Zhang, M. Yuksel, and H. Yanikomeroglu, “Faster-than-Nyquist +signaling for MIMO communications,” IEEE Transactions on Wireless +Communications, pp. 1–1, 2022. +[6] S. Verdu, “The capacity region of the symbol-asynchronous Gaussian +multiple-access channel,” IEEE Transactions on Information Theory, +vol. 35, no. 4, pp. 733–751, 1989. +[7] M. Ganji, X. Zou, and H. Jafarkhani, “Asynchronous transmission +for multiple access channels: Rate-region analysis and system design +for uplink NOMA,” IEEE Transactions on Wireless Communications, +vol. 20, no. 7, pp. 4364–4378, 2021. +[8] S. Li, Z. Wei, W. Yuan, J. Yuan, B. Bai, D. W. K. Ng, and L. Hanzo, +“Faster-than-Nyquist asynchronous NOMA outperforms synchronous +NOMA,” IEEE Journal on Selected Areas in Communications, vol. 40, +no. 4, pp. 1128–1145, 2022. +[9] T. M. Cover and J. A. Thomas, Elements of Information Theory. +USA: +Wiley-Interscience, 2006. +[10] R. M. Gray, “Toeplitz and circulant matrices: A review,” Foundations +and Trends in Communications and Information Theory, vol. 2, no. 3, +pp. 155–239, 2006. +[11] J. Gutierrez-Gutierrez and P. M. Crespo, “Asymptotically equivalent +sequences of matrices and Hermitian block Toeplitz matrices with con- +tinuous symbols: Applications to MIMO systems,” IEEE Transactions +on Information Theory, vol. 54, no. 12, pp. 5671–5680, 2008. +[12] L. H. Brandenburg and A. D. Wyner, “Capacity of the Gaussian channel +with memory: The multivariate case,” The Bell System Technical Journal, +vol. 53, no. 5, pp. 745–778, 1974. +[13] P. Lancaster and M. Tismenetsky, The Theory of Matrices: With Appli- +cations. +Elsevier Science, 1985. +[14] C. Therrien, Discrete Random Signals and Statistical Signal Processing. +Prentice Hall, 1992, no. v. 1. +[15] Y. J. D. Kim, “Properties of faster-than-Nyquist channel matrices and +folded-spectrum, and their applications,” in IEEE Wireless Communica- +tions and Networking Conference (WCNC), 2016, pp. 1–7. + diff --git a/qtE0T4oBgHgl3EQfaQD2/content/tmp_files/load_file.txt b/qtE0T4oBgHgl3EQfaQD2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aac7ecf4348f8882d49d3ba6ddf34efe914e1a5d --- /dev/null +++ b/qtE0T4oBgHgl3EQfaQD2/content/tmp_files/load_file.txt @@ -0,0 +1,567 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf,len=566 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='02334v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='IT] 5 Jan 2023 1 Capacity Region of Asynchronous Multiple Access Channels with FTN Zichao Zhang, Student Member, IEEE, Melda Yuksel, Senior Member, IEEE, Gokhan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Guvensen, Member, IEEE, and Halim Yanikomeroglu, Fellow, IEEE Abstract—This paper studies the capacity region of asyn- chronous multiple access channel (MAC) with faster-than- Nyquist (FTN) signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We first express the capacity region in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Next, we calculate an achievable rate region in time domain and prove that it is identical to the capacity region calculated in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Our analysis confirms that asynchronous transmission and FTN bring in significant gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Index Terms—Capacity, faster-than-Nyquist (FTN), multiple access channel (MAC), asynchronous transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' INTRODUCTION The rapid growth of need in rate and number of devices pro- poses a challenge to modern communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Multiple access communications is considered to be one of the potential solutions for 5G and beyond [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Compared to orthogonal mul- tiple access (OMA), multiple access performs non-orthogonal resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' For instance, one frequency band can be shared by more than one users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Besides increased connectivity, multiple access achieves rate pairs that OMA is not able to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Faster-than-Nyquist signaling is another promising physical layer technology for future communication systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' It improves spectral efficiency by increasing signaling rate while maintaining power consumption [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Since the groundbreaking work of Mazo in 1975 [2], there has been a substantial amount of research on FTN [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The information-theoretical study show that applying FTN to communication systems improves capacity [3] and this improvement becomes more favorable when FTN is applied to multi-antenna communication systems [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In order to support multiple devices sharing the same re- sources as well as satisfying rate requirements, it is beneficial to exploit the multiple access channel (MAC) with FTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' A realistic problem follows, in practice, that each device will experience a random time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' However, instead of being a hazard to the system, this asynchronism is analyzed in [6], [7] and [8] and is shown to be beneficial to multiple access transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In [6], the author explored the capacity This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, NSERC, under a Discovery Grant and in part by the Scientific and Technological Research Council of Turkey, TUBITAK, under Grant 122E248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Zhang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Yanikomeroglu are with the Department of Systems and Computer Engineering at Carleton University, Ottawa, ON, K1S 5B6, Canada e-mail: zichaozhang@cmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='ca, halim@sce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Yuksel and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Guvensen are with the Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, 06800, Turkey, e-mail: ymelda@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='tr, guvensen@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' region of asynchronous MAC with fixed or random time delay differences and showed that these differences bring in additional gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' However, the analysis in [6] is constrained to rectangular pulse shapes in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The authors of [7] extended this limitation and derived the capacity region for band-limited pulse shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In [8], the authors studied FTN in asynchronous MAC and obtained an achievable rate region with fixed power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In this paper we derive the capacity region of the asynchronous MAC with FTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The organization of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In Section II we establish the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In Section III we derive the capacity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In Section IV we show that the capacity region for discrete MAC with finite memory defined in [6] actually leads to the same region as in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In Section V we plot the rate regions for finite number of symbols and in Section VI we conclude the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' SYSTEM MODEL The MAC is composed of K transmitters and one receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Due to imperfect clock generation or different propagation delays, signals coming from each transmitter have differ- ent time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We denote them as τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , τK, τk ∈ [0, T ], k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Without loss of generality, we assume τ1 ≤ τ2 · · · ≤ τK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' All the transmitters use the same pulse shaping filter p(t) and the same acceleration factor δ for FTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The signal transmitted from the lth user, xl(t) then has the form xl(t) = N−1 � m=0 al[m]p(t − mδT − τl), (1) where al[m] are the symbols transmitted from the lth user and N is the number of symbols transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' At the receiver, the matched filter p∗(−t) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' An additive white Gaussian noise ξ(t) with power spectral density σ2 0 is added at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' After passing through the matched filter this white noise becomes correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We denote this noise as η(t) = ξ(t)⋆p∗(−t), where ⋆ denotes the convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The signal at the output of the matched filter is y(t) = ��K k=1 xl(t) + ξ(t) � ⋆ p∗(−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In order to obtain the sufficient statistics in this asyn- chronous MAC with FTN, we need to sample according to the time delay of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Thus, we sample at all t = nδT +τk, n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , N−1, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , K and obtain K sets of samples instead of a single set [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then, the samples yl[n] corresponding to user l are written by sampling the output of 2 the matched filter, y(t), at time nδT + τl, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , N − 1, and we write yl[n] = K � k=1 N−1 � m=0 ak[m]g � (n − m)δT + (τl − τk) � + ηl[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (2) Here g(t) = p(t) ⋆ p∗(−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Furthermore, ηl[n] = η(nδT + τl) = ξ(t) ⋆ p∗(−t)|t=nδT +τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (3) By defining the N × 1 vectors yl, al and ηl to represent respectively the output samples, data symbols and noise, the input-output relationship in (2) can be written in a compact matrix product form as \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 y1 y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' yK \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 G11 G12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' G1K G21 G22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' G2K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' GK1 GK2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' GKK \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 a1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' aK \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb+ \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 η1 η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' ηK \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (4) This expression can further be simplified as y = ˜Ga + η, (5) where y = [y⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , y⊤ K]⊤, a = [a⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , a⊤ K]⊤ and η = [η⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , η⊤ K]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The matrix ˜G in (5) is KN×KN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The matrix Glk in (4) is the N × N interference matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' It represents user k’s effect on the samples of user l and its (n, m)th entry, n, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , N, is (Glk)n,m = g � (n−m)δT +(τl−τk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In this paper, we focus on the special case of K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Note that, the matrix Glk is a Toeplitz matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' An N ×N Toeplitz matrix TN has the structure (TN)i,j = ti−j, i, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Its generating function is defined as G(TN) = ∞ � k=−∞ tkejkλ, λ ∈ � −1 2, 1 2 � (6) where we denote the operation of generating function compu- tation by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' THE CAPACITY REGION ANALYSIS In this section we derive the capacity region in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The capacity region C of the two-user multiple access channel with memory is defined as [7] C = � � 1 2 − 1 2 Sk(λ)dλ≤Pk Sk(λ)≥0,λ∈[− 1 2 , 1 2] k=1,2 � (R1, R2) : 0 ≤ R1 ≤ lim N→∞ 1 N IN(a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a2) 0 ≤ R2 ≤ lim N→∞ 1 N IN(a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a1) 0 ≤ R1 + R2 ≤ lim N→∞ 1 N IN(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y) � , (7) where S1(fn) and S2(fn) are the power spectral densities of user 1 and user 2, while P1 and P2 are the power constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In (7), IN is the mutual information between two random vectors with length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In FTN signaling, the input power spectrum to the physical channel contains the effect of both data symbols as well as FTN [5], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' This can be written as Sk(λ) = 1 δT Gδ(λ)Sak(λ), (8) where Gδ(λ) is the folded spectrum defined as Gδ(λ) = 1 δT ∞ � n=−∞ ����P �λ − n δT ����� 2 = 1 δT ∞ � n=−∞ G �λ − n δT � (9) and P(·) and G(·) are respectively the continuous time Fourier transforms of p(t) and g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The data power spectrum Sak(λ), k = 1, 2, is obtained by the discrete-time Fourier transform of the autocorrelation function of input symbols, Rak[n] = E[ak[m + n]a∗ k[m]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=', Sak(λ) = ∞ � n=−∞ Rak[n]e−jλn, k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (10) Therefore the power constraint of user k is 1 δT � 1 2 − 1 2 Gδ(fn)Sk(fn)dfn ≤ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (11) In order to obtain a closed-form expression for (7), we need to calculate the mutual information expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The differential entropy of a Gaussian vector y is h(y) = 1 2 log2((2π)2N det(Σy)), (12) where Σy = E[yy†], with † denoting the Hermitian conjuga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Define matrix G = G11 = G22, the (n, m)th entry of which is g((n−m)δT ), it is easy to see that G is a Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Notice that G† 12 = G21, thus ˜G is a Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' According to [6], for any non-zero vector a, a† ˜Ga is the energy of x1(t) + x2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' therefore, the quadratic form a† ˜Ga is guaranteed to be greater than zero, and ˜G is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The colored Gaussian noise vector η has the correla- tion E[ηi[n]ηj[m]] = σ2 0(Gij)n,m, i, j ∈ {1, 2}, n, m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=', N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' As this noise process is a stationary, zero mean, colored Gaussian process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' therefore, the optimal input is also a stationary Gaussian process [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' It is also reasonable to assume that data symbols from the two users a1 and a2 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then the covariance matrix of each user is E[aka† k] = Rk, k = 1, 2, and the covariance matrix Σy can be written as Σy = ˜G �R1 0 0 R2 � ˜G† + σ2 0 ˜G, (13) where 0 is an all-zero matrix of size N × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then, mutual information expressions for the single-user rate constraints in (7) can be calculated as IN(a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a2) = h(y1|a2) − h(y1|a1, a2) (14) ≤ 1 2N log2 det � E � (Ga1 + η1)(Ga1 + η1)†�� − 1 2N log2 det � E � η1η† 1 �� (15) = 1 2N log2 det � GR1G + σ2 0G � − 1 2N log2 det � σ2 0G � (16) = 1 2N log2 det � IN + σ−2 0 GR1 � , (17) 3 and IN(a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a1) = 1 2N log2 det � IN + σ−2 0 GR2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (18) Remark 1: In order to calculate (16), we need the matrix G to be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Theoretically, a matrix is invertible as long as it is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' However, this inversion may not be numerically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' For root raised cosine pulses p(t), numerical stability is achieved if δ(1+β) ≥ 1, where β is the roll-off factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Note that the matrices G, G12, G21, R1 and R2 are all Toeplitz matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In addition, comparing (6) and (10), we observe that Sak(−λ) is the generating function of the matrix Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Since Gδ(λ) in (9) is an even function, Sak(−λ) = Sak(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then, applying Szeg¨o’s theorem [10] and [11, Theo- rem 2] on the single-user rate constraints of (7), we have Ri ≤ 1 2 � 1 2 − 1 2 log2(1 + σ−2 0 Sai(λ)Gδ(λ))dλ, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (19) To find the sum-rate constraint, we first observe that ˜G and ˜R = � R1 0 0 R2 � are block Toeplitz matrices [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then, we derive the sum-rate constraint in (7) as R1 + R2 ≤ lim N→∞ � 1 2N log2 det � E � yy†�� − 1 2N log2 det � E � η1η† 1 �� � (20) = lim N→∞ 1 2N log2 det � I2N + σ−2 0 ˜G ˜R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (21) In (21), ˜G ˜R is a block Toeplitz matrix, because the product of block Toeplitz matrices is also block Toeplitz [11, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then applying [11, Theorem 6] on the sum-rate constraint (21) we write lim N→∞IN(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y) = 1 2 � 1 2 − 1 2 log2 σ−2 0 ���� 1 + Sa1(λ)Gδ(λ) Sa2(λ)G12,δ(−λ) Sa1(λ)G21,δ(−λ) 1 + Sa2(λ)Gδ(λ) ���� dλ (22) = 1 2 � 1 2 − 1 2 log2 � 1 + σ−2 0 Sa1(λ)Gδ(λ) + σ−2 0 Sa2(λ)Gδ(λ) + σ−4 0 Sa1(λ)Sa2(λ) � |Gδ(λ)|2 − |G12,δ(λ)|2� � dλ, (23) where G12,δ(λ) is the generating function of the matrix G12 obtained via (6) and written as G12,δ(λ) = ∞ � n=∞ g(nδT + (τ1 − τ2))ejλn (24) = 1 δT ∞ � n=∞ G(λ − n δT )ej(τ1−τ2) λ−n δT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (25) Similarly G21,δ(λ) is the generating function of the matrix G21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' It is easy to see that G12,δ(λ) = (G21,δ(λ))∗ and |G12,δ(λ)|2 = |G12,δ(λ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Theorem 1: The capacity region of the two-user asyn- chronous MAC with FTN is given as C = � � 1 2 − 1 2 Sk(λ)dλ≤Pi Sk(λ)≥0, ∀λ∈[− 1 2 , 1 2 ],k=1,2 � (R1, R2) : R1 ≤ 1 2 � 1 2 − 1 2 log2(1 + σ−2 0 S1(λ))dλ R2 ≤ 1 2 � 1 2 − 1 2 log2(1 + σ−2 0 S2(λ))dλ (26) R1 + R2 ≤ 1 2 � 1 2 − 1 2 log2 � 1 + σ−2 0 S1(λ) + σ−2 0 S2(λ) + σ−4 0 S1(λ)S2(λ) � 1 − ���� G12,δ(λ) Gδ(λ) ���� 2� � dλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' It is worth mentioning that this expression is more general than [7, Theorem 1] in the sense that when δ = 1, (26) degrades to [7, Theorem 1] for orthogonal signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' AN ALTERNATIVE CAPACITY CALCULATION The capacity region of the asynchronous MAC with finite memory is defined as [6] C = closure � lim inf N→∞ CN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (27) Here CN is the achievable region for N symbols defined as CN = � p(a1)p(a2) � (R1, R2) :0 ≤ R1 ≤ I(a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a2) 0 ≤ R2 ≤ I(a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a1) 0 ≤ R1 + R2 ≤ I(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y) � , (28) where p(a1) and p(a2) means the distribution of a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Capacity for an arbitrary MAC with infinite memory cannot be defined in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' However, we believe that this same expression is valid as long as the limit exists [9], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Therefore, in this section we calculate (27) and prove that it is equal to the capacity region in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' By combining (17), (18) and (21), CN for the asynchronous MAC with FTN can be written as CN = � 1 N tr(GRk)≤Pk Rk⪰0,k=1,2 � (R1, R2) : R1 ≤ 1 2N log2 ��IN + σ−2 0 GR1 �� R2 ≤ 1 2N log2 ��IN + σ−2 0 GR2 �� R1 + R2 ≤ 1 2N log2 ����I2N + σ−2 0 � G G12 G21 G � �R1 0 0 R2 ����� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (29) In order to further push the sum-rate upper bound, we suggest the novel derivation in (30)-(35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In order for (30) to be computable, we need Remark 1 to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In step (a), we define Φ ≜ G− 1 2 G12G− 1 2 , Ψ1 ≜ G 1 2 R1G 1 2 and Ψ2 ≜ G 1 2 R2G 1 2 , where Ψ1 and Ψ2 are Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In (b), we perform singular value decomposition on the matrix Φ = UΦΛΦV † Φ, where ΛΦ = diag{λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , λN} is a diagonal matrix and λi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , N are the singular values of 4 I(a1, a2;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='G− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='2 G12G− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='G− 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='2N log2 det ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='I + σ−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='ΛΦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='Λ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� �U† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='ΦΨ1UΦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='V † ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='ΦΨ2VΦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='(34) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='2N log2 det ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='I + σ−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='ΛΦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='Λ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='� � ˜Ψ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='˜Ψ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (35) Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In (c) we define ˜Ψ1 ≜ U† ΦΨ1UΦ and ˜Ψ2 ≜ V † ΦΨ2VΦ, where ψ1i and ψ2i are the diagonal entries of ˜Ψ1 and ˜Ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then we apply [6, Lemma 2] on (35) to upper bound the mutual information as I(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y) = 1 2N log2 � I + σ−2 0 � I ΛΦ Λ† Φ I � � ˜Ψ1 0 0 ˜Ψ2 �� ≤ 1 2N N−1 � i=0 log2 � 1 + ψ1i σ2 0 + ψ2i σ2 0 + ψ1iψ2i σ4 0 (1 − |λi|2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (36) The equality in (36) is achieved when ˜Ψ1 and ˜Ψ2 are diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Moreover, in order for [6, Lemma 2] to be valid or the upper bound to be achieved, we need the complex scalars λi to satisfy |λi| ≤ 1, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=', N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' As the matrix ˜G is positive definite, for any non-zero vector v, we have v† ˜Gv (a) = ˜v† �G 1 2 0 0 G 1 2 �† ˜G �G 1 2 0 0 G 1 2 � ˜v = ˜v† � I Φ Φ† I � ˜v > 0, where (a) is because v ≜ � G 1 2 0 0 G 1 2 � ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Thus, the matrix � I Φ Φ† I � is positive definite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Then according to [13], |λi| < 1, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=', N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Next, the upper bound for I(a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a2) and I(a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a1) can be obtained as in [6] as I(a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a2) ≤ 1 2N N−1 � i=0 log2 � 1 + ψ1i σ2 0 � (37) I(a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' y|a1) ≤ 1 2N N−1 � i=0 log2 � 1 + ψ2i σ2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (38) The transmit power constraint for each user in this N-block asynchronous multiple access channel with FTN is calculated as 1 NδT tr(GRk) = 1 NδT tr(G 1 2 RkG 1 2 ) = 1 NδT N−1 � i=0 ψki ≤ Pk, k = 1, 2 (39) where ψki ≥ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Therefore, the region CN in (29) is obtained using (36)-(39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' As in the asynchronous MAC with rectangular pulses and without FTN [6], the input distribution achieving the upper bound in (37) has a covari- ance matrix G−1 scaled according to the power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' However, this distribution does not achieve the upper bound in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Lemma 1: If we have �∞ n=−∞ |n|tn < ∞, then the discrete Fourier transform (DFT) vectors are asymptotically the eigenvectors of Toeplitz matrix TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Proof 1: Although this result is discussed in [14], it is not proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We omit the full proof due to page limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Corollary 1: The region defined in (27) with CN defined using (36), (37), and (38) with the power constraint in (39) is the same as the capacity region in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Proof 2: Since the raised cosine filter g[n] satisfies �∞ n=−∞ |n|g[n] < ∞, by Lemma 1, G− 1 2 is an asymp- totically Toeplitz matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Its generating function is G 1 2 δ (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We know that |λi|2’s are the eigenvalues of the Hermitian matrix Φ†Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' As G− 1 2 is asymptotically Toeplitz and the product of asymptotically Toeplitz matrices is also asymp- totically Toeplitz [10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='3], the product Φ†Φ = G− 1 2 G† 12G−1G12G− 1 2 is also asymptotically Toeplitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The generating function of Φ†Φ is ��� G12,δ(λ) Gδ(λ) ��� 2 , which is the product of the generating functions of the individual matrices in the above Φ†Φ expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The eigenvalues of a Toeplitz matrix asymptotically approximate the samples of its gener- ating function [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Thus we have |λi|2 ≈ ��� G12,δ(i/N) Gδ(i/N) ��� 2 , i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=', N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Moreover, the values |λi|2 are the samples from the constant spectrum ��� G12,δ(λ) Gδ(λ) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Hence the discussion in [6] about time and frequency domain capacity region comparison applies, and we conclude that the two regions are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we plot the capacity region for N = 10 for root raised cosine pulses p(t) with roll-off factor β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We set the SNR for both users to be 20 dB, and the signaling period T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 R1(bit/s/Hz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 R2(bit/s/Hz) aMAC,( , )=(1,0) aMAC,( , )=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25) aMAC,( , )=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25),iid aMAC,( , )=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='9,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25) aMAC,( , )=(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25) MAC,( , )=(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Capacity regions for asynchronous MAC with different (δ, β) pairs, asynchronous MAC with FTN with independent and identically distributed (iid) inputs, and synchronous MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' 1, we compare the capacity region (aMAC, (δ, β) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25)) with the performance of synchronous MAC (MAC, (δ, β) = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25)) [9], asynchronous MAC without FTN (aMAC, (δ, β) = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25)) [7], and asynchronous MAC with FTN without power optimization (aMAC, (δ, β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25), iid) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' For all the asynchronous simulations in this figure, we set the time difference τ to be δT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We can see that FTN brings a significant gain for both single-user rates as well as the sum-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' The region without power optimization (aMAC, (δ, β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25), iid) is obtained by setting ψki’s to be PkδT, ∀i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=', N − 1, k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We observe that both asynchronous transmission and FTN significantly improve the rate region and optimal power allocation is necessary to establish the capacity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In this figure, we also compare the results with asynchronous MAC with FTN (aMAC, (δ, β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25)) and with asynchronous MAC with Nyquist signaling (aMAC, (δ, β) = 1, 0)) as an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We observe that for a given β value, we should let δ(1 + β) as close to 1 as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' This also supports the discussion of choices of (δ, β) pairs in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Although impractical, the best choice is Nyquist transmission with ideal sinc pulses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' (δ, β) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We then study the influence of time difference τ between two users on the performance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' In this figure we set (δ, β) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We can see that when the time difference between two users is half the sampling period δT 2 , the performance is better than the performance with other values of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' This suggests that [7, Proposition 2] also holds in the presence of FTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' CONCLUSION In this paper we derive the capacity region of asynchronous multiple access channels with FTN signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' We calculate the capacity region both in frequency and time domains, and show that the two regions are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' As a side result, we 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 R1(bit/s/Hz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 R2(bit/s/Hz) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='5 T =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='3 T =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='8 T =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content='1 T Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' Capacity regions for asynchronous MAC with FTN with different time differences τ between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' prove that the DFT vectors are asymptotically the eigenvectors of the Toeplitz matrix TN as long as �∞ n=−∞ |n|tn < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfaQD2/content/2301.02334v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': 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In this paper we study exponential utility indifference pricing of pure endowment +policies in a stochastic-factor model for an insurance company, which can also invest in a financial +market. Specifically, we propose a modeling framework where the hazard rate is described by an +observable general diffusion process and the risky asset price evolves as a jump diffusion affected +by a continuous-time finite-state Markov chain representing regimes of the economy. Using the +classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we describe +the optimal investment strategies with and without the insurance derivative and characterize the +indifference price in terms of a classical solution to a linear PDE. We also provide its probabilistic +representation via an extension of the Feynman-Kac formula show that it satisfies a final value +problem. Furthermore, we also discuss the indifference price for a portfolio of insurance policies +and for a term life insurance. Finally, some numerical experiments are performed to address +sensitivity analyses. +Keywords: Pure endowment; regime-switching; jump processes; optimal investment; stochastic +control; indifference pricing. +JEL Classification: G22; C61; G11. +AMS Classification: 91B30; 91B25; 93E20; 60J27. +1. Introduction +The utility indifference pricing method, initially proposed by (Hodges and Neuberger, 1989) and +refined by (Davis et al., 1993), has gained much attention in the literature on pricing and hedging +contingent claims, see e.g. (Henderson and Hobson, 2009) for a survey. According to this tech- +nique, the indifference seller’s (insurer’s, in this framework) price is defined at the level where the +issuer of the contract is indifferent between entering the market on its own, or selling the claim +and entering the market with the collected premium. It can be determined by solving an equation +involving two value functions, resulting from the stochastic control problems with and without +Alessandra Cretarola(�), Department of Mathematics and Computer Science, University of +Perugia, Via Luigi Vanvitelli, 1, I-06123 Perugia, Italy. +Benedetta Salterini, Department of Mathematics and Computer Science, University of Firenze, +Viale Morgagni, 67/A, I-50134 Firenze, Italy. +E-mail addresses: alessandra.cretarola@unipg.it, benedetta.salterini@unifi.it. +1 + +2 +A. CRETAROLA AND B. SALTERINI +insurance liabilities. The indifference pricing approach has become a popular method for evalu- +ating derivatives in incomplete markets and has been successfully applied to price insurance con- +tracts in e.g. (Young and Zariphopoulou, 2002; Moore and Young, 2003; Ludkovski and Young, +2008; Delong, 2009; Eichler et al., 2017; Liang and Lu, 2017; Ceci et al., 2020). +Precisely, in +(Young and Zariphopoulou, 2002) explicit results are derived for an exponential utility function by +solving the Hamilton Jacobi equation in a market driven by a geometric Brownian motion when +the insurance risk is independent of the financial risk. A more general framework is studied in +(Moore and Young, 2003), where the payment amount of the endowment policy is a function of the +underlying risky asset. In (Ludkovski and Young, 2008), the authors investigate pricing of mor- +tality contingent claims under the effects of the stochastic hazard and interest rates. The pricing +and hedging problem for a group of life insurance contracts in the presence of systematic mortality +risks in a market model driven by a Levy process is considered in (Delong, 2009). In (Eichler et al., +2017), the authors analyze the valuation of catastrophe derivatives, while in (Liang and Lu, 2017) +they investigate the pricing problem for life insurance contracts with equity-indexed life contingent +payments, in a financial market which allows for shot-noise effects in the stock prices. Finally, +results on the valuation of pure endowment policies under partial information via backward sto- +chastic differential equations can be found in (Ceci et al., 2020). Pricing and hedging of unit-linked +life insurance contracts via other techniques has been studied e.g. in (Ceci et al., 2014, 2015, 2017), +where the authors apply the (local) risk-minimization approach in a partial information framework. +It is worth noting that the indifference pricing approch is widely used also in non-life insurance, +for instance to evaluate insurance-linked securities, see e.g. (Liu et al., 2020). +In this paper, we investigate the indifference pricing problem of pure endowment contracts for an +insurance company in a continuous-time financial market where the risky asset price dynamics can +exhibit jumps and is affected by regime changes, when the hazard rate governing the population +mortality is stochastic and driven by a general diffusion process. +A pure endowment is a life +insurance policy which yields a sum of money after a specified number of years, provided some +nominated person be alive at that time. Precisely, we consider a pure endowment with maturity of +T years for which the terminal survival benefit is given by a fixed amount, payable provided that +the insured person is still alive at time T. Our modeling framework takes into account financial +risk due to price fluctuations, economic risk (or regime-switching risk) arising from structural +changes in economic conditions and mortality risk. +Inspired by (Ludkovski and Young, 2008), +we consider a more sophisticated financial market introducing several jumps in the stock price +behavior. To the best of our knowledge, indifference pricing of life-insurance liabilities in a Markov- +modulated framework accounting for a market behavior affected by long-term macroeconomic +conditions described by the continuous-time Markov chain, possible jumps in the risky asset price +dynamics and stochastic hazard rate, is taken up for the first time. +In particular, we consider a financial market with a riskless asset and a risky asset. The latter +is described by a jump diffusion process where the appreciation rate and the volatility depend on +an observable continuous-time, finite-state Markov chain representing the regimes of the economy. +Taking a mixture of continuous and jump processes for the stock price dates back to (Merton, + +3 +1976) and it can also be found in more recent papers, see e.g. +(Ceci and Gerardi, 2009) and +(Xiao and Zhao, 2021). It seem reasonable to deal with this financial market model, indeed recent +research provides strong empirical evidence of jumps in stock prices, see (Jawadi et al., 2015). +Moreover, the stock price behavior could be also affected by long-term macroeconomic conditions +that should be included in the market modeling and represented by another stochastic process. +Therefore, the presence of an exogenous term affecting the risky asset makes the model even more +realistic. +This stochastic factor may represents some environmental conditions, social circum- +stances, economic crisis or natural phenomena, that can have a considerable impact on financial +returns. The economic effects of catastrophic events, climate changes and pandemics, as for in- +stance the COVID-19, on the financial market are recently analyzed, see, e.g., (Baek et al., 2020; +Just and Echaust, 2020; Tesselaar et al., 2020; Wang et al., 2020). Here, we address this modeling +issue by assuming that all these exogenous events are aggregated to create different regimes, as e.g. +in (Sotomayor and Cadenillas, 2009; Altay et al., 2018; Cretarola and Figà-Talamanca, 2020). +An additional feature of our model is to take the hazard rate of individuals as a general diffusion +process, in order to capture the unexpected changes in mortality. We are not the first to consider +stochastic mortality rates, see e.g. (Milevsky and Promislow, 2001; Dahl, 2004; Dahl and Møller, +2006; Biffis, 2005; Ludkovski and Young, 2008). Indeed, empirical evidence suggests that wars, +medical breakthroughs, developments in healthcare and improved lifestyles combine to affect hu- +man mortality in a fluctuating and unpredictable manner. The uncertainty given by minuscule and +continuous movements of the mortality intensity is usually represented by a Brownian motion, see +(Cairns et al., 2006) for an overview. As a consequence, it seems reasonable to require that in our +setting the exogenous stochastic factor, representing long-term environmental changes, does not +affect the mortality intensity; therefore the insurance market remains independent of the financial +market. +We price the policy through the principle of equivalent utility by comparing the maximal expected +utility functions with and without writing the life insurance contract. Under exponential utility +and using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman (in short +HJB) equation, we describe the optimal investment strategy and show verification results for the +value functions of the problems without and with insurance liabilities via classical solutions to a +linear partial differential equation (in short PDE) and a system of ordinary differential equations +(in short ODEs), see Theorem 4.6 and Theorem 4.11. Further, we characterize the indifference price +of the pure endowment in Proposition 5.2. We prove that it solves a proper final value problem and +we also obtain its probabilistic representation by means of an extended version of the Feynman- +Kac formula. We also discuss the indifference price for a group of insurance contracts and another +kind of mortality-contingent claim. Finally, numerical experiments are performed to investigate +some features of our model specification, emphasizing the impact of the regime-switching and the +randomness effect introduced by the stochastic hazard rate. +The paper is organized as follows. In Section 2 we introduce the mathematical framework and +describe the Markov-modulated financial-insurance market. The pricing problem formulation via +utility indifference pricing can be found in Section 3. In Section 4 we apply the HJB approach to + +4 +A. CRETAROLA AND B. SALTERINI +the resulting stochastic control problems and provide the Verification Theorems and the optimal +investment strategies. The characterization of the indifference price of the pure endowment policy +is given in Section 5. In Section 6 we illustrate some numerical results and sensitivity analyses. +Finally, technical proofs are collected in Appendix A and how to derive the HJB equation for the +problem with insurance liability is shown in Appendix B. +2. Modeling framework +We consider a complete probability space (Ω, F, P) endowed with a filtration G = {Gt, t ∈ [0, T]}, +satisfying the usual conditions of completeness and right continuity, where T > 0 is a fixed, finite +time horizon. Specifically, the filtration G is given by +G = F ∨ FI, +where the filtration F = {Ft, t ∈ [0, T]} models the information flow in the financial market and +FI = {F I +t , t ∈ [0, T]} contains information about the lifetime of the individual insured. We assume +that the subfiltrations F and FI are independent. +To describe some possible structural changes in economic conditions, we introduce an irreducible +and continuous-time Markov chain X = {Xt, t ∈ [0, T]} with finite state space X = {1, 2, . . . , M}, +whose transition probabilities satisfy +P(Xt+δt = j|Xt = i) = aijδt + o(δt), i ̸= j; +P(Xt+δt = i|Xt = i) = 1 + aiiδt + o(δt), +when δt −→ 0, where for each i ∈ X we have +aij ≥ 0 for each i ̸= j +and +aii = − +M +� +j=1 +aij. +Here, Xt represents the regime of the economy at time t, and M the number of regimes. Let +A = (aij)i,j∈X denote the generating Q-matrix of the Markov chain X. It is convenient to represent +X as a stochastic integral with respect to a Poisson random measure. Following the description of +(Basak et al., 2011), for i, j ∈ X , with i ̸= j, we denote by ∆ij the consecutive (with respect to the +lexicographic ordering on X ×X ) left-closed right-open intervals of the real line, each having length +aij and define a function h : X × R −→ RM by embedding {1, 2, . . . , M} into RM (identifying i +with ei ∈ RM), as follows +h(i, z) = +� j − i, +if z ∈ ∆ij +0, +otherwise. +Then, we get +Xt = X0 + +� t +0 +� +R +h(Xv−, z)P(dz, dv), +t ∈ [0, T], +(2.1) +where the integration is over the interval (0, t] and P(dz, dt) is a Poisson random measure with +intensity m(dz)dt, with m(dz) being the Lebesgue measure on R. Let ˆP(dz, dt) be the compensated +Poisson random measure, i.e. ˆP(dz, dt) = P(dz, dt) − m(dz)dt. + +5 +In this setting, we consider a financial market consisting of a locally risk-free money market account +and one stock as a risky asset. The price process B = {Bt, t ∈ [0, T]} of the locally risk-free asset +is described by +dBt = rBtdt, +B0 = 1, +where r is a positive constant denoting the risk-less interest rate. The risky asset price process +S = {St, t ∈ [0, T]} evolves over time according to the following Markov-modulated dynamics +dSt = St− +� +µ(t, Xt)dt + σ(t, Xt)dZS +t + K1(t, Xt−)dN1 +t − K2(t, Xt−)dN2 +t +� +, +S0 = s ∈ R+, +(2.2) +where R+ = (0, +∞). Here, ZS = {ZS +t , t ∈ [0, T]} is a standard Brownian motion independent +of X and N1 = {N1 +t , t ∈ [0, T]} and N2 = {N2 +t , t ∈ [0, T]} are independent Poisson processes +defined on (Ω, F, P; F). Furthermore, we suppose that N1, N2 are independent of ZS and X and +that the F-intensities of N1 and N2 are positive deterministic functions Θ1 : [0, T] −→ R+ and +Θ2 : [0, T] −→ R+, respectively. The coefficients µ : [0, T]×X −→ R+ and σ : [0, T]×X −→ R+ are +measurable functions which model the appreciation rate and the volatility of the stock, respectively, +such that µ(t, i) > r, for all (t, i) ∈ [0, T] × X and +� T +0 +� +µ(t, Xt) + σ2(t, Xt) +� +dt < ∞ +P-a.s.. +(2.3) +Moreover, K1 : [0, T] × X −→ R+ and K2 : [0, T] × X −→ R+ are measurable functions such +that Kl(t, i) > 0, l = 1, 2 K2(t, i) < 1, for every (t, i) ∈ [0, T] × X . From (2.1) and (2.2) it is +clear that the pair (S, X) is an (F, P)-Markov process. The main motivation for introducing a +regime-switching behavior is to have a model capable of describing the risky asset price dynamics +under different market conditions. +Remark 2.1. By the Doléans-Dade exponential formula, condition K2(t, i) < 1 allows us to write +St = seLt, +t ∈ [0, T], +where the logreturn process L = {Lt, t ∈ [0, T]} is given by +dLt = +� +µ(t, Xt) − 1 +2σ2(t, Xt) +� +dt + σ(t, Xt)dZS +t + ln(1 + K1(t, Xt−))dN1 +t + ln(1 − K2(t, Xt−))dN2 +t , +with L0 = 0. +Proposition 2.2. If we assume that +� T +0 +� +K2 +1(t, Xt−)Θ1(t) + K2 +2(t, Xt−)Θ2(t) +� +dt < ∞ +P-a.s., +(2.4) +then the process S is an F-semimartingale with decomposition +St = s + AS +t + MS +t , +where AS = {AS +t , t ∈ [0, T]} defined as +AS +t = +� t +0 +Sv− (µ(v, Xv−) + K1(v, Xv−)Θ1(v) + K2(v, Xv−)Θ2(v)) dv, + +6 +A. CRETAROLA AND B. SALTERINI +is an R-valued process with finite variation paths and AS +0 = 0, while MS = {MS +t , t ∈ [0, T]} given +by +MS +t = +� t +0 +Svσ(v, Xv)dZS +v + +� t +0 +Sv−K1(v, Xv−){dN1 +v −Θ1(v)dv}− +� t +0 +Sv−K2(v, Xv−){dN2 +v −Θ2(v)dv} +is an F-local martingale with MS +0 = 0. +Proof. Conditions (2.3) and (2.4) imply that the process R = {Rt, t ∈ [0, T]} defined as +Rt = +� t +0 +� +µ(v, Xv)dv + σ(v, Xv)dZS +v + K1(v, Xv−)dN1 +v − K2(v, Xv−)dN2 +v +� +is an F-semimartingale. Noting that +dSt = St−dRt, +we can conclude the proof. +□ +Now, we consider an individual to be insured and a stochastic model for the mortality of the +equivalent age cohort of the population. We assume that the hazard rate (or force of mortality) +is governed by a diffusion process, i.e. we describe the mortality intensity as a stochastic process +Λ = {λt, t ∈ [0, T]} that is given by the following stochastic differential equation (in short SDE) +dλt = b(t, λt)λtdt + c(t, λt)λtdZΛ +t , +λ0 = λ ∈ R+. +(2.5) +Here, ZΛ = {ZΛ +t , t ∈ [0, T]} is an additional standard Brownian motion on (Ω, F, P; FI). +Moreover, b : [0, T] × R −→ R and c : [0, T] × R −→ R are two measurable functions such that a +unique strong solution to (2.5) exists and the following conditions hold +E +�� T +0 +|b(t, λt)λt|dt + +� T +0 +c(t, λt)2λ2 +tdt +� +< ∞, +(2.6) +sup +t∈[0,T] +E +� +λ2 +t +� +< ∞. +(2.7) +These conditions are satisfied if, for instance, the coefficients of the SDE (2.5) fulfill the classical +Lipschitz and sublinear growth conditions, see e.g. (Gihman and Skorohod, 1972). We observe +that, the mortality rate of the insured is generally different from that of its age cohort. However, +to keep the framework tractable we consider individuals subjected to the same stochastic hazard +rate, as e.g. in (Ludkovski and Young, 2008). +Let τ be a non negative random variable on (Ω, F, P) which represents the remaining lifetime of the +given individual of the reference population with mortality rate Λ. Denote by D = {Dt, t ∈ [0, T]} +the death indicator process associated to τ by setting Dt := 1{τ≤t}, for every t ∈ [0, T]. We assume +that D is an FI-adapted process independent of Λ. + +7 +3. The indifference pricing problem formulation +Now, we assume that the insurance company issues a unit-linked life insurance policy, which is a +long term insurance contract whose payoff depends on the insured remaining lifetime and on the +underlying stock. In particular, we consider a pure endowment contract with maturity of T years, +which pays a fixed amount if the policyholder is still alive. Then, the associated payoff is given by +the random variable +GT := K1{τ>T} = K(1 − DT), +(3.1) +where K is a positive constant. The goal is to evaluate the pure endowment policy with payoff +given by (3.1) in the Markov-modulated model outlined in Section 2. Since the financial market +consists of two primary securities and several sources of random shocks due to mortality events +and structural changes in economic conditions, it turns out to be incomplete. +Therefore, we apply the indifference pricing approach assuming that the insurance company pref- +erences towards the risk are given by an exponential utility function of the form +u(w) = −e−αw, +w ∈ R, +where α is a positive parameter which measures the absolute risk aversion. In the underlying +financial market, the insurance company starts out with an initial wealth w, and then proceeds to +trade dynamically among the risky asset and the locally risk-free asset, following a self-financing +strategy. Let Π = {Πt, t ∈ [0, T]} be the total amount of wealth invested in the stock, with +the remainder of wealth in the money market account. The insurance company is also allowed +to short-sell and to borrow/lend any infinitesimal amount, so that Πt ∈ R, for each t ∈ [0, T]. +Precisely, given an initial wealth w ∈ R+ +0 , the insurance company wealth process {W Π +t , t ∈ [0, T]} +associated to a given strategy Π evolves over time as +dW Π +t = Πt +dSt +St− ++ (W Π +t − Πt)dBt +Bt += (rW π +t + Πt (µ(t, Xt) − r)) dt + Πtσ(t, Xt)dZS +t + Πt +� +K1(t, Xt−)dN1 +t − K2(t, Xt−)dN2 +t +� +, +(3.2) +with W Π +0 = w ∈ R+ +0 . +Remark 3.1. It can be checked that the solution to the SDE (3.2) is given by +W Π +t = W Π +0 ert + +� t +0 +er(t−s)Πs(µ(s, Xs) − r)ds + +� t +0 +er(t−s)Πsσ(s, Xs)dZS +s ++ +� t +0 +er(t−s)Πs +� +K1(s, Xs−)dN1 +s − K2(s, Xs−)dN2 +s +� +, +(3.3) +with W Π +0 = w. +In order to characterize the indifference price of the pure endowment, we introduce two optimal +investment problems, with and without insurance liabilities. We start by defining the class of +admissible strategies. + +8 +A. CRETAROLA AND B. SALTERINI +Definition 3.2. An admissible strategy is a self-financing portfolio identified by an R-valued G- +predictable process Π = {Πt, t ∈ [0, T]} such that +E +�� T +0 +|Πt|(µ(t, Xt) − r)dt +� +< ∞, +E +�� T +0 +Π2 +tσ2(t, Xt)dt +� +< ∞, +E +�� T +0 +|Πt| +� +K1(t, Xt−)Θ1(t) + K2(t, Xt−)Θ2(t) +� +dt +� +< ∞. +(3.4) +We denote by A the set of G-admissible strategies. Whenever the controls are restricted to the time +interval [t, T], we will use the notation At. +Now, we assume that the following assumptions are in force throughout the paper. +Assumption 3.3. +(i) There exist three positive constants M1, M2 and K such that +Θ1(t) ≤ M1, +Θ2(t) ≤ M2, +K1(t, i) ≤ K, +for every (t, i) ∈ [0, T] × X . +(ii) There is a constant C > 0 such that µ(t,i)−r +σ(t,i) +≤ C, for every (t, i) ∈ [0, T] × X . +In particular, Assumption 3.3(i) provides a sufficient condition for a strategy Π to be admissible +as shown in the next result. +Proposition 3.4. Let Π = {Πt, t ∈ [0, T]} be a G-predictable strategy with values in R. Assume +there exists a square-integrable function η : [0, T] × X → (0, +∞) such that +|Πt| ≤ η(t, Xt), +t ∈ [0, T], +P − a.s. +(3.5) +and +� T +0 +η(s, i) +� +(µ(s, i) − r) + η(s, i)σ2(s, i) +� +ds < ∞, +∀i ∈ X . +(3.6) +Then, Π is an admissible strategy, i.e. Π ∈ A. +Proof. We note that by (3.6), we have +E +�� T +0 +|Πs| +� +(µ(s, Xs) − r) + Πsσ2(s, Xs) +� +ds +� +≤ E +�� T +0 +η(s, Xs) +� +(µ(s, Xs) − r) + η(s, Xs)σ2(s, Xs) +� +ds +� +≤ +max +i=1,...,M +� T +0 +η(s, i) +� +(µ(s, i) − r) + η(s, i)σ2(s, i) +� +ds < ∞. +Finally, in view of Assumption 3.3(i), condition (3.4) is satisfied and this concludes the proof. +□ +We consider the case where the insurance company simply invests its wealth in the financial market, +without writing the insurance derivative. Then, the goal is the following. + +9 +Problem 3.5. To maximize the expected utility of its terminal wealth, i.e. to solve +sup +Π∈A +E +� +− e−αW Π +T +� +. +Let (t, w, i) ∈ [0, T] × R × X . In a dynamic framework, we define the corresponding value function +¯V by +¯V (t, w, i) := sup +Π∈At +Et,w,i +� +− e−αW Π +T (t,w)� +, +(3.7) +where Et,w,i denotes the conditional expectation given W Π +t = w and Xt = i, and {W Π +s (t, w), s ∈ +[t, T]} stands for the solution to equation (3.2) with initial condition W Π +t += w. Note that, since +the coefficients µ, σ, K1 and K2 only depend on t and i, it is possible to absorb the stock price in +the wealth and therefore to remove the variable corresponding to S. +Now, we suppose that the insurance company invests its wealth in the market, writing a pure +endowment contract with payoff given in (3.1). In this case, the goal of the insurance company is +the following. +Problem 3.6. To maximize the expected utility of its terminal wealth, i.e. to solve +sup +Π∈A +E +� +− e−α(W Π +T −GT )� +, +where GT is defined in (3.1). +Let (t, w, λ, i) ∈ [0, T] × R × R+ × X . We define the corresponding value function V as +V (t, w, λ, i) := sup +Π∈At +Et,w,λ,i +� +− e−α(W Π +T (t,w)−GT )� +, +(3.8) +where Et,w,λ,i denotes the conditional expectation given W Π +t += w, λt = λ and Xt = i and we +implicitly condition on Gt = K. +Remark 3.7. We note that the control Π = 0 is admissible and such that +Et,w,i +� +e−αW 0 +T (t,w)� +< ∞, +for each (t, w, i) ∈ [0, T] × R × X . +Et,w,λ,i +� +e−α(W 0 +T (t,w)−GT )� +< ∞, +for each (t, w, λ, i) ∈ [0, T] × R × R+ × X . This implies that +ess sup +Π∈At +E +� +−e−αW Π +T +� +> −∞, +ess sup +Π∈At +E +� +−e−α(W Π +T −GT )� +> −∞, +P − a.s., t ∈ [0, T], +and as a consequence that +sup +Π∈A +E +� +−e−αW Π +T +� +> −∞, +sup +Π∈A +E +� +−e−α(W Π +T −GT )� +> −∞. + +10 +A. CRETAROLA AND B. SALTERINI +4. The optimal investment problems +In this section, applying the classical stochastic control approach based on the Hamilton-Jacobi- +Bellman (in short HJB) equation, we characterize the optimal investment strategies and provide +verification results for the value functions ¯V and V given in (3.7) and (3.8), respectively. +4.1. The pure investment problem. Firstly, we consider the case where the insurance company +simply invests in the underlying financial market, so the corresponding value function ¯V is given +by (3.7). +Let us consider the HJB equation with final condition that the value function ¯V is expected to +solve, if sufficiently smooth: +� supΠ∈R ¯LΠ +i ¯V (t, w, i) = 0, +∀(t, w, i) ∈ [0, T) × R × X , +¯V (T, w, i) = −e−αw, +∀(w, i) ∈ R × X , +(4.1) +where ¯LΠ +i denotes the Markov generator of (W Π, X) associated with a constant control Π ∈ R, +given by +¯LΠ +i f(t, w, i) += ∂f +∂t (t, w, i) + +� +rw + (µ(t, i) − r)Π +� ∂f +∂w(t, w, i) + 1 +2σ2(t, i)Π2 ∂2f +∂w2(t, w, i) + +� +j∈X +aijf(t, w, j) ++ Θ1(t) +�¯V (t, w + ΠK1(t, i), i) − ¯V (t, w, i) +� ++ Θ2(t) +�¯V (t, w − ΠK2(t, i), i) − ¯V (t, w, i) +� +, +(4.2) +for every (t, w, i) ∈ [0, T] × R × X and for every function f(·, ·, i) in C1,2, given i ∈ X , which is +sufficiently integrable. +Remark 4.1. Since the pair (W Π, X) is a Markov process, any Markovian control is of the form +Πt = Π(t, W Π +t , Xt). The generator ¯LΠ +i f(t, w, i) associated to a general Markovian strategy can be +easily obtained by replacing Π with Π(t, w, i) in (4.2). +Now, we consider the ansatz ¯V (t, w, i) = −e−wαer(T −t)ϕ(t, i), with (t, w, i) ∈ [0, T] × R × X , for a +suitable function ϕ, which is motivated by the following result. +Proposition 4.2. Assume that there exists a unique function ϕ(·, i), for each i ∈ X , solution to +the following Cauchy problem: + + + +∂ϕ +∂t (t, i) = H(t, ϕ(t, i)), +t ∈ [0, T), +ϕ(T, i) = 1, +(4.3) +where +H(t, ϕ(t, i)) = − +� +j∈X +ϕ(t, j)aij − ϕ(t, i) inf +Π∈R +¯ΨΠ(t, i), +(4.4) + +11 +with the function ¯ΨΠ : [0, T] × X → R defined by +¯ΨΠ(t, i) = − αer(T−t)(µ(t, i) − r)Π + 1 +2α2e2r(T−t)σ2(t, i)Π2 + Θ1(t) +� +e−αΠK1(t,i)er(T −t) − 1 +� ++ Θ2(t) +� +eαΠK2(t,i)er(T −t) − 1 +� +. +(4.5) +Then, the function +¯V (t, w, i) = −e−wαer(T −t)ϕ(t, i), +(4.6) +solves the HJB problem given in (4.1). +Proof. From the expression (4.6), we can easily verify that the original HJB problem given in (4.1) +reads as follows +∂ϕ +∂t (t, i) + +� +j∈X +ϕ(t, j)aij + inf +Π∈R +� +− αer(T−t)ϕ(t, i)(µ(t, i) − r)Π + 1 +2α2e2r(T−t)ϕ(t, i)σ2(t, i)Π2 ++ ϕ(t, i)Θ1(t) +� +e−αΠK1(t,i)er(T −t) − 1 +� ++ ϕ(t, i)Θ2(t) +� +eαΠK2(t,i)er(T −t) − 1 +�� += 0, +t ∈ [0, T), +(4.7) +with final condition ϕ(T, i) = 1, for all i ∈ X . Thus, if we define the function ¯ΨΠ by means of +expression (4.5), equation (4.7) can be written as +∂ϕ +∂t (t, i) + +� +j∈X +ϕ(t, j)aij + ϕ(t, i) inf +Π∈R +¯ΨΠ(t, i) = 0 +and we find out the problem (4.3). +□ +The previous result suggests to focus on the minimization of the function (4.5), that is the aim of +the next subsection. +4.1.1. Optimal investment strategy without the insurance derivative. Now, we study the following +minimization problem +inf +Π∈R +¯ΨΠ(t, i), +(4.8) +where the function ¯ΨΠ is introduced in (4.5). +Proposition 4.3. The following equation +σ2(t, i)αer(T−t)Π − (µ(t, i) − r) += K1(t, i)Θ1(t)e−αΠK1(t,i)er(T −t) − K2(t, i)Θ2(t)eαΠK2(t,i)er(T −t). +(4.9) +admits at least a solution �Π(t, i) in R for any (t, i) ∈ [0, T] × X and the minimization problem +(4.8) has a unique solution Π∗(t, i) = �Π(t, i), for all (t, i) ∈ [0, T] × X . + +12 +A. CRETAROLA AND B. SALTERINI +Proof. Firstly, we observe that ¯ΨΠ(t, i) is continuous with respect to Π ∈ R, for every (t, i) ∈ +[0, T] × X and has continuous first and second order derivatives with respect to Π ∈ R, which are +respectively given by +∂ ¯ΨΠ +∂Π (t, i) = −αer(T−t)(µ(t, i) − r) + σ2(t, i)α2e2r(T−t)Π − αer(T−t)K1(t, i)Θ1(t)e−αΠK1(t,i)er(T −t) ++ αer(T−t)K2(t, i)Θ2(t)eαΠK2(t,i)er(T −t), +∂2 ¯ΨΠ +∂Π2 (t, i) = α2e2r(T−t)σ2(t, i) + α2e2r(T−t)K2 +1(t, i)Θ1(t)e−αΠK1(t,i)er(T −t) ++ α2e2r(T−t)K2 +2(t, i)Θ2(t)eαΠK2(t,i)er(T −t). +Note that these derivatives are well defined and ∂2 ¯ΨΠ +∂Π2 (t, i) > 0, for every (t, i) ∈ [0, T] × X ; +therefore, the function ¯ΨΠ(t, i) is strictly convex in Π ∈ R. Moreover, it is easy to check that, for +any (t, i) ∈ [0, T] × X , we have +lim +Π−→+∞ +∂ ¯ΨΠ +∂Π (t, i) −→ +∞, +while +lim +Π−→−∞ +∂ ¯ΨΠ +∂Π (t, i) −→ −∞. +As a consequence, being ∂ ¯ΨΠ +∂Π (t, i) a continuous function in Π ∈ R, there exists �Π(t, i) ∈ R such +that ∂ ¯ΨΠ +∂Π (t, i) = 0, for every (t, i) ∈ [0, T]×X , that is, (4.9) is satisfied. Since the function ¯ΨΠ(t, i) +is strictly convex, the stationary point �Π(t, i) ∈ R is unique and provides the unique minimizer +Π∗(t, i) = �Π(t, i) on R. +□ +Remark 4.4. We point out that Π∗ = Π∗(t, i), i.e. the solution of the problem (4.8) depends +on time and on the Markov chain, since it solves equation (4.9). This means that the optimal +investment strategy evolves over time and changes according to the different economic regimes. +Moreover, we note that Π∗ does not depend on wealth, as usually happens when the investor’s +preferences are described by an exponential utility function. +Proposition 4.5 (Properties of Π∗). The following condition is satisfied +min + + +0, +ln +� +µ(t,i)−r +M2 +� +αer(T−t) + + + ≤ Π∗(t, i) ≤ µ(t, i) − r + CM1 +σ2(t, i)αer(T−t) , +for all (t, i) ∈ [0, T] × X , where C, M1 ∈ R+ are the constants limiting the functions K1 and Θ1, +respectively. + +13 +Proof. By Proposition 4.3 (we omit the dependence in Π∗ on (t, i)), we get the upper limit and the +lower limit for Π∗. If Π∗0 is non-negative, we have +0 = σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − K1(t, i)Θ1(t)e−αΠ∗K1(t,i)er(T −t) ++ K2(t, i)Θ2(t)eαΠ∗K2(t,i)er(T −t) +> σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − K1(t, i)Θ1(t)e−αΠ∗K1(t,i)er(T −t) +≥ σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − CM1e−αΠ∗K1(t,i)er(T −t) +≥ σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − CM1, +which implies +Π∗(t, i) ≤ µ(t, i) − r + CM1 +σ2(t, i)αer(T−t) , +for all (t, i) ∈ [0, T] × X . Otherwise, if Π∗ is non-positive, we get +0 = σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − K1(t, i)Θ1(t)e−αΠ∗K1(t,i)er(T −t) ++ K2(t, i)Θ2(t)eαΠ∗K2(t,i)er(T −t) +< −(µ(t, i) − r) + K2(t, i)Θ2(t)eαΠ∗K2(t,i)er(T −t) +≤ −(µ(t, i) − r) − M2eαΠ∗er(T −t), +that leads to +Π∗(t, i) ≥ +ln +� +µ(t,i)−r +M2 +� +αer(T−t) +, +for all (t, i) ∈ [0, T] × X . +□ +4.1.2. Verification Theorem. Now, we are ready to state the verification result. +Theorem 4.6 (Verification Theorem). Suppose that the Cauchy problem (4.3) admits a classical +solution ϕ(·, i) ∈ C1� +(0, T[ +� +∩C +� +[0, T] +� +, for each i ∈ X . Then, the function ¯V : [0, T]×R×X −→ R +defined by +¯V (t, w, i) = −e−wαer(T −t)ϕ(t, i) +is the value function in (3.7). Consequently, the strategy Π∗ +t = Π∗(t, Xt) described in Proposition +4.3 is an optimal control. +Proof. The proof uses similar arguments as in that of Theorem 4.11 below for the problem with the +insurance derivative. Note that Problem 3.5 corresponds to a special case of Problem 3.6, choosing +GT = 0. Nevertheless, for the sake of clarity we trace the fundamental steps of the proof. +By Proposition 4.2, the function ¯V (t, w, i) defined in equation (4.6) solves the HJB problem (4.1). +Hence, for any (t, w, i) ∈ [0, T] × R+ +0 × X , we have +¯LΠ +i ¯V (s, W Π +s (t, w), Xs(t, i)) ≤ 0, +∀s ∈ [t, T], Π ∈ At, +(4.10) + +14 +A. CRETAROLA AND B. SALTERINI +where we recall that {W Π +s (t, w), s ∈ [t, T]} and {Xs(t, i), s ∈ [t, T]} denote the solutions to +equations (3.2) and (2.1) at time s ∈ [t, T], starting from (t, w) ∈ [0, T]×R+ +0 and (t, i) ∈ [0, T]×X , +respectively. Clearly, ¯V (·, ·, i) ∈ C1,2([0, T] × R), for each i ∈ X . +In view of (3.2), by applying Itô’s formula, we have +¯V (T, W Π +T (t, w), XT(t, i)) = ¯V (t, λ, i) + +� T +t +¯LΠ +i ¯V (v, W Π +v (t, w), Xv(t, i))dv + MT , +(4.11) +where M = {Mr, r ∈ [t, T]} is the stochastic process given by +Mr = +� r +t +Πvσ(v, Xv)∂ ¯V +∂w (v, W Π +v , Xv)dZS +v ++ +� r +t +� +R +�¯V +� +v, W Π +v , Xv− + h(Xv−, z) +� +− ¯V (v, W Π +v , Xv−) +� ˆP(dv, dz) ++ +� r +t +�¯V +� +v, W Π +v− + ΠvK1(v, Xv−), Xv−) +� +− ¯V (v, W Π +v−, Xv−) +� +{dN1 +v − Θ1(v)dv} ++ +� r +t +�¯V +� +v, W Π +v− − ΠvK2(v, Xv−), Xv−) +� +− ¯V (v, W Π +v−, Xv−) +� +{dN2 +v − Θ2(v)dv}. +In order to prove that M is a (G, P)-local martingale, we use a localization argument, taking +τn := inf{s ∈ [t, T] | W Π +s < −n}, +n ∈ N, +which defines a non-decreasing sequence of stopping times {τn}n∈N such that limn−→+∞ τn = +∞. +Therefore, taking the conditional expectation with respect to Wt = w and Xt = i on both sides of +(4.11), with T replaced by T ∧ τn, by (4.10) we obtain that +Et,w,i +� ¯V (T ∧ τn, W Π +T∧τn(t, w), XT∧τn(t, i)) +� +≤ ¯V (t, w, i), +for every Π ∈ At, t ∈ �0, T ∧ τn�, n ∈ N. Now, we note that +E +��¯V (T ∧ τn, W Π +T∧τn(t, w), XT∧τn(t, i)) +�2� += E +� +e−2αW Π +T ∧τner(T ∧τn−t)ϕ(T ∧ τn, i)2� +< ∞. +Consequently, { ¯V (T ∧ τn, W Π +T∧τn(t, w), XT∧τn(t, i))}n∈N is a family of uniformly integrable random +variables. +Hence, it converges almost surely. +Since {τn}n∈N is a bounded and non-decreasing +sequence of random times and P(|W Π +t | < +∞) = 1, see (3.3), we get +Et,w,i +�¯V (T, W Π +T (t, w), XT(t, i)) +� += +lim +n−→+∞ Et,w,i +�¯V (T ∧ τn, W Π +T∧τn(t, w), XT∧τn(t, i)) +� +≤ ¯V (t, w, i), +∀t ∈ [0, T], Π ∈ At. +As a byproduct, since Π∗(t, i) given in Proposition 4.3 realizes the infimum in (4.8), we have that +¯LΠ∗ +i +¯V (t, w, i) = 0 and, performing the computations above, we get the equality +Et,w,i +� +− e−αW Π∗ +T +(t,w)� += sup +Π∈At +Et,w,i +� +− e−αW Π +T (t,w)� += ¯V (t, w, i), +that is, Π∗ +t = Π∗ +t(t, Xt) is an optimal control. +□ + +15 +Remark 4.7. By Theorem 4.6, the value function (3.7) can be characterized as a transformation +of the solution ϕ to a certain system of ODEs with a particular terminal condition. As regards exis- +tence and uniqueness of a solution to this specific Cauchy problem (4.3), we refer to (Walter, 1998, +Theorem VII, Chapter II:6) or to (Baran et al., 2013, Section 6). According to (Walter, 1998), if +H given in (4.4) is a locally Lipschitz function with respect to the second variable, uniformly in t, +we get that there exists a unique solution ϕ(t, i), for every t ∈ [0, T], for all i ∈ X . Requiring that +µ, σ, K1 and K2 are continuous functions is a sufficient condition for the regularity of function +H and, as a consequence, the smoothness of ϕ. Otherwise, (4.3) can be seen as a trivial case of +the Cauchy problem faced by (Baran et al., 2013). Assuming that µ(·, i) and σ(·, i) are continuous +functions in t ∈ [0, T], for all i ∈ X , guarantees that infΠ∈R ¯Ψ(t, i) is bounded with respect to the +first variable and thus all required hypotheses are satisfied. +The next result provides the optimal investment strategy corresponding to Problem 3.5. +Proposition 4.8. Assume existence and uniqueness of a classical solution to the the HJB equation +with final condition (4.1). Moreover, suppose that for all (t, i) ∈ [0, T] × X , +σ(t, i) > σ > 0. +(4.12) +Then, the process {Π∗(t, i), t ∈ [0, T]} characterized in Proposition 4.3 provides the optimal in- +vestment strategy for Problem 3.5. +Proof. Let +η(t, i) = max + + + +���ln +� +µ(t,i)−r +M2 +���� +αer(T−t) +, µ(t, i) − r + CM1 +σ2(t, i)αer(T−t) + + +, +(t, i) ∈ [0, T] × X . +We show that conditions (3.5) and (3.6) in Proposition 3.4 are satisfied. By Proposition 4.5, we +immediately have Π∗(t, Xt) ≤ η(t, Xt) and Π∗(t, Xt) ≥ −η(t, Xt), for every t ∈ [0, T]. Moreover, +by (4.12) and Assumption 3.3, we get condition (3.6). Then, the process {Π∗(t, i), t ∈ [0, T]} is +an admissible investment strategy and the statement follows by applying the Verification Theorem +4.6 and Proposition 4.3. +□ +4.2. The investment problem with the insurance derivative. Now, we suppose that the +insurance company can write a pure endowment contract, whose payoff is given in (3.1). +The following result ensures that the financial-insurance model outlined in Section 2 has a Mar- +kovian structure, i.e. the vector process (W Π, Λ, X) is a (G, P)-Markov-process. Let LΠ +i denote +the Markov generator of (W Π, Λ, X) associated with a constant control Π ∈ R. + +16 +A. CRETAROLA AND B. SALTERINI +Definition 4.9. The set D(LΠ +i ) denotes the class of functions f(·, ·, ·, i) ∈ C1([0, T]) × C2(R × +(0, +∞)), for each i ∈ X , such that for every constant Π ∈ R, we have +E +� � T +0 +� +σ(v, Xv)Π ∂f +∂w(v, W Π +v , λv, Xv) +�2 +dv +� +< ∞, +(4.13) +E +� � T +0 +� +c(v, λv)λv +∂f +∂λ(v, W Π +v , λv, Xv) +�2 +dv +� +< ∞, +and +E +�� T +0 +� +R +��f +� +v, W Π +v , λv, Xv− + h(Xv−, z) +� +− f(v, W Π +v , λv, Xv−) +�� m(dz)dv +� +< ∞, +(4.14) +E +�� T +0 +��f +� +v, W Π +v− + ΠK1(v, Xv−), λv, Xv−) +� +− f(v, W Π +v−, λv, Xv−) +�� Θ1(v)dv +� +< ∞, +E +�� T +0 +��f +� +v, W Π +v− − ΠK2(v, Xv−), λv, Xv−) +� +− f(v, W Π +v−, λv, Xv−) +�� Θ2(v)dv +� +< ∞. +Lemma 4.10. The stochastic process (W Π, Λ, X) is a Markov process on (Ω, F, P; G), with infin- +itesimal generator LΠ +i for all constant strategies Π ∈ R given by +LΠ +i f(t, w, λ, i) =∂f +∂t (t, w, λ, i) + +� +rw + (µ(t, i) − r)Π +� ∂f +∂w(t, w, λ, i) + b(t, λ)λ∂f +∂λ(t, w, λ, i) ++ 1 +2σ2(t, i)Π2 ∂2f +∂w2(t, w, λ, i) + 1 +2c2(t, λ)λ2∂2f +∂λ2 (t, w, λ, i)+ +� +j∈X +aijf(t, w, λ, j) ++ Θ1(t) +� +V (t, w + ΠK1(t, i), λ, i) − V (t, w, λ, i) +� ++ Θ2(t) +� +V (t, w − ΠK2(t, i), λ, i) − V (t, w, λ, i) +� +, +for every i ∈ X . The domain of the generator LΠ +i is D(LΠ +i ), for each i ∈ X . +The proof is postponed to Appendix A. +Let us consider the HJB equation that the value function V is expected to solve, if sufficiently +smooth: + + + + + + + +supΠ∈R LΠ +i V (t, w, λ, i) + λ +� ¯V (t, w, i) − V (t, w, λ, i) +� += 0, +∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , +V (T, w, λ, i) = −e−α(w−K) +∀(w, λ, i) ∈ R × R+ × X , +(4.15) +How to derive the HJB equation in (4.15) is shown in Appendix B. +Now, let us introduce the following ansatz +V (t, w, λ, i) = −e−wαer(T −t)ϕ(t, i)φ(t, λ), +(4.16) +with (t, w, λ, i) ∈ [0, T] × R × R+ × X , where ϕ solves (4.3), while the function φ is non-negative +and does not depend on w. + +17 +From (4.16), replacing all the derivatives and performing some computations, problem (4.15) re- +duces to + + + +∂φ +∂t (t, λ) + b(t, λ)λ∂φ +∂λ(t, λ) + 1 +2c2(t, λ)λ2∂2φ +∂λ2 (t, λ) − λ(φ(t, λ) − 1) = 0, +∀(t, λ) ∈ [0, T) × R+, +φ(T, λ) = eαK, +∀λ ∈ R+. +(4.17) +We observe that the PDE in (4.17) is linear and a solution exists under suitable conditions on +model coefficients; see, e.g. (Pham, 1998, Theorem5.3) or (Colaneri and Frey, 2021, Theorem 1). +Clearly, if the function φ is a classical solution of the Cauchy problem (4.17), then V (·, ·, ·, i) ∈ +C1,2,2([0, T] × R × R+), for each i ∈ X and we have that V (t, w, λ, i) = −e−wαer(T −t)ϕ(t, i)φ(t, λ) +solves the original HJB equation given in (4.15). +Now, we can state the verification result. +Theorem 4.11 (Verification Theorem). Let ϕ(·, i) ∈ C1� +(0, T) +� +∩ C +� +[0, T] +� +and φ(·, ·) ∈ +C1� +(0, T) × R+� +∩ C +� +[0, T] × R+� +, for each i ∈ X , be classical solutions of the Cauchy prob- +lems (4.3) and (4.17), respectively. Then, the function V : [0, T] × R × R+ × X −→ R defined +by (4.16) is the value function in (3.8). Consequently, the strategy Π∗ +t = Π∗(t, Xt) described in +Proposition 4.3 is an optimal control. +Proof. Let ϕ : [0, T] × X −→ R be a function such that ϕ(·, i) ∈ C1� +(0, T) +� +∩ C +� +[0, T] +� +, for each +i ∈ X , and suppose that it is a solution of the problem (4.3). Moreover, let φ : [0, T] × R+ −→ R+ +be a function such that φ(·, ·) ∈ C1� +(0, T) × R+� +∩ C +� +[0, T] × R+� +, and suppose that it solves +the problem (4.17). Now, taking V defined in (4.16), we have that V is a solution of the problem +(4.15). This implies that, for every (t, w, λ, i) ∈ [0, T] × R × R+ × X +LΠ +i V (r, W Π +r (t, w), λr(t, λ), Xr(t, i)) ++ λr(t, λ) +�¯V (r, W Π +r (t, w), Xr(t, i)) − V (r, W Π +r (t, w), λr(t, λ), Xr(t, i)) +� +≤ 0, +r ∈ [t, T], +(4.18) +for all Π ∈ At, where {λr(t, λ), r ∈ [t, T]} denotes the solution to equation (2.5) with initial +condition λt = λ and ¯V is the value function of the pure investment problem given in (3.7). In +view of (3.2), by applying Itô’s formula, we have +V (T, W Π +T (t, w), λT(t, λ), XT(t, i)) = V (t, λ, i) + +� T +t +LΠ +i V (v, W Π +v (t, w), λv(t, λ), Xv(t, i))dv ++ +� T +t +λv(t, λ) +�¯V (v, W Π +v (t, w), Xv(t, i)) − V (v, W Π +v (t, w), λv(t, λ), Xv(t, i)) +� +dv + MT, +(4.19) + +18 +A. CRETAROLA AND B. SALTERINI +where M = {Mr, r ∈ [t, T]} is the stochastic process given by +Mr = +� r +t +Πvσ(v, Xv)∂V +∂w (v, W Π +v , λv, Xv)dZS +v + +� r +t +c(v, Yv)λv +∂V +∂y (v, W Π +v , λv, Xv)dZΛ +v ++ +� r +t +� +R +� +V +� +v, W Π +v , λv, Xv− + h(Xv−, z) +� +− V (v, W Π +v , λv, Xv−) +� ˆP(dv, dz) ++ +� r +t +� +V +� +v, W Π +v− + ΠvK1(v, Xv−), λv, Xv−) +� +− V (v, W Π +v−, λv, Xv−) +� +{dN1 +v − Θ1(v)dv} ++ +� r +t +� +V +� +v, W Π +v− − ΠvK2(v, Xv−), λv, Xv−) +� +− V (v, W Π +v−, λv, Xv−) +� +{dN2 +v − Θ2(v)dv}. +Now, we prove that M is a (G, P)-local martingale. Precisely, we need to show that +E +� � T∧τn +t +� +σ(v, Xv)Πv +∂V +∂w (v, W Π +v , λv, Xv) +�2 +dv +� +< ∞, +E +� � T∧τn +t +� +c(v, λv)λv +∂V +∂λ (v, W Π +v , λv, Xv) +�2 +dv +� +< ∞, +for a suitable, non-decreasing sequence of stopping times {τn}n∈N such that limn−→+∞ τn = +∞. +Taking expression (4.16) into account, we note that +∂V +∂w (t, w, λ, i) += αφ(t, λ)ϕ(t, i)er(T−t)−αwer(T −t), +∂V +∂y (t, w, λ, i) += −∂φ +∂λ(t, λ)ϕ(t, i)e−αwer(T −t). +Let us define a sequence of random times {τn}n∈N by setting +τn := inf{s ∈ [t, T] | W Π +s < −n, λs > n, φ(s, λs) > n, ∂φ +∂λ(s, λs) > n}, +n ∈ N. +Throughout the proof, we denote by Cn any constant depending on n ∈ N. Consequently, we get +E +� � T∧τn +0 +� +σ(v, Xv)Πv +∂V +∂w (v, W Π +v , λv, Xv) +�2 +dv +� += E +� � T∧τn +0 +σ2(v, Xv)Π2 +v +� +αφ(v, λv)ϕ(v, Xv)er(T−v)−αW Π +v er(T −v)�2 +dv +� +≤ CnE +� � T +0 +σ2(v, Xv)Π2 +vdv +� +< ∞ +∀n ∈ N, +since Π is admissible. Further, by (2.6) we have that +E +� � T∧τn +0 +� +c(v, λv)λv +∂V +∂λ (v, W Π +v , λv, Xv) +�2 +dv +� += E +� � T∧τn +0 +� +c(v, λv)λv +∂φ +∂λ(v, λv)ϕ(v, Xv)e−αW Π +v er(Xv)(T −t)�2 +dv +� +≤ CnE +� � T +0 +c(v, λv)2λ2 +vdv +� +< ∞ +∀n ∈ N. + +19 +Furthermore, due to the boundedness of function V until time τn, we have that the stopped process +�� r∧τn +t +� +R +� +V +� +v, W Π +v , λv, Xv− + h(Xv−, z) +� +− V +� +v, W Π +v , λv, Xv− +�� +ˆP(dv, dz), r ∈ [t, T] +� +is a (G, P)-martingale (see e.g. (Davis, 1993, Theorem 26.12(2))), for every n ∈ N. Finally, even +the stopped processes +�� r∧τn +t +� +V +� +v, W Π +v− + ΠvK1(v, Xv−), λv, Xv− +� +− V +� +v, W Π +v−, λv, Xv +�� +{dN1 +v − Θ1(v)dv}, r ∈ [t, T] +� +and +�� r∧τn +t +� +V +� +v, W Π +v− − ΠvK2(v, Xv−), λv, Xv− +� +− V +� +v, W Π +v−, λv, Xv +�� +{dN2 +v − Θ2(v)dv}, r ∈ [t, T] +� +are (G, P)-martingales, (see e.g. (Brémaud, 1981, Lemma L3, Ch.II)). Thus, the process {Mr, r ∈ +[t, T]} turns out to be a (G, P)-local martingale and {τn}n∈N is a localizing sequence for {Mr, r ∈ +[t, T]}. Therefore, taking the conditional expectation of both sides of (4.19) with respect to Wt = w, +λt = λ and Xt = i with T replaced by T ∧ τn, by (4.18) we obtain that +Et,w,λ,i +� +V (T ∧ τn, W Π +T∧τn(t, w), λT∧τn, XT∧τn(t, i)) +� +≤ V (t, w, λ, i), +for every Π ∈ At, t ∈ �0, T ∧ τn�, n ∈ N. Now, we note that +E +�� +V (T ∧ τn, W Π +T∧τn(t, w), λT∧τn(t, λ), XT∧τn(t, i)) +�2� += E +� +e−2αW Π +T ∧τner(T ∧τn−t)ϕ(T ∧ τn, XT∧τn)2φ(T ∧ τn, λT∧τn)2� +≤ �K, +for a positive constant �K. This means that {V (T ∧ τn, W Π +T∧τn(t, w), λT∧τn, XT∧τn(t, i))}n∈N is a +family of uniformly integrable random variables. Hence, it converges almost surely. Since {τn}n∈N +is a bounded and non-decreasing sequence of random times and P(|W Π +t | < +∞) = 1, see (3.3), in +view of (2.7), we can apply the dominated convergence theorem and, taking the limit for n −→ +∞, +we get +Et,w,λ,i +� +V (T, W Π +T (t, w), λT(t, λ), XT(t, i)) +� += +lim +n−→+∞ Et,w,λ,i +� +V (T ∧ τn, W Π +T∧τn(t, w), λT∧τn(t, λ), XT∧τn(t, i)) +� +≤ V (t, w, λ, i), +for every Π ∈ At, t ∈ [0, T]. By the final condition in (4.15) and the previous inequality, we get +Et,w,λ,i +� +− e−α(W Π +T (t,w)−K)� +≤ V (t, w, λ, i), +for every Π ∈ At, t ∈ [0, T]. Finally, since the insurance payment does not depend on the risky asset +price, we have that Π∗(t, w, λ, i) = Π∗(t, i) given in Proposition 4.3 yields that LΠ∗ +i V (t, w, λ, i) + +λ +�¯V (t, w, i) − V (t, w, λ, i) +� += 0; then, if we apply the above arguments to Π∗ and replacing LΠ +i +with LΠ∗ +i , we find the equality +sup +Π∈At +Et,w,λ,i +� +− e−α(W Π +T (t,w)−K)� += V (t, w, λ, i), + +20 +A. CRETAROLA AND B. SALTERINI +which implies that the process Π∗(t, Xt) is an optimal Markovian control. +□ +Remark 4.12. We observe that the optimal investment strategy Π∗(t, Xt) turns out to be the +same as the pure investment problem. This means that the optimal portfolio for the investment +problem with the insurance derivative equals the strategy without insurance risks when the insurance +payment is independent of the risky asset price process. This statement is the same as that given +in (Delong, 2009) and (Liang and Lu, 2017) when the risky asset price dynamics is driven by a +Lévy process and a shot-noise process, respectively. +5. Characterization of the indifference price +In this section we compute explicitly the indifference price for the pure endowment contract whose +payoff is given in (3.1). +To this aim, we provide the definition of the indifference price charged by an insurance company +which writes a pure endowment. Recall that ¯V and V are the value functions introduced in (3.7) +and (3.8), respectively. +Definition 5.1. Given Wt = w, λt = λ and Xt = i, the indifference price process or reservation +price process P = {Pt, t ∈ [0, T]} of the insurer related to the pure endowment contract is defined +at any time t ∈ [0, T] as the G-adapted process implicit solution to the equation +¯V (t, w, i) = V (t, w + Pt, λ, i). +(5.1) +In other words, P is the price that makes the insurer indifferent, in terms of expected utility, +between not selling and selling the policy for the price P now and paying the benefits at maturity. +We have the following explicit characterization of the indifference price process in our framework. +Proposition 5.2. Under the same hypotheses of Theorem 4.11, for every t ∈ [0, T], the indifference +price of the insurer related to the pure endowment with maturity T is given by +Pt = P(t, λ; T) = ln +� +φ(t, λ) +� +αer(T−t) , +(5.2) +for all (t, λ) ∈ [0, T] × R+, where the function φ solves the Cauchy problem (4.17). +Proof. By Theorem 4.6 and Theorem 4.11, equation (5.1) reads as +e−wαer(T −t)ϕ(t, i) = e−(w+Pt)αer(T −t)ϕ(t, i)φ(t, λ), +and then +ePtαer(T −t) = φ(t, λ), +from which, computing the logarithm of both members, we get (5.2). +□ + +21 +Remark 5.3. In the actuarial literature, exponential utility preferences imply that the indifference +price of a pure endowment depends on the risk-aversion coefficient, the stochastic interest rate and +the logarithm of the function that links the two value functions and is independent of wealth (see e.g. +(Young and Zariphopoulou, 2002; Young, 2003; Moore and Young, 2003; Ludkovski and Young, +2008)). Here, the indifference price shares the same features. Moreover, we note that it does not +explicitly depend on the current regime; meaning that in our model, the state of the economy does +not affect directly the indifference price of such type of insurance contracts but only the amount +invested in the financial market. +Under the indifference pricing principle, the premium solves a terminal value problem. +Corollary 5.4. For every (t, λ) ∈ [0, T] × R+ the indifference premium P = P(t, λ; T) satisfies +the following PDE +rP = ∂P +∂t + b(t, λ)λ∂P +∂λ + 1 +2c2(t, λ)λ2�∂2P +∂λ2 + αer(T−t) +�∂P +∂λ +�2 � ++ +λ +αer(T−t) +� +e−P αer(T−t) − 1 +� +, +with boundary condition P(T, λ; T) = K, for each λ ∈ R+. +Proof. It follows from a straightforward application of (4.17) and (5.2). +□ +Finally, we provide a probabilistic representation for the indifference price process P. Indeed, if +the function φ solves the Cauchy problem (4.17), we can represent φ as an expectation via an +extension of the Feynman-Kac formula. More precisely, using the linear PDE for φ − 1, it is easy +to see that +φ(t, λ) − 1 = Et,λ +� +e− +� T +t λvdv� +eαK − 1 +�� +, +for every (t, λ) ∈ [0, T] × R+. So, as a consequence, we have that +φ(t, λ) = 1 + +� +eαK − 1 +� +Et,λ +� +e− +� T +t λvdv� +, +where Et,λ denotes the conditional expectation given λt = λ, for every (t, λ) ∈ [0, T] × R+. We +outline that Et,λ +� +e− � T +t λvdv� +is the conditional probability that an individual will survive until time +T given that she/he is alive at time t. Hence, representing the function φ as +φ(t, λ) = eαKEt,λ +� +e− +� T +t λvdv� ++ +� +1 − Et,λ +� +e− +� T +t λvdv�� += Et,λ +� +eαGT � +, +for every (t, λ) ∈ [0, T] × R+, the indifference price of the insurer related to a pure endowment +contract can be written as +Pt = P(t, λ; T) = +ln +� +Et,λ +� +eαGT �� +αer(T−t) +, +for every (t, λ) ∈ [0, T] × R+. + +22 +A. CRETAROLA AND B. SALTERINI +5.1. Group of insured people. In this subsection, we evaluate a panel of insurance policies, +extending the previous results. +Put another way, we deal with a portfolio consisting of pure +endowments issued to a group of n ∈ N individuals, who are all the same age with indipendent +and identically distributed times until death. We assume that the loss payable at the maturity +T equals the amount K > 0 for each of the insured people who have not died yet. So, the value +function (3.8) is replaced by +V (n)(t, w, λ, i) := +sup +Π∈At(G) +Et,w,λ,i +� +− e−α(W Π +T −G(n) +T +)� +, +where G(n) +T += mK, with K > 0 constant, if there are exactly m individuals alive at time T out of +the group of n individuals alive at time t. Analogously to (4.15), V (n) solves the following HJB +problem: + + + + + + + +supΠ∈R LΠ +i V (n)(t, w, λ, i) + nλ +� +V (n−1)(t, w, i) − V (n)(t, w, λ, i) +� += 0, +∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , +V (n)(T, w, λ, i) = −e−α(w−nK) +∀(w, λ, i) ∈ R × R+ × X , +in which V (0) = ¯V . Note that V (1) = V in (4.15). One can easily shows that +V (n)(t, w, λ, i) = ¯V (t, w, i)φ(n)(t, λ), +where φ(n) : [0, T] × R+ −→ R+ solves the linear PDE + + + +∂φ +∂t +(n) +(t, λ) + b(t, λ)λ∂φ +∂λ +(n) +(t, λ) + 1 +2c(t, λ)2λ2∂2φ +∂λ2 +(n) +(t, λ) − nλ +� +φ(n)(t, λ) − φ(n−1)(t, λ) +� += 0 +φ(T, λ) = enαK, +(5.3) +with φ(0) ≡ 1. Thus, the indifference price of n pure endowments P (n) is an implicit solution of +the following equation +¯V (t, w, i) = V (n)(t, w + P (n) +t +, λ, i), +for every (t, w, λ, i) ∈ [0, T] × R × R+ × X . Similarly to (5.2), the reservation price of the insurer +related to n pure endowment contracts is given by +P (n) +t += P (n)(t, λ, i; T) = +ln +� +φ(n)(t, λ) +� +αer(T−t) +, +for all (t, λ, i) ∈ [0, T] × R+ × X , where the function φ(n) solves the Cauchy problem (5.3). +5.2. Term life insurance. Finally, we discuss the indifference price of another type of a mortality- +contingent claim, the so-called term life insurance that can be defined as follows. +Definition 5.5. A term life insurance contract with maturity T is a life insurance policy where +the amount is paid at time T if the policyholder dies before time T. The associated payoff is given +by the random variable +GT := K1{τ≤T}, +(5.4) + +23 +where K is a positive constant. +We determine the indifference price of the term life insurance policy whose payoff is given by (5.4) +in the Markov-modulated model outlined in Section 2. Then, we consider the problem with the +new kind of insurance derivative +sup +Π∈A(G) +E +� +− e−α(W Π +T −K)� +. +Thus, the corresponding value function is given by +V (t, w, λ, i) := +sup +Π∈At(G) +Et,w,λ,i +� +− e−α(W Π +T −K)� +, +for every (t, w, λ, i) ∈ [0, T] × R × R+ × X . Note that it equals the value function (3.8) of the +problem with the pure endowment. Thus, proceeding as above, the HJB problem for V is given +by + + + + + + + +supΠ∈R LΠ +i V (t, w, λ, i) + λ +� ¯V (t, w − Ke−r(T−t), i) − V (t, w, λ, i) +� += 0, +∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , +V (T, w, λ, i) = −e−α(w−K) +∀(w, λ, i) ∈ R × R+ × X , +where ¯V is introduced in (3.7). Note that the HJB equation corresponds to (4.15) with ¯V (t, w, i) +replaced by ¯V (t, w − Ke−r(T−t), i), since the insurer has to pay the amount K at time T for the +death of the policyholder and so she/he needs to charge Ke−r(T−t) at time t in order to cover this +payout. One can easily shows that +V (t, w, λ, i) = ¯V (t, w, i)ξ(t, λ), +where the function ξ : [0, T] × R+ −→ R+ solves the linear PDE + + + +∂ξ +∂t (t, λ) + b(t, λ)λ ∂ξ +∂λ(t, λ) + 1 +2c(t, λ)2λ2 ∂2ξ +∂λ2(t, λ) − λ(eαK − ξ(t, λ)) = 0 +φ(T, λ) = eαK. +(5.5) +Hence, the reservation price of the insurer related to a term life contract is given by +Pt = P(t, λ, i; T) = +ln +� +ξ(t, λ) +� +αer(T−t) +, +for all (t, λ, i) ∈ [0, T] × R+ × X , where the function ξ solves the Cauchy problem (5.5). +6. Numerical experiment +In this section, we present some numerical results based on the theoretical framework developed +previously, in order to illustrate certain qualitative features of the model that are difficult to verify +analytically. Our aim is to investigate how the regime-switching and the stochastic hazard rate +affect the decisions of the insurer. + +24 +A. CRETAROLA AND B. SALTERINI +To simplify the analysis, we suppose that the Markov chain X has two states, namely X = {1, 2}, +that can be interpreted as the ’Good’ and ’Bad’ economic regimes, respectively. For instance, the +good regime could represent a market in economic boom whereas the bad regime could be a market +in economic recession in which security prices are expected to fall. We also call these two regimes +of the market ’bull’ market and ’bear’ market, respectively. +First of all, we have to set values for the infinitesimal generator of the 2-state Markov chain. Since +aij represents the average of number of switches in an unit time, from state i to j, and since +empirical observations of the market suggest that it is more likely to pass from a good economic +state to a bad one than the opposite, we choose a12 > a21. In particular, we take a12 = 0.2 and +a21 = 0.1. +For the sake of simplicity, we assume that functions µ, σ, K1 and K2 depend only on the Markov +chain. So, by (2.2), the risky asset price dynamics is given by +dSt = St−{µidt + σidZS +t + K1,idN1 +t − K2,idN2 +t }, +S0 > 0, i = 1, 2, +where µi, σi, K1,i and K2,i denote the expected rate of return, the volatility and the jump coefficients +in the i-th regime. By way of example, we set the initial value of the stock price to be S0 = 1 and +the short-term interest rate to be r = 5%. As shown by (French et al., 1987), the appreciation +rate of the underlying risky asset is higher in a growing economy, so we assume that µ1 > µ2. +Moreover, in each economic regime, the return of the risky asset should be higher than that of the +risk-free rate, as required also in our modeling framework. (Hamilton and Gang, 1996) find that +economic recessions represent the main factor that drives fluctuations in the volatility of stock +returns, so we assume that volatility is lower in a good economy, i.e. σ1 < σ2. Furthermore, let +us assume that µ1 − r +σ2 +1 +> µ2 − r +σ2 +2 +. In fact, according to (French et al., 1987), even though the +expected market risk premium (defined as the expected return on the stock minus the risk-free +interest rate) is usually higher during a ’bear’ market than during a ’bull’ market, the volatility +of the stock offsets the effect of this quantity and, as a consequence, the ratio ’expected excess +return/return variance’ is greater when the economic conditions are good. As for the jump terms, +we consider two homogeneous Poisson processes N1 and N2 with constant intensities Θ1 = 0.3 +and Θ2 = 0.4. We observe that the higher are the values of function K1, the higher is the price of +the stock S. On the other hand, any increase in the coefficient K2 leads to smaller prices for the +risky stock. Moreover, we notice that large values of K2 cause dizzying upward or downward peaks +for the stock price, even though the intensity Θ2 is tiny. Therefore, since in a market with good +economic conditions stock prices are rising or are expected to rise, we suppose that K1,1 > K1,2 +and K2,1 < K2,2. +In particular, we choose the parameters as in Table 1. + +25 +Table 1. Simulation market parameters. +Regime +µ +σ +K1 +K2 +’Good’ +0.15 +0.15 +0.15 +0.3 +’Bad’ +0.12 +0.25 +0.1 +0.35 +Since the underlying market is a continuous-time model, we need to discretize it by Monte Carlo +simulation. The time horizon is taken to be T = 10 years and we discretize time with a total of +1000 time steps (that means that we take into account about two updates of S every workweek), +each of width ∆t = +1 +100). +In order to have an idea of our model, we simulate three trajectories of the risky asset S in Figure +1. We notice that the stock price is greater during a ’bull market’ rather than during a ’bear’ +market and it also exhibits jumps at switching times of the Markov chain. +0 +2 +4 +6 +8 +10 +Time t +0 +1 +2 +3 +4 +5 +Stock S +1 +2 +Markov chain X +Figure 1. The effect of the regime-switching on the stock price S. +Next, we compute the optimal investment strategy based on Proposition 4.3, in order to investigate +how it is sensitive to economic regimes. In Figure 2 we plot the optimal dynamic portfolio given +by (4.9), as a function of time. + +26 +A. CRETAROLA AND B. SALTERINI +0 +2 +4 +6 +8 +10 +Time t +-1 +0 +1 +Optimal investment strategy +* +1 +2 +Markov chain X +Figure 2. The effect of regime-switching on the optimal strategy Π∗. +We clearly notice that a regime switch leads to a sudden change in the optimal strategy. Moreover, +we note that in a good economy the amount invested in the stock is always positive and increasing +with respect to time; instead, if the market scenario is bad, the strategy is negative; this indicates +that when the economic conditions are bad, the insurer prefers to short-sell the risky asset. +After that, we take into account a life-insurance policy and we investigate its indifference price. +In the current toy example of our proposed model, we assume that the hazard rate follows a mean- +reverting Brownian Gompertz model, similar to the one proposed in (Milevsky and Promislow, +2001), i.e. +λt = λ0ec1t+c2Yt, +c1, c2, λ0 > 0, +dYt = −mYtdt + dZY +t , +Y0 = 0, +m ≥ 0, +with c1 = 0.083, c2 = 0.1, λ0 = 0.01 and m = 0.5. Let us observe that this choice corresponds to +(2.5), considering b(t, λ) = c1 + m ln(λ0) + 1 +2c2 +2 − m ln(λ) + mc1t and c(t, λ) = c2λ, for all (t, y). +This model guarantees that the hazard rate is kept positive and does not explode on [0, T], since it +is an exponential function that depends on a stochastic factor Y with a mean reversion behavior. +In this context, based on the results obtained above, we compute the indifference price of an insurer +related to a pure endowment contract that pays K = 1 if the policyholder is still alive after 10 +years from purchaising the policy. Thus, the payoff is easily given by the random variable +GT := 1{τ>T}, +recalling that τ represents the remaining lifetime of the insured. +Now, we want analyse the value function ¯V related to the insurer that simply invests her/his wealth +in the market and the value function V related to the insurer who also writes a pure endowment +contract. + +27 +0 +1 +2 +3 +4 +5 +Initial wealth w +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +Optimal value function +Good +Bad +0 +1 +2 +3 +4 +5 +Initial wealth w +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +Optimal value function with claim +Good +Bad +Figure 3. Optimal value at time 0 as a function of wealth when the economic regime is +i = 1 (solid line) or i = 2 (dashed line). Left panel: the pure investment problem. Right +panel: the investment problem with the insurance contract. +Figure 3 depicts the value functions ¯V (left panel) and V (right panel) at time t = 0, with respect +to the initial wealth w, associated to the optimal strategy computed above, when the market state +is good (solid line) or bad (dashed line). The two panels exhibit the same behavior: the optimal +value functions are increasing functions of wealth, in both regimes. It is worth noting that values +reached by functions ¯V and V are always higher in a ’bull’ market, as it is reasonable. Furthermore, +we can also point out that different economic conditions imply different value functions and that +this gap becomes greater when the insurer, beyond investing in the financial market, also writes +an insurance contract. +Next, we investigate the indifference price of a pure endowment policy, in order to highlight the +dependence of a life insurance contract on mortality force and time to expiration. In view of the +probabilistic representation provided in Section 5, the indifference price charged by the insurer is +determined as +Pt = P(t, λ; T) = +ln +� +1 + (eα − 1)Et,λ +� +e− +� T +t λvdv�� +αer(T−t) +, +(6.1) +for every (t, λ) ∈ [0, T] × R+. We employ this formula, using the standard Monte Carlo method +(with parameter M = 5000) to evaluate expectations with respect to the probability measure P. +From expression (6.1), we point out that economic regimes do not affect the price which instead +strongly depends on the risk aversion coefficient and the risk-free interest rate. In particular, it is +easy to see that the indifference price increases as risk aversion increases and, at the same time, + +28 +A. CRETAROLA AND B. SALTERINI +it decreases as long as the interest rate increases. Further, since the dependence on the mortality +rate λ is not explicit, we would like to analize numerically how the hazard rate affects the price +of the life insurance policy involved. First of all, we show the impact of changing initial mortality +rate on the indifference price charged at the beginning of the time interval. +0 +0.02 +0.04 +0.06 +0.08 +0.1 +Initial hazard rate +0 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +Indifference price P0 +Figure 4. The effect of the hazard rate on the indifference price at time t = 0. +In Figure 4 we observe that larger force of mortality decreases the indifference price P0 charged by +the insurer at time t = 0, as it is reasonable to expect for such type of insurance contracts. This is +consistent with common intuition as, under higher mortality, an endowment payout is less likely. +Finally, we investigate the evolution of indifference price over time. For the sake of simplicity, we +assume constant mortality (such as in some numerical experiments of (Moore and Young, 2003)). +In this framework, we calculate the indifference premium related to a pure endowment policy for +our insurer and we plot it as a function of time to maturity. + +29 +0 +2 +4 +6 +8 +10 +Time to maturity (T-t) +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Indifference price P(T-t) +=0.01 +=0.05 +=0.1 +Figure 5. The effect of the hazard rate on the indifference price for several different +deferral periods. +As before, in Figure 5, we can see that the higher is the hazard rate, the lower is the indifference +price for a pure endowment policy, whether the economic conditions are good or not; in other +terms, the price is more sensitive to variations of deferral periods, when the population mortality +intensity is more pronounced. Moreover, it is worth noting that the indifference price is a decreasing +function of time of maturity, i.e. the premium is bigger as time approaches to maturity, as usually +happens. +7. Conclusions +In this paper, we have analyzed indifference pricing of mortality contingent claims in a stochastic- +factor model for an insurance company endowed with exponential utility preferences. We have +considered a financial market model consisting of a riskless asset and a stock whose price is de- +scribed by a jump diffusion process affected by a continuous-time finite-state Markov chain repre- +senting the states of the economy. Moreover, we have assumed a stochastic hazard rate to describe +population mortality. In this framework, using the actuarial principle of equivalent utility, we have +characterized the indifference price for a pure endowment contract and provided its probabilistic +representation. In addition, we have also shown that the indifference price solves a final value +problem. Indeed, the price that makes the insurer indifferent, in terms of expected utility, between +not selling and selling the policy for that premium now and paying the benefits at maturity, is +linked to a classical solution of a specific linear PDE with a proper terminal condition. Indeed, the +indifference price has been determined by solving an equation involving two value functions, result- +ing from the stochastic control problems with and without insurance liabilities. Using the classical +control approach based on the HJB equation, we have found the optimal investment strategies and +shown verification results for the value functions of the problems with and without the policy via +classical solutions to a linear PDE and a system of ODEs. Moreover, we have briefly discussed the + +30 +A. CRETAROLA AND B. SALTERINI +indifference price of a portfolio of pure endowments and also for a term life insurance policy. A +sensitivity analysis in case of a two-state Markov chain has highlighted some interesting features of +the indifference price. We have investigated the effect of the hazard rate and the time to maturity +on the price. We have pointed out that, when the mortality intensity is low, an endowment payout +is more likely and so its premium is greater. Further, we have outlined that the indifference price +of a pure endowment contract decreases for longer deferral periods, as it is reasonable. Another +numerical result shows that the insurance company opts to short-sell the risky asset when the +financial market is in the bad state (i.e. when the stock price presents a low rate of return, big +fluctuations and a lot peaks). Applying the same methodology, it would be interesting to evaluate +more complex insurance products, such as equity-linked policies. This will be done in a future +work, also assuming that the insurance company preferences towards the risk are given by a utility +function of power (or logarithmic) type. +Acknowledgements +The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le +loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). +Statements and Declarations +Alessandra Cretarola declares that she has no conflict of interest. Benedetta Salterini declares that +she has no conflict of interest. +Appendix A. Technical proofs +Proof of Lemma 4.10. In view of (2.2), (2.5) and (3.2), by applying Itô’s formula to the stochastic +process f(t, W Π +t , λt, Xt), we have +f(t, W Π +t , λt, Xt) = f(0, W Π +0 , λ0, X0) + +� t +0 +LΠf(u, W Π +u , λu, Xu)du + mt, +where +mt = m0 + +� t +0 +Πvσ(v, Xv) ∂f +∂w(v, W Π +v , λv, Xv)dZS +v + +� t +0 +c(v, λv)λv +∂f +∂λ(v, W Π +v , λv, Xv)dZΛ +v ++ +� t +0 +� +R +� +f +� +v, W Π +v , λv, Xv− + h(Xv−, z) +� +− f(v, W Π +v , λv, Xv−) +� ˆP(dv, dz) ++ +� t +0 +� +f +� +v, W Π +v− + ΠvK1(v, Xv−), λv, Xv−) +� +− f(v, W Π +v−, λv, Xv−) +� +{dN1 +v − Θ1(v)dv} ++ +� t +0 +� +f +� +v, W Π +v− − ΠvK2(v, Xv−), λv, Xv−) +� +− f(v, W Π +v−, λv, Xv−) +� +{dN2 +v − Θ2(v)dv}. +(A.1) +We only need to prove that the process m = {mt, t ∈ [0, T]} is a (G, P)-martingale. By (4.13), the +first two integrals in (A.1) are well-defined and turn out to be (G, P)-martingales. Furthermore, +due to (4.14), we have that also the jump terms in (A.1) are (G, P)-martingales, (see e.g. (Davis, + +31 +1993, Theorem 26.12(2)) and (Brémaud, 1981, Lemma L3, Ch.II) for further details about the +martingale property related to a Poisson random measure and a Poisson process, respectively). +□ +Appendix B. Derivation of the HJB equation +For the sake of clarity, we show how to obtain a formal derivation of the HJB equation (4.15) +associated to the problem with the insurance derivative. To this aim, we apply the Bellman’s +dynamic programming principle that, in this context, it is formulated as follows. +Proposition B.1 (Bellman optimality principle). Let (t, w, λ, i) ∈ [0, T] ×R ×R+ ×X . Then, for +t ≤ t + h ≤ T and Π ∈ At, we have +V (t, w, λ, i) ≥ Et,w,λ,i +� +V (t + h, W Π +t+h, λt+h, Xt+h) +� +, +(B.1) +where V is the value function introduced in (3.8). Moreover, equality holds in (B.1) if, and only if, +the arbitrary control Π on the interval [t, t + h] is optimal. +The idea is that if the insurer follows the optimal strategy on [t, T], her/his expected utility is +at least as great as if she/he invests arbitrarily on [t, t + h[ and then optimally on [t + h, T], +for h sufficiently small such that t + h < T. In the application of the dynamic programming +principle, we must consider whether the policyholder survives from time t until time t + h, as in +(Young and Zariphopoulou, 2002), (Moore and Young, 2003), (Ludkovski and Young, 2008) and +(Young, 2003). Consider an individual aged l, who is seeking to buy a pure endowment policy. For +the rest of this section, we write (l) to refer to this individual. For each h such that t+h < T, if the +individual (l+t) survives for another h years until time t+h, which happens with probability hpl+t, +the insurer still faces the endowment risk on the time interval [t + h, T]. In this case, by (3.8), the +maximum expected utility derived by investing optimally on [t+h, T] is V (t+h, W Π +t+h, λt+h, Xt+h). +However, if the individual (l + t) dies in [t, t + h], an event that happens with probability hql+t, +then the insurer is not longer at risk for the endowment payout. Hence, by (3.7), the maximum +expected utility derived by investing optimally on [t + h, T] is ¯V (t + h, W Π +t+h, Xt+h). From (B.1), +we have +V (t, w, λ, i) ≥ hpl+tEt,w,λ,i +� +V (t + h, W Π +t+h, λt+h, Xt+h) +� ++ +hql+tEt,w,i +� +¯V (t + h, W Π +t+h, Xt+h) +� +. + +32 +A. CRETAROLA AND B. SALTERINI +If we assume enough regularity conditions and proper integrability on the value functions and their +derivatives, by applying Itô’s formula and conditioning on W Π +t = w, λt = λ and Xt = i, we get +V (t, w, λ, i) ≥ +hpl+tV (t, w, λ, i) +h ql+t ¯V (t, w, i) ++h pl+tEt,w,λ,i +� � t+h +t +�∂V +∂t + +� +rW Π +v + +� +µ(v, Xv) − r +� +Πv +�∂V +∂w dv +�� ++h pl+tEt,w,λ,i +� � t+h +t +� +b(v, λv)λv +∂V +∂λ + 1 +2σ2(v, Xv)Π2 +v +∂2V +∂w2 + 1 +2c2(v, λv)λ2 +v +∂2V +∂λ2 +� +dv +� ++h pl+tEt,w,λ,i +� � t+h +t +� � +j∈X +V (v, W Π +v , λv, j)av,j +� +dv +� ++h pl+tEt,w,λ,i +� � t+h +t +Θ1(v) +� +V (v, W Π +v + ΠvK1(v, i), λv, Xv) − V (v, W Π +v , λv, Xv) +� +dv +� ++h pl+tEt,w,λ,i +� � t+h +t +Θ2(v) +� +V (v, W Π +v − ΠvK2(v, i), λv, i) − V (v, W Π +v , λv, Xv) +� +dv +� ++h ql+tEt,w,i +� � t+h +t +�∂ ¯V +∂t + +� +rW Π +v + +� +µ(v, Xv) − r +� +Πv +�∂ ¯V +∂w +� +dv +� ++h ql+tEt,w,i +� � t+h +t +�1 +2σ(v, Xv)2Π2 +v +∂2V +∂w2 + +� +j∈X +¯V (v, Wv, j)av,j +� +dv +� ++h ql+tEt,w,i +� � t+h +t +Θ1(v) +�¯V (v, W Π +v + ΠvK1(v, i), Xv) − ¯V (v, W Π +v , Xv) +� +dv +� ++h ql+tEt,w,i +� � t+h +t +Θ2(v) +�¯V (v, W Π +v − ΠvK2(v, i), i) − ¯V (v, W Π +v , Xv) +� +dv +� +. +To keep the formulas readable, in the integrals above we have suppressed the independent variables +(v, Wv, λv, Xv) and (v, Wv, Xv) of the partial derivatives of V and ¯V , respectively. By subtracting + +33 +hpl+tV (t, w, λ, i) from both sides of inequality and dividing both sides by h, we obtain +hql+t +h V (t, w, λ, i) ≥ +hql+t +h +¯V (t, w, i) ++h pl+tEt,w,λ,i +� � t+h +t +1 +h +�∂V +∂t + +� +rW Π +v + +� +µ(v, Xv) − r +� +Πv +�∂V +∂w dv +�� ++h pl+tEt,w,λ,i +� � t+h +t +1 +h +� +b(v, λv)λv +∂V +∂λ + 1 +2σ2(v, Xv)Π2 +v +∂2V +∂w2 + 1 +2c2(v, λv)λ2 +v +∂2V +∂λ2 +� +dv +� ++h pl+tEt,w,λ,i +� � t+h +t +1 +h +� � +j∈X +V (v, Wv, λv, j)av,j +� +dv +� ++h pl+tEt,w,λ,i +� � t+h +t +1 +h +� +Θ1(v) +� +V (v, W Π +v + ΠvK1(v, i), λv, Xv) − V (v, W Π +v , λv, Xv) +�� +dv +� ++h pl+tEt,w,λ,i +� � t+h +t +1 +h +� +Θ2(v) +� +V (v, W Π +v − ΠvK2(v, i), λv, i) − V (v, W Π +v , λv, Xv) +�� +dv +� ++h ql+tEt,w,i +� � t+h +t +1 +h +�∂ ¯V +∂t + +� +rWv + +� +µ(v, Xv) − r +� +Πv +�∂ ¯V +∂w +� +dv +� ++h ql+tEt,w,i +� � t+h +t +1 +h +�1 +2σ(v, Xv)2Π2 +v +∂2V +∂w2 + +� +j∈X +¯V (v, W Π +v , j)av,j +� +dv +� ++h ql+tEt,w,i +� � t+h +t +1 +h +� +Θ1(v) +�¯V (v, W Π +v + ΠvK1(v, i), Xv) − ¯V (v, W Π +v , Xv) +�� +dv +� ++h ql+tEt,w,i +� � t+h +t +1 +h +� +Θ2(v) +�¯V (v, W Π +v − ΠvK2(v, i), i) − ¯V (v, W Π +v , Xv) +�� +dv +� +. +We observe that as h −→ 0+, we have +hpl+t −→ 1, +hql+t −→ 0 +and +hql+t +h +−→ λt, +for each t ∈ [0, T]. Consequently, taking the limit as h −→ 0+ yields +0 ≥λ +� ¯V (t, w, i) − V (t, w, λ, i) +� ++ ∂V +∂t + +� +rw + +� +µ(t, i) − r +� +Π +�∂V +∂w + b(t, λ)λ∂V +∂λ ++ 1 +2Π2σ2(t, i)∂2V +∂w2 + 1 +2c2(t, λ)λ2∂2V +∂λ2 + +� +j∈X +V (t, w, λ, j)aij ++ Θ1(t) +� +V (t, w + ΠK1(t, i), λ, i) − V (t, w, λ, i) +� ++ Θ2(t) +� +V (t, w − ΠK2(t, i), λ, i) − V (t, w, λ, i) +� +. + +34 +A. CRETAROLA AND B. SALTERINI +Finally, we note that along the optimum, we have +0 =λ +�¯V (t, w, i) − V (t, w, λ, i) +� ++ ∂V +∂t + rw∂V +∂w + b(t, λ)λ∂V +∂λ + 1 +2c2(t, λ)λ2∂2V +∂λ2 + +� +j∈X +V (t, w, λ, j)aij ++ sup +Π∈R +� +� +µ(t, i) − r +� +Π∂V +∂w + 1 +2σ2(t, i)Π2∂2V +∂w2 + Θ1(t) +� +V (t, w + ΠK1(t, i), λ, i) − V (t, w, λ, i) +� ++ Θ2(t) +� +V (t, w − ΠK2(t, i), λ, i) − V (t, w, λ, i) +� +� +, +∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , +with V (T, w, λ, i) = −e−α(w−K), for each (w, λ, i) ∈ R × R+ × X , which coincides with (4.15). +References +S. Altay, K. Colaneri, and Z. 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Zariphopoulou. +Pricing dynamic insurance risks using the principle of +equivalent utility. +Scandinavian Actuarial Journal, 2002(4):246–279, 2002. +doi: +10.1080/ +03461230110106327. + diff --git a/rdFRT4oBgHgl3EQffDfO/content/tmp_files/load_file.txt b/rdFRT4oBgHgl3EQffDfO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4114691474b9bb4b4d5a0290a43069ce2f7a6e7 --- /dev/null +++ b/rdFRT4oBgHgl3EQffDfO/content/tmp_files/load_file.txt @@ -0,0 +1,1581 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf,len=1580 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='13575v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='PM] 31 Jan 2023 UTILITY-BASED INDIFFERENCE PRICING OF PURE ENDOWMENTS IN A MARKOV-MODULATED MARKET MODEL ALESSANDRA CRETAROLA AND BENEDETTA SALTERINI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this paper we study exponential utility indifference pricing of pure endowment policies in a stochastic-factor model for an insurance company, which can also invest in a financial market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Specifically, we propose a modeling framework where the hazard rate is described by an observable general diffusion process and the risky asset price evolves as a jump diffusion affected by a continuous-time finite-state Markov chain representing regimes of the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we describe the optimal investment strategies with and without the insurance derivative and characterize the indifference price in terms of a classical solution to a linear PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We also provide its probabilistic representation via an extension of the Feynman-Kac formula show that it satisfies a final value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Furthermore, we also discuss the indifference price for a portfolio of insurance policies and for a term life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, some numerical experiments are performed to address sensitivity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Keywords: Pure endowment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' regime-switching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' jump processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' optimal investment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' stochastic control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' indifference pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' JEL Classification: G22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' C61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' G11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' AMS Classification: 91B30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 91B25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 93E20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 60J27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Introduction The utility indifference pricing method, initially proposed by (Hodges and Neuberger, 1989) and refined by (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 1993), has gained much attention in the literature on pricing and hedging contingent claims, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Henderson and Hobson, 2009) for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' According to this tech- nique, the indifference seller’s (insurer’s, in this framework) price is defined at the level where the issuer of the contract is indifferent between entering the market on its own, or selling the claim and entering the market with the collected premium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It can be determined by solving an equation involving two value functions, resulting from the stochastic control problems with and without Alessandra Cretarola(�), Department of Mathematics and Computer Science, University of Perugia, Via Luigi Vanvitelli, 1, I-06123 Perugia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Benedetta Salterini, Department of Mathematics and Computer Science, University of Firenze, Viale Morgagni, 67/A, I-50134 Firenze, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' E-mail addresses: alessandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='cretarola@unipg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='it, benedetta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='salterini@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI insurance liabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The indifference pricing approach has become a popular method for evalu- ating derivatives in incomplete markets and has been successfully applied to price insurance con- tracts in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Young and Zariphopoulou, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moore and Young, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Ludkovski and Young, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Delong, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Eichler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Liang and Lu, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Ceci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Precisely, in (Young and Zariphopoulou, 2002) explicit results are derived for an exponential utility function by solving the Hamilton Jacobi equation in a market driven by a geometric Brownian motion when the insurance risk is independent of the financial risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' A more general framework is studied in (Moore and Young, 2003), where the payment amount of the endowment policy is a function of the underlying risky asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In (Ludkovski and Young, 2008), the authors investigate pricing of mor- tality contingent claims under the effects of the stochastic hazard and interest rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The pricing and hedging problem for a group of life insurance contracts in the presence of systematic mortality risks in a market model driven by a Levy process is considered in (Delong, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In (Eichler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2017), the authors analyze the valuation of catastrophe derivatives, while in (Liang and Lu, 2017) they investigate the pricing problem for life insurance contracts with equity-indexed life contingent payments, in a financial market which allows for shot-noise effects in the stock prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, results on the valuation of pure endowment policies under partial information via backward sto- chastic differential equations can be found in (Ceci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Pricing and hedging of unit-linked life insurance contracts via other techniques has been studied e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' in (Ceci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2014, 2015, 2017), where the authors apply the (local) risk-minimization approach in a partial information framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It is worth noting that the indifference pricing approch is widely used also in non-life insurance, for instance to evaluate insurance-linked securities, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this paper, we investigate the indifference pricing problem of pure endowment contracts for an insurance company in a continuous-time financial market where the risky asset price dynamics can exhibit jumps and is affected by regime changes, when the hazard rate governing the population mortality is stochastic and driven by a general diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' A pure endowment is a life insurance policy which yields a sum of money after a specified number of years, provided some nominated person be alive at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Precisely, we consider a pure endowment with maturity of T years for which the terminal survival benefit is given by a fixed amount, payable provided that the insured person is still alive at time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Our modeling framework takes into account financial risk due to price fluctuations, economic risk (or regime-switching risk) arising from structural changes in economic conditions and mortality risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Inspired by (Ludkovski and Young, 2008), we consider a more sophisticated financial market introducing several jumps in the stock price behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To the best of our knowledge, indifference pricing of life-insurance liabilities in a Markov- modulated framework accounting for a market behavior affected by long-term macroeconomic conditions described by the continuous-time Markov chain, possible jumps in the risky asset price dynamics and stochastic hazard rate, is taken up for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In particular, we consider a financial market with a riskless asset and a risky asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The latter is described by a jump diffusion process where the appreciation rate and the volatility depend on an observable continuous-time, finite-state Markov chain representing the regimes of the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Taking a mixture of continuous and jump processes for the stock price dates back to (Merton, 3 1976) and it can also be found in more recent papers, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Ceci and Gerardi, 2009) and (Xiao and Zhao, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It seem reasonable to deal with this financial market model, indeed recent research provides strong empirical evidence of jumps in stock prices, see (Jawadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, the stock price behavior could be also affected by long-term macroeconomic conditions that should be included in the market modeling and represented by another stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Therefore, the presence of an exogenous term affecting the risky asset makes the model even more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This stochastic factor may represents some environmental conditions, social circum- stances, economic crisis or natural phenomena, that can have a considerable impact on financial returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The economic effects of catastrophic events, climate changes and pandemics, as for in- stance the COVID-19, on the financial market are recently analyzed, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', (Baek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Just and Echaust, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Tesselaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Here, we address this modeling issue by assuming that all these exogenous events are aggregated to create different regimes, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' in (Sotomayor and Cadenillas, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Altay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Cretarola and Figà-Talamanca, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' An additional feature of our model is to take the hazard rate of individuals as a general diffusion process, in order to capture the unexpected changes in mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We are not the first to consider stochastic mortality rates, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Milevsky and Promislow, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Dahl, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Dahl and Møller, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Biffis, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Ludkovski and Young, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Indeed, empirical evidence suggests that wars, medical breakthroughs, developments in healthcare and improved lifestyles combine to affect hu- man mortality in a fluctuating and unpredictable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The uncertainty given by minuscule and continuous movements of the mortality intensity is usually represented by a Brownian motion, see (Cairns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2006) for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As a consequence, it seems reasonable to require that in our setting the exogenous stochastic factor, representing long-term environmental changes, does not affect the mortality intensity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' therefore the insurance market remains independent of the financial market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We price the policy through the principle of equivalent utility by comparing the maximal expected utility functions with and without writing the life insurance contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Under exponential utility and using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman (in short HJB) equation, we describe the optimal investment strategy and show verification results for the value functions of the problems without and with insurance liabilities via classical solutions to a linear partial differential equation (in short PDE) and a system of ordinary differential equations (in short ODEs), see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Further, we characterize the indifference price of the pure endowment in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We prove that it solves a proper final value problem and we also obtain its probabilistic representation by means of an extended version of the Feynman- Kac formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We also discuss the indifference price for a group of insurance contracts and another kind of mortality-contingent claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, numerical experiments are performed to investigate some features of our model specification, emphasizing the impact of the regime-switching and the randomness effect introduced by the stochastic hazard rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In Section 2 we introduce the mathematical framework and describe the Markov-modulated financial-insurance market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The pricing problem formulation via utility indifference pricing can be found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In Section 4 we apply the HJB approach to 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI the resulting stochastic control problems and provide the Verification Theorems and the optimal investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The characterization of the indifference price of the pure endowment policy is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In Section 6 we illustrate some numerical results and sensitivity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, technical proofs are collected in Appendix A and how to derive the HJB equation for the problem with insurance liability is shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Modeling framework We consider a complete probability space (Ω, F, P) endowed with a filtration G = {Gt, t ∈ [0, T]}, satisfying the usual conditions of completeness and right continuity, where T > 0 is a fixed, finite time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Specifically, the filtration G is given by G = F ∨ FI, where the filtration F = {Ft, t ∈ [0, T]} models the information flow in the financial market and FI = {F I t , t ∈ [0, T]} contains information about the lifetime of the individual insured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We assume that the subfiltrations F and FI are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To describe some possible structural changes in economic conditions, we introduce an irreducible and continuous-time Markov chain X = {Xt, t ∈ [0, T]} with finite state space X = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' , M}, whose transition probabilities satisfy P(Xt+δt = j|Xt = i) = aijδt + o(δt), i ̸= j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' P(Xt+δt = i|Xt = i) = 1 + aiiδt + o(δt), when δt −→ 0, where for each i ∈ X we have aij ≥ 0 for each i ̸= j and aii = − M � j=1 aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Here, Xt represents the regime of the economy at time t, and M the number of regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let A = (aij)i,j∈X denote the generating Q-matrix of the Markov chain X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It is convenient to represent X as a stochastic integral with respect to a Poisson random measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Following the description of (Basak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2011), for i, j ∈ X , with i ̸= j, we denote by ∆ij the consecutive (with respect to the lexicographic ordering on X ×X ) left-closed right-open intervals of the real line, each having length aij and define a function h : X × R −→ RM by embedding {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' , M} into RM (identifying i with ei ∈ RM), as follows h(i, z) = � j − i, if z ∈ ∆ij 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, we get Xt = X0 + � t 0 � R h(Xv−, z)P(dz, dv), t ∈ [0, T], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) where the integration is over the interval (0, t] and P(dz, dt) is a Poisson random measure with intensity m(dz)dt, with m(dz) being the Lebesgue measure on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let ˆP(dz, dt) be the compensated Poisson random measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' ˆP(dz, dt) = P(dz, dt) − m(dz)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 5 In this setting, we consider a financial market consisting of a locally risk-free money market account and one stock as a risky asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The price process B = {Bt, t ∈ [0, T]} of the locally risk-free asset is described by dBt = rBtdt, B0 = 1, where r is a positive constant denoting the risk-less interest rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The risky asset price process S = {St, t ∈ [0, T]} evolves over time according to the following Markov-modulated dynamics dSt = St− � µ(t, Xt)dt + σ(t, Xt)dZS t + K1(t, Xt−)dN1 t − K2(t, Xt−)dN2 t � , S0 = s ∈ R+, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) where R+ = (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Here, ZS = {ZS t , t ∈ [0, T]} is a standard Brownian motion independent of X and N1 = {N1 t , t ∈ [0, T]} and N2 = {N2 t , t ∈ [0, T]} are independent Poisson processes defined on (Ω, F, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Furthermore, we suppose that N1, N2 are independent of ZS and X and that the F-intensities of N1 and N2 are positive deterministic functions Θ1 : [0, T] −→ R+ and Θ2 : [0, T] −→ R+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The coefficients µ : [0, T]×X −→ R+ and σ : [0, T]×X −→ R+ are measurable functions which model the appreciation rate and the volatility of the stock, respectively, such that µ(t, i) > r, for all (t, i) ∈ [0, T] × X and � T 0 � µ(t, Xt) + σ2(t, Xt) � dt < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='. (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) Moreover, K1 : [0, T] × X −→ R+ and K2 : [0, T] × X −→ R+ are measurable functions such that Kl(t, i) > 0, l = 1, 2 K2(t, i) < 1, for every (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) it is clear that the pair (S, X) is an (F, P)-Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The main motivation for introducing a regime-switching behavior is to have a model capable of describing the risky asset price dynamics under different market conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By the Doléans-Dade exponential formula, condition K2(t, i) < 1 allows us to write St = seLt, t ∈ [0, T], where the logreturn process L = {Lt, t ∈ [0, T]} is given by dLt = � µ(t, Xt) − 1 2σ2(t, Xt) � dt + σ(t, Xt)dZS t + ln(1 + K1(t, Xt−))dN1 t + ln(1 − K2(t, Xt−))dN2 t , with L0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' If we assume that � T 0 � K2 1(t, Xt−)Θ1(t) + K2 2(t, Xt−)Θ2(t) � dt < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) then the process S is an F-semimartingale with decomposition St = s + AS t + MS t , where AS = {AS t , t ∈ [0, T]} defined as AS t = � t 0 Sv− (µ(v, Xv−) + K1(v, Xv−)Θ1(v) + K2(v, Xv−)Θ2(v)) dv, 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI is an R-valued process with finite variation paths and AS 0 = 0, while MS = {MS t , t ∈ [0, T]} given by MS t = � t 0 Svσ(v, Xv)dZS v + � t 0 Sv−K1(v, Xv−){dN1 v −Θ1(v)dv}− � t 0 Sv−K2(v, Xv−){dN2 v −Θ2(v)dv} is an F-local martingale with MS 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) imply that the process R = {Rt, t ∈ [0, T]} defined as Rt = � t 0 � µ(v, Xv)dv + σ(v, Xv)dZS v + K1(v, Xv−)dN1 v − K2(v, Xv−)dN2 v � is an F-semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Noting that dSt = St−dRt, we can conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ Now, we consider an individual to be insured and a stochastic model for the mortality of the equivalent age cohort of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We assume that the hazard rate (or force of mortality) is governed by a diffusion process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we describe the mortality intensity as a stochastic process Λ = {λt, t ∈ [0, T]} that is given by the following stochastic differential equation (in short SDE) dλt = b(t, λt)λtdt + c(t, λt)λtdZΛ t , λ0 = λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) Here, ZΛ = {ZΛ t , t ∈ [0, T]} is an additional standard Brownian motion on (Ω, F, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, b : [0, T] × R −→ R and c : [0, T] × R −→ R are two measurable functions such that a unique strong solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) exists and the following conditions hold E �� T 0 |b(t, λt)λt|dt + � T 0 c(t, λt)2λ2 tdt � < ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6) sup t∈[0,T] E � λ2 t � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) These conditions are satisfied if, for instance, the coefficients of the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) fulfill the classical Lipschitz and sublinear growth conditions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Gihman and Skorohod, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We observe that, the mortality rate of the insured is generally different from that of its age cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' However, to keep the framework tractable we consider individuals subjected to the same stochastic hazard rate, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' in (Ludkovski and Young, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let τ be a non negative random variable on (Ω, F, P) which represents the remaining lifetime of the given individual of the reference population with mortality rate Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Denote by D = {Dt, t ∈ [0, T]} the death indicator process associated to τ by setting Dt := 1{τ≤t}, for every t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We assume that D is an FI-adapted process independent of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The indifference pricing problem formulation Now, we assume that the insurance company issues a unit-linked life insurance policy, which is a long term insurance contract whose payoff depends on the insured remaining lifetime and on the underlying stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In particular, we consider a pure endowment contract with maturity of T years, which pays a fixed amount if the policyholder is still alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, the associated payoff is given by the random variable GT := K1{τ>T} = K(1 − DT), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) where K is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The goal is to evaluate the pure endowment policy with payoff given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) in the Markov-modulated model outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Since the financial market consists of two primary securities and several sources of random shocks due to mortality events and structural changes in economic conditions, it turns out to be incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Therefore, we apply the indifference pricing approach assuming that the insurance company pref- erences towards the risk are given by an exponential utility function of the form u(w) = −e−αw, w ∈ R, where α is a positive parameter which measures the absolute risk aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In the underlying financial market, the insurance company starts out with an initial wealth w, and then proceeds to trade dynamically among the risky asset and the locally risk-free asset, following a self-financing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let Π = {Πt, t ∈ [0, T]} be the total amount of wealth invested in the stock, with the remainder of wealth in the money market account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The insurance company is also allowed to short-sell and to borrow/lend any infinitesimal amount, so that Πt ∈ R, for each t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Precisely, given an initial wealth w ∈ R+ 0 , the insurance company wealth process {W Π t , t ∈ [0, T]} associated to a given strategy Π evolves over time as dW Π t = Πt dSt St− + (W Π t − Πt)dBt Bt = (rW π t + Πt (µ(t, Xt) − r)) dt + Πtσ(t, Xt)dZS t + Πt � K1(t, Xt−)dN1 t − K2(t, Xt−)dN2 t � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) with W Π 0 = w ∈ R+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It can be checked that the solution to the SDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) is given by W Π t = W Π 0 ert + � t 0 er(t−s)Πs(µ(s, Xs) − r)ds + � t 0 er(t−s)Πsσ(s, Xs)dZS s + � t 0 er(t−s)Πs � K1(s, Xs−)dN1 s − K2(s, Xs−)dN2 s � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) with W Π 0 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In order to characterize the indifference price of the pure endowment, we introduce two optimal investment problems, with and without insurance liabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We start by defining the class of admissible strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' An admissible strategy is a self-financing portfolio identified by an R-valued G- predictable process Π = {Πt, t ∈ [0, T]} such that E �� T 0 |Πt|(µ(t, Xt) − r)dt � < ∞, E �� T 0 Π2 tσ2(t, Xt)dt � < ∞, E �� T 0 |Πt| � K1(t, Xt−)Θ1(t) + K2(t, Xt−)Θ2(t) � dt � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) We denote by A the set of G-admissible strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Whenever the controls are restricted to the time interval [t, T], we will use the notation At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we assume that the following assumptions are in force throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (i) There exist three positive constants M1, M2 and K such that Θ1(t) ≤ M1, Θ2(t) ≤ M2, K1(t, i) ≤ K, for every (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (ii) There is a constant C > 0 such that µ(t,i)−r σ(t,i) ≤ C, for every (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In particular, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3(i) provides a sufficient condition for a strategy Π to be admissible as shown in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let Π = {Πt, t ∈ [0, T]} be a G-predictable strategy with values in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Assume there exists a square-integrable function η : [0, T] × X → (0, +∞) such that |Πt| ≤ η(t, Xt), t ∈ [0, T], P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) and � T 0 η(s, i) � (µ(s, i) − r) + η(s, i)σ2(s, i) � ds < ∞, ∀i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6) Then, Π is an admissible strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Π ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We note that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6), we have E �� T 0 |Πs| � (µ(s, Xs) − r) + Πsσ2(s, Xs) � ds � ≤ E �� T 0 η(s, Xs) � (µ(s, Xs) − r) + η(s, Xs)σ2(s, Xs) � ds � ≤ max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=',M � T 0 η(s, i) � (µ(s, i) − r) + η(s, i)σ2(s, i) � ds < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, in view of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3(i), condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) is satisfied and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ We consider the case where the insurance company simply invests its wealth in the financial market, without writing the insurance derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, the goal is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 9 Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To maximize the expected utility of its terminal wealth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' to solve sup Π∈A E � − e−αW Π T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let (t, w, i) ∈ [0, T] × R × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In a dynamic framework, we define the corresponding value function ¯V by ¯V (t, w, i) := sup Π∈At Et,w,i � − e−αW Π T (t,w)� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) where Et,w,i denotes the conditional expectation given W Π t = w and Xt = i, and {W Π s (t, w), s ∈ [t, T]} stands for the solution to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) with initial condition W Π t = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Note that, since the coefficients µ, σ, K1 and K2 only depend on t and i, it is possible to absorb the stock price in the wealth and therefore to remove the variable corresponding to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we suppose that the insurance company invests its wealth in the market, writing a pure endowment contract with payoff given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this case, the goal of the insurance company is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To maximize the expected utility of its terminal wealth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' to solve sup Π∈A E � − e−α(W Π T −GT )� , where GT is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let (t, w, λ, i) ∈ [0, T] × R × R+ × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We define the corresponding value function V as V (t, w, λ, i) := sup Π∈At Et,w,λ,i � − e−α(W Π T (t,w)−GT )� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8) where Et,w,λ,i denotes the conditional expectation given W Π t = w, λt = λ and Xt = i and we implicitly condition on Gt = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We note that the control Π = 0 is admissible and such that Et,w,i � e−αW 0 T (t,w)� < ∞, for each (t, w, i) ∈ [0, T] × R × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Et,w,λ,i � e−α(W 0 T (t,w)−GT )� < ∞, for each (t, w, λ, i) ∈ [0, T] × R × R+ × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This implies that ess sup Π∈At E � −e−αW Π T � > −∞, ess sup Π∈At E � −e−α(W Π T −GT )� > −∞, P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', t ∈ [0, T], and as a consequence that sup Π∈A E � −e−αW Π T � > −∞, sup Π∈A E � −e−α(W Π T −GT )� > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The optimal investment problems In this section, applying the classical stochastic control approach based on the Hamilton-Jacobi- Bellman (in short HJB) equation, we characterize the optimal investment strategies and provide verification results for the value functions ¯V and V given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The pure investment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Firstly, we consider the case where the insurance company simply invests in the underlying financial market, so the corresponding value function ¯V is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let us consider the HJB equation with final condition that the value function ¯V is expected to solve, if sufficiently smooth: � supΠ∈R ¯LΠ i ¯V (t, w, i) = 0, ∀(t, w, i) ∈ [0, T) × R × X , ¯V (T, w, i) = −e−αw, ∀(w, i) ∈ R × X , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) where ¯LΠ i denotes the Markov generator of (W Π, X) associated with a constant control Π ∈ R, given by ¯LΠ i f(t, w, i) = ∂f ∂t (t, w, i) + � rw + (µ(t, i) − r)Π � ∂f ∂w(t, w, i) + 1 2σ2(t, i)Π2 ∂2f ∂w2(t, w, i) + � j∈X aijf(t, w, j) + Θ1(t) �¯V (t, w + ΠK1(t, i), i) − ¯V (t, w, i) � + Θ2(t) �¯V (t, w − ΠK2(t, i), i) − ¯V (t, w, i) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) for every (t, w, i) ∈ [0, T] × R × X and for every function f(·, ·, i) in C1,2, given i ∈ X , which is sufficiently integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Since the pair (W Π, X) is a Markov process, any Markovian control is of the form Πt = Π(t, W Π t , Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The generator ¯LΠ i f(t, w, i) associated to a general Markovian strategy can be easily obtained by replacing Π with Π(t, w, i) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we consider the ansatz ¯V (t, w, i) = −e−wαer(T −t)ϕ(t, i), with (t, w, i) ∈ [0, T] × R × X , for a suitable function ϕ, which is motivated by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Assume that there exists a unique function ϕ(·, i), for each i ∈ X , solution to the following Cauchy problem: \uf8f1 \uf8f2 \uf8f3 ∂ϕ ∂t (t, i) = H(t, ϕ(t, i)), t ∈ [0, T), ϕ(T, i) = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) where H(t, ϕ(t, i)) = − � j∈X ϕ(t, j)aij − ϕ(t, i) inf Π∈R ¯ΨΠ(t, i), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) 11 with the function ¯ΨΠ : [0, T] × X → R defined by ¯ΨΠ(t, i) = − αer(T−t)(µ(t, i) − r)Π + 1 2α2e2r(T−t)σ2(t, i)Π2 + Θ1(t) � e−αΠK1(t,i)er(T −t) − 1 � + Θ2(t) � eαΠK2(t,i)er(T −t) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) Then, the function ¯V (t, w, i) = −e−wαer(T −t)ϕ(t, i), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6) solves the HJB problem given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' From the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6), we can easily verify that the original HJB problem given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) reads as follows ∂ϕ ∂t (t, i) + � j∈X ϕ(t, j)aij + inf Π∈R � − αer(T−t)ϕ(t, i)(µ(t, i) − r)Π + 1 2α2e2r(T−t)ϕ(t, i)σ2(t, i)Π2 + ϕ(t, i)Θ1(t) � e−αΠK1(t,i)er(T −t) − 1 � + ϕ(t, i)Θ2(t) � eαΠK2(t,i)er(T −t) − 1 �� = 0, t ∈ [0, T), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) with final condition ϕ(T, i) = 1, for all i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Thus, if we define the function ¯ΨΠ by means of expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5), equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) can be written as ∂ϕ ∂t (t, i) + � j∈X ϕ(t, j)aij + ϕ(t, i) inf Π∈R ¯ΨΠ(t, i) = 0 and we find out the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ The previous result suggests to focus on the minimization of the function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5), that is the aim of the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Optimal investment strategy without the insurance derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we study the following minimization problem inf Π∈R ¯ΨΠ(t, i), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8) where the function ¯ΨΠ is introduced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The following equation σ2(t, i)αer(T−t)Π − (µ(t, i) − r) = K1(t, i)Θ1(t)e−αΠK1(t,i)er(T −t) − K2(t, i)Θ2(t)eαΠK2(t,i)er(T −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='9) admits at least a solution �Π(t, i) in R for any (t, i) ∈ [0, T] × X and the minimization problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8) has a unique solution Π∗(t, i) = �Π(t, i), for all (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we observe that ¯ΨΠ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) is continuous with respect to Π ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' for every (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T] × X and has continuous first and second order derivatives with respect to Π ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' which are respectively given by ∂ ¯ΨΠ ∂Π (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) = −αer(T−t)(µ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − r) + σ2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i)α2e2r(T−t)Π − αer(T−t)K1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i)Θ1(t)e−αΠK1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i)er(T −t) + αer(T−t)K2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i)Θ2(t)eαΠK2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i)er(T −t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' ∂2 ¯ΨΠ ∂Π2 (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) = α2e2r(T−t)σ2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) + α2e2r(T−t)K2 1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i)Θ1(t)e−αΠK1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i)er(T −t) + α2e2r(T−t)K2 2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i)Θ2(t)eαΠK2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i)er(T −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Note that these derivatives are well defined and ∂2 ¯ΨΠ ∂Π2 (t, i) > 0, for every (t, i) ∈ [0, T] × X ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' therefore, the function ¯ΨΠ(t, i) is strictly convex in Π ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, it is easy to check that, for any (t, i) ∈ [0, T] × X , we have lim Π−→+∞ ∂ ¯ΨΠ ∂Π (t, i) −→ +∞, while lim Π−→−∞ ∂ ¯ΨΠ ∂Π (t, i) −→ −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As a consequence, being ∂ ¯ΨΠ ∂Π (t, i) a continuous function in Π ∈ R, there exists �Π(t, i) ∈ R such that ∂ ¯ΨΠ ∂Π (t, i) = 0, for every (t, i) ∈ [0, T]×X , that is, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='9) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Since the function ¯ΨΠ(t, i) is strictly convex, the stationary point �Π(t, i) ∈ R is unique and provides the unique minimizer Π∗(t, i) = �Π(t, i) on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We point out that Π∗ = Π∗(t, i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' the solution of the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8) depends on time and on the Markov chain, since it solves equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This means that the optimal investment strategy evolves over time and changes according to the different economic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, we note that Π∗ does not depend on wealth, as usually happens when the investor’s preferences are described by an exponential utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5 (Properties of Π∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The following condition is satisfied min \uf8f1 \uf8f2 \uf8f30, ln � µ(t,i)−r M2 � αer(T−t) \uf8fc \uf8fd \uf8fe ≤ Π∗(t, i) ≤ µ(t, i) − r + CM1 σ2(t, i)αer(T−t) , for all (t, i) ∈ [0, T] × X , where C, M1 ∈ R+ are the constants limiting the functions K1 and Θ1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 (we omit the dependence in Π∗ on (t, i)), we get the upper limit and the lower limit for Π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' If Π∗0 is non-negative, we have 0 = σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − K1(t, i)Θ1(t)e−αΠ∗K1(t,i)er(T −t) + K2(t, i)Θ2(t)eαΠ∗K2(t,i)er(T −t) > σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − K1(t, i)Θ1(t)e−αΠ∗K1(t,i)er(T −t) ≥ σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − CM1e−αΠ∗K1(t,i)er(T −t) ≥ σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − CM1, which implies Π∗(t, i) ≤ µ(t, i) − r + CM1 σ2(t, i)αer(T−t) , for all (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Otherwise, if Π∗ is non-positive, we get 0 = σ2(t, i)αer(T−t)Π∗ − (µ(t, i) − r) − K1(t, i)Θ1(t)e−αΠ∗K1(t,i)er(T −t) + K2(t, i)Θ2(t)eαΠ∗K2(t,i)er(T −t) < −(µ(t, i) − r) + K2(t, i)Θ2(t)eαΠ∗K2(t,i)er(T −t) ≤ −(µ(t, i) − r) − M2eαΠ∗er(T −t), that leads to Π∗(t, i) ≥ ln � µ(t,i)−r M2 � αer(T−t) , for all (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Verification Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we are ready to state the verification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 (Verification Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Suppose that the Cauchy problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) admits a classical solution ϕ(·, i) ∈ C1� (0, T[ � ∩C � [0, T] � , for each i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, the function ¯V : [0, T]×R×X −→ R defined by ¯V (t, w, i) = −e−wαer(T −t)ϕ(t, i) is the value function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Consequently, the strategy Π∗ t = Π∗(t, Xt) described in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 is an optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The proof uses similar arguments as in that of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11 below for the problem with the insurance derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Note that Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5 corresponds to a special case of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6, choosing GT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Nevertheless, for the sake of clarity we trace the fundamental steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2, the function ¯V (t, w, i) defined in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6) solves the HJB problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Hence, for any (t, w, i) ∈ [0, T] × R+ 0 × X , we have ¯LΠ i ¯V (s, W Π s (t, w), Xs(t, i)) ≤ 0, ∀s ∈ [t, T], Π ∈ At, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='10) 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI where we recall that {W Π s (t, w), s ∈ [t, T]} and {Xs(t, i), s ∈ [t, T]} denote the solutions to equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) at time s ∈ [t, T], starting from (t, w) ∈ [0, T]×R+ 0 and (t, i) ∈ [0, T]×X , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Clearly, ¯V (·, ·, i) ∈ C1,2([0, T] × R), for each i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2), by applying Itô’s formula, we have ¯V (T, W Π T (t, w), XT(t, i)) = ¯V (t, λ, i) + � T t ¯LΠ i ¯V (v, W Π v (t, w), Xv(t, i))dv + MT , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11) where M = {Mr, r ∈ [t, T]} is the stochastic process given by Mr = � r t Πvσ(v, Xv)∂ ¯V ∂w (v, W Π v , Xv)dZS v + � r t � R �¯V � v, W Π v , Xv− + h(Xv−, z) � − ¯V (v, W Π v , Xv−) � ˆP(dv, dz) + � r t �¯V � v, W Π v− + ΠvK1(v, Xv−), Xv−) � − ¯V (v, W Π v−, Xv−) � {dN1 v − Θ1(v)dv} + � r t �¯V � v, W Π v− − ΠvK2(v, Xv−), Xv−) � − ¯V (v, W Π v−, Xv−) � {dN2 v − Θ2(v)dv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In order to prove that M is a (G, P)-local martingale, we use a localization argument, taking τn := inf{s ∈ [t, T] | W Π s < −n}, n ∈ N, which defines a non-decreasing sequence of stopping times {τn}n∈N such that limn−→+∞ τn = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Therefore, taking the conditional expectation with respect to Wt = w and Xt = i on both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11), with T replaced by T ∧ τn, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='10) we obtain that Et,w,i � ¯V (T ∧ τn, W Π T∧τn(t, w), XT∧τn(t, i)) � ≤ ¯V (t, w, i), for every Π ∈ At, t ∈ �0, T ∧ τn�, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we note that E ��¯V (T ∧ τn, W Π T∧τn(t, w), XT∧τn(t, i)) �2� = E � e−2αW Π T ∧τner(T ∧τn−t)ϕ(T ∧ τn, i)2� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Consequently, { ¯V (T ∧ τn, W Π T∧τn(t, w), XT∧τn(t, i))}n∈N is a family of uniformly integrable random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Hence, it converges almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Since {τn}n∈N is a bounded and non-decreasing sequence of random times and P(|W Π t | < +∞) = 1, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3), we get Et,w,i �¯V (T, W Π T (t, w), XT(t, i)) � = lim n−→+∞ Et,w,i �¯V (T ∧ τn, W Π T∧τn(t, w), XT∧τn(t, i)) � ≤ ¯V (t, w, i), ∀t ∈ [0, T], Π ∈ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As a byproduct, since Π∗(t, i) given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 realizes the infimum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8), we have that ¯LΠ∗ i ¯V (t, w, i) = 0 and, performing the computations above, we get the equality Et,w,i � − e−αW Π∗ T (t,w)� = sup Π∈At Et,w,i � − e−αW Π T (t,w)� = ¯V (t, w, i), that is, Π∗ t = Π∗ t(t, Xt) is an optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ 15 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6, the value function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) can be characterized as a transformation of the solution ϕ to a certain system of ODEs with a particular terminal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As regards exis- tence and uniqueness of a solution to this specific Cauchy problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3), we refer to (Walter, 1998, Theorem VII, Chapter II:6) or to (Baran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2013, Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' According to (Walter, 1998), if H given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) is a locally Lipschitz function with respect to the second variable, uniformly in t, we get that there exists a unique solution ϕ(t, i), for every t ∈ [0, T], for all i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Requiring that µ, σ, K1 and K2 are continuous functions is a sufficient condition for the regularity of function H and, as a consequence, the smoothness of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Otherwise, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) can be seen as a trivial case of the Cauchy problem faced by (Baran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Assuming that µ(·, i) and σ(·, i) are continuous functions in t ∈ [0, T], for all i ∈ X , guarantees that infΠ∈R ¯Ψ(t, i) is bounded with respect to the first variable and thus all required hypotheses are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The next result provides the optimal investment strategy corresponding to Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Assume existence and uniqueness of a classical solution to the the HJB equation with final condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, suppose that for all (t, i) ∈ [0, T] × X , σ(t, i) > σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='12) Then, the process {Π∗(t, i), t ∈ [0, T]} characterized in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 provides the optimal in- vestment strategy for Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let η(t, i) = max \uf8f1 \uf8f2 \uf8f3 ���ln � µ(t,i)−r M2 ���� αer(T−t) , µ(t, i) − r + CM1 σ2(t, i)αer(T−t) \uf8fc \uf8fd \uf8fe, (t, i) ∈ [0, T] × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We show that conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5, we immediately have Π∗(t, Xt) ≤ η(t, Xt) and Π∗(t, Xt) ≥ −η(t, Xt), for every t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='12) and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3, we get condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, the process {Π∗(t, i), t ∈ [0, T]} is an admissible investment strategy and the statement follows by applying the Verification Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The investment problem with the insurance derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we suppose that the insurance company can write a pure endowment contract, whose payoff is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The following result ensures that the financial-insurance model outlined in Section 2 has a Mar- kovian structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' the vector process (W Π, Λ, X) is a (G, P)-Markov-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let LΠ i denote the Markov generator of (W Π, Λ, X) associated with a constant control Π ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The set D(LΠ i ) denotes the class of functions f(·, ·, ·, i) ∈ C1([0, T]) × C2(R × (0, +∞)), for each i ∈ X , such that for every constant Π ∈ R, we have E � � T 0 � σ(v, Xv)Π ∂f ∂w(v, W Π v , λv, Xv) �2 dv � < ∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='13) E � � T 0 � c(v, λv)λv ∂f ∂λ(v, W Π v , λv, Xv) �2 dv � < ∞, and E �� T 0 � R ��f � v, W Π v , λv, Xv− + h(Xv−, z) � − f(v, W Π v , λv, Xv−) �� m(dz)dv � < ∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='14) E �� T 0 ��f � v, W Π v− + ΠK1(v, Xv−), λv, Xv−) � − f(v, W Π v−, λv, Xv−) �� Θ1(v)dv � < ∞, E �� T 0 ��f � v, W Π v− − ΠK2(v, Xv−), λv, Xv−) � − f(v, W Π v−, λv, Xv−) �� Θ2(v)dv � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The stochastic process (W Π, Λ, X) is a Markov process on (Ω, F, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' G), with infin- itesimal generator LΠ i for all constant strategies Π ∈ R given by LΠ i f(t, w, λ, i) =∂f ∂t (t, w, λ, i) + � rw + (µ(t, i) − r)Π � ∂f ∂w(t, w, λ, i) + b(t, λ)λ∂f ∂λ(t, w, λ, i) + 1 2σ2(t, i)Π2 ∂2f ∂w2(t, w, λ, i) + 1 2c2(t, λ)λ2∂2f ∂λ2 (t, w, λ, i)+ � j∈X aijf(t, w, λ, j) + Θ1(t) � V (t, w + ΠK1(t, i), λ, i) − V (t, w, λ, i) � + Θ2(t) � V (t, w − ΠK2(t, i), λ, i) − V (t, w, λ, i) � , for every i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The domain of the generator LΠ i is D(LΠ i ), for each i ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The proof is postponed to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let us consider the HJB equation that the value function V is expected to solve, if sufficiently smooth: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 supΠ∈R LΠ i V (t, w, λ, i) + λ � ¯V (t, w, i) − V (t, w, λ, i) � = 0, ∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , V (T, w, λ, i) = −e−α(w−K) ∀(w, λ, i) ∈ R × R+ × X , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15) How to derive the HJB equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15) is shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, let us introduce the following ansatz V (t, w, λ, i) = −e−wαer(T −t)ϕ(t, i)φ(t, λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='16) with (t, w, λ, i) ∈ [0, T] × R × R+ × X , where ϕ solves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3), while the function φ is non-negative and does not depend on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 17 From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='16), replacing all the derivatives and performing some computations, problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15) re- duces to \uf8f1 \uf8f2 \uf8f3 ∂φ ∂t (t, λ) + b(t, λ)λ∂φ ∂λ(t, λ) + 1 2c2(t, λ)λ2∂2φ ∂λ2 (t, λ) − λ(φ(t, λ) − 1) = 0, ∀(t, λ) ∈ [0, T) × R+, φ(T, λ) = eαK, ∀λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17) We observe that the PDE in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17) is linear and a solution exists under suitable conditions on model coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Pham, 1998, Theorem5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) or (Colaneri and Frey, 2021, Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Clearly, if the function φ is a classical solution of the Cauchy problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17), then V (·, ·, ·, i) ∈ C1,2,2([0, T] × R × R+), for each i ∈ X and we have that V (t, w, λ, i) = −e−wαer(T −t)ϕ(t, i)φ(t, λ) solves the original HJB equation given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we can state the verification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11 (Verification Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let ϕ(·, i) ∈ C1� (0, T) � ∩ C � [0, T] � and φ(·, ·) ∈ C1� (0, T) × R+� ∩ C � [0, T] × R+� , for each i ∈ X , be classical solutions of the Cauchy prob- lems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, the function V : [0, T] × R × R+ × X −→ R defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='16) is the value function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Consequently, the strategy Π∗ t = Π∗(t, Xt) described in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 is an optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let ϕ : [0, T] × X −→ R be a function such that ϕ(·, i) ∈ C1� (0, T) � ∩ C � [0, T] � , for each i ∈ X , and suppose that it is a solution of the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, let φ : [0, T] × R+ −→ R+ be a function such that φ(·, ·) ∈ C1� (0, T) × R+� ∩ C � [0, T] × R+� , and suppose that it solves the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, taking V defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='16), we have that V is a solution of the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This implies that, for every (t, w, λ, i) ∈ [0, T] × R × R+ × X LΠ i V (r, W Π r (t, w), λr(t, λ), Xr(t, i)) + λr(t, λ) �¯V (r, W Π r (t, w), Xr(t, i)) − V (r, W Π r (t, w), λr(t, λ), Xr(t, i)) � ≤ 0, r ∈ [t, T], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='18) for all Π ∈ At, where {λr(t, λ), r ∈ [t, T]} denotes the solution to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) with initial condition λt = λ and ¯V is the value function of the pure investment problem given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2), by applying Itô’s formula, we have V (T, W Π T (t, w), λT(t, λ), XT(t, i)) = V (t, λ, i) + � T t LΠ i V (v, W Π v (t, w), λv(t, λ), Xv(t, i))dv + � T t λv(t, λ) �¯V (v, W Π v (t, w), Xv(t, i)) − V (v, W Π v (t, w), λv(t, λ), Xv(t, i)) � dv + MT, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='19) 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI where M = {Mr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' r ∈ [t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T]} is the stochastic process given by Mr = � r t Πvσ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)∂V ∂w (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)dZS v + � r t c(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Yv)λv ∂V ∂y (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)dZΛ v + � r t � R � V � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv− + h(Xv−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' z) � − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � ˆP(dv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' dz) + � r t � V � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v− + ΠvK1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � {dN1 v − Θ1(v)dv} + � r t � V � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v− − ΠvK2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � {dN2 v − Θ2(v)dv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we prove that M is a (G, P)-local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Precisely, we need to show that E � � T∧τn t � σ(v, Xv)Πv ∂V ∂w (v, W Π v , λv, Xv) �2 dv � < ∞, E � � T∧τn t � c(v, λv)λv ∂V ∂λ (v, W Π v , λv, Xv) �2 dv � < ∞, for a suitable, non-decreasing sequence of stopping times {τn}n∈N such that limn−→+∞ τn = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Taking expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='16) into account, we note that ∂V ∂w (t, w, λ, i) = αφ(t, λ)ϕ(t, i)er(T−t)−αwer(T −t), ∂V ∂y (t, w, λ, i) = −∂φ ∂λ(t, λ)ϕ(t, i)e−αwer(T −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let us define a sequence of random times {τn}n∈N by setting τn := inf{s ∈ [t, T] | W Π s < −n, λs > n, φ(s, λs) > n, ∂φ ∂λ(s, λs) > n}, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Throughout the proof, we denote by Cn any constant depending on n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Consequently, we get E � � T∧τn 0 � σ(v, Xv)Πv ∂V ∂w (v, W Π v , λv, Xv) �2 dv � = E � � T∧τn 0 σ2(v, Xv)Π2 v � αφ(v, λv)ϕ(v, Xv)er(T−v)−αW Π v er(T −v)�2 dv � ≤ CnE � � T 0 σ2(v, Xv)Π2 vdv � < ∞ ∀n ∈ N, since Π is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Further, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6) we have that E � � T∧τn 0 � c(v, λv)λv ∂V ∂λ (v, W Π v , λv, Xv) �2 dv � = E � � T∧τn 0 � c(v, λv)λv ∂φ ∂λ(v, λv)ϕ(v, Xv)e−αW Π v er(Xv)(T −t)�2 dv � ≤ CnE � � T 0 c(v, λv)2λ2 vdv � < ∞ ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 19 Furthermore, due to the boundedness of function V until time τn, we have that the stopped process �� r∧τn t � R � V � v, W Π v , λv, Xv− + h(Xv−, z) � − V � v, W Π v , λv, Xv− �� ˆP(dv, dz), r ∈ [t, T] � is a (G, P)-martingale (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Davis, 1993, Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='12(2))), for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, even the stopped processes �� r∧τn t � V � v, W Π v− + ΠvK1(v, Xv−), λv, Xv− � − V � v, W Π v−, λv, Xv �� {dN1 v − Θ1(v)dv}, r ∈ [t, T] � and �� r∧τn t � V � v, W Π v− − ΠvK2(v, Xv−), λv, Xv− � − V � v, W Π v−, λv, Xv �� {dN2 v − Θ2(v)dv}, r ∈ [t, T] � are (G, P)-martingales, (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Brémaud, 1981, Lemma L3, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='II)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Thus, the process {Mr, r ∈ [t, T]} turns out to be a (G, P)-local martingale and {τn}n∈N is a localizing sequence for {Mr, r ∈ [t, T]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Therefore, taking the conditional expectation of both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='19) with respect to Wt = w, λt = λ and Xt = i with T replaced by T ∧ τn, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='18) we obtain that Et,w,λ,i � V (T ∧ τn, W Π T∧τn(t, w), λT∧τn, XT∧τn(t, i)) � ≤ V (t, w, λ, i), for every Π ∈ At, t ∈ �0, T ∧ τn�, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we note that E �� V (T ∧ τn, W Π T∧τn(t, w), λT∧τn(t, λ), XT∧τn(t, i)) �2� = E � e−2αW Π T ∧τner(T ∧τn−t)ϕ(T ∧ τn, XT∧τn)2φ(T ∧ τn, λT∧τn)2� ≤ �K, for a positive constant �K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This means that {V (T ∧ τn, W Π T∧τn(t, w), λT∧τn, XT∧τn(t, i))}n∈N is a family of uniformly integrable random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Hence, it converges almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Since {τn}n∈N is a bounded and non-decreasing sequence of random times and P(|W Π t | < +∞) = 1, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3), in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7), we can apply the dominated convergence theorem and, taking the limit for n −→ +∞, we get Et,w,λ,i � V (T, W Π T (t, w), λT(t, λ), XT(t, i)) � = lim n−→+∞ Et,w,λ,i � V (T ∧ τn, W Π T∧τn(t, w), λT∧τn(t, λ), XT∧τn(t, i)) � ≤ V (t, w, λ, i), for every Π ∈ At, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By the final condition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15) and the previous inequality, we get Et,w,λ,i � − e−α(W Π T (t,w)−K)� ≤ V (t, w, λ, i), for every Π ∈ At, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, since the insurance payment does not depend on the risky asset price, we have that Π∗(t, w, λ, i) = Π∗(t, i) given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 yields that LΠ∗ i V (t, w, λ, i) + λ �¯V (t, w, i) − V (t, w, λ, i) � = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' then, if we apply the above arguments to Π∗ and replacing LΠ i with LΠ∗ i , we find the equality sup Π∈At Et,w,λ,i � − e−α(W Π T (t,w)−K)� = V (t, w, λ, i), 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI which implies that the process Π∗(t, Xt) is an optimal Markovian control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We observe that the optimal investment strategy Π∗(t, Xt) turns out to be the same as the pure investment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This means that the optimal portfolio for the investment problem with the insurance derivative equals the strategy without insurance risks when the insurance payment is independent of the risky asset price process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This statement is the same as that given in (Delong, 2009) and (Liang and Lu, 2017) when the risky asset price dynamics is driven by a Lévy process and a shot-noise process, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Characterization of the indifference price In this section we compute explicitly the indifference price for the pure endowment contract whose payoff is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To this aim, we provide the definition of the indifference price charged by an insurance company which writes a pure endowment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Recall that ¯V and V are the value functions introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Given Wt = w, λt = λ and Xt = i, the indifference price process or reservation price process P = {Pt, t ∈ [0, T]} of the insurer related to the pure endowment contract is defined at any time t ∈ [0, T] as the G-adapted process implicit solution to the equation ¯V (t, w, i) = V (t, w + Pt, λ, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) In other words, P is the price that makes the insurer indifferent, in terms of expected utility, between not selling and selling the policy for the price P now and paying the benefits at maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We have the following explicit characterization of the indifference price process in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Under the same hypotheses of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11, for every t ∈ [0, T], the indifference price of the insurer related to the pure endowment with maturity T is given by Pt = P(t, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) = ln � φ(t, λ) � αer(T−t) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2) for all (t, λ) ∈ [0, T] × R+, where the function φ solves the Cauchy problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='11, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) reads as e−wαer(T −t)ϕ(t, i) = e−(w+Pt)αer(T −t)ϕ(t, i)φ(t, λ), and then ePtαer(T −t) = φ(t, λ), from which, computing the logarithm of both members, we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ 21 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In the actuarial literature, exponential utility preferences imply that the indifference price of a pure endowment depends on the risk-aversion coefficient, the stochastic interest rate and the logarithm of the function that links the two value functions and is independent of wealth (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Young and Zariphopoulou, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Young, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moore and Young, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Ludkovski and Young, 2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Here, the indifference price shares the same features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, we note that it does not explicitly depend on the current regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' meaning that in our model, the state of the economy does not affect directly the indifference price of such type of insurance contracts but only the amount invested in the financial market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Under the indifference pricing principle, the premium solves a terminal value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' For every (t, λ) ∈ [0, T] × R+ the indifference premium P = P(t, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) satisfies the following PDE rP = ∂P ∂t + b(t, λ)λ∂P ∂λ + 1 2c2(t, λ)λ2�∂2P ∂λ2 + αer(T−t) �∂P ∂λ �2 � + λ αer(T−t) � e−P αer(T−t) − 1 � , with boundary condition P(T, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) = K, for each λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It follows from a straightforward application of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ Finally, we provide a probabilistic representation for the indifference price process P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Indeed, if the function φ solves the Cauchy problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='17), we can represent φ as an expectation via an extension of the Feynman-Kac formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' More precisely, using the linear PDE for φ − 1, it is easy to see that φ(t, λ) − 1 = Et,λ � e− � T t λvdv� eαK − 1 �� , for every (t, λ) ∈ [0, T] × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' So, as a consequence, we have that φ(t, λ) = 1 + � eαK − 1 � Et,λ � e− � T t λvdv� , where Et,λ denotes the conditional expectation given λt = λ, for every (t, λ) ∈ [0, T] × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We outline that Et,λ � e− � T t λvdv� is the conditional probability that an individual will survive until time T given that she/he is alive at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Hence, representing the function φ as φ(t, λ) = eαKEt,λ � e− � T t λvdv� + � 1 − Et,λ � e− � T t λvdv�� = Et,λ � eαGT � , for every (t, λ) ∈ [0, T] × R+, the indifference price of the insurer related to a pure endowment contract can be written as Pt = P(t, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) = ln � Et,λ � eαGT �� αer(T−t) , for every (t, λ) ∈ [0, T] × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Group of insured people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this subsection, we evaluate a panel of insurance policies, extending the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Put another way, we deal with a portfolio consisting of pure endowments issued to a group of n ∈ N individuals, who are all the same age with indipendent and identically distributed times until death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We assume that the loss payable at the maturity T equals the amount K > 0 for each of the insured people who have not died yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' So, the value function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8) is replaced by V (n)(t, w, λ, i) := sup Π∈At(G) Et,w,λ,i � − e−α(W Π T −G(n) T )� , where G(n) T = mK, with K > 0 constant, if there are exactly m individuals alive at time T out of the group of n individuals alive at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Analogously to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15), V (n) solves the following HJB problem: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 supΠ∈R LΠ i V (n)(t, w, λ, i) + nλ � V (n−1)(t, w, i) − V (n)(t, w, λ, i) � = 0, ∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , V (n)(T, w, λ, i) = −e−α(w−nK) ∀(w, λ, i) ∈ R × R+ × X , in which V (0) = ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Note that V (1) = V in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' One can easily shows that V (n)(t, w, λ, i) = ¯V (t, w, i)φ(n)(t, λ), where φ(n) : [0, T] × R+ −→ R+ solves the linear PDE \uf8f1 \uf8f2 \uf8f3 ∂φ ∂t (n) (t, λ) + b(t, λ)λ∂φ ∂λ (n) (t, λ) + 1 2c(t, λ)2λ2∂2φ ∂λ2 (n) (t, λ) − nλ � φ(n)(t, λ) − φ(n−1)(t, λ) � = 0 φ(T, λ) = enαK, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3) with φ(0) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Thus, the indifference price of n pure endowments P (n) is an implicit solution of the following equation ¯V (t, w, i) = V (n)(t, w + P (n) t , λ, i), for every (t, w, λ, i) ∈ [0, T] × R × R+ × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Similarly to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2), the reservation price of the insurer related to n pure endowment contracts is given by P (n) t = P (n)(t, λ, i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) = ln � φ(n)(t, λ) � αer(T−t) , for all (t, λ, i) ∈ [0, T] × R+ × X , where the function φ(n) solves the Cauchy problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Term life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, we discuss the indifference price of another type of a mortality- contingent claim, the so-called term life insurance that can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' A term life insurance contract with maturity T is a life insurance policy where the amount is paid at time T if the policyholder dies before time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The associated payoff is given by the random variable GT := K1{τ≤T}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) 23 where K is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We determine the indifference price of the term life insurance policy whose payoff is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4) in the Markov-modulated model outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, we consider the problem with the new kind of insurance derivative sup Π∈A(G) E � − e−α(W Π T −K)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Thus, the corresponding value function is given by V (t, w, λ, i) := sup Π∈At(G) Et,w,λ,i � − e−α(W Π T −K)� , for every (t, w, λ, i) ∈ [0, T] × R × R+ × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Note that it equals the value function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8) of the problem with the pure endowment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Thus, proceeding as above, the HJB problem for V is given by \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 supΠ∈R LΠ i V (t, w, λ, i) + λ � ¯V (t, w − Ke−r(T−t), i) − V (t, w, λ, i) � = 0, ∀(t, w, λ, i) ∈ [0, T) × R × R+ × X , V (T, w, λ, i) = −e−α(w−K) ∀(w, λ, i) ∈ R × R+ × X , where ¯V is introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Note that the HJB equation corresponds to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15) with ¯V (t, w, i) replaced by ¯V (t, w − Ke−r(T−t), i), since the insurer has to pay the amount K at time T for the death of the policyholder and so she/he needs to charge Ke−r(T−t) at time t in order to cover this payout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' One can easily shows that V (t, w, λ, i) = ¯V (t, w, i)ξ(t, λ), where the function ξ : [0, T] × R+ −→ R+ solves the linear PDE \uf8f1 \uf8f2 \uf8f3 ∂ξ ∂t (t, λ) + b(t, λ)λ ∂ξ ∂λ(t, λ) + 1 2c(t, λ)2λ2 ∂2ξ ∂λ2(t, λ) − λ(eαK − ξ(t, λ)) = 0 φ(T, λ) = eαK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) Hence, the reservation price of the insurer related to a term life contract is given by Pt = P(t, λ, i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) = ln � ξ(t, λ) � αer(T−t) , for all (t, λ, i) ∈ [0, T] × R+ × X , where the function ξ solves the Cauchy problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Numerical experiment In this section, we present some numerical results based on the theoretical framework developed previously, in order to illustrate certain qualitative features of the model that are difficult to verify analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Our aim is to investigate how the regime-switching and the stochastic hazard rate affect the decisions of the insurer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI To simplify the analysis, we suppose that the Markov chain X has two states, namely X = {1, 2}, that can be interpreted as the ’Good’ and ’Bad’ economic regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' For instance, the good regime could represent a market in economic boom whereas the bad regime could be a market in economic recession in which security prices are expected to fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We also call these two regimes of the market ’bull’ market and ’bear’ market, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' First of all, we have to set values for the infinitesimal generator of the 2-state Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Since aij represents the average of number of switches in an unit time, from state i to j, and since empirical observations of the market suggest that it is more likely to pass from a good economic state to a bad one than the opposite, we choose a12 > a21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In particular, we take a12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2 and a21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' For the sake of simplicity, we assume that functions µ, σ, K1 and K2 depend only on the Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' So, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2), the risky asset price dynamics is given by dSt = St−{µidt + σidZS t + K1,idN1 t − K2,idN2 t }, S0 > 0, i = 1, 2, where µi, σi, K1,i and K2,i denote the expected rate of return, the volatility and the jump coefficients in the i-th regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By way of example, we set the initial value of the stock price to be S0 = 1 and the short-term interest rate to be r = 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As shown by (French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 1987), the appreciation rate of the underlying risky asset is higher in a growing economy, so we assume that µ1 > µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, in each economic regime, the return of the risky asset should be higher than that of the risk-free rate, as required also in our modeling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Hamilton and Gang, 1996) find that economic recessions represent the main factor that drives fluctuations in the volatility of stock returns, so we assume that volatility is lower in a good economy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' σ1 < σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Furthermore, let us assume that µ1 − r σ2 1 > µ2 − r σ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In fact, according to (French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=', 1987), even though the expected market risk premium (defined as the expected return on the stock minus the risk-free interest rate) is usually higher during a ’bear’ market than during a ’bull’ market, the volatility of the stock offsets the effect of this quantity and, as a consequence, the ratio ’expected excess return/return variance’ is greater when the economic conditions are good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As for the jump terms, we consider two homogeneous Poisson processes N1 and N2 with constant intensities Θ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 and Θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We observe that the higher are the values of function K1, the higher is the price of the stock S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' On the other hand, any increase in the coefficient K2 leads to smaller prices for the risky stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, we notice that large values of K2 cause dizzying upward or downward peaks for the stock price, even though the intensity Θ2 is tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Therefore, since in a market with good economic conditions stock prices are rising or are expected to rise, we suppose that K1,1 > K1,2 and K2,1 < K2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In particular, we choose the parameters as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 25 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Simulation market parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Regime µ σ K1 K2 ’Good’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 ’Bad’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='35 Since the underlying market is a continuous-time model, we need to discretize it by Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The time horizon is taken to be T = 10 years and we discretize time with a total of 1000 time steps (that means that we take into account about two updates of S every workweek), each of width ∆t = 1 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In order to have an idea of our model, we simulate three trajectories of the risky asset S in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We notice that the stock price is greater during a ’bull market’ rather than during a ’bear’ market and it also exhibits jumps at switching times of the Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 0 2 4 6 8 10 Time t 0 1 2 3 4 5 Stock S 1 2 Markov chain X Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The effect of the regime-switching on the stock price S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Next, we compute the optimal investment strategy based on Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3, in order to investigate how it is sensitive to economic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In Figure 2 we plot the optimal dynamic portfolio given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='9), as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI 0 2 4 6 8 10 Time t 1 0 1 Optimal investment strategy 1 2 Markov chain X Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The effect of regime-switching on the optimal strategy Π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We clearly notice that a regime switch leads to a sudden change in the optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, we note that in a good economy the amount invested in the stock is always positive and increasing with respect to time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' instead, if the market scenario is bad, the strategy is negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' this indicates that when the economic conditions are bad, the insurer prefers to short-sell the risky asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' After that, we take into account a life-insurance policy and we investigate its indifference price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In the current toy example of our proposed model, we assume that the hazard rate follows a mean- reverting Brownian Gompertz model, similar to the one proposed in (Milevsky and Promislow, 2001), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λt = λ0ec1t+c2Yt, c1, c2, λ0 > 0, dYt = −mYtdt + dZY t , Y0 = 0, m ≥ 0, with c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='083, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1, λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='01 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let us observe that this choice corresponds to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5), considering b(t, λ) = c1 + m ln(λ0) + 1 2c2 2 − m ln(λ) + mc1t and c(t, λ) = c2λ, for all (t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This model guarantees that the hazard rate is kept positive and does not explode on [0, T], since it is an exponential function that depends on a stochastic factor Y with a mean reversion behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this context, based on the results obtained above, we compute the indifference price of an insurer related to a pure endowment contract that pays K = 1 if the policyholder is still alive after 10 years from purchaising the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Thus, the payoff is easily given by the random variable GT := 1{τ>T}, recalling that τ represents the remaining lifetime of the insured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Now, we want analyse the value function ¯V related to the insurer that simply invests her/his wealth in the market and the value function V related to the insurer who also writes a pure endowment contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 27 0 1 2 3 4 5 Initial wealth w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1 0 Optimal value function Good Bad 0 1 2 3 4 5 Initial wealth w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1 0 Optimal value function with claim Good Bad Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Optimal value at time 0 as a function of wealth when the economic regime is i = 1 (solid line) or i = 2 (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Left panel: the pure investment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Right panel: the investment problem with the insurance contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Figure 3 depicts the value functions ¯V (left panel) and V (right panel) at time t = 0, with respect to the initial wealth w, associated to the optimal strategy computed above, when the market state is good (solid line) or bad (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The two panels exhibit the same behavior: the optimal value functions are increasing functions of wealth, in both regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' It is worth noting that values reached by functions ¯V and V are always higher in a ’bull’ market, as it is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Furthermore, we can also point out that different economic conditions imply different value functions and that this gap becomes greater when the insurer, beyond investing in the financial market, also writes an insurance contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Next, we investigate the indifference price of a pure endowment policy, in order to highlight the dependence of a life insurance contract on mortality force and time to expiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In view of the probabilistic representation provided in Section 5, the indifference price charged by the insurer is determined as Pt = P(t, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) = ln � 1 + (eα − 1)Et,λ � e− � T t λvdv�� αer(T−t) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) for every (t, λ) ∈ [0, T] × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We employ this formula, using the standard Monte Carlo method (with parameter M = 5000) to evaluate expectations with respect to the probability measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' From expression (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1), we point out that economic regimes do not affect the price which instead strongly depends on the risk aversion coefficient and the risk-free interest rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In particular, it is easy to see that the indifference price increases as risk aversion increases and, at the same time, 28 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI it decreases as long as the interest rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Further, since the dependence on the mortality rate λ is not explicit, we would like to analize numerically how the hazard rate affects the price of the life insurance policy involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' First of all, we show the impact of changing initial mortality rate on the indifference price charged at the beginning of the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1 Initial hazard rate 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 Indifference price P0 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The effect of the hazard rate on the indifference price at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In Figure 4 we observe that larger force of mortality decreases the indifference price P0 charged by the insurer at time t = 0, as it is reasonable to expect for such type of insurance contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This is consistent with common intuition as, under higher mortality, an endowment payout is less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Finally, we investigate the evolution of indifference price over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' For the sake of simplicity, we assume constant mortality (such as in some numerical experiments of (Moore and Young, 2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this framework, we calculate the indifference premium related to a pure endowment policy for our insurer and we plot it as a function of time to maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 29 0 2 4 6 8 10 Time to maturity (T-t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='9 1 Indifference price P(T-t) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='01 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='05 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The effect of the hazard rate on the indifference price for several different deferral periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' As before, in Figure 5, we can see that the higher is the hazard rate, the lower is the indifference price for a pure endowment policy, whether the economic conditions are good or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' in other terms, the price is more sensitive to variations of deferral periods, when the population mortality intensity is more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, it is worth noting that the indifference price is a decreasing function of time of maturity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' the premium is bigger as time approaches to maturity, as usually happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Conclusions In this paper, we have analyzed indifference pricing of mortality contingent claims in a stochastic- factor model for an insurance company endowed with exponential utility preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We have considered a financial market model consisting of a riskless asset and a stock whose price is de- scribed by a jump diffusion process affected by a continuous-time finite-state Markov chain repre- senting the states of the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, we have assumed a stochastic hazard rate to describe population mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this framework, using the actuarial principle of equivalent utility, we have characterized the indifference price for a pure endowment contract and provided its probabilistic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In addition, we have also shown that the indifference price solves a final value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Indeed, the price that makes the insurer indifferent, in terms of expected utility, between not selling and selling the policy for that premium now and paying the benefits at maturity, is linked to a classical solution of a specific linear PDE with a proper terminal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Indeed, the indifference price has been determined by solving an equation involving two value functions, result- ing from the stochastic control problems with and without insurance liabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Using the classical control approach based on the HJB equation, we have found the optimal investment strategies and shown verification results for the value functions of the problems with and without the policy via classical solutions to a linear PDE and a system of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, we have briefly discussed the 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI indifference price of a portfolio of pure endowments and also for a term life insurance policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' A sensitivity analysis in case of a two-state Markov chain has highlighted some interesting features of the indifference price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We have investigated the effect of the hazard rate and the time to maturity on the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We have pointed out that, when the mortality intensity is low, an endowment payout is more likely and so its premium is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Further, we have outlined that the indifference price of a pure endowment contract decreases for longer deferral periods, as it is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Another numerical result shows that the insurance company opts to short-sell the risky asset when the financial market is in the bad state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' when the stock price presents a low rate of return, big fluctuations and a lot peaks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Applying the same methodology, it would be interesting to evaluate more complex insurance products, such as equity-linked policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' This will be done in a future work, also assuming that the insurance company preferences towards the risk are given by a utility function of power (or logarithmic) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Acknowledgements The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Statements and Declarations Alessandra Cretarola declares that she has no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Benedetta Salterini declares that she has no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Technical proofs Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' by applying Itô’s formula to the stochastic process f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we have f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xt) = f(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' X0) + � t 0 LΠf(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xu)du + mt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' where mt = m0 + � t 0 Πvσ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) ∂f ∂w(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)dZS v + � t 0 c(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv)λv ∂f ∂λ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)dZΛ v + � t 0 � R � f � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv− + h(Xv−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' z) � − f(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � ˆP(dv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' dz) + � t 0 � f � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v− + ΠvK1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � − f(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � {dN1 v − Θ1(v)dv} + � t 0 � f � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v− − ΠvK2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � − f(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv−) � {dN2 v − Θ2(v)dv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) We only need to prove that the process m = {mt, t ∈ [0, T]} is a (G, P)-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='13), the first two integrals in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) are well-defined and turn out to be (G, P)-martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Furthermore, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='14), we have that also the jump terms in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) are (G, P)-martingales, (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' (Davis, 31 1993, Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='12(2)) and (Brémaud, 1981, Lemma L3, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='II) for further details about the martingale property related to a Poisson random measure and a Poisson process, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Derivation of the HJB equation For the sake of clarity, we show how to obtain a formal derivation of the HJB equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15) associated to the problem with the insurance derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To this aim, we apply the Bellman’s dynamic programming principle that, in this context, it is formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1 (Bellman optimality principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Let (t, w, λ, i) ∈ [0, T] ×R ×R+ ×X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Then, for t ≤ t + h ≤ T and Π ∈ At, we have V (t, w, λ, i) ≥ Et,w,λ,i � V (t + h, W Π t+h, λt+h, Xt+h) � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) where V is the value function introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Moreover, equality holds in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1) if, and only if, the arbitrary control Π on the interval [t, t + h] is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' The idea is that if the insurer follows the optimal strategy on [t, T], her/his expected utility is at least as great as if she/he invests arbitrarily on [t, t + h[ and then optimally on [t + h, T], for h sufficiently small such that t + h < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In the application of the dynamic programming principle, we must consider whether the policyholder survives from time t until time t + h, as in (Young and Zariphopoulou, 2002), (Moore and Young, 2003), (Ludkovski and Young, 2008) and (Young, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Consider an individual aged l, who is seeking to buy a pure endowment policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' For the rest of this section, we write (l) to refer to this individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' For each h such that t+h < T, if the individual (l+t) survives for another h years until time t+h, which happens with probability hpl+t, the insurer still faces the endowment risk on the time interval [t + h, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' In this case, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='8), the maximum expected utility derived by investing optimally on [t+h, T] is V (t+h, W Π t+h, λt+h, Xt+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' However, if the individual (l + t) dies in [t, t + h], an event that happens with probability hql+t, then the insurer is not longer at risk for the endowment payout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='7), the maximum expected utility derived by investing optimally on [t + h, T] is ¯V (t + h, W Π t+h, Xt+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' From (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1), we have V (t, w, λ, i) ≥ hpl+tEt,w,λ,i � V (t + h, W Π t+h, λt+h, Xt+h) � + hql+tEt,w,i � ¯V (t + h, W Π t+h, Xt+h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI If we assume enough regularity conditions and proper integrability on the value functions and their derivatives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' by applying Itô’s formula and conditioning on W Π t = w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λt = λ and Xt = i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we get V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) ≥ hpl+tV (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) +h ql+t ¯V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t �∂V ∂t + � rW Π v + � µ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − r � Πv �∂V ∂w dv �� +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t � b(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv)λv ∂V ∂λ + 1 2σ2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)Π2 v ∂2V ∂w2 + 1 2c2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv)λ2 v ∂2V ∂λ2 � dv � +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t � � j∈X V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' j)av,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='j � dv � +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t Θ1(v) � V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v + ΠvK1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) � dv � +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t Θ2(v) � V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v − ΠvK2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) � dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t �∂ ¯V ∂t + � rW Π v + � µ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − r � Πv �∂ ¯V ∂w � dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t �1 2σ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)2Π2 v ∂2V ∂w2 + � j∈X ¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Wv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' j)av,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='j � dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t Θ1(v) �¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v + ΠvK1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − ¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) � dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t Θ2(v) �¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v − ΠvK2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − ¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) � dv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' To keep the formulas readable, in the integrals above we have suppressed the independent variables (v, Wv, λv, Xv) and (v, Wv, Xv) of the partial derivatives of V and ¯V , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' By subtracting 33 hpl+tV (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) from both sides of inequality and dividing both sides by h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we obtain hql+t h V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) ≥ hql+t h ¯V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h �∂V ∂t + � rW Π v + � µ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − r � Πv �∂V ∂w dv �� +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h � b(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv)λv ∂V ∂λ + 1 2σ2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)Π2 v ∂2V ∂w2 + 1 2c2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv)λ2 v ∂2V ∂λ2 � dv � +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h � � j∈X V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Wv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' j)av,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='j � dv � +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h � Θ1(v) � V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v + ΠvK1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) �� dv � +h pl+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h � Θ2(v) � V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v − ΠvK2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) �� dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h �∂ ¯V ∂t + � rWv + � µ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − r � Πv �∂ ¯V ∂w � dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h �1 2σ(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv)2Π2 v ∂2V ∂w2 + � j∈X ¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' j)av,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='j � dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h � Θ1(v) �¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v + ΠvK1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) − ¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) �� dv � +h ql+tEt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='i � � t+h t 1 h � Θ2(v) �¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v − ΠvK2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − ¯V (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' W Π v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Xv) �� dv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' We observe that as h −→ 0+, we have hpl+t −→ 1, hql+t −→ 0 and hql+t h −→ λt, for each t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Consequently, taking the limit as h −→ 0+ yields 0 ≥λ � ¯V (t, w, i) − V (t, w, λ, i) � + ∂V ∂t + � rw + � µ(t, i) − r � Π �∂V ∂w + b(t, λ)λ∂V ∂λ + 1 2Π2σ2(t, i)∂2V ∂w2 + 1 2c2(t, λ)λ2∂2V ∂λ2 + � j∈X V (t, w, λ, j)aij + Θ1(t) � V (t, w + ΠK1(t, i), λ, i) − V (t, w, λ, i) � + Θ2(t) � V (t, w − ΠK2(t, i), λ, i) − V (t, w, λ, i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 34 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' CRETAROLA AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' SALTERINI Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we note that along the optimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' we have 0 =λ �¯V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) � + ∂V ∂t + rw∂V ∂w + b(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ)λ∂V ∂λ + 1 2c2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ)λ2∂2V ∂λ2 + � j∈X V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' j)aij + sup Π∈R � � µ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − r � Π∂V ∂w + 1 2σ2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i)Π2∂2V ∂w2 + Θ1(t) � V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w + ΠK1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) � + Θ2(t) � V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w − ΠK2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) − V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' ∀(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' T) × R × R+ × X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' with V (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) = −e−α(w−K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' for each (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' i) ∈ R × R+ × X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' which coincides with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' References S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Altay, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Colaneri, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Eksi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Pairs trading under drift uncertainty and risk penaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Intnternational Journal of Theoretical and Applied Finance, 21, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1142/ 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='frl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='101748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Baran, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Yin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Feynman-kac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' Advances in Difference Equations, 2013, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFRT4oBgHgl3EQffDfO/content/2301.13575v1.pdf'} +page_content=' 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Lodygensky2,3 +Jose Dolz1 +1ÉTS Montreal +2CHU Sainte-Justine, University of Montreal +3Canadian Neonatal Brain Platform, Montreal +∗farzad.beizaee.1@ens.etsmtl.ca +Abstract +In this paper, we propose an unsupervised framework based on normalizing flows +that harmonizes MR images to mimic the distribution of the source domain. The +proposed framework consists of three steps. First, a shallow harmonizer network +is trained to recover images of the source domain from their augmented versions. +A normalizing flow network is then trained to learn the distribution of the source +domain. Finally, at test time, a harmonizer network is modified so that the output +images match the source domain’s distribution learned by the normalizing flow +model. Our unsupervised, source-free and task-independent approach is evaluated +on cross-domain brain MRI segmentation using data from four different sites. +Results demonstrate its superior performance compared to existing methods. The +code is available at https://github.com/farzad-bz/Harmonizing-Flows +1 +Introduction +Deep learning models have become the de facto solution for most image-based problems, including +those in the medical domain. Despite significant progress, these models still suffer under distributional +drift, and their performance largely degrades when they are applied to data obtained in different +conditions. +Clinical studies using magnetic resonance imaging (MRI) often have to deal with such large domain +shifts. Due to the qualitative nature of the MRI acquisition process, generated images are sensitive +to imaging devices, acquisition protocols, scanner artifacts, as well as to patient populations [28]. +For instance, images from the same modality (e.g., T1-w) acquired from two different scanners with +separate configurations will likely present noticeable differences, which can be considered a domain +shift. Consequently, collecting a multi-center MRI dataset to address a particular clinical question +does not guarantee a greater statistical power, as the increase in variance comes from a non-clinical +source. Furthermore, this data heterogeneity can also hamper the generalizability of deep learning +models, preventing their large dissemination. In particular, when trained on a specific site, such +models are typically unable to provide similar performance for other centers. +To alleviate this issue, image harmonization addresses the distributional shift problem from an image- +to-image mapping perspective, where the objective is to transfer image contrasts across different +domains. Nevertheless, most harmonization methods in the literature make strong assumptions that +might hamper their scalability and usability in real-life scenarios. First, some methods must have +access to source images during the adaptation, which may no longer be available. Labels associated +with the downstream task may also be required in other approaches. Finally, most harmonization +techniques need to know the target domains during training, while these domains are often unknown. +In this work, we make the following contributions: +Preprint. Under review. +arXiv:2301.11551v1 [cs.CV] 27 Jan 2023 + +• We relax all these assumptions and present a novel MR harmonization method that is source- +free (SF), task-agnostic (T A) and can handle unknown-domains (UD) without requiring to +be retrained for each target distribution. Indeed, our method only needs one domain and +modality at training time, as opposed to existing approaches. +• In particular, we propose to use a novel family of generative models, i.e., normalizing flows, +which have shown to be a powerful method to model data distributions in generative tasks. +We stress that leveraging normalizing flows to guide the adaptation of a harmonizer network +has not been explored. +• In addition to the methodological novelty, our empirical results demonstrate that the our +approach brings substantial improvements compared to existing techniques, while alleviating +their weaknesses. +• Furthermore, due to its task-agnostic nature and its capability to work under the unknown- +domains scenario, the proposed method can also be employed in the task of test-time +adaptation. In this setting, our method largely outperforms a popular task-agnostic test-time +adaptation strategy. +2 +Related work +Image harmonization. Several techniques have been proposed for the harmonization of images +in the medical domain, and particularly for MRI data. Classical post-processing steps, such as +intensity histogram matching [24, 26], reduce the influence of biases across scanners, but may also +remove informative local variations in intensity. Statistical approaches can model image intensity +and dataset bias at the voxel level [11, 12, 2], however they must often be adjusted each time images +from new sites are provided. Modern strategies for image harmonization, which are based on deep +learning models, have shown to be a promising alternative for this problem [5, 35, 21, 36, 4, 8]. +Nevertheless, they make unrealistic assumptions that hamper the scalability of existing approaches to +large scale multi-site harmonization tasks. First, images of the same target anatomy across multiple +sites, commonly referred to as traveling subjects are employed to identify intensity transformations +between different sites [5]. This involves that a given number of subjects are scanned at every +site or scanner required for training, a condition rarely met in practice. Second, another group +of methods is limited to two domains [35] and requires target domains to be known at training +time [35, 21]. In addition, each time a new domain is added, these approaches must be fine-tuned +in order to accommodate the characteristics of each domain. Calamity [21] further needs paired +multi-modal MR sequences, limiting even more its applicability to single modality scenarios. Last, +task-dependent approaches leverage labels associated to each image for a given down-stream task +[4, 8], thus optimizing the harmonization for this specific problem. Nevertheless, having access to +large labeled datasets might be impractical due to the underlying labeling cost. +Test-time Adaptation. +Our method also relates to the problem of test-time domain adaptation +(TTA) [30, 27, 3] which aims to quickly adapt a pre-trained deep network to domain shifts during +inference on test examples. One key difference between TTA and the well-known unsupervised +domain adaption (UDA) problem is that, in TTA, the source examples are no longer available. One +of the earliest TTA approaches, called TENT [30], updates the affine transformation parameters of +normalization layers by minimizing the Shannon entropy of predictions for test examples. In [23], this +strategy is improved by optimizing a log-likelihood ratio instead of entropy, as well as by considering +the normalization statistics of the test batch. The method named SHOT [20] fine-tunes the entire +feature extractor with a mutual information loss and uses pseudo-labels to provide additional test-time +guidance. Instead of updating the network parameters, LAME [3] uses Laplacian regularization to do +a post-hoc adaptation of the softmax predictions. +Normalizing flows. +Recently, normalizing Flows (NFs) have emerged as a popular approach +for constructing probabilistic and generative models with tractable distributions [19]. NFs aim at +transforming unknown complex distributions into simpler ones, for instance, a standard normal +distribution. This is achieved by applying a sequence of invertible and differentiable transformations. +While most existing literature has leveraged NFs for generative tasks (e.g., image generation [15, 17], +noise modeling [1], graph modeling [33]) and anomaly detection [14, 18], recent evidence also +suggests their usefulness for aligning a given set of source domains [13, 29]. To our knowledge, +a single work has investigated NFs in the context of harmonization [32]. However, it aimed at +2 + +𝑥!~𝑝𝑥′(𝑥′) +𝜒! +"# +𝜒$ +"# +𝜒! +%&' +𝜒! +%&' +⊙ ++ +Variational dequantization +Squeezing function +Affine coupling layer +Channel masking +Checkerboard masking +UNet +𝑥~𝑝𝑥(𝑥) +𝑧~𝒩(0, 𝐼) +𝛼 +𝛽 +𝛼𝑥+𝛽 +Step2: training the NF model with source domain images +Step1: pre-training the harmonizer network +Gradient +Step3: adapting the harmonizer network to map the input images from target to source domain +Harmonized images +NF model: +Harmonizer network: +Figure 1: Pipeline of the proposed Harmonizing Flows method. Our approach consists of two +steps. First, we employ normalizing flows (NFs) to capture the distribution of the source domain. +During the second stage, the trained NFs are leveraged to update the parameters of a harmonizer +network, which are updated in order to maximize the similarity between the harmonized outputs and +the distribution learned by the NF. Note that steps 1 and 2 are not dependent on each other, and can +therefore be performed in any order. +performing causal inference on pre-extracted features (brain ROI volume measures), and not image +harmonization as in our work. Moreover, since extracting ROIs requires pixel-wise labels, the method +in [32] is not task-agnostic. +3 +Methodology +We first define the problem addressed in our work. Let XS = {xn}N +n=1 be a set of unlabeled images +in the source domain S, where a given image i is represented by xi ∈ R|Ω| and Ω denotes its spatial +domain (i.e., W ×H). Similarly, we denote as XT = {xn}M +n=1 the set of unlabeled images in +a potential target domain T 1. The goal of unsupervised data harmonization is to find a mapping +function fθ : S →T without having access to labeled images for any of the domains. In what follows, +we present our NF-based solution for this problem, whose framework is depicted in Figure 1. +3.1 +Learning the source domain distribution +We leverage Normalizing Flows (NFs) [7] to model the distribution of the source domain. NFs are a +recent family of generative methods that can model a complex probability density px(x) (i.e., the +source) as a series of transformation functions, denoted as gφ = g1 ◦ g2 ◦ . . . gT , applied on simpler +and tractable probability density pu(u) (e.g., a standard multi-variate Gaussian distribution). We can +express a source image as x = gφ(u), where u ∼ pu(u) and pu(u) is the base distribution of the +flow model. An important requirement of the transformation function gφ is that it must be invertible, +and both gφ and g−1 +φ +should be differentiable. Under these conditions, the density of the original +variable x is well-defined and its likelihood can be computed exactly using the change of variables +rule as: +log px(x) = log pz +� +g−1 +φ (x) +� ++ log +���det +� +Jg−1 +φ (x) +���� += log pz +� +g−1 +φ (x) +� ++ +T +� +t=1 +log +���det +� +Jg−1 +t (ut−1) +���� +(1) +where the first term on the right-hand side is the log-likelihood under the simple distribution, and +Jg−1 +t (ut−1) is the Jacobian matrix of the inverse transformation gt. To train the NF model and learn +the source data distribution, the model parameters φ are typically optimized so to minimize the +negative log-likelihood in Eq. 1. This results in the following loss function: +LNF = − log px(x) +(2) +1Note that for simplicity, we assume here that there exists only a single domain. Nevertheless, our formulation +is directly applied to T different domains. +3 + +EBuilding the Normalizing Flow. To build a bijective transformation function for the NF model, +stacking a sequence of affine coupling layers [7, 17] has been demonstrated to be an efficient +strategy. Because flows based on coupling layers are computationally symmetric, i.e., equally fast +to evaluate or invert, they can overcome the usability issues of asymmetric flows such as masked +autoregressive flows, making them a popular choice. Let us consider z ∈ RD as the input to the +coupling layer, which is split into a disjoint partition: (zA, zB) ∈ Rd × RD−d. The transformation +function g(·) : RD → RD can then be defined as: +yA = zA, +yB = zB ⊙ exp +� +s +� +zA�� ++ t +� +zA� +(3) +This setting offers simplicity for calculating the Jacobian determinant, which makes it possible to +use complex neural networks as shift s(·) and scale t(·) networks. Note that the transformation +in Eq. 3 is invertible and therefore allows for efficient Jacobian computation in Eq. 1. The work +in [7] presented coupling flows on simpler tasks and datasets, e.g., CIFAR, which required less +enriched representations. In contrast, the problem at hand requires pixel-to-pixel mappings on more +challenging images. Thus, we replace the simple convolutional blocks in [7] with shallow U-shaped +convolutional neural networks to find the shift and scale parameters of the affine transformation, as +they capture more global context and provide higher representation power. Furthermore, as NFs are +based on the change of variables rule, which is defined in continuous space, it is crucial to make the +input continuous. Dequantization of the input can be achieved by adding a uniform noise u∈U[0, 1] +to the discrete values. However, it might result in a hypercube representation of the images with +sharp borders. These sharp borders are hard to model for a flow as it uses smooth transformations. +Recently, a variational framework was proposed [15] to extend dequantization to more sophisticated +distributions, by replacing the uniform distribution with a learnable distribution. +Constraining the source-distribution learning. Optimizing the objective in Eq. 2 with only source +images might bias the model to focus on characteristics of subjects, such as age and gender, rather +than on source-specific features like contrast and brightness. To overcome this issue, we propose a +strategy that facilitates the learning of the source-domain distribution. This technique consists in +randomly selecting N ′ images from the original dataset XS and applying a series of augmentations +faug(·) such that the resulting image has a dissimilarity to the original image (measured by mean +squared distance) higher than a specified threshold. In particular, we employ contrast augmentation, +brightness changes, multiplication, and random monotonically increasing mapping functions to +augment these images. Then, the total learning objective of our model can be defined as: +LT = − +N−N ′ +� +n=1 +log px(xn) +� +�� +� +Source distribution modeling +− +N ′ +� +n=1 +min (c, − log px(faug(xn))) +� +�� +� +Guiding term +. +(4) +The first term is the learning objective in Eq. 2 over the original source images, whereas the second +one forces the NF model to decrease the likelihood on the augmented images, which facilitates the +learning of domain-specific characteristics (e.g., contrast or brightness) instead of subject-related +features (e.g., sex or age). Furthermore, we use a constant margin c in the second term to prevent the +negative log-likelihood of an augmented sample from diverging to infinity. +3.2 +Achieving image harmonization +Harmonizer network. A simple solution to perform image-to-image translation is to employ a +harmonizer network hθ(·), such that MRIs from the target domain are translated to the source domain. +This can be expressed as px(x) = px′(hθ(x′)), where θ is the set of learnable parameters of the +harmonizer network, and x and x′ are images from the source and target domains, respectively. +To train this model, we can simply use a standard reconstruction loss over images across different +domains. However, we want the proposed method to follow a domain-free paradigm, where target +domains remain unknown at training time. Toward this goal, we train the harmonizer network to +reconstruct the original source images from their augmented versions. As in the previous step, we +augment the original images by using different types of contrast augmentation, brightness changes, +multiplication, or random monotonically increasing mapping functions. Contrary to the first step, there +is no constraint on the magnitude of the augmentations. The learning objective for the harmonizer +4 + +network thus becomes: +θinit = argmin +θ +1 +N +N +� +n=1 +∥(xn − hθ (faug(xn))∥2 +(5) +We stress that the performed augmentations are not reliable representations of potential unseen target +domains. Consequently, the direct application of the learned parameters θinit for image-to-image +mapping will result in suboptimal domain transformations. Nevertheless, they can serve as the initial +model for the subsequent step. A simple UNet is considered for the harmonizer network, which +learns two values. First, the last layer of the network (β) is employed as a bias value having the same +dimension as the input image. Second, a scalar α from the middle layer of the network is used as a +coefficient value. In this way, the output of the harmonizer can be defined as hθ(x) = α ∗ x + β. +Guiding the harmonizer network with the Normalizing Flow. The final step involves updating +the harmonizer network so that images from the target domain are mapped into the source domain +distribution. To achieve this, we propose to leverage the trained NF, which is stacked at the output +of the harmonizer network. Note that the NF model has already learned the distribution of source +data, and therefore its parameters remain frozen during the adaptation of the harmonizer. Thus, the +learning objective of the adaptation stage consists in increasing the likelihood of the harmonizer +outputs for images from the target domain, based on the NF model’s density estimation. This loss +function can be formally defined as follows: +LAdap = − +M +� +m=1 +log px +� +gφ +� +hθ(xm) +�� +(6) +As stopping criterion for updating the harmonizer, we evaluate two possible alternatives. First, we +measure the Shannon entropy of the predictions for the target task (e.g., segmentation or classification), +stopping the adaptation when the entropy plateaus. We also consider the bits per dimension (bpd), a +scaled version of the negative log-likelihood widely used for evaluating generative models: bpd = +− log px(x) · (log 2 · � +i Ωi)−1 where Ω1, ..., ΩT , is the spatial dimension of the input images. More +concretely, we can stop updating the harmonizer parameters when the reached bpd value is the same +as the one observed for the source domain using the NF model. In practice, this value can be obtained +at training time using a validation set. +4 +Experiments +4.1 +Experimental setting +We evaluate the proposed method on the task of brain MRI segmentation across multiple sites. The +reason behind this choice stems from the fact that the segmentation performance is a reliable indicator +of whether the structural information is well preserved during the mapping. +Datasets. Four sites of the Autism Brain Imaging Data Exchange (ABIDE) [6] dataset are employed: +California Institute of Technology (CALTECH), Kennedy Krieger Institute (KKI), University of +Pittsburgh School of Medicine (PITT) and NYU Langone Medical Center (NYU). The selection +of these sites is based on their cross-site difference, as these datasets present the most distinct +histogram from each other, which better highlights the impact of harmonization. These sites are +denoted as D1, D2, D3, and D4, respectively. From each site, we selected 20 T1-weighted MRIs from +the healthy control population (19 from CALTECH), which were skull-stripped, motion-corrected, +and quantized to 256 levels of intensity. 2D coronal slices of 60% of these images are used for +training, 15% for validation, and the remaining 25% for testing. Furthermore, the segmentation +labels are obtained from FreeSurfer [10], following other large-scale studies [9], and grouped into 15 +labels: background, cerebellum gray matter, cerebellum WM, cerebral GM, cerebral WM, thalamus, +hippocampus, amygdala, ventricles, caudate, putamen, pallidum, ventral DC, CSF, and brainstem. +Harmonization baselines. The proposed approach is benchmarked against a set of relevant har- +monization and image-to-image translation methods. We first consider a simple Baseline applying +the segmentation network directly on non-harmonized images, in order to assess the impact of +each harmonization approach. Our comparison also includes: Histogram Matching [24], aleatoric +5 + +uncertainty estimation (AUE) [31], Combat [25], BigAug [34] (which uses heavy augmentations for +generalization of the segmentation networks), and two popular generative-based approaches, i.e., +Cycle-GAN [22] and Style-Transfer [21]. +Evaluation protocol. To assess the performance of our harmonization approach, we resort to a +segmentation task as it requires the preservation of fine-grained structural details. First, a segmentation +network SΦ(·) is trained on the images from the source domain, whose parameters remain frozen +thereafter. The harmonized images from each method are then employed to evaluate segmentation +performance, which is measured with the Dice Similarity Coefficient (DSC) and modified Hausdorff +distance (HD). To evaluate the robustness of tested methods, we repeat the experiments four times, +each employing a different source and set of target domains. These different settings are denoted as A : +D1 →{D2, D3, D4}; B : D2 →{D1, D3, D4}; C : D3 →{D1, D2, D4}; D : D4 →{D1, D2, D3}. +Implementation details. +The Normalizing flow model is trained for 1600 epochs using Adam +optimizer with an initial learning rate of 1 × 10−3, a weight decay of 0.5 every 200 epochs and a +batch-size of 32. We use a U-shaped network inside the coupling layers, which consists of four +levels of different scales with a scaling factor of 2. Each level includes a modified version of the +ELU activation function, i.e., concat(ELU(x), ELU(−x)), and a convolutional layer followed by a +normalizing layer. To construct the NF model, we first cascade four coupling layers with checkerboard +masking to learn the noise distribution using variational dequantization. After applying four of the +same coupling layers, features are squeezed as explained in [7] to have a lower spatial dimension +and more channels. We then add four coupling layers using a channel-masking strategy, another +feature squeezing function, and a final set of four coupling layers with channel-masking. The overall +architecture of the flow model is shown in Fig. 1. The margin c used for guiding the flow is set +empirically to 1.2. The harmonizer has five levels of different scales with a scaling factor of 2, each +level including two layers of the modified ELU activation function followed by a convolutional layer. +The number of kernels of each level is 16, 32, 48, 64, and 64, respectively. The harmonizer is trained +for 200 epochs using Adam optimizer with a learning rate starting at 1 × 10−3, a weight decay of 0.5 +every 30 epochs and a batch-size of 32. The segmentation network is trained for 200 epochs using +Adam optimizer with an initial learning rate of 4 × 10−3, a weight decay of 0.5 every 30 epochs and +a batch-size of 32. All the models were implemented in PyTorch and were run on NVIDIA RTX +A6000 GPU cards. +4.2 +Results +Comparison to state-of-the-art. +Segmentation results obtained on the images harmonized by +different methods are reported in Table 1. We can observe that the proposed approach consistently +outperforms compared methods by a noticeable margin, across datasets and for both segmentation +metrics. In particular, the average improvement gain is nearly 4% in terms of DSC, and 0.7 mm in +terms of HD, compared to the second best performing method, CycleGAN. +Impact of normalizing flows. This section assesses the impact of each component of the proposed +method. In particular, we evaluate the segmentation performance when images are: i) not normalized, +ii) normalized with the pre-trained harmonizer θinit, or iii) normalized with the proposed harmonizing +flow. The results from this ablation study (Table 2) empirically motivate the proposed NF model as +a powerful mechanism to guide the harmonizer network. First, the strategy proposed to pre-train +the harmonizer brings a substantial improvement over non-harmonized images, yet it is very simple +and does not require access to images from the target domain. Secondly, driving the adaptation of +the harmonizer with the proposed NF further improves the segmentation results by a large margin, +demonstrating the benefits of our model. +Adaptation stopping criterion. In this section, we address the important question of when to stop +the adaptation. The first alternative is to stop it when the Shannon entropy of the segmentation +predictions reaches its minimum point. As this objective does not require any labeled data, it gives +a valid stopping point for adapting the harmonizer. As a second criterion, we stop adapting when +the output bpd of the NF model reaches the observed source domain bpd. As opposed to entropy, +this criterion is not task-dependent and is suitable for unsupervised tasks or tasks where entropy is +not applicable. Last, we resort to the segmentation performance, and stop the adaptation when it +reaches the best DSC score, which we define as Oracle. Note that this criterion is unrealistic, and its +purpose is just to demonstrate how a good stopping criterion can improve harmonization. As shown +6 + +Table 1: Performance overview. Main results for the compared methods across different settings +(A, B, C, D). The best results are highlighted in bold. +SF +T A +UD +A +B +C +D +Average +DSC (%) +Baseline +– +– +– +54.6 ±7.5 +60.8 ±4.6 +62.9 ±5.8 +72.6 ±4.5 +62.7 ±5.6 +AUE [31] + + + +54.7 ±7.4 +60.7 ±4.7 +62.6 ±5.7 +72.4 ±4.5 +62.6 ±5.6 +Hist matching[24] + + + +55.7 ±8.6 +58.1 ±5.1 +62.2 ±4.8 +69.5 ±4.9 +61.4 ±5.9 +Combat [25] + + + +75.7 ±9.2 +79.9 ±6.0 +79.5 ±8.1 +79.9 ±7.8 +78.7 ±7.8 +BigAug [34] + + + +54.2 ±7.6 +67.9 ±3.6 +61.5 ±4.5 +78.0 ±3.7 +65.4 ±4.8 +Cycle-GAN [22] + + + +74.5 ±3.0 +78.8 ±2.9 +80.1 ±2.2 +83.1 ±2.0 +79.1 ±2.5 +Style-transfer [21] + + + +56.9 ±7.1 +80.0 ±1.7 +67.8 ±4.9 +73.4 ±4.0 +69.5 ±4.4 +Ours + + + +80.8 ±3.2 +82.3 ±2.2 +83.2 ±3.3 +85.2 ±1.5 +82.9 ±2.6 +HD (mm) +Baseline +– +– +– +18.20 ±8.27 +9.57 ±3.23 +9.07 ±2.78 +5.73 ±1.81 +10.64 ±4.03 +AUE [31] + + + +17.57 ±8.18 +9.67 ±3.39 +9.03 ±2.87 +5.57 ±1.85 +10.46 ±4.08 +Hist matching[24] + + + +17.40 ±8.10 +10.47 ±3.77 +12.00 ±4.56 +6.73 ±2.40 +11.65 ±4.71 +Combat [25] + + + +5.23 ±3.87 +3.67 ±2.47 +3.30 ±1.80 +3.17 ±2.14 +3.84 ±2.57 +BigAug [34] + + + +19.53 ±10.51 +8.43 ±3.40 +18.87 ±7.76 +3.70 ±1.07 +12.63 ±5.69 +Cycle-GAN [22] + + + +4.63 ±2.89 +3.63 ±1.93 +2.63 ±0.62 +2.30 ±0.55 +3.30 ±1.50 +Style-transfer [21] + + + +14.23 ±7.20 +2.93 ±0.78 +7.53 ±2.38 +4.27 ±1.35 +7.24 ±2.92 +Ours + + + +3.10 ±1.63 +2.77 ±0.87 +2.37 ±0.77 +2.30 ±0.50 +2.63 ±0.94 +Table 2: Ablation study on the different components in terms of DSC. +A +B +C +D +Average +Without harmonization 54.6 ±7.5 60.8 ±4.6 62.9 ±5.8 72.6 ±4.5 62.7 ±5.6 +Pre-trained harmonizer 71.9 ±5.1 77.0 ±3.0 76.0 ±5.2 75.3 ±4.5 75.1 ±4.5 +Adapting using NF +80.8 ±3.2 82.3 ±2.2 83.2 ±3.3 85.2 ±1.5 82.9 ±2.6 +in Table 3, although minimum entropy is a better criterion compared to bpd, both achieve comparable +performances. In addition, both stopping criteria are a suitable choice, as their results are very close +to the Oracle. +Table 3: Impact of the adaptation stopping criterion (in terms of DSC). +A +B +C +D +Average +Minimum Entropy +80.8 ±3.2 82.3 ±2.2 83.2 ±3.3 85.2 ±1.5 82.9 ±2.6 +Source BPD +80.5 ±3.1 82.6 ±2.3 82.7 ±3.6 84.8 ±1.6 82.6 ±2.6 +Oracle (best epoch) 81.0 ±3.0 82.7 ±2.3 84.0 ±2.9 85.2 ±1.5 83.2 ±2.4 +Qualitative results. +Figure 2 depicts several examples of harmonized images produced by the +proposed approach. These results illustrate that, regardless of the target domain, our method produces +reliable image-to-image mappings to the source distribution. +Results when N4 bias correction is applied. In previous sections, we used the original MRIs of +the ABIDE dataset without bias correction to evaluate the proposed harmonization method on more +challenging scenarios, where pre-processing steps to enhance the images might not be applicable. +Compared to bias-corrected MRIs, original MRIs have arguably more complex distributions, which +makes it more difficult for harmonization methods to map MRIs from a target domain to the source +one. To demonstrate that our method also achieves satisfactory performance when the initial domain +shifts are reduced, we repeated the previous steps with N4-biased corrected MRIs. These results, +shown in Table 4, also showcase the advantage of our method in this different setting. For conciseness, +we report here the average results for the HD metric (in mm) across different methods: Baseline +(5.04±1.41), Hist matching (4.45±1.33), Combat (3.28±0.75), BigAUG (2.64±0.47), Cycle-GAN +(2.55 ± 0.31), Style-transfer (3.77 ± 1.07) and ours (2.36 ± 0.62). +7 + +Figure 2: Examples of harmonized images produced by the proposed method. +Table 4: Quantitative results on bias corrected images (in terms of DSC). +A +B +C +D +Average +Baseline +71.9 ±3.7 78.2 ±3.6 82.9 ±1.7 82.6 ±2.9 78.9 ±3.0 +AUE [31] +71.4 ±3.7 78.3 ±3.6 82.7 ±1.7 82.7 ±2.8 78.8 ±2.9 +Hist matching [24] 78.7 ±1.9 80.6 ±2.4 83.1 ±2.0 81.3 ±2.6 80.9 ±2.2 +Combat [25] +77.4 ±1.6 80.2 ±1.3 79.8 ±2.0 78.6 ±2.0 79.0 ±1.7 +BigAug [34] +81.7 ±2.1 82.4 ±1.5 85.2 ±0.9 84.6 ±1.3 83.5 ±1.4 +Cycle-GAN [22] +78.3 ±2.0 79.9 ±1.0 82.9 ±1.3 83.4 ±1.0 81.1 ±1.3 +Style-transfer [21] +74.1 ±3.0 80.0 ±1.3 80.6 ±1.5 81.1 ±1.5 79.0 ±1.8 +Ours +83.0 ±1.8 84.4 ±1.5 85.4 ±1.2 85.6 ±1.3 84.6 ±1.5 +Experiments on Test-Time Adaptation (TTA). Our model can also be employed in a TTA scenario, +where the model needs to be updated at inference for a given image, or set of images. To motivate +this assumption, we compare the performance of our approach to the popular TENT model [30]. +Adapting the segmentation network SΦ(·) with TENT yields 65.1 ± 5.0 of DSC, which represents a +considerable gap compared to our model, i.e., 82.9 ± 2.6. Note that there exist other TTA methods +for segmentation in the medical field, e.g., [16], however, they require segmentation masks for the +adaptation. +5 +Conclusion +In this paper, we proposed a novel harmonization method which leverages Normalizing Flows to +guide the adaptation of a harmonizer network. Our approach is source-free, task-agnostic, and works +with unseen domains. These characteristics make our model applicable in real-life problems where +the source domain might not accessible during adaptation, target domains are unknown at training +time, and harmonization is not dependent on a specified target task. The proposed method achieves +state-of-the-art harmonization performance based on the segmentation task, yet relaxes the strong +assumptions made by existing harmonization strategies. Thus, we believe that our model is a powerful +alternative for MRI multi-site harmonization. +References +[1] Abdelhamed, A., Brubaker, M.A., Brown, M.S.: Noise flow: Noise modeling with conditional +normalizing flows. In: ICCV. pp. 3165–3173 (2019) +[2] Beer, J.C., et al.: Longitudinal Combat: A method for harmonizing longitudinal multi-scanner +imaging data. Neuroimage 220, 117129 (2020) +[3] Boudiaf, M., et al.: Parameter-free Online Test-time Adaptation. 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In: IPMI. pp. 346–359 (2021) +10 + diff --git a/udFJT4oBgHgl3EQfeCzc/content/tmp_files/load_file.txt b/udFJT4oBgHgl3EQfeCzc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee37243533da59421d39e3363ef42ab9b23352f5 --- /dev/null +++ b/udFJT4oBgHgl3EQfeCzc/content/tmp_files/load_file.txt @@ -0,0 +1,745 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf,len=744 +page_content='Harmonizing Flows: Unsupervised MR harmonization based on normalizing flows Farzad Beizaee1,2∗ Christian Desrosiers1 Gregory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Lodygensky2,3 Jose Dolz1 1ÉTS Montreal 2CHU Sainte-Justine, University of Montreal 3Canadian Neonatal Brain Platform, Montreal ∗farzad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='beizaee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1@ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='etsmtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='ca Abstract In this paper, we propose an unsupervised framework based on normalizing flows that harmonizes MR images to mimic the distribution of the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The proposed framework consists of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, a shallow harmonizer network is trained to recover images of the source domain from their augmented versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' A normalizing flow network is then trained to learn the distribution of the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Finally, at test time, a harmonizer network is modified so that the output images match the source domain’s distribution learned by the normalizing flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Our unsupervised, source-free and task-independent approach is evaluated on cross-domain brain MRI segmentation using data from four different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Results demonstrate its superior performance compared to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='com/farzad-bz/Harmonizing-Flows 1 Introduction Deep learning models have become the de facto solution for most image-based problems, including those in the medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Despite significant progress, these models still suffer under distributional drift, and their performance largely degrades when they are applied to data obtained in different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Clinical studies using magnetic resonance imaging (MRI) often have to deal with such large domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Due to the qualitative nature of the MRI acquisition process, generated images are sensitive to imaging devices, acquisition protocols, scanner artifacts, as well as to patient populations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' For instance, images from the same modality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', T1-w) acquired from two different scanners with separate configurations will likely present noticeable differences, which can be considered a domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Consequently, collecting a multi-center MRI dataset to address a particular clinical question does not guarantee a greater statistical power, as the increase in variance comes from a non-clinical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Furthermore, this data heterogeneity can also hamper the generalizability of deep learning models, preventing their large dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In particular, when trained on a specific site, such models are typically unable to provide similar performance for other centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To alleviate this issue, image harmonization addresses the distributional shift problem from an image- to-image mapping perspective, where the objective is to transfer image contrasts across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Nevertheless, most harmonization methods in the literature make strong assumptions that might hamper their scalability and usability in real-life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, some methods must have access to source images during the adaptation, which may no longer be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Labels associated with the downstream task may also be required in other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Finally, most harmonization techniques need to know the target domains during training, while these domains are often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In this work, we make the following contributions: Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='11551v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='CV] 27 Jan 2023 We relax all these assumptions and present a novel MR harmonization method that is source- free (SF), task-agnostic (T A) and can handle unknown-domains (UD) without requiring to be retrained for each target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Indeed, our method only needs one domain and modality at training time, as opposed to existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In particular, we propose to use a novel family of generative models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', normalizing flows, which have shown to be a powerful method to model data distributions in generative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We stress that leveraging normalizing flows to guide the adaptation of a harmonizer network has not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In addition to the methodological novelty, our empirical results demonstrate that the our approach brings substantial improvements compared to existing techniques, while alleviating their weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Furthermore, due to its task-agnostic nature and its capability to work under the unknown- domains scenario, the proposed method can also be employed in the task of test-time adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In this setting, our method largely outperforms a popular task-agnostic test-time adaptation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 2 Related work Image harmonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Several techniques have been proposed for the harmonization of images in the medical domain, and particularly for MRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Classical post-processing steps, such as intensity histogram matching [24, 26], reduce the influence of biases across scanners, but may also remove informative local variations in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Statistical approaches can model image intensity and dataset bias at the voxel level [11, 12, 2], however they must often be adjusted each time images from new sites are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Modern strategies for image harmonization, which are based on deep learning models, have shown to be a promising alternative for this problem [5, 35, 21, 36, 4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Nevertheless, they make unrealistic assumptions that hamper the scalability of existing approaches to large scale multi-site harmonization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, images of the same target anatomy across multiple sites, commonly referred to as traveling subjects are employed to identify intensity transformations between different sites [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This involves that a given number of subjects are scanned at every site or scanner required for training, a condition rarely met in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Second, another group of methods is limited to two domains [35] and requires target domains to be known at training time [35, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In addition, each time a new domain is added, these approaches must be fine-tuned in order to accommodate the characteristics of each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Calamity [21] further needs paired multi-modal MR sequences, limiting even more its applicability to single modality scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Last, task-dependent approaches leverage labels associated to each image for a given down-stream task [4, 8], thus optimizing the harmonization for this specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Nevertheless, having access to large labeled datasets might be impractical due to the underlying labeling cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Test-time Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Our method also relates to the problem of test-time domain adaptation (TTA) [30, 27, 3] which aims to quickly adapt a pre-trained deep network to domain shifts during inference on test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' One key difference between TTA and the well-known unsupervised domain adaption (UDA) problem is that, in TTA, the source examples are no longer available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' One of the earliest TTA approaches, called TENT [30], updates the affine transformation parameters of normalization layers by minimizing the Shannon entropy of predictions for test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In [23], this strategy is improved by optimizing a log-likelihood ratio instead of entropy, as well as by considering the normalization statistics of the test batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The method named SHOT [20] fine-tunes the entire feature extractor with a mutual information loss and uses pseudo-labels to provide additional test-time guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Instead of updating the network parameters, LAME [3] uses Laplacian regularization to do a post-hoc adaptation of the softmax predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Recently, normalizing Flows (NFs) have emerged as a popular approach for constructing probabilistic and generative models with tractable distributions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' NFs aim at transforming unknown complex distributions into simpler ones, for instance, a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This is achieved by applying a sequence of invertible and differentiable transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' While most existing literature has leveraged NFs for generative tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', image generation [15, 17], noise modeling [1], graph modeling [33]) and anomaly detection [14, 18], recent evidence also suggests their usefulness for aligning a given set of source domains [13, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To our knowledge, a single work has investigated NFs in the context of harmonization [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' However, it aimed at 2 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='~𝑝𝑥′(𝑥′) 𝜒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' "# 𝜒$ "# 𝜒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=" %&' 𝜒!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=" %&' ⊙ + Variational dequantization Squeezing function Affine coupling layer Channel masking Checkerboard masking UNet 𝑥~𝑝𝑥(𝑥) 𝑧~𝒩(0, 𝐼) 𝛼 𝛽 𝛼𝑥+𝛽 Step2: training the NF model with source domain images Step1: pre-training the harmonizer network Gradient Step3: adapting the harmonizer network to map the input images from target to source domain Harmonized images NF model: Harmonizer network: Figure 1: Pipeline of the proposed Harmonizing Flows method." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Our approach consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, we employ normalizing flows (NFs) to capture the distribution of the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' During the second stage, the trained NFs are leveraged to update the parameters of a harmonizer network, which are updated in order to maximize the similarity between the harmonized outputs and the distribution learned by the NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Note that steps 1 and 2 are not dependent on each other, and can therefore be performed in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' performing causal inference on pre-extracted features (brain ROI volume measures), and not image harmonization as in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Moreover, since extracting ROIs requires pixel-wise labels, the method in [32] is not task-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 3 Methodology We first define the problem addressed in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Let XS = {xn}N n=1 be a set of unlabeled images in the source domain S, where a given image i is represented by xi ∈ R|Ω| and Ω denotes its spatial domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', W ×H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Similarly, we denote as XT = {xn}M n=1 the set of unlabeled images in a potential target domain T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The goal of unsupervised data harmonization is to find a mapping function fθ : S →T without having access to labeled images for any of the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In what follows, we present our NF-based solution for this problem, whose framework is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 Learning the source domain distribution We leverage Normalizing Flows (NFs) [7] to model the distribution of the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' NFs are a recent family of generative methods that can model a complex probability density px(x) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', the source) as a series of transformation functions, denoted as gφ = g1 ◦ g2 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' gT , applied on simpler and tractable probability density pu(u) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', a standard multi-variate Gaussian distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We can express a source image as x = gφ(u), where u ∼ pu(u) and pu(u) is the base distribution of the flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' An important requirement of the transformation function gφ is that it must be invertible, and both gφ and g−1 φ should be differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Under these conditions, the density of the original variable x is well-defined and its likelihood can be computed exactly using the change of variables rule as: log px(x) = log pz � g−1 φ (x) � + log ���det � Jg−1 φ (x) ���� = log pz � g−1 φ (x) � + T � t=1 log ���det � Jg−1 t (ut−1) ���� (1) where the first term on the right-hand side is the log-likelihood under the simple distribution, and Jg−1 t (ut−1) is the Jacobian matrix of the inverse transformation gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To train the NF model and learn the source data distribution, the model parameters φ are typically optimized so to minimize the negative log-likelihood in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This results in the following loss function: LNF = − log px(x) (2) 1Note that for simplicity, we assume here that there exists only a single domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Nevertheless, our formulation is directly applied to T different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 3 EBuilding the Normalizing Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To build a bijective transformation function for the NF model, stacking a sequence of affine coupling layers [7, 17] has been demonstrated to be an efficient strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Because flows based on coupling layers are computationally symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', equally fast to evaluate or invert, they can overcome the usability issues of asymmetric flows such as masked autoregressive flows, making them a popular choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Let us consider z ∈ RD as the input to the coupling layer, which is split into a disjoint partition: (zA, zB) ∈ Rd × RD−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The transformation function g(·) : RD → RD can then be defined as: yA = zA, yB = zB ⊙ exp � s � zA�� + t � zA� (3) This setting offers simplicity for calculating the Jacobian determinant, which makes it possible to use complex neural networks as shift s(·) and scale t(·) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Note that the transformation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 3 is invertible and therefore allows for efficient Jacobian computation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The work in [7] presented coupling flows on simpler tasks and datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', CIFAR, which required less enriched representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In contrast, the problem at hand requires pixel-to-pixel mappings on more challenging images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Thus, we replace the simple convolutional blocks in [7] with shallow U-shaped convolutional neural networks to find the shift and scale parameters of the affine transformation, as they capture more global context and provide higher representation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Furthermore, as NFs are based on the change of variables rule, which is defined in continuous space, it is crucial to make the input continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Dequantization of the input can be achieved by adding a uniform noise u∈U[0, 1] to the discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' However, it might result in a hypercube representation of the images with sharp borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' These sharp borders are hard to model for a flow as it uses smooth transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Recently, a variational framework was proposed [15] to extend dequantization to more sophisticated distributions, by replacing the uniform distribution with a learnable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Constraining the source-distribution learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Optimizing the objective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 2 with only source images might bias the model to focus on characteristics of subjects, such as age and gender, rather than on source-specific features like contrast and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To overcome this issue, we propose a strategy that facilitates the learning of the source-domain distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This technique consists in randomly selecting N ′ images from the original dataset XS and applying a series of augmentations faug(·) such that the resulting image has a dissimilarity to the original image (measured by mean squared distance) higher than a specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In particular, we employ contrast augmentation, brightness changes, multiplication, and random monotonically increasing mapping functions to augment these images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Then, the total learning objective of our model can be defined as: LT = − N−N ′ � n=1 log px(xn) � �� � Source distribution modeling − N ′ � n=1 min (c, − log px(faug(xn))) � �� � Guiding term .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' (4) The first term is the learning objective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 2 over the original source images, whereas the second one forces the NF model to decrease the likelihood on the augmented images, which facilitates the learning of domain-specific characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', contrast or brightness) instead of subject-related features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', sex or age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Furthermore, we use a constant margin c in the second term to prevent the negative log-likelihood of an augmented sample from diverging to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 Achieving image harmonization Harmonizer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' A simple solution to perform image-to-image translation is to employ a harmonizer network hθ(·), such that MRIs from the target domain are translated to the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This can be expressed as px(x) = px′(hθ(x′)), where θ is the set of learnable parameters of the harmonizer network, and x and x′ are images from the source and target domains, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To train this model, we can simply use a standard reconstruction loss over images across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' However, we want the proposed method to follow a domain-free paradigm, where target domains remain unknown at training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Toward this goal, we train the harmonizer network to reconstruct the original source images from their augmented versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' As in the previous step, we augment the original images by using different types of contrast augmentation, brightness changes, multiplication, or random monotonically increasing mapping functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Contrary to the first step, there is no constraint on the magnitude of the augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The learning objective for the harmonizer 4 network thus becomes: θinit = argmin θ 1 N N � n=1 ∥(xn − hθ (faug(xn))∥2 (5) We stress that the performed augmentations are not reliable representations of potential unseen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Consequently, the direct application of the learned parameters θinit for image-to-image mapping will result in suboptimal domain transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Nevertheless, they can serve as the initial model for the subsequent step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' A simple UNet is considered for the harmonizer network, which learns two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, the last layer of the network (β) is employed as a bias value having the same dimension as the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Second, a scalar α from the middle layer of the network is used as a coefficient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In this way, the output of the harmonizer can be defined as hθ(x) = α ∗ x + β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Guiding the harmonizer network with the Normalizing Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The final step involves updating the harmonizer network so that images from the target domain are mapped into the source domain distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To achieve this, we propose to leverage the trained NF, which is stacked at the output of the harmonizer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Note that the NF model has already learned the distribution of source data, and therefore its parameters remain frozen during the adaptation of the harmonizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Thus, the learning objective of the adaptation stage consists in increasing the likelihood of the harmonizer outputs for images from the target domain, based on the NF model’s density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This loss function can be formally defined as follows: LAdap = − M � m=1 log px � gφ � hθ(xm) �� (6) As stopping criterion for updating the harmonizer, we evaluate two possible alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, we measure the Shannon entropy of the predictions for the target task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', segmentation or classification), stopping the adaptation when the entropy plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We also consider the bits per dimension (bpd), a scaled version of the negative log-likelihood widely used for evaluating generative models: bpd = − log px(x) · (log 2 · � i Ωi)−1 where Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', ΩT , is the spatial dimension of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' More concretely, we can stop updating the harmonizer parameters when the reached bpd value is the same as the one observed for the source domain using the NF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In practice, this value can be obtained at training time using a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 Experimental setting We evaluate the proposed method on the task of brain MRI segmentation across multiple sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The reason behind this choice stems from the fact that the segmentation performance is a reliable indicator of whether the structural information is well preserved during the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Four sites of the Autism Brain Imaging Data Exchange (ABIDE) [6] dataset are employed: California Institute of Technology (CALTECH), Kennedy Krieger Institute (KKI), University of Pittsburgh School of Medicine (PITT) and NYU Langone Medical Center (NYU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The selection of these sites is based on their cross-site difference, as these datasets present the most distinct histogram from each other, which better highlights the impact of harmonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' These sites are denoted as D1, D2, D3, and D4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' From each site, we selected 20 T1-weighted MRIs from the healthy control population (19 from CALTECH), which were skull-stripped, motion-corrected, and quantized to 256 levels of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 2D coronal slices of 60% of these images are used for training, 15% for validation, and the remaining 25% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Furthermore, the segmentation labels are obtained from FreeSurfer [10], following other large-scale studies [9], and grouped into 15 labels: background, cerebellum gray matter, cerebellum WM, cerebral GM, cerebral WM, thalamus, hippocampus, amygdala, ventricles, caudate, putamen, pallidum, ventral DC, CSF, and brainstem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Harmonization baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The proposed approach is benchmarked against a set of relevant har- monization and image-to-image translation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We first consider a simple Baseline applying the segmentation network directly on non-harmonized images, in order to assess the impact of each harmonization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Our comparison also includes: Histogram Matching [24], aleatoric 5 uncertainty estimation (AUE) [31], Combat [25], BigAug [34] (which uses heavy augmentations for generalization of the segmentation networks), and two popular generative-based approaches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', Cycle-GAN [22] and Style-Transfer [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Evaluation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To assess the performance of our harmonization approach, we resort to a segmentation task as it requires the preservation of fine-grained structural details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, a segmentation network SΦ(·) is trained on the images from the source domain, whose parameters remain frozen thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The harmonized images from each method are then employed to evaluate segmentation performance, which is measured with the Dice Similarity Coefficient (DSC) and modified Hausdorff distance (HD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To evaluate the robustness of tested methods, we repeat the experiments four times, each employing a different source and set of target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' These different settings are denoted as A : D1 →{D2, D3, D4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' B : D2 →{D1, D3, D4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' C : D3 →{D1, D2, D4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' D : D4 →{D1, D2, D3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The Normalizing flow model is trained for 1600 epochs using Adam optimizer with an initial learning rate of 1 × 10−3, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 every 200 epochs and a batch-size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We use a U-shaped network inside the coupling layers, which consists of four levels of different scales with a scaling factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Each level includes a modified version of the ELU activation function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', concat(ELU(x), ELU(−x)), and a convolutional layer followed by a normalizing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To construct the NF model, we first cascade four coupling layers with checkerboard masking to learn the noise distribution using variational dequantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' After applying four of the same coupling layers, features are squeezed as explained in [7] to have a lower spatial dimension and more channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We then add four coupling layers using a channel-masking strategy, another feature squeezing function, and a final set of four coupling layers with channel-masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The overall architecture of the flow model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The margin c used for guiding the flow is set empirically to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The harmonizer has five levels of different scales with a scaling factor of 2, each level including two layers of the modified ELU activation function followed by a convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The number of kernels of each level is 16, 32, 48, 64, and 64, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The harmonizer is trained for 200 epochs using Adam optimizer with a learning rate starting at 1 × 10−3, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 every 30 epochs and a batch-size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The segmentation network is trained for 200 epochs using Adam optimizer with an initial learning rate of 4 × 10−3, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 every 30 epochs and a batch-size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' All the models were implemented in PyTorch and were run on NVIDIA RTX A6000 GPU cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 Results Comparison to state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Segmentation results obtained on the images harmonized by different methods are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' We can observe that the proposed approach consistently outperforms compared methods by a noticeable margin, across datasets and for both segmentation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In particular, the average improvement gain is nearly 4% in terms of DSC, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 mm in terms of HD, compared to the second best performing method, CycleGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Impact of normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' This section assesses the impact of each component of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In particular, we evaluate the segmentation performance when images are: i) not normalized, ii) normalized with the pre-trained harmonizer θinit, or iii) normalized with the proposed harmonizing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The results from this ablation study (Table 2) empirically motivate the proposed NF model as a powerful mechanism to guide the harmonizer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' First, the strategy proposed to pre-train the harmonizer brings a substantial improvement over non-harmonized images, yet it is very simple and does not require access to images from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Secondly, driving the adaptation of the harmonizer with the proposed NF further improves the segmentation results by a large margin, demonstrating the benefits of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Adaptation stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In this section, we address the important question of when to stop the adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The first alternative is to stop it when the Shannon entropy of the segmentation predictions reaches its minimum point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' As this objective does not require any labeled data, it gives a valid stopping point for adapting the harmonizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' As a second criterion, we stop adapting when the output bpd of the NF model reaches the observed source domain bpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' As opposed to entropy, this criterion is not task-dependent and is suitable for unsupervised tasks or tasks where entropy is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Last, we resort to the segmentation performance, and stop the adaptation when it reaches the best DSC score, which we define as Oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Note that this criterion is unrealistic, and its purpose is just to demonstrate how a good stopping criterion can improve harmonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' As shown 6 Table 1: Performance overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Main results for the compared methods across different settings (A, B, C, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The best results are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' SF T A UD A B C D Average DSC (%) Baseline – – – 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 AUE [31] \x13 \x17 \x13 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 Hist matching[24] \x13 \x13 \x13 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 Combat [25] \x13 \x13 \x13 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 BigAug [34] \x13 \x17 \x13 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 Cycle-GAN [22] \x17 \x13 \x17 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 Style-transfer [21] \x13 \x13 \x17 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 Ours \x13 \x13 \x13 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 HD (mm) Baseline – – – 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='20 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='57 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='23 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='07 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='73 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='81 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='64 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='03 AUE [31] \x13 \x17 \x13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='57 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='67 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='03 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='57 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='85 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='46 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='08 Hist matching[24] \x13 \x13 \x13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='40 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='47 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='77 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='00 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='73 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='40 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='65 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='71 Combat [25] \x13 \x13 \x13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='23 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='67 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='30 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='17 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='84 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='57 BigAug [34] \x13 \x17 \x13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='53 ±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='51 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='43 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='40 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='87 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='70 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='07 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='63 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='69 Cycle-GAN [22] \x17 \x13 \x17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='63 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='63 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='63 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='30 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='30 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='50 Style-transfer [21] \x13 \x13 \x17 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='23 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='93 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='78 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='53 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='27 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='24 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='92 Ours \x13 \x13 \x13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='10 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='77 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='37 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='30 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='63 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='94 Table 2: Ablation study on the different components in terms of DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' A B C D Average Without harmonization 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 Pre-trained harmonizer 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 Adapting using NF 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 in Table 3, although minimum entropy is a better criterion compared to bpd, both achieve comparable performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In addition, both stopping criteria are a suitable choice, as their results are very close to the Oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Table 3: Impact of the adaptation stopping criterion (in terms of DSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' A B C D Average Minimum Entropy 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 Source BPD 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 Oracle (best epoch) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Figure 2 depicts several examples of harmonized images produced by the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' These results illustrate that, regardless of the target domain, our method produces reliable image-to-image mappings to the source distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Results when N4 bias correction is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In previous sections, we used the original MRIs of the ABIDE dataset without bias correction to evaluate the proposed harmonization method on more challenging scenarios, where pre-processing steps to enhance the images might not be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Compared to bias-corrected MRIs, original MRIs have arguably more complex distributions, which makes it more difficult for harmonization methods to map MRIs from a target domain to the source one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To demonstrate that our method also achieves satisfactory performance when the initial domain shifts are reduced, we repeated the previous steps with N4-biased corrected MRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' These results, shown in Table 4, also showcase the advantage of our method in this different setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' For conciseness, we report here the average results for the HD metric (in mm) across different methods: Baseline (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='04±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='41), Hist matching (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='45±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='33), Combat (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='75), BigAUG (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='47), Cycle-GAN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='31), Style-transfer (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='77 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='07) and ours (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 7 Figure 2: Examples of harmonized images produced by the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Table 4: Quantitative results on bias corrected images (in terms of DSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' A B C D Average Baseline 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 AUE [31] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 Hist matching [24] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 Combat [25] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 BigAug [34] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='7 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 Cycle-GAN [22] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 Style-transfer [21] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 Ours 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='5 Experiments on Test-Time Adaptation (TTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Our model can also be employed in a TTA scenario, where the model needs to be updated at inference for a given image, or set of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' To motivate this assumption, we compare the performance of our approach to the popular TENT model [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Adapting the segmentation network SΦ(·) with TENT yields 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='1 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='0 of DSC, which represents a considerable gap compared to our model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Note that there exist other TTA methods for segmentation in the medical field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', [16], however, they require segmentation masks for the adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 5 Conclusion In this paper, we proposed a novel harmonization method which leverages Normalizing Flows to guide the adaptation of a harmonizer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Our approach is source-free, task-agnostic, and works with unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' These characteristics make our model applicable in real-life problems where the source domain might not accessible during adaptation, target domains are unknown at training time, and harmonization is not dependent on a specified target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' The proposed method achieves state-of-the-art harmonization performance based on the segmentation task, yet relaxes the strong assumptions made by existing harmonization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Thus, we believe that our model is a powerful alternative for MRI multi-site harmonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' References [1] Abdelhamed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', Brubaker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', Brown, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content='S.' metadata={'source': 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multi-site diffusion tensor imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' Neuroimage 161, 149–170 (2017) [13] Grover, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' : Alignflow: Cycle consistent learning from multiple domains via normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' In: AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFJT4oBgHgl3EQfeCzc/content/2301.11551v1.pdf'} +page_content=' 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b/wdAyT4oBgHgl3EQfnfgZ/content/tmp_files/2301.00488v1.pdf.txt @@ -0,0 +1,1362 @@ +arXiv:2301.00488v1 [cs.HC] 1 Jan 2023 +Information Transfer Rate in BCIs: Towards +Tightly Integrated Symbiosis +Suayb S. Arslan and Pawan Sinha +Department of Brain and Cognitive Sciences, +Massachusetts Institute of Technology, +Cambridge, MA, USA, 02139. +E-mail: sarslan@mit.edu, psinha@mit.edu +Version 1.0 – January 2023 +Abstract. +Objective. The information transmission rate (ITR), or effective bit rate, is a popular +and widely used information measurement metric, particularly popularized for SSVEP- +based Brain-Computer (BCI) interfaces. +By combining speed and accuracy into a +single-valued parameter, this metric aids in the evaluation and comparison of various +target identification algorithms across different BCI communities. In order to calculate +ITR, it is customary to assume a uniform input distribution and an oversimplified +channel model that is memoryless, stationary, and symmetrical in nature with discrete +alphabet sizes. To accurately depict performance and inspire an end-to-end design +for futuristic BCI designs, a more thorough examination and definition of ITR is +therefore required. +Approach. +We model the symbiotic communication medium, +hosted by the retinogeniculate visual pathway, as a discrete memoryless channel and +use the modified capacity expressions to redefine the ITR. We use graph theory +to characterize the relationship between the asymmetry of the transition statistics +and the ITR gain with the new definition, leading to potential bounds on data rate +performance. Main Results. On two well-known SSVEP datasets, we compared two +cutting-edge target identification methods. +Results indicate that the induced DM +channel asymmetry has a greater impact on the actual perceived ITR than the change +in input distribution. +Moreover, it is demonstrated that the ITR gain under the +new definition is inversely correlated with the asymmetry in the channel transition +statistics. Individual input customizations are further shown to yield perceived ITR +performance improvements. Moreover, an algorithm is proposed to find the capacity of +binary classification and further discussions are given to extend such results to ensemble +techniques. Significance. We anticipate that the results of our study will contribute +to the characterization of the highly dynamic BCI channel capacities, performance +thresholds, and improved BCI stimulus designs for a tighter symbiosis between the +human brain and computer systems while enhancing the efficiency of the underlying +communication resources. + +2 +1. Introduction +The primary goal of brain-computer interfaces (BCIs) is to provide new channel +formations for communication and control between the human brain and its surrounding +objects with computational capabilities [1]. +The majority of BCI research efforts +are focused on developing effective stimuli, novel protocol developments and target +identification algorithms to boost information transfer and eventually help novel +communication paradigms to emerge such as found in recent semantic communications +[2]. Different types of BCIs are employed in various applications nowadays, ranging from +clinical deployments [3] to entertainment world, including but not limited to gaming +[4], [5]. +Such commonplace applications has become quite encouraging and pushed +current state-of-the-art BCI research forward, enabling more coupling and enhanced +communication links. +A generic BCI system typically consists of three main parts: (1) the stimulation +generation, (2) the communication channel hosted by the participating subject, and +(3) the target identification system. To be able to deliver fast communication rates +and form a close symbiosis between the human brain and a computing device, these +components must work in concert. For a truly symbiotic system design in which the +stimulus generation and TIs co-adapt to each other [6], better understanding of the +performance evaluations, controlling measures and the underlying statistical channel +formed is required. Once established, new research directions can be explored such as +developing useful performance bounds and manage what is needed to enhance end-user +experience. +Assessments of BCI systems are typically performed at two levels of evaluation, +namely user-level and system-level. +User performance is measured by the degree of +congruence between user intent and the signal feature(s) the BCI uses to identify +the intent. +The user-level quality control heavily depends on the visual setup, the +selection and presentation of stimuli, and how the stimulation is carried out (usually +forming a protocol). However, system-level performance evaluations are often done in +terms of target identification speed and classification accuracy. Fair comparisons are +difficult since these two criteria, when articulated independently, are affected by the +program’s capability as well as how well the system combines the user’s control with +that application. Information transfer rate (ITR), cited primarily in [7] and [8], is one of +a number of measuring tools developed in response to the demand for a single theoretical +measure that combines accuracy and speed. Information transfer is quantified according +to information theory measures such as entropy, pioneered by Shannon’s seminal work +back in 1948 [9]. +ITR is a measure that can be used to measure the magnitude of coupling in a +communication setting as well as the levels of attention and counciousness which can +be leveraged heavily in passive-BCI settings [10]. ITR is used in BCIs both using P300 +paradigm [11] and in particular steady-state visual evoked potentials (SSVEPs) due +to proven high communication rates. For instance, recent progression towards task- + +3 +related component analysis is shown to achieve rates up to 325 bits/min ITR in a +cue-guided 40-character spelling task [12]. Such performance has been possible due to +carefully designed stimulus generation and Target Identification (TI) techniques. Stimuli +generation involves embedding information into frequencies and phases of the signals +(modulation) in an SSVEP paradigm [13]. TI requires compensating for the noise and +degradations imposed on the information passing through a channel induced by the BCI +and the physical medium of the retinogeniculate visual pathway. It turns out that many +information processing strategies are used throughout the visual system and the basic +approach is to lump them into a coarse description of an information link [14]. The main +objective of all the past work has been to maximize the information transfer rate that can +be transmitted over this induced channel [15]. Although the BCI channel properties are +determined by the choice of stimulation and the target identification methods, accurate +measurement of the ITR performance is also of critical importance for the assessment +of user experience and developing successful stimuli design techniques. +Preconditions and deficiencies of the conventional ITR definition are brought up +in a number of past studies. +For instance, [16] summarizes the problems with the +conventional definition and particularly emphasizes the significance of the channel +parameter estimations (accuracy or false alarm rates) in online synchronous BCIs. +The same study proposed a task-oriented online BCI test in the hope to help with +the real-world applications. Moreover, recent works such as [17] proposed to use an +alternative closed-form formulation derived in [18] for the ITR computation via making +approximations such as removing negativity constraint on the input distribution using +only a fairly limited dataset. In fact, this closed form’s usefulness is restricted to square +and non-singular channel transition characterizations [19]. Moreover, no further analysis +or intuition is presented in both studies in terms of the channel transition characteristics, +the stimuli design, and means and strategies for achieving the capacity of the underlying +BCI channel. On the other hand, one of the objectives of this study is to outline the +basic principles of conventional ITR definition and in order to re-express its deficiencies, +highlight the challenges of channel characterization problems between the human brain +and the computer system. We note that performance characterization of any technique +for an asymmetric and non-stationary channel requires a careful computation procedure +to accurately determine the practical ITR the subjects experience as well as directions +for tighter symbiosis within the context of joint system design. We will demonstrate +that this study may also help design better input for BCIs to maximize the information +flow in the induced communication channel. +The rest of the paper is organized as follows. +In Section II, we introduce the +generalized DM channel model and rephrased the conventional ITR definition. +We +report the deficiencies as well as workarounds through integrating the algorithmic +capacity calculations into the ITR definition subject to information theoretic bounds. +In Section III, we present a few results to distinguish different ITR definitions, and +explore the asymmetry in channel transition statistics and establish the ties with the +conditional entropy. We discuss some of the important implications of our results in + +4 +Section IV before concluding the paper with future directions. +2. Methods +2.1. Channel Model and Conventional ITR Definition +Let us consider a discrete BCI system where one of the M symbols from the set +X = {x1, . . . , xM} is to be communicated at a given time. It is quite typical to express +BCI system performance in terms of the information transfer rate (ITR) [7]. This is +expressed in bits per trial observation window T [22], +ITR = log2(M) + P(T) log2(P(T)) + (1 − P(T)) log2 +�1 − P(T) +M − 1 +� +(1) +where M is the number of targets and P(T) is the aggregate average accuracy of the +target identification algorithm. Note that the trial time dependency of the accuracy is +crucial in this formulation. Equality (1) is derived from the popular mutual information +measure defined for two random variables X and Y as +I(X; Y ) = H(Y ) − H(Y |X) +(2) += +� +y∈Y +PY (y) log2 +� +1 +PY (y) +� +− +� +x +PX(x)H(Y |X = x) +(3) +where H(.) is the entropy, X ∈ X represents the discrete source taking on one of the +M target classes and Y ∈ Y (typically Y = X ) is the predicted output at the other end +of the BCI system with distribution PY (y). Capacity is defined to be the supremum of +the mutual information over all input (probability) distributions PX(x). In the case of +perfect communication (P(T) → 1) the ITR will simply be log2(M), the number of bits +used to represent all targets assuming these targets have equal probability of occurring +i.e., 1/M (uniform input distribution i.e., PX(x) = +1 +M ). +Equality (1) is based on the capacity of a symmetric Discrete Memoryless Channel +(DMC) that errs with an equal probability +1−p +M−1 in favor of all other M − 1 classes. In +addition, T is expressed in terms of seconds and the ITR is usually scaled with 60/T +and expressed in terms of bits/min. On the other hand, we define the channel transition +matrix of the induced DMC and express it as follows, +P = + + +p1,1 +p1,2 +. . . +p1,M +p2,1 +p2,2 +. . . +p2,M +... +... +... +... +pM,1 +pM,2 +. . . +pM,M + + +with � +j pi,j = 1 for i ∈ {1, 2, . . . , M}. We use the short-cut notation PY |X(yj|xi) = pi,j +as each entry of P to refer to the channel transition/conditional probability distribution. +The generic channel models are illustrated in Fig. 1. Note that symmetry assumption +in the channel transition statistics i.e., the distribution of the probability of erring over + +5 +Figure 1: Typical discrete BCI Channel Models for symbol/character communications. +A discrete set of characters are communicated. In case, the communication reliability +is below a threshold, the communicated character can be assumed to be erased. In the +figure, e is used to represent erasure. +all non-target values make the uniform input distribution achieve the DMC capacity. +Hence, +max +PX(.) I(X, Y ) = +� +log2(M) − +� +j +pi,j log2 +� 1 +pi,j +�� += ITR +(4) +Although we have assumed the possible number of outcomes to be M i.e., |Y| = M, +the channel outputs can be expanded to include “erasures” i.e., making no decision on +the final output, leading to the channel transition matrix P to be of size M × (M + 1). +This extension is also illustrated in Fig. 1. +2.2. Deficiencies and Workarounds +Some of the major deficiencies of the conventional ITR definition have already been +articulated in [16]. +One of these deficiencies is the assumption that the source has +uniform distribution. However, this assumption is not necessarily true and optimal. +For instance, in a speller application, the characters of the English alphabet are +not necessarily used equally frequently in everyday language and hence in an online +experiment, some of the characters would naturally be intended to spell more often. In +addition, transition probabilities of the underlying channel are not necessarily symmetric +and stationary [20]. In an SSVEP-based BCI, the distinguishability of the two targets + +preprocessing +Target +preprocessing +Target +Identification +Identification +Stimuli +Stimuli +XO +Source Set +Target Set +Source Set +Target Set +P1,1 +P1,1 +x1 +1 +x1 +X1 +P1,2 +P1,2 +2 +×2 +X2 +X2 +P1,M +P1,M +e +PM,M +PM,M +XM +XM +XM +XM +Communication Channel +Communication Channel +with defintive decisions +with erasures6 +depends on the frequency and phase selections or even spatial distance in which these +targets are presented to the subjects. +Therefore, the assumption pi,j = +1−p +M−1 for all +i, j satisfying i ̸= j (i.e., equal probability transitions to all non-target classes) is not +necessarily totally accurate. +One of the challenges of the capacity computation for such a highly dynamic channel +is that the transition matrix is a function of both the stimulus design (the encoding of +information into visual pathway) and also the target identification methods, unlike in +classical digital communications. In addition, as the observation window (T) widens, +accuracy is reported to increase since the identification is performed via observation of +a wider signal window. However, the observation window being larger will change the +stochastic nature of the channel (and its parameters) i.e., the channel transition matrix +indeed is a function of time. The extent of the introduced channel memory is yet another +challenge to tackle. Thus, instead of considering the entire timeline, we consider specific +time points such that the dependency of the channel transition matrix at those points is +sufficiently eliminated. Some of the previous works proposed closed form expressions [21] +under certain assumptions about Px(x) and non-singularity assumption for P, which is +highly unlikely for larger window lengths. In this work, we do not make any assumptions +about the statistical nature of the channel and treat it as a DMC, calculate the capacity +numerically and report our final ITR results along with the conventional formulation. +2.3. Capacity for Asymmetric DMC +2.3.1. Exemplary Case: “Binary Classification”: +Let us suppose X = Y = {x1, x2} i.e., +the input is one of the two possible symbols with PX(x1) = px. This would correspond +to differentiation of two different classes such as face/non-face or familiar/non- +familiar (target/non-target paradigm) dichotomies. Thus, we can express the mutual +information +I(X; Y ) = H(Y ) − H(Y |X) +(5) += h(px(1 − p1,2) + (1 − px)p2,1) − pxh(p1,2) − (1 − px)h(p2,1) (6) +where h(x) = −x log(x) − (1 − x) log(1 − x) is the binary entropy function. Setting the +derivative with respect to px to zero, we obtain +1 +px(1 − p1,2 − p2,1) + p2,1 +− 1 = 2 +h(p1,2)−h(p2,1) +1−p1,2−p2,1 +(7) +Subsequently, px that satisfies this equality can be plugged into (6) and following some +algebraic operations, the final capacity can be expressed as +C2(p1,2, p2,1) = log2 +� +1 + 2 +h(p1,2)−h(p2,1) +1−p1,2−p2,1 +� +− (1 − p2,1)h(p1,2) + p1,2h(p2,1) +1 − p1,2 − p2,1 +(8) +Finally, the ITR in bits/min can be found by 60 +T C2(p1,2, p2,1). +2.3.2. +General Case “M symbols”: Having more than two classes complicates the +above computations (by requiring the computation of partial derivatives and solving + +7 +transcendental equations) which precludes closed form results. +There have been +successful attempts in the past that iteratively compute the capacity for discrete +stationary channel models. +For memoryless channels (independent choice of symbols) with finite input and +output alphabets X and Y respectively, the capacity can be computed by the Blahut- +Arimoto (BA) algorithm [19, 23]. On the other hand, in a typical speller task, due +to the formation of language and words, the source will inherently have memory. The +Blahut-Arimoto algorithm was also extended to channels with memory and finite input +alphabets and state spaces [24] such as Hidden Markov Models (HMM). However, +modeling language with an HMM is quite challenging [26] and can result in inordinate +computation time and/or an approximation for the capacity. +The BA algorithm is an iterative procedure which assumes an arbitrary input +probability distribution function P 0 +X(x) in the beginning and optimizes it over multiple +iterations [25]. Let us express the non-normalized input distribution for xi ∈ X at the +(k − 1)-th step (k ≥ 1) as +Dk−1(xi) = P k−1 +X +(xi) exp +� +D +� +pi,j||P k−1 +Y +(yj) +�� +(9) += P k−1 +X +(xi) exp +� N +� +i=0 +pi,j log +� +pi,j +P k−1 +Y +(yj) +�� +(10) += P k−1 +X +(xi) +N +� +i=0 +� +pi,j +P k−1 +Y +(yj) +�pi,j +(11) +where D[.||.] is the relative entropy (also known as Kullback-Leibler or KL divergence) +and P k−1 +Y +(yj) = � +i P k−1 +X +(xi)pi,j. Then, the input distribution is normalized to be +P k +X(xi) = +Dk−1(xi) +� +j Dk−1(xj). +(12) +Finally, iterations cease if +max +xi D +� +pi,j||P k +Y (yj) +� +− +� +i +P k +X(xi)D +� +pi,j||P k +Y (yj) +� +< γth +(13) +holds for a given error threshold γth (e.g., ǫth = 10−5). +For the rest of our discussions, we refer to Blahut-Arimoto algorithm as follows +[C, P ∗ +X(x)] = BA(P) +(14) +where BA(.) refers to the algorithm itself, C is the capacity and P ∗ +X(x) is the optimal +input distribution that maximizes the mutual information. We note that this two-step +process can be shown to converge through iterations of two different convex optimization +problems and the convergence rate can be shown to improve in a later study [27]. +2.4. Fano’s Inequality +Fano’s inequality provides a lower bound for the probability of target identification +error ǫM (= 1 − � +i PX(xi)pi,i) due to information degradation via the channel induced + +8 +by the BCI. The channel transition (conditional) probabilities PY |X(yj|xi), emprically +estimated by the confusion matrices, appear in the bound as follows, +ǫM ≥ H(Y |X) − h(ǫM) +log2(M − 1) +≥ H(Y ) − I(X; Y ) − 1 +log2(M) +(15) +where h(ǫM) = −ǫM log(ǫM) − (1 − ǫM) log(1 − ǫM) is the binary entropy function. +Accordingly, for a fixed ǫM, we can upper bound the conditional entropy as +H(Y |X) ≤ ǫM log2 +�M − 1 +ǫM +� ++ (1 − ǫM) log2 +� +1 +1 − ǫM +� +(16) += h(ǫM) + ǫM log2(M − 1) +(17) +Since in typical telecommunication applications the channel transition probabilities +are given as part of the model, the symbol detection error is bounded. However as can +be seen, ǫM appears in both sides of the inequality (15). Note that the bound in (15) +does not apply to M = 2 case (zero denominator) and countably infinite sets. However +later studies extended this bound to such corner cases [28, 29]. On the other hand, the +bound on the conditional entropy given in (16) becomes h(ǫM) when M = 2 and can be +shown to be looser for larger ǫM (see our experimental results). +2.5. Target Identification +It is conventional to divide TI methods into two broad overarching categories, namely +supervised and unsupervised (training-free). Unsupervised techniques are particularly +attractive since their use does not involve user-specific-calibration phase (long training +cycles) and provides more versatility in everyday practices. +On the other hand, +supervised methods are shown to outperform the unsupervised early strategies such +as Cannonical Correlation Analysis (CCA) [30]. +One of the most effective frequency recognition performance has been demonstrated +by a number of spatial filtering techniques to isolate task-specific source activities +from EEG signals. The task-related component analysis (TRCA) is one of prominent +techniques proposed in literature which hypothesizes that there are distinct cortical +sources in the brain which generates potentials upon the presentation of flickering +stimuli. +This idea originally applied to near-infrared spectroscopy (NIRS) [31], [32] +and later proven useful for multivariate EEG data [12]. Assuming s(t) ∈ R to be the +task-related, n(t) to be the task-unrelated (noise and some other background brain +activity) components, the multivariate EEG signal x(t) ∈ RNc is formed as a result of a +linear generative model as follows, +xj(t) = ajs(t) + bjn(t) for j = 1, 2, . . . , Nc +(18) +where Nc is the number of channels. Spatial filtering is about extracting the task-related +component s(t) from a linear combinations of multiple channel output signals x(t), i.e., +˜s(t) = +Nc +� +j +wjxj(t) = +Nc +� +j +wjajs(t) + wjbjn(t) +(19) + +9 +Main idea behind TRCA is to optimize weight coefficients (wj) so as to maximize +inter-trial covariance (reproducibility) of time-locked biomedical data. If we denote the +h-th trial of y(t) as y(h)(t), then the sum of covariances of all possible combinations of +trials can be expressed as +Nt +� +h1,h2=1 +h1̸=h2 +Cov +� +y(h1)(t), y(h2)(t) +� += wTSw +(20) +where Nt is the total number of trials, Cov(.) is the covariance operator, w = (wj)1≤j≤Nc +and S = (Sj1,j2)1≤j1,j2≤Nc is given by +Sj1,j2 = +Nt +� +h1,h2=1 +h1̸=h2 +Cov +� +x(h1) +j1 (t), x(h2) +j2 (t) +� +(21) +For a finite solution, TRCA maximizes wTSw subject to variance constraint, Var(y(t)) += wTQw ≤ 1. +The solution is given by the following unconstrained optimization +problem +w∗ = arg max +w +wTSw +wTQw +(22) +which can be recognized as a generalized eigenvalue problem. +By creating a spatial mapping that projects the multivariate EEG data onto +a standard SSVEP representation space, the sum of squared correlations (SSCOR) +framework [38] seeks to identify a session independent representation of SSVEP response. +We express the optimization problem as +w∗ +X, (w∗ +i ) = arg max +wX ,wi +Nt +� +i=1 +� +wT +XCov +� +x(X)(t), x(i)(t) +� +wi +�2 +(23) +where +x(X)(t) = 1 +Nt +Nt +� +i=1 +x(i)(t) +(24) +is the template signal calculated for each target frequency separately. Again for the +sake of obtaining a finite solution and put the optimization problem into a generalized +eigenvalue framework, we use the set of constraints for ∀i, wT +i Cov +� +x(i)(t), x(i)(t) +� +wi = 1. +Note that there could be other spatial filtering techniques that can generate the +filter weights (w) as an application of generalized Eigenvalue problem [36]. However, +the arguments for the detection logic is common to all. After determining the weights, +for a given single-trial test sample X, the classification decision is made in favor of the +frequency fn ∈ {f1, . . . , fNf} based on the Pearson’s correlation coefficient (ρ) as +τ = arg max +n +ρ +� +XTwn, X T +n wn +� +, +(25) +where Nf is the total number and n is the index of the stimulation frequency. Finally, +if we concatenate all weights to construct W = [w1 w2 . . . wNf] and replace it with +wn in Equation (25), we obtain an ensamble TI algorithm. Same idea can be applied +to TRCA method. + +10 +3. Experimental Results +3.1. Datasets +We have used two well known datasets in the literature, namely Benchmark [34] and Beta +datasets [35] to analyze/compare two known target identification algorithms, namely +TRCA and SSCOR, based on the conventional as well as proposed ITR definitions. +Benchmark dataset was collected from 35 subjects with normal/corrected-to-normal +vision on a 40-character (26 English alphabet letters, 10 digits, and 4 other symbols) +speller task using a 64-channel EEG recorder. Each subject was shown target characters +in distinct trials, where characters flicker at frequencies 8-15.8 Hz with 0.2 Hz increments +and phases 0-1.5π with 0.5π increments, where both increments are proportional to +stimulus index. Each trial began with a visual cue that was shown on the screen for +0.5 seconds to direct the subject’s gaze to the intended target, followed by 5 seconds +of stimulation and a final trailing 0.5-second offset, respectively. +The observation +window includes gaze length as well as the signal length after the stimuli onset. At +the preprocessing stage, the recorded signals were downsampled to 250Hz. We have +assumed 130 ms visual pathway delay for both datasets. In this study, the following +set of 9 channels (OZ, O1, O2, PZ, POZ, PO3 PO4, PO5 and PO6) are considered +since they are reported to be most reflective of neural activity due to tasks performed +in the experiment [34]. In Beta dataset on the other hand, 70 subjects participated in +four blocks of a cued-spelling task on a QWERTY virtual keyboard. The stimulation +duration is 2 seconds for the first 15 subjects whereas the rest of the other subjects are +stimulated for 3 seconds. Sixty-four channels of EEG data were collected by SynAmps2 +(Neuroscan Inc.) data acquisition/amplifier system at a sampling rate of 1000Hz, which +were later downsampled to 250Hz. The rest of the settings are the same (channels used, +electrode locations, gaze-shift times, visual pathway latency, etc.). Since the trials were +carried out outside a laboratory setting, the Signal-to-Noise Ratio (SNR) of EEG in +Beta dataset is measured to be lower than that of the Benchmark dataset. +3.2. Filter Banks +In our work, we have leveraged filter banks to decompose the EEG signals into +five overlapping sub-bands to make use of the independent information found in the +harmonics. A target detection technique is used independently for each of the sub-bands. +The cut-off frequency range for the sub-bands based on the EEG data bandwidth is set +between b × 8 Hz and 90 Hz, where b ∈ {1, 2, 3, 4, 5} [37]. As the Target Identification +(TI) algorithm, we have employed two competing approaches, namely the TRCA [12] +and SSCOR [38] due to their high performance relative to other methods with reasonable +complexity (in terms of the number of parameters to optimize). +As mentioned before, the design methodology behind TRCA was the hypothesis +that the single-trial EEG data can be reconstructed as a linear sum of multiple time +series from different cortical sources [39]. Therefore, TRCA is used as a technique for + +11 +0 +1 +2 +3 +4 +5 +6 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +Figure 2: Capacity achieving distributions for different signal lengths (sec) as well as +their entropies for TRCA method using Benchmark dataset. +0 +1 +2 +3 +4 +5 +6 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +Figure 3: Capacity achieving distributions for different signal lengths (sec) as well as +their entropies for SSCOR method using Benchmark dataset. +highlighting the task-related elements embedded in the EEG signals through enhancing +repeatability among different time-locked activities across trials. +The covariance +between different trials is maximized to ensure such repeatability. +On the other +hand, SSCOR transforms SSVEP signals to a common representation space through +the optimization of the individual SSVEP templates. Similar to TRCA, SSCOR also +achieves space transformation. In both of these methods, we learn a spatial filter for each +frequency (character) [36]. In our work, we have also employed an ensemble technique for +both methods where all spatial filters belonging to different frequencies are concatenated +for a performance boost. To determine a final score (a single correlation coefficient) for +classification, the correlation coefficients of the sub-band components are combined using +the weighted sum of squares approach [37]. + +12 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +100 +120 +140 +160 +180 +200 +220 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +80 +90 +100 +110 +120 +130 +140 +150 +Figure 4: +Average ITR as a function of signal Length (sec) for different TIs for +Benchmark dataset (left) and Beta dataset (right). The plots clearly show the difference +between the conventional definition as well as the proposed definitions. +Since input +distribution depends on the window length T, we have also expressed which time window +the ID is optimal for. +3.3. Evaluation Process +For each subject, we evaluated performance in a leave-one-trial-out fashion as in the +past literature i.e., the TI algorithms are trained over u − 1 trials and tested on the +remaining trial for all u different combinations for u = 6 in Benchmark and u = 4 in +Beta datasets. When we present average subject performance, test performances are +combined (classification outcomes) in a single confusion matrix, which is normalized to +obtain the estimate of the channel transition probability matrix ˜P with M = 40. In some +of the past works c.f. [12], [38], using conventional definition, ITR is calculated for each +subject and the average of these values (with the standard error) is reported despite +the nonlinear dependency of the definition on the accuracy. +This type of averaging +typically leads to larger ITR values. +In our work, we report all ITR results after +averaging the accuracy values (similarly false positive and negatives etc.) +over all +subjects before calculating and reporting the final ITR. This way, we also ensure that +the aggregate human performance is translated into an ITR metric. Note that with the +proposed definition, it would be unrealistic to expect the system to optimize the input +distributions for each subject separately in order to attain higher ITRs. To address this +point however, we have also demonstrated via violin plots by calculating the individual +ITRs using both definitions for all observation windows. +In that, we use individual +data to estimate the channel transition statistics for each subject and observation +window, separately. +We have leveraged BA(.) +algorithm to compute the capacity +maximizing input distributions and the corresponding capacity value for signal lengths +of 0.18, 0.2, 0.25, 0.3, . . ., 1 seconds, whereby the conditional entropy of ˜P complies with +the Fano’s inequality. + +13 +3.4. Numerical Results +3.4.1. Performance averaged over subjects: +In Figs. 2 and 3, using Benchmark dataset, +we demonstrate capacity achieving distributions (with channel transition statistics +averaged over all subjects) as bar plots for each signal length as well as corresponding +input entropy H(X), output entropy H(Y ) and the conditional entropy H(Y |X) that +characterizes the channel transition statistics for both TI methods, respectively. +As +can be seen, with larger observation window, H(X) and H(Y ) increases. On the other +hand, the channel transition probabilities begin to show more structure (such as reduced +symmetry and balanced distribution of probabilities) and hence less (conditional) +entropy. As a result, the capacity of the induced channel H(Y ) − H(Y |X) increases +with growing signal length in both techniques. +As can be seen, particularly at low +signal lengths, SSCOR fails to enhance the channel capacity due to poorer reduction in +H(Y |X). However as the observation window increases, the reduction in H(Y |X) for +both TIs become on par, almost equating the capacity gain at around an observation +window of one seconds. We observed similar trends using the Beta dataset. +We have also provided the ITR results using the BA algorithms and confusion +matrices in Fig. 4 for both datasets. Since the ITR is maximized at the signal length +of ≈ 0.5 secs for TRCA and ≈ 0.65 secs for SSCOR for Benchmark dataset (and ≈ 0.45 +secs for TRCA and ≈ 0.55 secs for SSCOR for Beta dataset), we use the optimal input +distribution (ID) of the signal lengths 0.5 (0.45) and 0.65 (0.55) secs, respectively, for +all signal lengths to obtain the ITR for the TI algorithms (Asym.+Optimal ID). For +comparison purposes, we have also included the conventional ITR definition in Fig. 4, +which neither takes the input distribution nor the asymmetry the channel introduces +into account and consequently underestimates the maximum information transfer rate +that can be achieved over the induced channel. To investigate it statistically, we have +carried out paired t-test and f-test (two-tailed) to determine whether the proposed ITR +definition is different from the conventional. Using Benchmark dataset, both methods +showed dramatic mean differences (p ≈ 3.97 × 10−8 for SSCOR and p ≈ 3.7 × 10−7 for +TRCA). However, there were no significant variational differences between the different +ITR definitions (F(17, 17) = 1.25, p > 0.05 for SSCOR and F(17, 17) = 1.47, p > 0.05 +for TRCA). This clearly demonstrates that although the trend is almost the same +for both TI algorithms, the actual ITR experience is meaningfully different than the +reported average ITR results in the literature. +To be able to demonstrate the effect of asymmetry of the channel on the ITR, +we have used the optimal ID while keeping the accuracy intact and divide the error +probabilities equally over all other non-target classes for each class (character) i.e., +pi,j = +1−p +M−1 for all i, j satisfying i ̸= j. +We have named this scheme “Balanced +transition probability matrix” and used it with the optimal ID for all signal lengths +(Balanced+Optimal ID). As can be seen, rather than the non-uniformity of the input +distribution, the asymmetry in the probability transition characteristics has much more +significant effect on the final ITR results. + +14 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Figure 5: On the left, we show asymmetry score ∆asm as a function of the ratio of change +in the ITR with the new definition for all T. On the right, we have also depicted the +asymmetry score ∆asm as a function of the conditional entropy of the induced channel. +We have varied the transparency of the markings from dark to light as we change the +observation time window from 0.18 seconds to 1 seconds. +On the other hand, to quantify the degree of asymmetry, we have used the +largest singular value of the skewed Laplacian of the channel transition matrix, namely +∆asm = (Γ − ΓT)/2 [40], where Γ = Φ1/2(I − P)Φ−1/2, I is the identity matrix, +Φ1/2 = + + +√φ1 +0 +. . . +0 +0 +√φ2 +. . . +0 +... +... +... +... +0 +0 +. . . +√φM + + , +and [φ]1≤i≤M are the stationary probabilities of a Markovian process whose transitions +are governed by the channel transition statistics. In Fig. 5 (left), we depicted the degree +of asymmetry ∆asm as a function of the ratio of change in the ITR (∆ITR) using the +proposed definition. More specifically, +∆ITR ≜ (Asym.+Optimal ID) − (Conventional) +(Conventional) +. +(26) +As can be clearly seen from Fig. +5 (left), asymmetry increases the ITR gain, +and it turns out to satisfy a logarithmic relationship. The parameters of this estimate +is explicitly given in the same figure using regression. +This suggests that although +the asymmetry in the channel transition statistics helps with increasing the rate of +communication, it also quickly saturates due to increased conditional entropy of the +channel and reduced accuracy. +Note that the conditional entropy is bounded above +by the Fano’s inequality as expressed in (16) which forms an upper bound on ∆asm. +One of the interesting conclusions is that particularly at short window lengths, stimuli +design that generates more asymmetric channel transition characteristics is likely to + +0.08 +P 0.07180ms +Benchmark +-Benchmark +Beta +-Beta +4 +4.50.06 +Score +0.05 +Asymmetry +0.04 +TRCA- +0.03 +X +SSCOR +TRCA- +1000ms +SSCOR +0.02 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +H(YIX)15 +Figure 6: Proposed ITR v.s. conventional ITR for subject-specific input customization +using two TIs and the Beta Dataset. Input customization can be shown to be effective +even with longer observation windows. +improve ∆ITR, higher improvement ratio with respect to the conventional definition. +However, the relationship between ∆asm and H(Y |X) clearly suggests that as we have +larger observation windows, to be able to maximize the mutual information, system +prefers to have less asymmetry in the channel transition characteristics (to minimize +the conditional entropy) and maximize the input entropy via uniform distribution. +On the other hand in Fig. +5 (right), we have illustrated ∆asm as a function +of conditional entropy H(Y |X). +As anticipated, there is a monotonic relationship +(presented with a linear regression) in between and both the degree of asymmetry +and conditional entropy increase as the observation window length grows. One of the +interesting observations is that the dataset has more significant effect on the slope of +the relationship compared to target identification algorithms. Different slopes can also +be interpreted as an indicator of the difficulty of the datasets (TIs perform worse with +Beta dataset compared to Benchmark) and reduced SNR while recording these EEG +datasets. +3.4.2. Performance customized to each subject: +We have also investigated via violin +plots in Fig. 6 (considering all 70 subjects individually as data points) whether the +proposed ITR computed per subject (finding the optimal input distribution for each +subject) differs significantly compared to conventional definition for the two different + +400 +Proposed ITRSSCOR+BetaDataset +400 +400 +400 +400 +400 +400 +400 +400300 +100 +1s1s300 +300 +300 +300 +300 +300 +300 +300 +(wd +0200 +200 +200 +200 +200 +200 +200 +200 +R +100 +100 +100 +100 +100 +100 +100 +100400 +Conventional ITRTRCA+BetaDataset +400 +400 +400 +400 +400 +400 +400 +400300 +100 +1S +1s300 +300 +300 +300 +300 +300 +300 +300 +(wd +0200 +200 +200 +200 +200 +200 +200 +200 +R +100 +100 +100 +100 +100 +100 +100 +10016 +Proposed ITR +Differences +Conventional ITR +Differences +Window +Length T (sec) +p value +t-test +CV +F(69, 69) +p value +t-test +CV +F(69, 69) +0.2 +2.87e-11 +8.1 +0.83 +1.49e-17 +24.41 +0.865 +0.3 +1.45e-9 +6.61 +0.59 +2.84e-11 +17 +0.614 +0.4 +3.68e-6 +3.08 +0.577 +3.48e-07 +8.61 +0.58 +0.5 +1.49e-3 +1.24 +0.602 +5.6e-06 +6.14 +0.57 +0.6 +6.21e-5 +1.31 +0.552 +1.98e-05 +4.23 +0.57 +0.7 +9.2e-05 +1.13 +0.566 +1.92e-05 +3.59 +0.568 +0.8 +3.4e-3 +0.52 +0.579 +7.38e-4 +2.13 +0.559 +0.9 +2.51e-3 +0.55 +0.588 +1.43e-3 +1.903 +0.573 +1.0 +8.04e-3 +0.32 +0.572 +7.36e-4 +1.905 +0.567 +Table 1: Critical and p values for the right-tailed paired t-tests (95% confidence) for +70 subjects of Beta data set using both definitions of ITR. The table also shows the +F-statistic for p ≥ 0.041 for the same confidence. CV: Critical Value. +TIs using Beta dataset. There are two important observations about the performance +of TIs as a function of observation window (0.2, 0.3, . . . , 1 seconds). First, customizing +the input distribution can tremendously help with the experienced ITR as compared +to conventional definition. Particularly, at shorter observation windows, the difference +is quite significant for both TIs. Second observation is that even though the two TIs +seem to differ in performance, more so with shorter observation windows, using the +conventional definition, their ITR performance do not show significant difference using +the proposed ITR definition with input distribution optimized across all observation +windows. To show that statistically, we have conducted right-tailed paired t-test with +the alternative hypothesis being the mean of TRCA performance is greater than that +of the SSCOR performance. With 95% confidence interval, we have presented in Table +1 the corresponding critical as well as p values. As can be seen, lower critical value +and higher p values for the proposed ITR difference indicates the minor variation in the +performance of TRCA and SSCOR algorithms. In addition, we have also conducted +f-test to measure the variational difference of both algorithm’s performances. As can +be seen in Table 1, our f-statistic (F(69, 69)) demonstrates (for all T and p ≥ 0.041) +that the variational differences are not significant. +4. Discussion +In this work, we have investigated the conventional definition of ITR, highlighted the +deficiencies and proposed to use an algorithmic approach for more accurate computation +for information transfer. One conclusion we derive from this study is that the effect of the +input distribution changes on the experienced ITR is quite minor when the performance + +17 +is averaged over all subjects. This implies that in an online speller task, for instance, we +typically would not have the option to change the ID and yet the TI algorithm would +be operating close to the capacity–maximum ITR. However, the asymmetry (proportion +of false positives and negatives) in the channel significantly changes the ITR in a +typical BCI-based communication setting. Therefore, target identification algorithms, +while in the design phase, should not solely optimize the accuracy performance. Note +that “accuracy” and ITR have long been used interchangeably in the SSVEP-based +BCI literature and it is the only performance indicator of a TI that appears in the +conventional ITR definition (see Equation (1)). +On the other hand, our results lead us to consider the possibility of alternative +(better) channel transition (confusion) matrices. In other words, given an average target +accuracy level (1 − ǫM) and a fixed ID (Px(x)), one can optimize the channel transition +probability matrix such that the channel mutual information is maximized i.e. to achieve +the maximum ITR subject to conditional entropy bound of the channel statistics, given +by the Fano’s inequality. With this study, we can show the potential of this approach +for the binary classification scenario as follows. +Let us consider binary character transmission i.e. M = 2 case. First, from (8), +we observe the symmetry C2(p1,2, p2,1) = C2(p2,1, p1,2) which is minimized for a given +average accuracy target A < 1 (or a classification error ǫM > 0) when p1,2 = p2,1 = +1 − A. +On the other hand, capacity is maximized when (p1,2, p2,1) ≅ (2(1 − A), 0) +or (p2,1, p1,2) ≅ (0, 2(1 − A)) with equality if the input distribution is uniform i.e., +PX(x) = +1 +M . In other words, the ITR is maximized when either Precision or Recall is +unity which can be obtained by playing with the parameters of the TI algorithm i.e., +replacing the separating hyperplanes of the classifiers. For a given classification error ǫM, +an iterative algorithm is provided in Algorithm 1 to optimize the input and channel +statistics at the same time. Both optimizations are subject to postulates of probability +as well as Fano’s inequality. Capacity results for accuracy targets 0.99, 9.95, 0.9, 0.85, 0.8 +and 0.75 are provided in Table 1. If a TI is able to achieve 0.99 accuracy with T = 0.2 +seconds, our capacity results imply that the upper bound on ITR can be calculated as +0.9277 × 60 +0.2 = 278.3bpm. We finally remark that the mapping between the achievable +accuracy and the observation window T is a function of the structure of the data manifold +as well as the dimensional reduction techniques used before the application of TIs. +Avg. Accuracy Target +0.99 +0.95 +0.9 +0.85 +0.8 +0.75 +Capacity +0.9277 +0.746 +0.5787 +0.4412 +0.3219 +0.2155 +Conditional Entropy +0.0703 +0.2271 +0.3367 +0.3908 +0.4001 +0.3655 +Fano’s Bound +0.0932 +0.3627 +0.6343 +0.8644 +1.0613 +1.2276 +Table 2: Optimization of the channel statistics (M = 2) to maximize the mutual +information (capacity) given the target average accuracy (classification error) rate. + +18 +Algorithm 1 Joint calculation of P∗ = {p∗ +i,j} and P ∗ +X(x) in an iterative manner. +Require: � +j pi,j = 1, 0 ≤ pi,j ≤ 1, 0 ≤ PX(xi) ≤ 1. +Ensure: ǫM = 1 − � +xi PX(xi)pi,i. +1: PX(xi) ⇐ +1 +M for all i ∈ {1, 2, . . . , M}. +2: pi,j ⇐ RAND(0,1) +3: pi,j ⇐ pi,j/ � +j pi,j {⊲ Normalization} +4: while � +xi PX(xi)pi,i ≤ 1 − ǫM and H(Y |X) ≤ h(ǫM) + ǫM log2(M − 1) do +5: +ˆP = arg maxP I(X, Y ) {⊲ Fix PX(x) and optimize P} +6: +[CDMC, PX(x)] = BA(ˆP) {⊲ Fix P, optimize PX(x)} +7: end while +8: P∗ ⇐ ˆP, P ∗ +X(x) ⇐ PX(x) +On the other hand, for M > 2, such optimizations can be carried out based on +the separability of the input data using a multi-class classification algorithm. However, +as the number of classes increase, the possibility of hitting a local minimum grows, +making the outcome unstable and vary at each run of the proposed algorithm (line +3, Algorithm 1). However, multi-class classification is typically implemented as an +ensemble of multiple binary (weak and unstable) classifications (such as one-vs-one or +one-vs-all) [41], thus making these results directly applicable. +Note that channel parameter optimizations are indeed not typical in telecommunica- +tion systems (regarding Shannon’s channel coding theorem [9]) where the channel model +is usually a given quantity defined by the transmission medium and capacity-achieving +IDs are found via solving an optimization problem to characterize the maximum informa- +tion transfer rate over this medium. Therefore, the practice of this paper will hopefully +help us understand what is achievable using TI algorithms or classification techniques +subject to a target accuracy threshold. Such an upper bound on the performance will +also help compare the performances of different TI algorithms under equal settings and +highlight how much improvement they can bring into the ITR enhancements for future +BCI systems. +Finally we remark that most offline BCI signal detections carry out subject- +dependent optimizations such as training the TI based on the individual template signals +for each observation window. We have also recognized if such optimizations are to be +extended to subject-level stimuli design and necessary input formations, the experienced +ITR can be tremendously boosted, closing the major performance gaps of the published +TI schemes in the literature. +This result merely shows the importance of the joint +design in SSVEP-based BCIs when the system is geared towards tight symbiosis and +hence stronger individual calibrations might be needed for each user of the system. + +19 +5. Conclusion +In this study, we have considered a more realistic ITR definition without making +extra assumptions about the channel transition and input statistics and numerically +supported it using two well known datasets along with two prominent TI algorithms in an +SSVEP-based BCI context. This numerical definition characterizes the communication +rate between the computer and the brain as if sending symbols over a discrete, +asymmetric, non-stationary memoryless channel. This definition also provided a set of +intriguing ways of comparing all previously developed TIs and highlight where they suffer +information-theoretically in achieving better or worse ITRs. Our findings also imply that +the proposed ITR definition is particularly important for subject-level customizations, +enabling a more realistic measurement. +Moreover, the proposed definition is shown +to help with the design of the channel transitions (binary classification use case) and +potentially lead to future work for finding upper bounds on TI performances with large +alphabet sizes in SSVEP-based BCI settings. +Acknowledgments +This research was supported by Intelligence Advanced Research Projects Activity +(IARPA) and Scientific and Technological Research Council of Turkey (TUBITAK) +under grant number 1059B192100830. +References +[1] J. R. Wolpaw, N. Birbaumer, D. J. McFarland, G. Pfurtscheller, and T. M. Vaughan, “Brain- +computer interfaces for communication and control,” Electroenceph. Clin. Neurophysiol., vol. +113, no. 6, pp. 767–791, June 2002. +[2] Geuze, J., Farquhar, J. and Desain, P. Towards a Communication Brain Computer Interface Based +on Semantic Relations. 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Herrera, “An overview of ensemble +methods for binary classifiers in multi-class problems: Experimental study on one-vs-one and +one-vs-all schemes,” Pattern Recog., vol. 44, no. 8, pp. 1761–1776, 2011. + diff --git a/wdAyT4oBgHgl3EQfnfgZ/content/tmp_files/load_file.txt b/wdAyT4oBgHgl3EQfnfgZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19efbc7522236c2b7f2bbafbd16c68d61dcd0972 --- /dev/null +++ b/wdAyT4oBgHgl3EQfnfgZ/content/tmp_files/load_file.txt @@ -0,0 +1,1074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf,len=1073 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='00488v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='HC] 1 Jan 2023 Information Transfer Rate in BCIs: Towards Tightly Integrated Symbiosis Suayb S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Arslan and Pawan Sinha Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA, 02139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' E-mail: sarslan@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='edu, psinha@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='edu Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='0 – January 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The information transmission rate (ITR), or effective bit rate, is a popular and widely used information measurement metric, particularly popularized for SSVEP- based Brain-Computer (BCI) interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' By combining speed and accuracy into a single-valued parameter, this metric aids in the evaluation and comparison of various target identification algorithms across different BCI communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In order to calculate ITR, it is customary to assume a uniform input distribution and an oversimplified channel model that is memoryless, stationary, and symmetrical in nature with discrete alphabet sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To accurately depict performance and inspire an end-to-end design for futuristic BCI designs, a more thorough examination and definition of ITR is therefore required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We model the symbiotic communication medium, hosted by the retinogeniculate visual pathway, as a discrete memoryless channel and use the modified capacity expressions to redefine the ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We use graph theory to characterize the relationship between the asymmetry of the transition statistics and the ITR gain with the new definition, leading to potential bounds on data rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On two well-known SSVEP datasets, we compared two cutting-edge target identification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Results indicate that the induced DM channel asymmetry has a greater impact on the actual perceived ITR than the change in input distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Moreover, it is demonstrated that the ITR gain under the new definition is inversely correlated with the asymmetry in the channel transition statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Individual input customizations are further shown to yield perceived ITR performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Moreover, an algorithm is proposed to find the capacity of binary classification and further discussions are given to extend such results to ensemble techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We anticipate that the results of our study will contribute to the characterization of the highly dynamic BCI channel capacities, performance thresholds, and improved BCI stimulus designs for a tighter symbiosis between the human brain and computer systems while enhancing the efficiency of the underlying communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Introduction The primary goal of brain-computer interfaces (BCIs) is to provide new channel formations for communication and control between the human brain and its surrounding objects with computational capabilities [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The majority of BCI research efforts are focused on developing effective stimuli, novel protocol developments and target identification algorithms to boost information transfer and eventually help novel communication paradigms to emerge such as found in recent semantic communications [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Different types of BCIs are employed in various applications nowadays, ranging from clinical deployments [3] to entertainment world, including but not limited to gaming [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Such commonplace applications has become quite encouraging and pushed current state-of-the-art BCI research forward, enabling more coupling and enhanced communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' A generic BCI system typically consists of three main parts: (1) the stimulation generation, (2) the communication channel hosted by the participating subject, and (3) the target identification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To be able to deliver fast communication rates and form a close symbiosis between the human brain and a computing device, these components must work in concert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For a truly symbiotic system design in which the stimulus generation and TIs co-adapt to each other [6], better understanding of the performance evaluations, controlling measures and the underlying statistical channel formed is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Once established, new research directions can be explored such as developing useful performance bounds and manage what is needed to enhance end-user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Assessments of BCI systems are typically performed at two levels of evaluation, namely user-level and system-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' User performance is measured by the degree of congruence between user intent and the signal feature(s) the BCI uses to identify the intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The user-level quality control heavily depends on the visual setup, the selection and presentation of stimuli, and how the stimulation is carried out (usually forming a protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, system-level performance evaluations are often done in terms of target identification speed and classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Fair comparisons are difficult since these two criteria, when articulated independently, are affected by the program’s capability as well as how well the system combines the user’s control with that application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Information transfer rate (ITR), cited primarily in [7] and [8], is one of a number of measuring tools developed in response to the demand for a single theoretical measure that combines accuracy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Information transfer is quantified according to information theory measures such as entropy, pioneered by Shannon’s seminal work back in 1948 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' ITR is a measure that can be used to measure the magnitude of coupling in a communication setting as well as the levels of attention and counciousness which can be leveraged heavily in passive-BCI settings [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' ITR is used in BCIs both using P300 paradigm [11] and in particular steady-state visual evoked potentials (SSVEPs) due to proven high communication rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For instance, recent progression towards task- 3 related component analysis is shown to achieve rates up to 325 bits/min ITR in a cue-guided 40-character spelling task [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Such performance has been possible due to carefully designed stimulus generation and Target Identification (TI) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Stimuli generation involves embedding information into frequencies and phases of the signals (modulation) in an SSVEP paradigm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' TI requires compensating for the noise and degradations imposed on the information passing through a channel induced by the BCI and the physical medium of the retinogeniculate visual pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' It turns out that many information processing strategies are used throughout the visual system and the basic approach is to lump them into a coarse description of an information link [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The main objective of all the past work has been to maximize the information transfer rate that can be transmitted over this induced channel [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Although the BCI channel properties are determined by the choice of stimulation and the target identification methods, accurate measurement of the ITR performance is also of critical importance for the assessment of user experience and developing successful stimuli design techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Preconditions and deficiencies of the conventional ITR definition are brought up in a number of past studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For instance, [16] summarizes the problems with the conventional definition and particularly emphasizes the significance of the channel parameter estimations (accuracy or false alarm rates) in online synchronous BCIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The same study proposed a task-oriented online BCI test in the hope to help with the real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Moreover, recent works such as [17] proposed to use an alternative closed-form formulation derived in [18] for the ITR computation via making approximations such as removing negativity constraint on the input distribution using only a fairly limited dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In fact, this closed form’s usefulness is restricted to square and non-singular channel transition characterizations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Moreover, no further analysis or intuition is presented in both studies in terms of the channel transition characteristics, the stimuli design, and means and strategies for achieving the capacity of the underlying BCI channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, one of the objectives of this study is to outline the basic principles of conventional ITR definition and in order to re-express its deficiencies, highlight the challenges of channel characterization problems between the human brain and the computer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We note that performance characterization of any technique for an asymmetric and non-stationary channel requires a careful computation procedure to accurately determine the practical ITR the subjects experience as well as directions for tighter symbiosis within the context of joint system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We will demonstrate that this study may also help design better input for BCIs to maximize the information flow in the induced communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In Section II, we introduce the generalized DM channel model and rephrased the conventional ITR definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We report the deficiencies as well as workarounds through integrating the algorithmic capacity calculations into the ITR definition subject to information theoretic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In Section III, we present a few results to distinguish different ITR definitions, and explore the asymmetry in channel transition statistics and establish the ties with the conditional entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We discuss some of the important implications of our results in 4 Section IV before concluding the paper with future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Channel Model and Conventional ITR Definition Let us consider a discrete BCI system where one of the M symbols from the set X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' , xM} is to be communicated at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' It is quite typical to express BCI system performance in terms of the information transfer rate (ITR) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This is expressed in bits per trial observation window T [22], ITR = log2(M) + P(T) log2(P(T)) + (1 − P(T)) log2 �1 − P(T) M − 1 � (1) where M is the number of targets and P(T) is the aggregate average accuracy of the target identification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that the trial time dependency of the accuracy is crucial in this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Equality (1) is derived from the popular mutual information measure defined for two random variables X and Y as I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Y ) = H(Y ) − H(Y |X) (2) = � y∈Y PY (y) log2 � 1 PY (y) � − � x PX(x)H(Y |X = x) (3) where H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') is the entropy, X ∈ X represents the discrete source taking on one of the M target classes and Y ∈ Y (typically Y = X ) is the predicted output at the other end of the BCI system with distribution PY (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Capacity is defined to be the supremum of the mutual information over all input (probability) distributions PX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In the case of perfect communication (P(T) → 1) the ITR will simply be log2(M), the number of bits used to represent all targets assuming these targets have equal probability of occurring i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', 1/M (uniform input distribution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', PX(x) = 1 M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Equality (1) is based on the capacity of a symmetric Discrete Memoryless Channel (DMC) that errs with an equal probability 1−p M−1 in favor of all other M − 1 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In addition, T is expressed in terms of seconds and the ITR is usually scaled with 60/T and expressed in terms of bits/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, we define the channel transition matrix of the induced DMC and express it as follows, P = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 p1,1 p1,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' p1,M p2,1 p2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' p2,M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' pM,1 pM,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' pM,M \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb with � j pi,j = 1 for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We use the short-cut notation PY |X(yj|xi) = pi,j as each entry of P to refer to the channel transition/conditional probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The generic channel models are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that symmetry assumption in the channel transition statistics i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', the distribution of the probability of erring over 5 Figure 1: Typical discrete BCI Channel Models for symbol/character communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' A discrete set of characters are communicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In case, the communication reliability is below a threshold, the communicated character can be assumed to be erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In the figure, e is used to represent erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' all non-target values make the uniform input distribution achieve the DMC capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Hence, max PX(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') I(X, Y ) = � log2(M) − � j pi,j log2 � 1 pi,j �� = ITR (4) Although we have assumed the possible number of outcomes to be M i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', |Y| = M, the channel outputs can be expanded to include “erasures” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', making no decision on the final output, leading to the channel transition matrix P to be of size M × (M + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This extension is also illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Deficiencies and Workarounds Some of the major deficiencies of the conventional ITR definition have already been articulated in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' One of these deficiencies is the assumption that the source has uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, this assumption is not necessarily true and optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For instance, in a speller application, the characters of the English alphabet are not necessarily used equally frequently in everyday language and hence in an online experiment, some of the characters would naturally be intended to spell more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In addition, transition probabilities of the underlying channel are not necessarily symmetric and stationary [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In an SSVEP-based BCI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' the distinguishability of the two targets preprocessing Target preprocessing Target Identification Identification Stimuli Stimuli XO Source Set Target Set Source Set Target Set P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1 P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1 x1 1 x1 X1 P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 2 ×2 X2 X2 P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='M P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='M e PM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='M PM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='M XM XM XM XM Communication Channel Communication Channel with defintive decisions with erasures6 depends on the frequency and phase selections or even spatial distance in which these targets are presented to the subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Therefore, the assumption pi,j = 1−p M−1 for all i, j satisfying i ̸= j (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', equal probability transitions to all non-target classes) is not necessarily totally accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' One of the challenges of the capacity computation for such a highly dynamic channel is that the transition matrix is a function of both the stimulus design (the encoding of information into visual pathway) and also the target identification methods, unlike in classical digital communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In addition, as the observation window (T) widens, accuracy is reported to increase since the identification is performed via observation of a wider signal window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, the observation window being larger will change the stochastic nature of the channel (and its parameters) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', the channel transition matrix indeed is a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The extent of the introduced channel memory is yet another challenge to tackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Thus, instead of considering the entire timeline, we consider specific time points such that the dependency of the channel transition matrix at those points is sufficiently eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Some of the previous works proposed closed form expressions [21] under certain assumptions about Px(x) and non-singularity assumption for P, which is highly unlikely for larger window lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In this work, we do not make any assumptions about the statistical nature of the channel and treat it as a DMC, calculate the capacity numerically and report our final ITR results along with the conventional formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Capacity for Asymmetric DMC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Exemplary Case: “Binary Classification”: Let us suppose X = Y = {x1, x2} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', the input is one of the two possible symbols with PX(x1) = px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This would correspond to differentiation of two different classes such as face/non-face or familiar/non- familiar (target/non-target paradigm) dichotomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Thus, we can express the mutual information I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Y ) = H(Y ) − H(Y |X) (5) = h(px(1 − p1,2) + (1 − px)p2,1) − pxh(p1,2) − (1 − px)h(p2,1) (6) where h(x) = −x log(x) − (1 − x) log(1 − x) is the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Setting the derivative with respect to px to zero, we obtain 1 px(1 − p1,2 − p2,1) + p2,1 − 1 = 2 h(p1,2)−h(p2,1) 1−p1,2−p2,1 (7) Subsequently, px that satisfies this equality can be plugged into (6) and following some algebraic operations, the final capacity can be expressed as C2(p1,2, p2,1) = log2 � 1 + 2 h(p1,2)−h(p2,1) 1−p1,2−p2,1 � − (1 − p2,1)h(p1,2) + p1,2h(p2,1) 1 − p1,2 − p2,1 (8) Finally, the ITR in bits/min can be found by 60 T C2(p1,2, p2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' General Case “M symbols”: Having more than two classes complicates the above computations (by requiring the computation of partial derivatives and solving 7 transcendental equations) which precludes closed form results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' There have been successful attempts in the past that iteratively compute the capacity for discrete stationary channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For memoryless channels (independent choice of symbols) with finite input and output alphabets X and Y respectively, the capacity can be computed by the Blahut- Arimoto (BA) algorithm [19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, in a typical speller task, due to the formation of language and words, the source will inherently have memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The Blahut-Arimoto algorithm was also extended to channels with memory and finite input alphabets and state spaces [24] such as Hidden Markov Models (HMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, modeling language with an HMM is quite challenging [26] and can result in inordinate computation time and/or an approximation for the capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The BA algorithm is an iterative procedure which assumes an arbitrary input probability distribution function P 0 X(x) in the beginning and optimizes it over multiple iterations [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Let us express the non-normalized input distribution for xi ∈ X at the (k − 1)-th step (k ≥ 1) as Dk−1(xi) = P k−1 X (xi) exp � D � pi,j||P k−1 Y (yj) �� (9) = P k−1 X (xi) exp � N � i=0 pi,j log � pi,j P k−1 Y (yj) �� (10) = P k−1 X (xi) N � i=0 � pi,j P k−1 Y (yj) �pi,j (11) where D[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='] is the relative entropy (also known as Kullback-Leibler or KL divergence) and P k−1 Y (yj) = � i P k−1 X (xi)pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Then, the input distribution is normalized to be P k X(xi) = Dk−1(xi) � j Dk−1(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' (12) Finally, iterations cease if max xi D � pi,j||P k Y (yj) � − � i P k X(xi)D � pi,j||P k Y (yj) � < γth (13) holds for a given error threshold γth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', ǫth = 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For the rest of our discussions, we refer to Blahut-Arimoto algorithm as follows [C, P ∗ X(x)] = BA(P) (14) where BA(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') refers to the algorithm itself, C is the capacity and P ∗ X(x) is the optimal input distribution that maximizes the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We note that this two-step process can be shown to converge through iterations of two different convex optimization problems and the convergence rate can be shown to improve in a later study [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Fano’s Inequality Fano’s inequality provides a lower bound for the probability of target identification error ǫM (= 1 − � i PX(xi)pi,i) due to information degradation via the channel induced 8 by the BCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The channel transition (conditional) probabilities PY |X(yj|xi), emprically estimated by the confusion matrices, appear in the bound as follows, ǫM ≥ H(Y |X) − h(ǫM) log2(M − 1) ≥ H(Y ) − I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Y ) − 1 log2(M) (15) where h(ǫM) = −ǫM log(ǫM) − (1 − ǫM) log(1 − ǫM) is the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Accordingly, for a fixed ǫM, we can upper bound the conditional entropy as H(Y |X) ≤ ǫM log2 �M − 1 ǫM � + (1 − ǫM) log2 � 1 1 − ǫM � (16) = h(ǫM) + ǫM log2(M − 1) (17) Since in typical telecommunication applications the channel transition probabilities are given as part of the model, the symbol detection error is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However as can be seen, ǫM appears in both sides of the inequality (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that the bound in (15) does not apply to M = 2 case (zero denominator) and countably infinite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However later studies extended this bound to such corner cases [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, the bound on the conditional entropy given in (16) becomes h(ǫM) when M = 2 and can be shown to be looser for larger ǫM (see our experimental results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Target Identification It is conventional to divide TI methods into two broad overarching categories, namely supervised and unsupervised (training-free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Unsupervised techniques are particularly attractive since their use does not involve user-specific-calibration phase (long training cycles) and provides more versatility in everyday practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, supervised methods are shown to outperform the unsupervised early strategies such as Cannonical Correlation Analysis (CCA) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' One of the most effective frequency recognition performance has been demonstrated by a number of spatial filtering techniques to isolate task-specific source activities from EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The task-related component analysis (TRCA) is one of prominent techniques proposed in literature which hypothesizes that there are distinct cortical sources in the brain which generates potentials upon the presentation of flickering stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This idea originally applied to near-infrared spectroscopy (NIRS) [31], [32] and later proven useful for multivariate EEG data [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Assuming s(t) ∈ R to be the task-related, n(t) to be the task-unrelated (noise and some other background brain activity) components, the multivariate EEG signal x(t) ∈ RNc is formed as a result of a linear generative model as follows, xj(t) = ajs(t) + bjn(t) for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' , Nc (18) where Nc is the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Spatial filtering is about extracting the task-related component s(t) from a linear combinations of multiple channel output signals x(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', ˜s(t) = Nc � j wjxj(t) = Nc � j wjajs(t) + wjbjn(t) (19) 9 Main idea behind TRCA is to optimize weight coefficients (wj) so as to maximize inter-trial covariance (reproducibility) of time-locked biomedical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' If we denote the h-th trial of y(t) as y(h)(t), then the sum of covariances of all possible combinations of trials can be expressed as Nt � h1,h2=1 h1̸=h2 Cov � y(h1)(t), y(h2)(t) � = wTSw (20) where Nt is the total number of trials, Cov(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') is the covariance operator, w = (wj)1≤j≤Nc and S = (Sj1,j2)1≤j1,j2≤Nc is given by Sj1,j2 = Nt � h1,h2=1 h1̸=h2 Cov � x(h1) j1 (t), x(h2) j2 (t) � (21) For a finite solution, TRCA maximizes wTSw subject to variance constraint, Var(y(t)) = wTQw ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The solution is given by the following unconstrained optimization problem w∗ = arg max w wTSw wTQw (22) which can be recognized as a generalized eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' By creating a spatial mapping that projects the multivariate EEG data onto a standard SSVEP representation space, the sum of squared correlations (SSCOR) framework [38] seeks to identify a session independent representation of SSVEP response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We express the optimization problem as w∗ X, (w∗ i ) = arg max wX ,wi Nt � i=1 � wT XCov � x(X)(t), x(i)(t) � wi �2 (23) where x(X)(t) = 1 Nt Nt � i=1 x(i)(t) (24) is the template signal calculated for each target frequency separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Again for the sake of obtaining a finite solution and put the optimization problem into a generalized eigenvalue framework, we use the set of constraints for ∀i, wT i Cov � x(i)(t), x(i)(t) � wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that there could be other spatial filtering techniques that can generate the filter weights (w) as an application of generalized Eigenvalue problem [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, the arguments for the detection logic is common to all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' After determining the weights, for a given single-trial test sample X, the classification decision is made in favor of the frequency fn ∈ {f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' , fNf} based on the Pearson’s correlation coefficient (ρ) as τ = arg max n ρ � XTwn, X T n wn � , (25) where Nf is the total number and n is the index of the stimulation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Finally, if we concatenate all weights to construct W = [w1 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' wNf] and replace it with wn in Equation (25), we obtain an ensamble TI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Same idea can be applied to TRCA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Experimental Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Datasets We have used two well known datasets in the literature, namely Benchmark [34] and Beta datasets [35] to analyze/compare two known target identification algorithms, namely TRCA and SSCOR, based on the conventional as well as proposed ITR definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Benchmark dataset was collected from 35 subjects with normal/corrected-to-normal vision on a 40-character (26 English alphabet letters, 10 digits, and 4 other symbols) speller task using a 64-channel EEG recorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Each subject was shown target characters in distinct trials, where characters flicker at frequencies 8-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 Hz with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 Hz increments and phases 0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5π with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5π increments, where both increments are proportional to stimulus index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Each trial began with a visual cue that was shown on the screen for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 seconds to direct the subject’s gaze to the intended target, followed by 5 seconds of stimulation and a final trailing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5-second offset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The observation window includes gaze length as well as the signal length after the stimuli onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' At the preprocessing stage, the recorded signals were downsampled to 250Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We have assumed 130 ms visual pathway delay for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In this study, the following set of 9 channels (OZ, O1, O2, PZ, POZ, PO3 PO4, PO5 and PO6) are considered since they are reported to be most reflective of neural activity due to tasks performed in the experiment [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In Beta dataset on the other hand, 70 subjects participated in four blocks of a cued-spelling task on a QWERTY virtual keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The stimulation duration is 2 seconds for the first 15 subjects whereas the rest of the other subjects are stimulated for 3 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Sixty-four channels of EEG data were collected by SynAmps2 (Neuroscan Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') data acquisition/amplifier system at a sampling rate of 1000Hz, which were later downsampled to 250Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The rest of the settings are the same (channels used, electrode locations, gaze-shift times, visual pathway latency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Since the trials were carried out outside a laboratory setting, the Signal-to-Noise Ratio (SNR) of EEG in Beta dataset is measured to be lower than that of the Benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Filter Banks In our work, we have leveraged filter banks to decompose the EEG signals into five overlapping sub-bands to make use of the independent information found in the harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' A target detection technique is used independently for each of the sub-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The cut-off frequency range for the sub-bands based on the EEG data bandwidth is set between b × 8 Hz and 90 Hz, where b ∈ {1, 2, 3, 4, 5} [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As the Target Identification (TI) algorithm, we have employed two competing approaches, namely the TRCA [12] and SSCOR [38] due to their high performance relative to other methods with reasonable complexity (in terms of the number of parameters to optimize).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As mentioned before, the design methodology behind TRCA was the hypothesis that the single-trial EEG data can be reconstructed as a linear sum of multiple time series from different cortical sources [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Therefore, TRCA is used as a technique for 11 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='12 Figure 2: Capacity achieving distributions for different signal lengths (sec) as well as their entropies for TRCA method using Benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='12 Figure 3: Capacity achieving distributions for different signal lengths (sec) as well as their entropies for SSCOR method using Benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' highlighting the task-related elements embedded in the EEG signals through enhancing repeatability among different time-locked activities across trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The covariance between different trials is maximized to ensure such repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, SSCOR transforms SSVEP signals to a common representation space through the optimization of the individual SSVEP templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Similar to TRCA, SSCOR also achieves space transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In both of these methods, we learn a spatial filter for each frequency (character) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In our work, we have also employed an ensemble technique for both methods where all spatial filters belonging to different frequencies are concatenated for a performance boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To determine a final score (a single correlation coefficient) for classification, the correlation coefficients of the sub-band components are combined using the weighted sum of squares approach [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9 1 100 120 140 160 180 200 220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9 1 80 90 100 110 120 130 140 150 Figure 4: Average ITR as a function of signal Length (sec) for different TIs for Benchmark dataset (left) and Beta dataset (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The plots clearly show the difference between the conventional definition as well as the proposed definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Since input distribution depends on the window length T, we have also expressed which time window the ID is optimal for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Evaluation Process For each subject, we evaluated performance in a leave-one-trial-out fashion as in the past literature i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', the TI algorithms are trained over u − 1 trials and tested on the remaining trial for all u different combinations for u = 6 in Benchmark and u = 4 in Beta datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' When we present average subject performance, test performances are combined (classification outcomes) in a single confusion matrix, which is normalized to obtain the estimate of the channel transition probability matrix ˜P with M = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In some of the past works c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' [12], [38], using conventional definition, ITR is calculated for each subject and the average of these values (with the standard error) is reported despite the nonlinear dependency of the definition on the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This type of averaging typically leads to larger ITR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In our work, we report all ITR results after averaging the accuracy values (similarly false positive and negatives etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') over all subjects before calculating and reporting the final ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This way, we also ensure that the aggregate human performance is translated into an ITR metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that with the proposed definition, it would be unrealistic to expect the system to optimize the input distributions for each subject separately in order to attain higher ITRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To address this point however, we have also demonstrated via violin plots by calculating the individual ITRs using both definitions for all observation windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In that, we use individual data to estimate the channel transition statistics for each subject and observation window, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We have leveraged BA(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=') algorithm to compute the capacity maximizing input distributions and the corresponding capacity value for signal lengths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', 1 seconds, whereby the conditional entropy of ˜P complies with the Fano’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Numerical Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Performance averaged over subjects: In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2 and 3, using Benchmark dataset, we demonstrate capacity achieving distributions (with channel transition statistics averaged over all subjects) as bar plots for each signal length as well as corresponding input entropy H(X), output entropy H(Y ) and the conditional entropy H(Y |X) that characterizes the channel transition statistics for both TI methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As can be seen, with larger observation window, H(X) and H(Y ) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, the channel transition probabilities begin to show more structure (such as reduced symmetry and balanced distribution of probabilities) and hence less (conditional) entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As a result, the capacity of the induced channel H(Y ) − H(Y |X) increases with growing signal length in both techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As can be seen, particularly at low signal lengths, SSCOR fails to enhance the channel capacity due to poorer reduction in H(Y |X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However as the observation window increases, the reduction in H(Y |X) for both TIs become on par, almost equating the capacity gain at around an observation window of one seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We observed similar trends using the Beta dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We have also provided the ITR results using the BA algorithms and confusion matrices in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 4 for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Since the ITR is maximized at the signal length of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 secs for TRCA and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='65 secs for SSCOR for Benchmark dataset (and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='45 secs for TRCA and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='55 secs for SSCOR for Beta dataset), we use the optimal input distribution (ID) of the signal lengths 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='45) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='55) secs, respectively, for all signal lengths to obtain the ITR for the TI algorithms (Asym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='+Optimal ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For comparison purposes, we have also included the conventional ITR definition in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 4, which neither takes the input distribution nor the asymmetry the channel introduces into account and consequently underestimates the maximum information transfer rate that can be achieved over the induced channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To investigate it statistically, we have carried out paired t-test and f-test (two-tailed) to determine whether the proposed ITR definition is different from the conventional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Using Benchmark dataset, both methods showed dramatic mean differences (p ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='97 × 10−8 for SSCOR and p ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='7 × 10−7 for TRCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, there were no significant variational differences between the different ITR definitions (F(17, 17) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='25, p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='05 for SSCOR and F(17, 17) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='47, p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='05 for TRCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This clearly demonstrates that although the trend is almost the same for both TI algorithms, the actual ITR experience is meaningfully different than the reported average ITR results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To be able to demonstrate the effect of asymmetry of the channel on the ITR, we have used the optimal ID while keeping the accuracy intact and divide the error probabilities equally over all other non-target classes for each class (character) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', pi,j = 1−p M−1 for all i, j satisfying i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We have named this scheme “Balanced transition probability matrix” and used it with the optimal ID for all signal lengths (Balanced+Optimal ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As can be seen, rather than the non-uniformity of the input distribution, the asymmetry in the probability transition characteristics has much more significant effect on the final ITR results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='08 Figure 5: On the left, we show asymmetry score ∆asm as a function of the ratio of change in the ITR with the new definition for all T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the right, we have also depicted the asymmetry score ∆asm as a function of the conditional entropy of the induced channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We have varied the transparency of the markings from dark to light as we change the observation time window from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='18 seconds to 1 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, to quantify the degree of asymmetry, we have used the largest singular value of the skewed Laplacian of the channel transition matrix, namely ∆asm = (Γ − ΓT)/2 [40], where Γ = Φ1/2(I − P)Φ−1/2, I is the identity matrix, Φ1/2 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 √φ1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 0 0 √φ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' √φM \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb , and [φ]1≤i≤M are the stationary probabilities of a Markovian process whose transitions are governed by the channel transition statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 5 (left), we depicted the degree of asymmetry ∆asm as a function of the ratio of change in the ITR (∆ITR) using the proposed definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' More specifically, ∆ITR ≜ (Asym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='+Optimal ID) − (Conventional) (Conventional) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' (26) As can be clearly seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 5 (left), asymmetry increases the ITR gain, and it turns out to satisfy a logarithmic relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The parameters of this estimate is explicitly given in the same figure using regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This suggests that although the asymmetry in the channel transition statistics helps with increasing the rate of communication, it also quickly saturates due to increased conditional entropy of the channel and reduced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that the conditional entropy is bounded above by the Fano’s inequality as expressed in (16) which forms an upper bound on ∆asm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' One of the interesting conclusions is that particularly at short window lengths, stimuli design that generates more asymmetric channel transition characteristics is likely to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='08 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='07180ms Benchmark Benchmark Beta Beta 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='06 Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='05 Asymmetry 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='04 TRCA- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='03 X SSCOR TRCA- 1000ms SSCOR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5 H(YIX)15 Figure 6: Proposed ITR v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' conventional ITR for subject-specific input customization using two TIs and the Beta Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Input customization can be shown to be effective even with longer observation windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' improve ∆ITR, higher improvement ratio with respect to the conventional definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, the relationship between ∆asm and H(Y |X) clearly suggests that as we have larger observation windows, to be able to maximize the mutual information, system prefers to have less asymmetry in the channel transition characteristics (to minimize the conditional entropy) and maximize the input entropy via uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 5 (right), we have illustrated ∆asm as a function of conditional entropy H(Y |X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As anticipated, there is a monotonic relationship (presented with a linear regression) in between and both the degree of asymmetry and conditional entropy increase as the observation window length grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' One of the interesting observations is that the dataset has more significant effect on the slope of the relationship compared to target identification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Different slopes can also be interpreted as an indicator of the difficulty of the datasets (TIs perform worse with Beta dataset compared to Benchmark) and reduced SNR while recording these EEG datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Performance customized to each subject: We have also investigated via violin plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 6 (considering all 70 subjects individually as data points) whether the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='proposed ITR computed per subject (finding the optimal input distribution for each ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='subject) differs significantly compared to conventional definition for the two different ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='Proposed ITRSSCOR+BetaDataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='400 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='04e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='572 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='36e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='567 Table 1: Critical and p values for the right-tailed paired t-tests (95% confidence) for 70 subjects of Beta data set using both definitions of ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' The table also shows the F-statistic for p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='041 for the same confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' CV: Critical Value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' TIs using Beta dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' There are two important observations about the performance of TIs as a function of observation window (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' , 1 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' First, customizing the input distribution can tremendously help with the experienced ITR as compared to conventional definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Particularly, at shorter observation windows, the difference is quite significant for both TIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Second observation is that even though the two TIs seem to differ in performance, more so with shorter observation windows, using the conventional definition, their ITR performance do not show significant difference using the proposed ITR definition with input distribution optimized across all observation windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' To show that statistically, we have conducted right-tailed paired t-test with the alternative hypothesis being the mean of TRCA performance is greater than that of the SSCOR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' With 95% confidence interval, we have presented in Table 1 the corresponding critical as well as p values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As can be seen, lower critical value and higher p values for the proposed ITR difference indicates the minor variation in the performance of TRCA and SSCOR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In addition, we have also conducted f-test to measure the variational difference of both algorithm’s performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' As can be seen in Table 1, our f-statistic (F(69, 69)) demonstrates (for all T and p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='041) that the variational differences are not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Discussion In this work, we have investigated the conventional definition of ITR, highlighted the deficiencies and proposed to use an algorithmic approach for more accurate computation for information transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' One conclusion we derive from this study is that the effect of the input distribution changes on the experienced ITR is quite minor when the performance 17 is averaged over all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This implies that in an online speller task, for instance, we typically would not have the option to change the ID and yet the TI algorithm would be operating close to the capacity–maximum ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, the asymmetry (proportion of false positives and negatives) in the channel significantly changes the ITR in a typical BCI-based communication setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Therefore, target identification algorithms, while in the design phase, should not solely optimize the accuracy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that “accuracy” and ITR have long been used interchangeably in the SSVEP-based BCI literature and it is the only performance indicator of a TI that appears in the conventional ITR definition (see Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, our results lead us to consider the possibility of alternative (better) channel transition (confusion) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In other words, given an average target accuracy level (1 − ǫM) and a fixed ID (Px(x)), one can optimize the channel transition probability matrix such that the channel mutual information is maximized i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' to achieve the maximum ITR subject to conditional entropy bound of the channel statistics, given by the Fano’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' With this study, we can show the potential of this approach for the binary classification scenario as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Let us consider binary character transmission i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' M = 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' First, from (8), we observe the symmetry C2(p1,2, p2,1) = C2(p2,1, p1,2) which is minimized for a given average accuracy target A < 1 (or a classification error ǫM > 0) when p1,2 = p2,1 = 1 − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' On the other hand, capacity is maximized when (p1,2, p2,1) ≅ (2(1 − A), 0) or (p2,1, p1,2) ≅ (0, 2(1 − A)) with equality if the input distribution is uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', PX(x) = 1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' In other words, the ITR is maximized when either Precision or Recall is unity which can be obtained by playing with the parameters of the TI algorithm i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=', replacing the separating hyperplanes of the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' For a given classification error ǫM, an iterative algorithm is provided in Algorithm 1 to optimize the input and channel statistics at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Both optimizations are subject to postulates of probability as well as Fano’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Capacity results for accuracy targets 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='99, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='75 are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' If a TI is able to achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='99 accuracy with T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 seconds, our capacity results imply that the upper bound on ITR can be calculated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9277 × 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2 = 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3bpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We finally remark that the mapping between the achievable accuracy and the observation window T is a function of the structure of the data manifold as well as the dimensional reduction techniques used before the application of TIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Accuracy Target 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='75 Capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='9277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='5787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2155 Conditional Entropy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='0703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='4001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3655 Fano’s Bound 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='0932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='3627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='6343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='8644 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='0613 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content='2276 Table 2: Optimization of the channel statistics (M = 2) to maximize the mutual information (capacity) given the target average accuracy (classification error) rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 18 Algorithm 1 Joint calculation of P∗ = {p∗ i,j} and P ∗ X(x) in an iterative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Require: � j pi,j = 1, 0 ≤ pi,j ≤ 1, 0 ≤ PX(xi) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Ensure: ǫM = 1 − � xi PX(xi)pi,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 1: PX(xi) ⇐ 1 M for all i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 2: pi,j ⇐ RAND(0,1) 3: pi,j ⇐ pi,j/ � j pi,j {⊲ Normalization} 4: while � xi PX(xi)pi,i ≤ 1 − ǫM and H(Y |X) ≤ h(ǫM) + ǫM log2(M − 1) do 5: ˆP = arg maxP I(X, Y ) {⊲ Fix PX(x) and optimize P} 6: [CDMC, PX(x)] = BA(ˆP) {⊲ Fix P, optimize PX(x)} 7: end while 8: P∗ ⇐ ˆP, P ∗ X(x) ⇐ PX(x) On the other hand, for M > 2, such optimizations can be carried out based on the separability of the input data using a multi-class classification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, as the number of classes increase, the possibility of hitting a local minimum grows, making the outcome unstable and vary at each run of the proposed algorithm (line 3, Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' However, multi-class classification is typically implemented as an ensemble of multiple binary (weak and unstable) classifications (such as one-vs-one or one-vs-all) [41], thus making these results directly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Note that channel parameter optimizations are indeed not typical in telecommunica- tion systems (regarding Shannon’s channel coding theorem [9]) where the channel model is usually a given quantity defined by the transmission medium and capacity-achieving IDs are found via solving an optimization problem to characterize the maximum informa- tion transfer rate over this medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Therefore, the practice of this paper will hopefully help us understand what is achievable using TI algorithms or classification techniques subject to a target accuracy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Such an upper bound on the performance will also help compare the performances of different TI algorithms under equal settings and highlight how much improvement they can bring into the ITR enhancements for future BCI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Finally we remark that most offline BCI signal detections carry out subject- dependent optimizations such as training the TI based on the individual template signals for each observation window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' We have also recognized if such optimizations are to be extended to subject-level stimuli design and necessary input formations, the experienced ITR can be tremendously boosted, closing the major performance gaps of the published TI schemes in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This result merely shows the importance of the joint design in SSVEP-based BCIs when the system is geared towards tight symbiosis and hence stronger individual calibrations might be needed for each user of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Conclusion In this study, we have considered a more realistic ITR definition without making extra assumptions about the channel transition and input statistics and numerically supported it using two well known datasets along with two prominent TI algorithms in an SSVEP-based BCI context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This numerical definition characterizes the communication rate between the computer and the brain as if sending symbols over a discrete, asymmetric, non-stationary memoryless channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' This definition also provided a set of intriguing ways of comparing all previously developed TIs and highlight where they suffer information-theoretically in achieving better or worse ITRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Our findings also imply that the proposed ITR definition is particularly important for subject-level customizations, enabling a more realistic measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Moreover, the proposed definition is shown to help with the design of the channel transitions (binary classification use case) and potentially lead to future work for finding upper bounds on TI performances with large alphabet sizes in SSVEP-based BCI settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Acknowledgments This research was supported by Intelligence Advanced Research Projects Activity (IARPA) and Scientific and Technological Research Council of Turkey (TUBITAK) under grant number 1059B192100830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Wolpaw, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfnfgZ/content/2301.00488v1.pdf'} +page_content=' Birbaumer, D.' metadata={'source': 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Jenkins,3 and Enrico Barausse4, 5 +1Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, +Campus UAB, 08193 Bellaterra (Barcelona), Spain +2Grup de Física Teòrica, Departament de Física, +Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain +3Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom +4SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste +5IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy +Besides the transient effect, the passage of a gravitational wave also causes a persistent displacement +in the relative position of an interferometer’s test masses through the nonlinear memory effect. This +effect is generated by the gravitational backreaction of the waves themselves, and encodes additional +information about the source. In this work, we explore the implications of using this information for +the parameter estimation of massive binary black holes with LISA. Based on a Fisher analysis, our +results show that the memory can help to reduce the degeneracy between the luminosity distance +and the inclination for binaries observed only for a short time (∼ few hours) before merger. To +assess how many such short signals will be detected, we utilized state-of-the-art predictions for the +population of massive black hole binaries and models for the gaps expected in the LISA data. We +forecast from tens to few hundreds of binaries with observable memory, but only ∼ O(0.1) events +in 4 years for which the memory helps to reduce the degeneracy between distance and inclination. +Based on this, we conclude that the new information from the non-linear memory, while promising +for testing general relativity in the strong field regime, has probably a limited impact on further +constraining the uncertainty on massive black hole binary parameters with LISA. +I. +INTRODUCTION +The direct detection of gravitational waves (GWs), pre- +dicted by Einstein in 1916 [1], is one of the greatest +accomplishments in modern physics, showing (at present) +a spectacular agreement with the theory of general relativ- +ity (GR) [2–4]. By now, almost a hundred GW signals +have been observed and interpreted as resulting from the +coalescence of compact binaries by LIGO/Virgo [5–7]. +As the sensitivity of current detectors improves and new +detectors become available, it will be possible to estim- +ate the binary parameters more accurately and to find +GWs from new types of sources. This will allow us to +not only better test GR in its strong-field regime, but +also to probe astrophysics, cosmology and fundamental +physics [8–10]. The future space-borne detector Laser +Interferometer Space Antenna (LISA) [11] will play a key +role in this quest, due to both its expected high signal-to- +noise ratio (SNR) measurements and the rich population +of sources expected to inhabit its frequency band (from +0.1 mHz to 0.1 Hz) [12–14]. +Progress may still be hindered by the fact that some +binary parameters — such as the luminosity distance dL +and inclination of the orbital plane with respect to the +line of sight ι — may be highly correlated in GW sig- +nals, limiting our ability to accurately estimate them [15– +17]. +This is especially important in the context of +standard sirens [18, 19], where the precision with which +∗ sgasparotto@ifae.es +one can estimate the present-day Hubble parameter H0 +depends primarily on how accurately one can meas- +ure dL [17, 20, 21]. +Indeed, this was the main con- +tribution to the large uncertainty on the first estim- +ate of H0 from GW170817 in Ref. [22]. +The distance- +inclination degeneracy can be simply understood by noting +that, at leading (Newtonian) order, an inspiralling binary +sources the two GW polarisations h+ ∝ (1 + cos2 ι)/dL +and h× ∝ cos ι/dL [23]; if the detector network is mostly +sensitive to one particular combination of h+ and h×, the +luminosity distance and inclination are therefore degener- +ate [15, 17] (c.f. App. B). The degree of this degeneracy +depends on the sky location of the binary and the specific +detector network [17], and can be greatly reduced by the +observation of the afterglow light curve of an electromag- +netic counterpart (which critically depends on ι) [24, 25]. +Interestingly, the degeneracy may also be mitigated by us- +ing subleading effects in the waveform. Examples include +the effect of spins misaligned with the orbital angular +momentum [26–28] (which lead to the precession of the +orbital plane), of higher multipole modes (HMs) [29–34] +(in particular, for unequal component masses), or using +binary Love relations [35] (for neutron star binaries).1 +Another subleading effect in the GW signal with the +potential to break the distance-inclination degeneracy is +1 Nevertheless, in the ∼ 100 GW signals observed to date there +is only limited evidence for higher multipole content (with no +evidence at all beyond ℓ = 3) [36] and only one measurement of +strong-field precession has been claimed [37] (though some doubt +has been cast on this claim due to data-quality issues [38, 39]). +arXiv:2301.13228v1 [gr-qc] 30 Jan 2023 + +2 +the nonlinear (Christodoulou) GW memory [40]. This is +a well-grounded prediction of GR which originates from a +change in the radiative multipole moments of the gravita- +tional field sourced by the flux of gravitational radiation +itself, resulting in a permanent displacement of free-falling +test masses upon the passage of GWs [40–46].2 While +essentially any source of GWs will generate nonlinear +memory,3 our focus here (and in much of the relevant +literature) is on binary black holes (BBHs). The reason +for this is twofold: firstly, binaries are the one source of +detectable GWs we definitively know to exist; and secondly, +the amplitude of the memory scales with the total GW en- +ergy radiated, which for binaries is ∼ 1 − 10% of the total +mass [52], favouring BBHs over lighter binaries containing +neutron stars or white dwarfs. +The gravitational wave memory modifies the BBH wave- +form by introducing a slowly-growing offset of the oscilla- +tions that builds over the whole coalescence and whose +time evolution follows that of the instantaneous orbital +frequency. +This shift rises over the radiation-reaction +timescale during the inspiral [44, 45, 53] and accumulates +rapidly during the merger before saturating to its final +value during the ringdown [54]. Although the memory +arises from a 2.5 PN nonlinear interaction in a post- +Newtonian (PN) expansion of Einstein equations, because +it accumulates over the whole coalescence, it affects the +gravitational waveform at leading (Newtonian) order [54], +increasing the prospects of observing it in the near future. +Several searches for memory from BBHs have been per- +formed using LIGO/Virgo data, returning only null results +thus far [55–58]. This is in agreement with forecasts for +LIGO/Virgo, which show that the detection of memory +from a single event would require a much more massive +and nearby binary than any yet observed, and that to +find collective evidence of memory in the total population +of observed binaries one would need ∼ 5 yr of collected +data [59–61] (or ∼ 2.5 yr taking into account the expected +improvement of detector sensitivities [62]). The difficulty +in detecting the memory with current ground-based inter- +ferometers resides mostly in the fact that, besides being +responsible for only a small amplitude offset, its power is +larger at lower frequencies where the detectors sensitivity +is limited by several sources of noise [63]. However, the +prospects for memory detection in single events are consid- +erably better for third-generation ground-based detectors +(e.g. Einstein Telescope [64] and Cosmic Explorer [65]) +and for the future space-based detectors LISA and Tian- +Qin, due to their better sensitivity and low frequency +coverage [54, 62, 66–69].4 +2 A similar effect sourced by the flux of matter or non-gravitational +radiation was actually discovered first and is called the linear GW +memory [47–49]. Hereafter, we use “memory” to refer exclusively +to Christodoulou’s GW memory. +3 See, e.g., Refs. [50, 51], which studied the nonlinear memory +generated by cosmic string loops. +4 Memory from the merger of supermassive BHs is also a target of +In this work we investigate the impact of the nonlinear +memory on parameter estimation via a Fisher matrix +analysis. In particular, we focus on how the memory +signal breaks the distance-inclination degeneracy, which +is crucially important if these binaries are to be used +as standard sirens. We found that the information of +the memory can indeed reduce the uncertainty on the +luminosity distance by reducing its correlation with the +inclination angle, whereas it has almost no impact on the +uncertainty of the other parameters. For LISA sources +its greatest effect involves cases where (i) the constituent +BHs are light enough [MBH ≲ 105 M⊙/(1 + z), with z +the source’s cosmological redshift] that the merger takes +place near the upper edge of the LISA band, and (ii) the +information from the primary waveform is limited to a +few cycles before the merger. +The presence of gaps in the data stream and confusion +noise from other sources will reduce the effective dura- +tion of usable LISA data, and thus the observed number +of cycles for BBHs [74], making the memory potentially +useful for the distance estimation of some BBH events. +Therefore, using the state-of-art astrophysical BBH popu- +lation models described in Refs. [75, 76] (and based upon +previous work presented in [77–79]), we assess quantitat- +ively the impact of the memory in the distance estima- +tion of LISA sources, taking into account the presence of +gaps in the data stream. Considering the particular gap +model used in Ref. [80] we did not find any significant +enhancement in the distance estimation by the inclusion +of memory on the BBH waveforms. We do however find a +greater number of events with detectable memory at LISA +as compared to previous forecasts [67, 69], especially in +our models with heavy BH seeds. +This paper is organised as follows. In Sec. II we re- +view the computation of the memory and describe the +phenomenology of the signal. In Sec. III we describe our +Fisher forecasting analysis. In Sec. IV we present our +results for the distance-inclination inference of individual +BBHs, and discuss the impact of the binary parameters +and signal duration. In Sec. V we perform population- +level forecasts for LISA using synthetic BBH catalogues, +and assess the impact of including the memory on the +luminosity distance estimation in the presence of gaps in +the data stream. We conclude in Sec. VI. Some technical +material is discussed in the appendices. We use geometric +units throughout (c = G = 1). +II. +GW MEMORY WAVEFORM +II.1. +Computation scheme: Thorne’s formula +The most direct way to compute the memory contri- +bution to waveforms would be to extract it directly from +Pulsar Timing Arrays (PTAs) [70, 71], but searches in PTA data +thus far have returned only null results [72, 73]. + +3 +numerical relativity simulations. However, most simula- +tions to date have struggled to accurately capture this +information for a number of reasons (see e.g. [44]). Some +exceptions are Ref. [81], where the dominant memory +mode (ℓ, m) = (2, 0) was first resolved, and the recent +work of Ref. [82], which used a Cauchy-characteristic +extraction (CCE) technique to extract the waveform. Al- +ternatively, the Bondi, van der Burg, Metzner, and Sachs +(BMS) balance laws [83] have recently been used to add +the memory to waveforms [84, 85] (see also [57, 69]). +Instead, in this work we use a perturbative approach +to evaluate the memory [44, 46], which we now briefly +review. A GW strain h0 (which we call the “primary” +signal) sources an additional memory strain δh, which +can be expressed in the transverse-traceless (TT) gauge +using Thorne’s formula [86], +δhTT +ij (u) = 4 +R +� u +−∞ +du′ +� +R +dΩ d2EGW +du′dΩ +� +ninj +1 − nkN k +�TT +, +(1) +where the angular integral is over the solid angle dΩ of a +(large) sphere of radius R surrounding the source, and ni +and N i are the unit radial vectors pointing, respectively, +to dΩ and the detector. The TT superscript represents a +TT projection with respect to the direction of the detector. +The time integral is over the entire history of the source +up to retarded time u, which shows that the memory is +a hereditary effect. The GW energy flux carried by the +primary GW is [44] +d2EGW +dtdΩ += R2 +16π (˙h2 +0,+ + ˙h2 +0,×), +(2) +where ˙h ≡ dh/dt and h+,× ≡ hTT +ij eij ++,×. We use the same +choice of TT-polarisation tensors eij ++,× as Ref. [87]. From +the spin-weighted spherical harmonic decomposition +h+ − ih× ≡ +� +ℓ≥2 +� +|m|≤ℓ +hℓm(u, r) −2Yℓm(ι, φ), +(3) +it is possible to show that the sourced memory can be +expressed as [56] +δhℓm(u) = −R +� +ℓ′,ℓ′′≥2 +� +m′,m′′ +� +(ℓ − 2)! +(ℓ + 2)! +× +� +dΩ Y ∗ +ℓm −2Y ∗ +ℓ′m′ −2Yℓ′′m′′ +� u +−∞ +du′ ˙h∗ℓ′m′ +0 +˙hℓ′′m′′ +0 +, (4) +which allows us to straightforwardly compute the memory +modes δhℓm from the primary waveform modes hℓm +0 . We +can then reconstruct δh+ and δh× from Eq. (3) to give +the total strain h ≈ h0 + δh. In principle, this process +should be iterated to give higher-order contributions (the +“memory of the memory” [59]). In practice, these extra +terms are subleading, and it is sufficient for our purposes +to consider just the leading-order memory effect, δh. +To generate our primary waveforms, we use the surrog- +ate NRHybSur3dq8 model [88], which includes all higher +spherical harmonic modes up to (ℓ, |m|) = (4, 4) and is con- +siderably more accurate than other often-used phenomen- +ological models in modelling the merger stage of BBH +coalescences [61]. We use the publicly available GWmemory +package [89] to implement the calculation scheme de- +scribed above for the corresponding memory signal. +Equation (4) is valid on a background Minkowski space- +time. It can be extended to a spatially flat Friedmann- +Lemaître-Robertson-Walker (FLRW) spacetime using the +fact that, for sources at the same luminosity distance dL, +the memory amplitude in FLRW is enhanced over the +Minkowski case by the redshift factor (1 + z) [90, 91]. +Additionally, we shall use the time at the detector t ≡ +tpeak − (1 + z)(upeak − u), where tpeak is the instant when +the primary strain reaches its peak amplitude. Summar- +izing, in this work we use +δhℓm +FLRW(t) = (1 + z)δhℓm +Mink +� +u(t) +� +R→dL. +(5) +This can be shown to be equivalent to using redshifted +component masses Mi,z ≡ (1 + z)Mi, with i ∈ {1, 2}, and +luminosity distance dL to generate the primary signal (e.g., +with NRHybSur3dq8), plugging this primary directly into +Eq. (4), and identifying (R, upeak − u) → (dL, tpeak − +t). +We omit the subscript “FLRW” throughout, but +a spatially flat FLRW is implicitly assumed in all our +expressions. +II.2. +Phenomenology of the signal +Equation (4) shows how the memory modes are sourced +by pairs of primary modes. For a BBH coalescence oc- +curring in the x-y plane, the primary modes have the +form hℓm +0 +∝ e−imϕ(t) with ϕ(t) the orbital phase. So, +from the time integral in Eq. (4) it is clear that the lead- +ing contribution to the memory modes at low frequency +(i.e., u → +∞) comes from the non-oscillatory (DC) terms +with m′−m′′ = 0 which accumulate in time [44, 45]; these +source m = 0 memory modes (from the angular integ- +ral in Eq. (4)). Note, however, that oscillatory m ̸= 0 +memory modes do become dominant at high frequencies +(c.f. Fig. 1). +The memory sourced in the quasi-circular inspiral +of non-spinning BBHs is known analytically up to +3 PN [44, 45]. +Due to the accumulation over the in- +spiral, the non-oscillatory contribution to the memory +enters at Newtonian (0 PN) order in the waveform, +δh(0PN) ++ += ηMz +48 dL +[Mω(t)] +2 +3 sin2 ι (17 + cos2 ι), +(6) +and δh(0PN) +× += 0, with the conventional choice of po- +larization triad [44]. +The orbital frequency is ω(t) ≡ +˙ϕ(t) and the symmetric reduced mass η ≡ M1M2/M 2, +where M ≡ M1 + M2 is the total mass. The memory +has the same scaling as the Newtonian primary waveform +(c.f. Eq. (B1)), but rather than the main time depend- +ence coming from the oscillatory term, its time evolution + +4 +10-5 +10-4 +10-3 +10-2 +10-1 +f[Hz] +10-23 +10-22 +10-21 +10-20 +10-19 +hc(f) +(2, 0) +(4, 0) +(2, 1) +(3, 1) +(4, 1) +(2, 2) +(3, 2) +(4, 2) +1/60Mz +fpeak +2fpeak +QNM1 +QNM2 +Figure 1. Memory characteristic strain hℓm +c (f) ≡ 2f| � +δhℓm| of the most important modes computed from Eq. (4). We consider +the non-spinning “heavy” BBH studied in Figs. 2 and 3, with total mass M = 2 × 105 M⊙, mass ratio q = 1.2 and redshift z = 2. +The non-oscillatory (ℓ, m) = (2, 0) mode dominates at low frequencies, but is suppressed at f ≳ 1/60Mz, where oscillatory m ̸= 0 +modes start becoming important (in particular, at their maxima f ∼ mfpeak). We can also see the presence of the ringdown in +the memory spectrum in the form of high-frequency peaks [QNM1 is at f = (τ −1 +221 + τ −1 +222)/2πMz, and QNM2 at f = τ −1 +222/πMz]. +(QNMs were computed using the Python package qnm [92], and the final mass and spin of the remnant BH via surfinBH [93], +whose fitting procedure is described in [94].) +is captured by the instantaneous orbital frequency ω(t). +This explains the typical step shape of the memory, which +has a steep increase in the merger-plunge phase and a +saturation during the ringdown. Moreover, the memory is +characterized by a different dependence on the inclination +angle ι and an overall amplitude ∼ 20 times weaker than +the primary. In particular, the two signals have oppos- +ite monotonic dependence on the inclination angle, and +while the primary signal is maximised for face-on binaries +(ι = 0), the memory is instead maximised for edge-on bin- +aries (ι = π/2). This behaviour is maintained when using +primary waveforms generated by NRHybSur3dq8. The dif- +ferent dependence on ι is what makes the memory helpful +in reducing the (ι, dL) correlation (c.f. Fig. 4). +Figure 1 shows the spectral shape of memory mode char- +acteristic strains hℓm +c (f) ≡ 2f|� +δhℓm(f)|, with � +δh(f) ≡ +� dt e−i2πftδh(t) the Fourier transform (FT) of δh. All +modes exhibit a plateau at low frequencies, but the spec- +tral content of the non-oscillatory (m = 0) modes is +clearly distinct from the oscillatory (m ̸= 0) modes. The +plateau of the m = 0 modes is easily understood from +the approximation δhℓ0 ≈ ∆hℓ0H(t − tpeak), with H the +Heaviside step function and where ∆hℓ0 scales with the +fraction of radiated EGW that sources δhℓ0; this results +in a constant hc ≈ ∆hℓ0/π [53]. +Taking into account that the memory growth is not in- +stantaneous, but happens in τ ∼ 60Mz (the timescale over +which most of EGW is radiated [95]), one can understand +the suppression at f ≳ 1/60Mz in Fig. 1. For the m ̸= 0 +modes the low-frequency plateau has a similar origin, but +its value is always subdominant because, due to the oscil- +lations in the integral in Eq. (4), the memory does not ac- +cumulate, averaging out to a net small value that depends +strongly on the value of the orbital phase at which the +BHs merge; this is also expected from PN results [44, 45]. +We note that for m ̸= 0 the maximum of the strain scales +as mfpeak where the peak frequency fpeak ∼ 0.1/πMz +roughly corresponds to the moment in which most of +the energy is radiated [95]. The oscillations at the left +of these maxima are numerically stable (in particular, +they are not artefacts of our FT) and come from inter- +ference between different ℓ-modes in Eq. (4). On the +other hand, the high-frequency peaks seen in all modes +are associated with the ringdown stage and are located at +f ≈ (τ −1 +ℓ′m′n′ + τ −1 +ℓ′′m′′n′′)/2πMz, where the complex quasi- +normal modes are σℓmn ≡ (ωℓmn+iτ −1 +ℓmn)/Mz [96]. It’s in- +teresting to note that these peaks occur at a frequency set +by the QNM decay rate τ −1 +ℓmn (i.e. imaginary frequency), +not the oscillatory part ωℓmn (i.e. real frequency). This +can be confirmed analytically using Favata’s minimal +waveform model (MWM; Eq. (14) of [54]). +Figure 2 shows the characteristic strain hc(f, ι, ϕc) ≡ +2f|�h+ − i�h×| of the primary and memory – containing +all modes up to (ℓ, |m|) = (4, 4) – for two binaries with +different total masses and redshifts, seen from a fixed dir- +ection ι = 40 deg and ϕc = 0. At this particular direction, +the primary characteristic strain is O(102) greater than + +5 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +f [Hz] +10-22 +10-20 +10-18 +10-16 +hc(f) +2 × 105M ⊙ +2 × 104M ⊙ +Figure 2. Characteristic strain hc(f, ι, ϕc) ≡ 2f|�h+ − i�h×| +of the primary (solid curve) and memory (dashed curve) +computed from Eqs. (3) and (4), seen from a fixed direc- +tion ι = 40 deg and ϕc = 0. We consider two fiducial non- +spinning BBHs, which will also be used in the following sections: +a “light” binary (in violet) with total mass M = 2 × 104M⊙ at +redshift z = 0.5, and an “heavy” binary (in blue) with M = +2 × 105M⊙ at z = 2. Both BBH have mass ratio q = 1.2. We +consider the last 25 cycles before merger for both BBHs (which +corresponds to ∼ 6 minutes for the “light” source, and ∼ 2 +hours for the “heavy”). +the memory. Thus, the memory adds information to para- +meter estimation only if the number of cycles that can be +observed during inspiral is limited (c.f. Fig. 5); this may +happen due to gaps in the data stream and/or confusion +noise from other sources. +Indeed, whereas truncating +the primary waveform at some minimum fin (related to +the time/cycles prior to merger) significantly reduces the +SNR of the primary, the SNR of the memory is almost +unchanged (as also noted in Refs. [62, 69]). +As long +as the memory is observed in LISA for a period of at +least 103 s ≈ 15 min after merger, its SNR is approxim- +ately independent of the observation time (c.f. Fig. 6). +In this work we focus on quasi-circular and non-spinning +BBHs. For the dependence of the memory on the spins, +mass ratio and eccentricity, we refer the reader to Refs. [57, +68, 81, 89, 97, 98]. +III. +FISHER MATRIX AND PARAMETER +ESTIMATION +We use a Fisher matrix analysis (commonly used in +GW astronomy [15, 99–103]) to quantify the impact of +GW memory on the parameter estimation of the BBH +source. We only consider events with merger occurring +during the mission lifetime of LISA, since the memory +is mainly generated during this phase of the binary evol- +ution. +The total signal h = h0 + δh is generated in +time-domain and is given by the sum of the (primary) sur- +rogate NRHybSur3dq8 waveform h0, and the memory δh +computed through the GWMemory package [89] from the +primary waveform. In this work we focus on non-spinning +binaries, so the input parameters for the waveforms are +the redshifted total mass Mz = (1+z)M, the mass ratio q, +the luminosity distance dL, the inclination angle ι, and +the coalescence phase ϕc; thus, Θ = {ln Mz, q, ln dL, ι, ϕc}. +We only investigate the dependence on these parameters, +meaning that we ignore the particular sky position of the +source and the LISA response function. Accounting for +them should not affect our results, since the LISA orbital +motion is a year time scale, whereas the longest inspirals +where we see the effect of the memory are of the order of +hours. +In the strong-signal limit the probability distribution +of the parameters is a multivariate Gaussian distribution +centred on the true values Θ = ¯Θ, with the covariance +matrix Σij described by the inverse of the Fisher inform- +ation matrix Γij, up to corrections of order of the inverse +signal-to-noise ratio, +Σij = (Γ−1)ij[1 + O(SNR−1)]. +(7) +The SNR ρ and the Fisher matrix are computed as +ρ2 = (˜h|˜h), +Γij ≡ +� ∂˜h +∂Θi +��� ∂˜h +∂Θj +���� +Θ= ¯Θ, +(8) +where we have used the standard inner product +(˜a|˜b) ≡ 4Re +� fmax +fmin +df ˜a∗(f)˜b(f) +Sn(f) +. +(9) +The Sn(f) is the sky-position- and polarisation-averaged, +but not inclination-averaged, LISA power spectral density, +as found in [104], with the confusion noise corresponding +to Tobs = 4 years. Computing the Fisher matrix thus +allows us to find the variance of each parameter and +correlation of each pair of parameters due to measurement +errors, +σ2 +i = (Γ−1)ii, +cij = (Γ−1)ij +σiσj +, +(10) +(with no summation implied over repeated indices here). +The total frequency-domain signal ˜h = ˜h0 + δ˜h is given +by the sum of the FT of the primary and the memory +signals, which we compute numerically via the fast Fourier +transform (FFT) implemented in NumPy [105]. In App. A +we explain in detail our choices for manipulating the +signal, such as the padding and the window function +applied, while in App. C we elaborate on the numerical +computation of the Fisher matrix. +IV. +RESULTS FOR THE +DISTANCE-INCLINATION ESTIMATION +The goal of this section is to study the impact of the +memory on the parameter estimation of intermediate-mass + +6 +Figure 3. The 1σ confidence ellipses computed for the primary waveform without memory (in magenta), with memory (in +blue), and with only the memory (in green). For each of those we compare the effect of including higher modes (solid lines) and +excluding them (dashed lines). The left panel shows the result for the “light” binary, whereas the right panel shows it for the +“heavy” binary (c.f. Fig. 2); in both cases we consider the last 25 cycles before merger and a line of sight ι = 40 deg. For the +“heavy” binary, the memory has negligible impact, since we cannot distinguish the purple and the blue lines (the green contours +of the memory fall beyond the panel). +to supermassive BBHs with LISA, with a particular fo- +cus on breaking the distance-inclination degeneracy. Our +motivation is twofold: (i) although nonlinear memory +is expected to be observed in single events at LISA, no +attention (to our knowledge) has been paid on its impact +on the parameter estimation, and (ii) the memory signal +has an opposite correlation between distance and inclina- +tion cln dL,ι with respect to the primary signal, which can +make it useful to break the distance-inclination degener- +acy, thus improving the accuracy of distance estimations +(c.f. Fig. 3). +Firstly, using our Fisher analysis we noted that the +primary signal alone already constrains the binary in- +trinsic parameters quite well, and that the additional +information provided by the memory does not constrain +them further. On the other hand, as already anticip- +ated, we found that the dependence of the memory on +the extrinsic parameters is complementary to that of the +primary signal, and can mitigate the uncertainty on the +luminosity distance dL and the inclination ι estimations. +This is demonstrated in Fig. 3 where we explicitly show +the constraints coming from the primary signal and the +memory, as well as the effect of including all higher modes +(HMs) up to (ℓ, |m|) = (4, 4). +These results are also +summarized in Tab. I. +As it can be seen in Fig. 3, the ellipses computed from +the memory signal and the primary waveform are ortho- +gonal, this can be intuitively expected from the opposite +monotonic dependence on the inclination of the two com- +ponents, as explained in Sec. II.2. We analytically derive +the opposite correlation for the two signals in App. B, +where we compute the 2 × 2 Fisher matrix of this pair +of parameters {ι, ln dL} for the dominant mode (DM) of +the primary waveform and of the memory. This property +leads to a lower correlation cι,ln dL of the overall Fisher +matrix, as shown in Fig. 4, which can mitigate the error +on the luminosity distance estimation. The estimation of +the other parameters is less affected, thus justifying our +approach to focus on this particular pair of parameters. +For signals with sufficiently large SNR or whose sources +are at sufficiently high redshifts, the uncertainty on the +luminosity distance estimation becomes dominated by +weak lensing effects (see, e.g., Ref. [21, 106]), and our +Fisher analysis (which neglects lensing) ceases to be valid. +However, as we show below, the memory is helpful to +parameter estimation only for binaries whose primary +signal is not very loud, and whose total redshifted mass +Light +h0,DM +h0,HM +δhDM +δhHM +SNR +20.5 +20.6 +1.9 +2.3 +h0,DM +h0,HM +hDM +hHM +σdL/dL +0.56 +0.55 +0.20 +0.18 +σι [rad] +0.75 +0.73 +0.27 +0.23 +Heavy +h0,DM +h0,HM +δhDM +δhHM +SNR +1001.4 +1006.3 +3.5 +4.3 +h0,DM +h0,HM +hDM +hHM +σdL/dL +0.0116 +0.0108 +0.0115 +0.0107 +σι [rad] +0.0158 +0.0140 +0.0157 +0.0139 +Table I. Summary of the results shown in Fig. 3. The first two +columns correspond to the results coming from the primary +signal alone, whereas the last two correspond to the total +(including the memory) waveform. While the memory improves +significantly the estimation of the distance-inclination of the +“light” binary, it does not help much with the “heavy” binary +parameters. + +1.75 +1o, ho +1o, Sh +1.50 +10,h +1.25 +1.00 +[rad] +0.75 +0.50 +0.25 +0.00 +-0.25 +1000 +2000 +3000 +4000 +5000 +dL [Mpc]1o, ho +1o,h +0.72 +0.71 +[rad] +0.70 +0.69 +0.68 +0.67 +15600 +15800 +16000 +dL [Mpc]7 +q +lnMz +lndL +ι +ϕc +q +lnMz +lndL +ι +ϕc +1.0 +0.73 +1.0 +0.05 +0.02 +1.0 +-0.07 +-0.04 +-0.99 +1.0 +0.78 +0.25 +0.04 +-0.06 +1.0 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +q +lnMz +lndL +ι +ϕc +q +lnMz +lndL +ι +ϕc +1.0 +0.73 +1.0 +-0.08 +-0.06 +1.0 +0.01 +0.01 +-0.9 +1.0 +0.78 +0.25 +-0.06 +0.01 +1.0 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Figure 4. Correlation matrices cij for the “light” binary of +Fig. 2. The upper panel shows the results for the primary +signal without memory, while the lower panel is for the total +waveform including memory. The major effect of including the +memory is decreasing the (dL, ι) correlation. +is in the range 104M⊙ ≲ Mz ≲ 105M⊙. The memory of +these sources will only be observed if they are sufficiently +close z ≲ 1.5, where lensing effects are not too strong. +Therefore, our results indicate that the memory has an +impact on the distance estimation only in cases where its +intrinsic uncertainty (even after adding the memory) is +much larger than the contribution due to lensing. +It is natural to compare the effect of the memory with +that of HMs, as the latter can also improve the BBH para- +meter estimation and partially break the aforementioned +degeneracy [29–32, 107–113]. Their main contribution to +the SNR comes from extending the signal to higher fre- +quencies as compared to the dominant (2, 2)-mode, thus +being more relevant when the merger falls well inside the +sensitivity curve of the detector. Importantly, HMs have +a different dependence on the inclination angle than the +DM, which is again why they may be useful in breaking +the distance-inclination degeneracy. The memory signal, +on the other hand, is always subdominant, but can play +a significant role, as we will see, when the information +from the primary at lower frequencies (i.e., from the in- +spiral) is absent/degraded, and the memory becomes the +only contribution at those frequencies. In the following +we explain in detail the dependence of memory-assisted +parameter estimation on the binary mass, duration of the +signal, and line of sight, and we discuss the impact of +HMs. +IV.1. +Dependence on the binary mass +By evaluating the SNR defined in Eq. (8), we find +that LISA can detect memory (more precisely, its SNR +is > 1 [114, 115]) from binary mergers with total redshifted +mass in the range [104, 108] M⊙, thus confirming previous +results [54, 67, 69]. For lower masses, the memory strain +is too weak to be detected, whereas for larger masses the +turnover frequency at which the memory drops off falls +below the LISA band. Because of the dependence of the +memory characteristic strain on the binary mass, LISA +will be most sensitive to the low-frequency plateau part of +the signal for light binaries, and to the subsequent high- +frequency features associated with the merger/ringdown +stage for more massive binaries (c.f. Fig. 1).5 We found +that nearly all the SNR of the memory accumulates during +the merger and it is maximised for Mz ∼ 106M⊙, in which +case the memory can be detectable up to redshift z ∼ 14. +For short/degraded primary signals (with fixed number +of cycles), we found that the memory is most helpful in +parameter estimation for binaries with total redshifted +mass Mz ≲ 105M⊙, in which case the memory falls in the +most sensitive part of the LISA sensitivity band, whereas +the merger covers only the high-frequency edge of the +spectrum. +For this reason, the hierarchy of the SNR +between the memory and the primary signals is less severe +than for more massive binaries Mz ≳ 106M⊙, whose +mergers occur in the middle/low-frequency part of the +LISA band. +The difference between “light” and “heavy” binaries is +clear from Fig. 3, which shows the effect of considering +short signals (in this case, the last 25 cycles) for two differ- +ent binary masses. In the left panel, for the “light” binary, +there is a manifest reduction in the parameter uncertain- +ties, which is due to the intersection of the primary and +memory confidence ellipses that results in the shrinking +of the total signal’s confidence ellipse. Such an effect is +5 See also Fig. 4 of Ref. [66]. + +8 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Ncycles +0.25 +0.50 +0.75 +1.00 +σdL, wm/σdL +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +ρm/ρ0 +0.25 +0.50 +0.75 +1.00 +σdL, wm/σdL +2 ×104M ⊙ +5 ×104M ⊙ +8 ×104M ⊙ +Figure 5. Effect of the memory on the luminosity distance estimation when the primary signal is truncated at increasing number +of cycles prior to merger. The memory becomes less important for more massive binaries, for which the merger occurs before or +close to the peak of LISA sensitivity and the memory adds negligible contribution to the total SNR. The sources are kept at +redshift z = 0.5 and line of sight ι = 40 deg. +not present in the right panel, for the “heavy” binary, +because, due to the large SNR of the primary signal, its +confidence ellipse is already entirely enclosed within the +memory’s confidence ellipse. The main factor controlling +the relative sizes of the confidence ellipses, and thus the +impact of the memory in parameter estimation, is the +ratio between the memory and the primary signal SNRs +(c.f. Fig. 5). +IV.2. +Dependence on the signal duration +As just discussed, our results show that the most reli- +able indicator for how much the memory contributes to +constrain the binary parameters (for a fixed line of sight) +is the ratio between the SNR of the memory and of the +primary signal, ρm/ρ0. This ratio depends on the total +mass of the binary (c.f. Tab. I), but it is also a function +of the inclination and the time duration of the data taken +before the merger. Decreasing the mass and the duration +of the signal prior to merger tends to increase this ratio. +We quantify the improvement in parameter estimation +in terms of the ratio between the standard deviation of +the luminosity distance with and without the memory, +σdL,wm/σdL. +Our results are presented in Fig. 5 for different masses +in the range [104, 105]M⊙. In the upper panel, we show +the σ-ratio as a function of the number of cycles Ncycles ob- +served prior to merger. We note that for any given number +of observed cycles, the memory of more massive binaries +leads to a smaller improvement in distance estimation as +compared to lighter binaries (as explained in Sec. IV.1). +In the lower panel, we show that there exists a mono- +tonic, one-to-one relationship between the σ-ratio and +the ρ-ratio, for a fixed inclination, which is approximately +insensitive to the mass of the binary. This observation +will allows us to determine, in the next section, the “crit- +ical” ρ-ratio needed to achieve a given improvement in +the distance estimation as a function of the inclination. +IV.3. +Dependence on the inclination +Fixing the other parameters, the impact of the memory +depends strongly on the inclination of the binary, as we +show in the upper panel of Fig. 6. In this plot we present +the relative uncertainty in the luminosity distance at +different inclination angles, considering the “light” binary +(discussed previously) with mass-ratio q = 1.2 and fixing +the observed signal to 25 cycles prior to merger. +We +restrict the plot to values of ι ∈ [0, π/2], since its behavior +is symmetric with respect to ι = π/2. Note that, in the +absence of the memory and the HMs, the uncertainty on +the luminosity distance diverges for face-on ι → 0 (and +face-off ι → π) configurations. As we discuss in App. B, +this is due to the fact that the Fisher matrix is singular + +9 +20 +30 +40 +50 +60 +70 +80 +90 +ι[deg] +0.2 +0.4 +0.6 +0.8 +1.0 +σdL/dL +h0, HM +hHM +h0, DM +hDM +30 +40 +50 +60 +70 +80 +90 +ι[deg] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +ρm/ρ0 +q = 3 +q = 1.2 +Figure 6. +Upper panel: the effect of the memory on the +estimation of the luminosity distance as a function of the +inclination. The parameters of the source are those of the +“light” binary of Fig. 3 with q = 1.2, and the number of +cycles before merger is Ncycles = 25. +Lower panel: the ρ- +ratio required to achieve a 10% improvement in the luminosity +distance estimation (i.e., such that σdL,wm/σdL = 0.9) as a +function of inclination for two different values of the mass- +ratio q = {1.2, 3}. +at this inclination and, thus, the Fisher analysis for the +primary DM ceases to be valid in a neighborhood of ι = 0 +(and of ι = π). However, for unequal-mass binaries the +HMs regularize the Fisher matrix, and even a slightly +asymmetric binary as q = 1.2 has a relative uncertainty in +the distance smaller than ∼ 0.8 for all inclination angles. +In the upper panel of Fig. 6 we compare the impact of +HMs on the distance estimation with that of the memory +for different inclination angles, keeping the mass-ratio +q = 1.2 fixed. However, we also verified that HMs become +more relevant the more asymmetric the binary is, whereas +the impact of the memory is larger for smaller q ∼ 1.6 +The large effect of the memory observed in the plot – +improving the distance estimation by more than a factor +of 4 for some inclinations – is partially related to the short +duration of the signal considered (truncated at 6 minutes +prior to merger); as discussed in Sec. IV.2, for longer +signals the information in the primary (which accumulates +over the inspiral) eventually becomes dominant, with the +memory contributing negligibly to parameter estimation +(c.f. Fig. 5). +In the lower panel of Fig. 6 we show the critical ρ- +ratio needed to achieve a 10% reduction in σdL (i.e., such +that σdL,wm/σdL = 0.9) as function of the inclination, for +two different mass-ratios q = {1.2, 3}. From the discussion +in Sec. IV.2, these curves are approximately independent +of the binary’s total mass. Moreover, we see from this plot +that they are also only mildly dependent on the mass- +ratio q. +Another important observation is that closer +to face-on a smaller ρ-ratio is needed to achieve a given +improvement in the distance estimation compared to edge- +on. So, close to face-on the memory can be relevant even if +the primary is observed for relatively long periods (∼ few +hours). For example, if our “light” binary (with q = 1.2) +is seen close to face-on, the inclusion of the memory +information leads to a 10% reduction in σdL if the inspiral +is observed over less than 6 hours prior to merger. +IV.4. +Impact of higher modes +Here we discuss the contribution of HMs to the SNR +of the primary and memory signals, and its consequent +influence on the estimation of the luminosity distance. +We consider the “light” and “heavy” binaries of Fig. 2, +both of them with mass ratio q = 1.2 and seen at an +intermediate inclination angle ι = 40 deg. The results are +shown in Fig. 3 and Tab. I. +Our results indicate that HMs have a much greater +impact on the loudness of the memory than on the primary +signal, boosting the memory SNR by ∼ 19%, as opposed to +just ∼ 0.5% for the primary. This is mainly caused by HMs +increasing the total radiated EGW, which pushes the low- +frequency plateau to larger amplitudes. The increase in +the memory SNR with HMs has only a mild dependence on +the inclination, but it depends strongly on the mass ratio, +since for more asymmetric configurations more energy is +released into HMs. +6 The influence of the mass-ratio on the impact of the memory can +be understood from Eq. (6), δh(0PN) ∝ q/(1 + q)2, which results +in the memory characteristic strain +f � +δh +(0PN) ∝ q/(1 + q)2, +whereas the primary characteristic strain is [15] +f�h(0PN) +0 +∝ √q/(1 + q). + +10 +Regarding parameter estimation, Tab. I shows that +HMs have a greater impact for the “heavy” binary than +for the “light” one. This can be understood from the fact +that the HMs add information mostly close to merger (i.e., +at high frequencies), which is therefore masked for the +“light” binary. This is opposed to the memory which, as we +have seen, contributes the most to parameter estimation +for the “light” binary. As discussed in the last section, +the impact of HMs increases with the mass-ratio; indeed +we verified that for q ≲ 8 the relative error in the distance +estimation may be reduced by up to a factor of ∼ 2 − 3 +with the inclusion of HMs, as compared to the values of +Tab. I. The information contained in the memory and in +the HMs is complementary for parameter estimation, due +to their different dependence on the mass-ratio and total +mass. +V. +POPULATION FORECASTS +In this section we study the detectability of the non- +linear memory for realistic population models of massive +black holes, and assess its potential impact on parameter +estimation considering the presence of gaps in the data +stream. We update the previous forecasts of Refs. [67, 69] +for the measurability of the memory in single events with +space-based detectors, by using the recent population +models described in Refs. [75, 76] (and recently used in +Ref. [80]). +V.1. +Framework +In our analysis we consider all 8 models described in +Refs. [75, 76], each of those corresponding to different as- +sumptions about some of the main uncertainties in the cos- +mological evolution of massive BHs. These involve [75, 76]: +(i) the high-redshift mass function of the “seeds” of the +massive black hole population [“light seeds” (LS) origin- +ating from population III stars, or or “heavy seeds” (HS) +originating from direct collapse of protogalactic gaseous +disks], (ii) the time delay between the galaxy merger and +the corresponding BBH mergers (realistic “delays”, or +“short delays” neglecting the contribution from scales of +the order of hundreds of pc), and (iii) the presence or not +of supernova feedback (“SN” or “noSN”) on the accretion +disk of massive black holes. For each of the 8 models +we compute the memory random realizations of mergers +corresponding to 4 years of LISA mission, and present +average results over many such realizations. +We consider only the final 20 cycles prior to merger, +which is enough to give us a reliable estimate of the SNR of +the memory (c.f. Sec. II.2). Due to the limited parameter +space covered by the waveforms NRHybSur3dq8, we restrict +the mass-ratio to q ≤ 8 (i.e., we artificially fix q = 8 for +every merger with higher values of q). This assumption is +stronger for the LS case than for the HS one, since the two +cases have quite different mass-ratio distributions, with +the latter having a sharper peak close to equal mass (see +Fig. 11 of Ref. [75]). Since the waveforms NRHybSur3dq8 +do not cover precessing BBHs, we consider only the spin +components orthogonal to the orbital plane and restrict +them to |χi,ˆz| ≤ 0.8, i ∈ {1, 2}.7 We removed from the +catalogues the sources with Mz ≥ 108 M⊙, because evalu- +ating their SNR is computationally very expensive and, +as discussed in Sec. IV.1, they fall outside the parameter +space of interest for memory observation with LISA. We +take the inclination angle and the coalescence phase to be +uniformly distributed in cos ι ∈ [−1, 1] and ϕc ∈ [0, 2π]. +V.2. +Detectability of memory in single events +In the different 4-year realisations, we looked for events +with SNR of the memory above the threshold value ρth ≡ +1. Indeed, it has been shown that this is the condition +to claim distinguishability of two waveforms (that dif- +fer by δh) over the detector noise [114, 115].8 Table II +summarises our results. For each population model we +Astrophysical Catalogues +Light seeds +Heavy seeds +SN-delays +Ntot = 47 +Ntot = 27.3 +Nth = 0.4 (0.1) +Nth = 21.2 (10) +⟨ρ⟩ = 0.04 +⟨ρ⟩ = 6 +ρmax = 7 +ρmax = 97 +noSN-delay +Ntot = 191 +Ntot = 10 +Nth = 6 (1) +Nth = 7.5 (4) +⟨ρ⟩ = 0.17 +⟨ρ⟩ = 6.9 +ρmax = 11.64 +ρmax = 68.7 +SN-short +Ntot = 149 +Ntot = 1245 +Delays +Nth = 1 (1) +Nth = 418 (33) +⟨ρ⟩ = 0.04 +⟨ρ⟩ = 1 +ρmax = 5.01 +ρmax = 43 +noSN-short +Ntot = 1203 +Ntot = 1251 +Delays +Nth = 12 (2) +Nth = 392 (29) +⟨ρ⟩ = 0.06 +⟨ρ⟩ = 1.1 +ρmax = 17 +ρmax = 51 +Table II. SNR of the memory for the astrophysical models of +Refs. [75, 76]. For each model we consider a random realisation +of 4 years of events. We denote the total number of events +by Ntot and the number of those with SNR above the threshold +value ρm > 1 (5) by Nth. The ⟨ρ⟩ and ρmax are, respectively, +the average and the maximum ρm in the particular realisation +of events. For the SN-delays models and the NoSN-delay HS +model, the reported values are the averages over 10 realisations +of 4-year events, since for these models the total number of +mergers are much smaller than for the others. +7 Using the waveforms NrSur7dq2 which allow for precessing BBHs, +we found that introducing the spin components along the orbital +plane generally leads to an increase of the memory SNR. +8 This is equivalent to require that, if we parametrize the model +waveform as h = h0 + λδh with the fudge parameter λ ∈ [0, 1], +the statistical error of the fudge parameter is σλ < 1. + +11 +0 +2 +4 +6 +8 +10 +12 +14 +redshift +0.0 +0.2 +0.4 +0.6 +PDF +SN delays +noSN delays +SN short delays +noSN short delays +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +7.0 +log10(Mtot/M ⊙ ) +0.0 +0.5 +1.0 +1.5 +PDF +SN delays +noSN delays +SN short delays +noSN short delays +Figure 7. The redshift and total mass distributions of mergers with observable memory ρm > 1 for the various HS population +models. The models with “delays” have mergers typically at lower redshift, explaining the corresponding higher fraction of +events with observable memory (c.f. Tab. II). The different redshift distribution translates into slightly different peak locations +in the total mass distributions. +1 +2 +3 +4 +5 +6 +7 +8 +q +0.00 +0.25 +0.50 +0.75 +1.00 +PDF +SN delays +noSN delays +SN short delays +noSN short delays +0 +20 +40 +60 +80 100 120 140 160 180 +ι [deg] +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +PDF +memory events +prior +Figure 8. Left panel: the mass-ratio distribution for the events of Fig. 7. Right panel: the inclination angle distribution of the +events with detectable memory of the “SN short delays” model (with Nth = 418) as compared to the prior isotropic (cosine) +distribution. +denote by Ntot the total number of events considered in +the 4-year realisation, and by Nth the number of those +with memory SNR above the threshold ρth = 1 (inside +parentheses ρth = 5). We also indicate the average SNR +of the memory ⟨ρ⟩ and its maximum value ρmax in the +particular 4-year realisation of events. +The number of events with significant memory SNR +depends strongly on the astrophysical population model, +with clear differences between the LS and the HS models. +Our results are especially promising for the HS scenario, +as it suggests that about 75−78% of events for the “delays” +model and 31 − 33% for the “short-delays” model will +have observable memory with ρm > 1. +The numbers +inside the parenthesis can be directly compared with the +results of Ref. [69], where the “Q3d” and “Q3nod” models +considered there (and presented in Ref. [116], based on +Refs. [77–79]) can be compared, respectively, with “delays” +and “short-delays” HS models. For those, we found that +about 36 − 40% and 25%, respectively, of the total events +have detectable memory with ρm > 5, as opposed to 3.7% +and 1% for the “Q3d” and “Q3nod” models. The main +reason for this large mismatch is that the new population +models of Refs. [75, 76] that we used in this work have +a different (more realistic) delay model, which shifts the +mergers to lower redshift. +Figure 7 shows the distribution of the redshift and the +total mass of the events with ρm > 1 for the various HS +models. The distribution of models with “delays” peaks +at lower redshift, while that of “short-delays” models +extends up to z ∼ 13, which explains why the former +have a bigger fraction of events with memory SNR above +the threshold than the latter. Interestingly, we found +some events with particularly high SNR (ρm ≳ 50), as +can be seen in Tab. II; these belong to the low redshift + +12 +tail of the distributions (z ≲ 1). The location of the +peak of the total mass distribution changes slightly for +the various models, but it is such that the total redshifted +mass is about ∼ 106M⊙. In Fig. 8 we show the mass-ratio +distribution for the same events of Fig. 7. Most of the +events with detectable memory have a mass-ratio close to +unity with a sharp suppression at higher values, so that +the restriction to q < 8 turns out not to affect our results +for the HS models. The “noSN delays” model is the only +one presenting a mild accumulation of events with q = 8 +due to this restriction. +The situation is quite different for the LS models, which +have a much broader distribution in the mass-ratio, res- +ulting in a substantial (fictitious) accumulation of events +at q = 8. So, we repeated our analysis removing directly +the binaries with q > 8 from the catalogue. In this more +conservative approach, for “noSN-short delays” we noted +a reduction from 9 to 12 events with ρm > 1, but no +change in the number of events with ρm > 5. For “SN- +short delays” and “SN delays” models we found no events +with detectable memory, and for “noSN-short delays” a +reduction from 6 to 4 events with ρm > 1, and no events +at all with ρm > 5. Therefore, for LS population models +our results indicate that the prospects of observing the +memory with LISA do not seem promising. +We repeated our analysis of LS population models also +for the future generation ground-based detectors Cosmic +Explorer (CE) [65] and Einstein Telescope (ET) [117], +which have better sensitivity at higher frequencies, and +thus to lower masses.9 We have found almost no events +with observable memory ρm > 1 in 4 years of observation +(there was just one event with ρm ≃ 2 for the “SN short +delays” model with ET), and an average memory SNR +within 10−1 − 10−3. +V.3. +Impact of the memory on distance estimation +Given that HS models predict such a large number of +events with observable memory at LISA (which can be +almost up to 80% of the total number of events, in the +most optimistic scenario), we consider here the impact +of the memory on the luminosity distance estimation for +these sources. As we have shown in Sec. IV.2 the impact of +including the memory is highly dependent on the ratio of +the SNR of the memory and the primary signals, with the +memory helping substantially to constrain the distance +when the information (or the duration) of the primary +signal is somehow limited. Among other possibilities, this +could happen due to the presence of gaps in the data +stream, which causes a partial loss of signal. +9 We used the sensitivity curve of the configurations ET-D of +Ref. [118] and CE_40km_lf of https://cosmicexplorer.org/ +sensitivity.html, which have the best sensitivity at low fre- +quencies. +Two kinds of gaps are expected at LISA: the sched- +uled ones, related to the regular maintenance of the de- +tector, and the unscheduled ones, due to unexpected prob- +lems/events. In Ref. [80] it was shown that the scheduled +gaps have little or no impact at all on the parameter estim- +ation of massive BBHs, but the unscheduled ones could +degrade significantly the parameter estimation. Thus, +there is the intriguing possibility that, in the presence of +unscheduled gaps, the memory may add useful informa- +tion to constrain the binary parameters.10 +To quantify this effect we consider the particular gap +model used in Ref. [80], which is consistent with a 75% +duty cycle, as expected for LISA [74]. We simulate the +presence of gaps by windowing the signals as in Ref. [80], +considering scheduled gaps with a typical duration of 3.5 +hours every week and unscheduled gaps with a duration +of 3 days. The time interval between two unscheduled +gaps is treated as a random variable following an expo- +nential probability distribution p(∆T) = λ exp (−λ∆T) +with 1/λ = 9 days. With these choices, we simulate the +effective data taking of the mission and we distribute the +merger times uniformly over the 4-year mission duration. +From our study in Sec. IV, we expect the memory to be +helpful in constraining the binary (extrinsic) parameters +for a particular chunk of data if the merger happens within +the first few hours from the last gap. +For concreteness, let us compute the (average) total +number of mergers occurring within 6 hours from the last +gap. The number of unscheduled gaps can be estimated +by Ngap ∼ Tmission/Tgap, where Tgap is the sum of the +average time interval between gaps and the gap dura- +tion, ⟨∆T⟩ + 3 ≃ 12 days, thus Ngap ∼ 120. We focus +on the “SN-short delays” HS population model, the most +optimistic scenario with the highest number of events +with observable memory, Nth = 418; note that due to the +presence of gaps this number is reduced by 75%. Thus, +we can estimate the number of mergers by multiplying +the probability of having at least one merger in 6 hours +by 0.75NthNgap, which gives ∼ 6.4 events. We checked +this result numerically by simulating 50 times the distri- +bution of mergers over the gap realisation and we found +consistent results. +To find the number of events for which the inclusion of +the memory decreases by more than 5% the uncertainty +on the luminosity distance, we computed the ρ−ratio +(i.e., ρm/ρ0) for each event occurring within 6 hours from +the last gap in 50 numerical realisations, neglecting the +information accumulated in the inspiral before the gap. +Subsequently, we compared those ρ-ratios with the critical +values needed to achieve a 5% improvement on the lumin- +osity distance estimation, which depend on the particular +binary inclination (as in the lower panel of Fig. 6, but +for σdLwm/σdL = 0.95). We found that, on average, only +0.14 events of the 6.4 occurring close after a gap have an +10 Note that the LISA data analysis will be further complicated by +the presence of many overlapping signals [119]. + +13 +improvement of more than 5% on the distance estimation +from including the memory; this corresponds to 0.04% +of the total number of events with observable memory in +this population model. +We believe that this low value is due to the fact that +most of the BBH mergers with observable memory cor- +respond to configurations relatively close to edge-on (c.f. +right panel of Fig. 8), where the critical ρ−ratio is much +higher. Another reason is that the majority of the BBHs +with observable memory in the population considered +have a redshifted total mass ≳ 106M⊙, whereas as dis- +cussed in sec IV.1 the memory is more helpful for lighter +binaries Mz ≲ 105M⊙. +The “noSN-short delays” HS +population model, which is the second most optimistic in +terms of number of events with detectable memory, suffers +from these same issues and is, thus, expected to give a +similarly small result. The other population models have +far fewer events with observable memory, thus it is very +unlikely that any of these mergers will happen sufficiently +close to a gap to have a sufficiently large ρ−ratio. +In summary, applying our Fisher analysis to state-of- +the-art synthetic catalogues of massive BBHs indicates +that the memory will not help constraining further the +binary parameters at LISA, even in the presence of gaps in +the data stream. However, there is substantial uncertainty +on the assumptions adopted in this analysis, in particular, +regarding the population and gap models, and we cannot +exclude the possibility that there may exist additional +effects leading to a larger degradation of the primary +signal than those considered here. +VI. +CONCLUSION +In this work we have investigated the prospects of using +the nonlinear GW memory to help infer the parameters of +merging BBHs. In particular, we have focused on massive +BBHs detections with the future space-based interfero- +meter LISA, as these are the most promising individual +sources of memory. Our motivation is to use the additional +source of information provided by the memory signal to +break the degeneracy between inclination ι and lumin- +osity distance dL, which is present in the leading-order +GW signal. This is especially important for attempts to +use these BBHs as standard sirens (either via statistical +identification of the host galaxy [120], or possibly using +an electromagnetic counterpart due to the merger taking +place in a gas-rich environment [21, 24, 25]), as the un- +certainty on the Hubble constant H0 crucially depends +on the uncertainty on dL. +We find that the memory can indeed play a significant +role in breaking this inclination–luminosity distance de- +generacy. This occurs in cases where the redshifted total +mass is relatively small (≲ 105 M⊙), the binary is seen +not very close to edge-on, and the observation time is +limited to a few hours prior to merger. The limitation on +the observation time could occur due to, e.g., gaps in the +data stream caused by interferometer downtime, or confu- +sion noise from the presence of many other simultaneous +signals in the LISA frequency band. +In order to understand the relevance of these results for +the LISA mission, we started by performing a population +study using new synthetic catalogues of massive BBHs +to forecast the number of BBH events with observable +memory (ρm > 1). While there are currently large theor- +etical uncertainties on the astrophysical processes leading +to these mergers, we find a substantially larger number +of events with significant memory as compared to previ- +ous forecasts. The prospects are particularly bright for +the heavy seed model with “short delays” [75, 76], which +presents about 400 memory events for a 4-year mission +time. On the other hand, most of the mergers coming +from light seed models [75, 76] are undetectable by LISA +(and so is their memory). +Finally, we considered a commonly used gap model, +which includes both the scheduled and unscheduled types, +to quantify the benefit of the memory in the estimation +of the luminosity distance. For the most optimistic “short +delays” heavy seed models [75, 76], we found that, out +of the ∼ 0.75 × 400 observable memory events in a 4- +year mission time, just 0.14 events will produce a larger +than 5% decrease in σdL. Thus, our analysis indicates +that the information in the memory signal will not help +constraining further the binary parameters at LISA, even +in the presence of gaps in the data stream. This is due to +the fact that most of the events with observable memory +are seen close to edge-on, in which case the luminosity +distance and inclination are only slightly correlated in the +primary signal and, thus, the information added by the +memory is negligible for parameter estimation. +Our study, based on a Fisher matrix analysis, could be +further improved by performing a full Bayesian analysis +and by investigating the effect of priors on the lumin- +osity distance estimation, which is especially important +when the parameters are not well constrained. Another +interesting extension of our work would be to consider +the impact of the memory on parameter estimation for +binaries with precession. However, we expect that our key +finding — that the memory signal can only play an im- +portant role in BBH parameter estimation when there is +limited information from the inspiral — holds generically, +due to the different orders of magnitude of the primary +and memory signal characteristic strains. We also leave +open the possibility that some currently unforeseen effects +may lead to a much larger degradation of the primary +signal than the one due to the presence of gaps, which +could make the memory information more relevant to +parameter estimation. +As a final remark, we note that even if the informa- +tion in the memory turns out not to be very useful in +constraining the binary parameters, the amount of events +with detectable memory we found for LISA (c.f. Table II) +suggests that it may still play a significant role as a test +of GR in the strong-gravity nonlinear regime, since most +of the memory is generated close to merger. We leave +these questions for future work. + +14 +ACKNOWLEDGMENTS +The authors would like to thank Juan Calderon Bustillo, +Xisco Jimenez Forteza, Giada Caneva and Marc An- +drés for their technical help in the first stage of this +project. +We are also grateful to Neil Cornish for his +valuable comments on a draft of the paper. +In this +study we used the software packages matplotlib [121], +numpy [105], +scipy [122], +LISA Sensitivity [104], +gwmemory [59], GWsurrogate [123], surfinBH [93], and +qnm [92]. RV is supported by grant no. FJC2021-046551-I +funded by MCIN/AEI/10.13039/501100011033 and by +the European Union NextGenerationEU/PRTR. +RV +also acknowledges support by grant no. +CERN/FIS- +PAR/0023/2019. DB is supported by a ‘Ayuda Beatriz +Galindo Senior’ from the Spanish ‘Ministerio de Uni- +versidades’, grant BG20/00228. +The research leading +to these results has received funding from the Spanish +Ministry of Science and Innovation (PID2020-115845GB- +I00/AEI/10.13039/501100011033). +IFAE is partially +funded by the CERCA program of the Generalitat de +Catalunya. This work was partly enabled by the UCL Cos- +moparticle Initiative. EB acknowledges support from the +European Union’s H2020 ERC Consolidator Grant “GRav- +ity from Astrophysical to Microscopic Scales” (Grant No. +GRAMS-815673) and the EU Horizon 2020 Research and +Innovation Programme under the Marie Sklodowska-Curie +Grant Agreement No. 101007855. +Appendix A: Signal processing +In this section we provide more details about our choices +in manipulating the BBH waveforms. We first generate +the primary signal with a sampling time ∆t = 1/4 s, and +we subsequently generate its memory via the GWmemory +package [89]. As explained in Sec. III, we compute the +total signal in frequency domain, summing the individual +FFTs of the primary waveform and of the memory. How- +ever, we find that a spurious contribution of the primary +waveform at frequencies f < fin generates cross-terms +between the primary and the memory signal of order O(hc) +which affect the computation of the SNR and the Fisher +matrix. To prevent these artefacts from affecting our +results, we removed the contribution from f < fin of the +primary signal before summing the individual FFTs. +We follow a standard procedure to manipulate the +primary waveform, namely, applying a window function, +padding the signal, and taking the FFT. We apply the +following window function to the primary signal: +z(t) = 1 +4 +� +1 + tanh +� t−t0 +σ0/4 +��� +1 − tanh +� t−th +σh/4 +�� +, +(A1) +with Mt0 = 150 and Mth = 110, respectively, at the +beginning and at the end of the time series. The duration +of the windowing is set by Mσ0 = 50 and Mσh = 20. +For the memory signal we follow a different procedure. +We first extend the generated δh(t) (evaluated numerically +for t0 ≤ t ≤ tf) at the beginning and the end with the con- +stant values δh(t0) and δh(tf), respectively, using the same +padding length as for the primary signal. Subsequently, +we apply the following window (as in Ref. [124]): +w(t) = +� +� +� +� +� +1, +t − td < 0 +1 +2 +� +1 + cos[2πfd(t − td)] +� +, +t − td ≥ 0 +0, +t − td ≥ +1 +2fd +(A2) +and choose Mtd = 140. +The choice of the decay fre- +quency of the window function fd greatly impacts the +spectral shape of the memory at low frequencies, as +can be seen in Fig. 9, where we show the character- +istic strain hc of the memory for different values of fd ∈ +{10−2, 10−3, 10−4, 10−5} Hz. Note that higher values of fd +inject spurious power at the frequencies for which LISA +is most sensitive, thus leading to an artificial increase of +the respective memory SNR ρm = {7.37, 5.35, 5.13, 5.12}. +Thus, while taking a lower fd is more reliable, in the +sense that it does not overestimate the memory SNR, it +implies a corresponding longer observation time of the +memory, which can become inconsistent with the max- +imum observation time considered for the primary signal. +However, this does not pose a real problem since most of +our analysis applies to cases where the primary signal is +observed for a few hours, whereas the SNR of the memory +does not greatly change as long as the memory is observed +for more than 15 minutes (fd ≲ 10−3 Hz). +In computing the SNR and the Fisher matrix we +take fmin = 1/T, where T is the total length of the +signal, and fmax = min{1 Hz, f440}, since we find that +the QNM f440 ≡ ω440/2πMz is a good measure of the +maximum frequency present in the signal (note that our +waveforms include modes up to ℓmax = 4). +For total +masses between [104, 105]M⊙ we find that fixing the min- +imum frequency at fmin = 10−4 Hz does not change our +SNR and Fisher forecasts. +Appendix B: Analytic considerations on the +distance-inclination Fisher matrix +Here we review the Fisher matrix derivation of the +(dL, ι) degeneracy by computing the relative 2 × 2 matrix +analytically. Including other parameters have little effect +close to the degenerate points, since the main source of +error comes from this submatrix. Subsequently, we show +that the results represented in Fig. 3 can be understood by +taken into account simply the main angular dependence +of the primary and the memory signals. +In order to +estimate the effect of the memory on this degeneracy it is +enough to focus on the part of the Fisher matrix regarding +the extrinsic parameters {dL, cos(ι), φ, ϕc} since, at linear +order in SNR−1, it is decoupled from the one associated to +the intrinsic parameters {Mz, q, tc, ψ, Spins} [15], where ψ +is the polarization angle. +Indeed, while the intrinsic +parameters are mainly extracted from the phase evolution +of the waveform, the extrinsic parameters depend on + +15 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +f [Hz] +10-22 +10-20 +10-18 +10-16 +hc(f) +fd= 0.01 +fd= 0.001 +fd= 0.0001 +fd= 1e-05 +Figure 9. Characteristic strain hc(f, ι, Φ) of the memory for dif- +ferent choices of decay frequency of the window function (A2). +The parameters of the binary are the same as those of the +“light” binary in Fig. 2, but with an inclination ι = 90 deg. +the amplitudes h+ and h× [16]. +Here we ignore the +dependence on the coalescence phase ϕc, since (at leading +order) it does not affect the memory. At Newtonian (0 +PN) order the primary waveform is +h+,0 = 2ηMz +dL +[Mω(t)] +2 +3 (1 + cos2 ι) cos[2ϕ(t)], +(B1) +h×,0 = 4ηMz +dL +[Mω(t)] +2 +3 cos ι sin[2ϕ(t)], +(B2) +using the polarisation conventions of Ref. [87]. The GW +amplitude in the detector can be written in the frequency +domain as [104] +˜h(f) = F +(f)˜h+(f) + F ×(f)˜h×(f), +(B3) +where F +,×(ι, φ, ψ, f) are the frequency-dependent de- +tector response functions, which also depend on the source +sky-location and polarization angle. +Substituting the +primary waveform FTs in the last expression we find +˜h0 = κ0 +dL +� +F +(1 + cos2 ι) − 2iF × cos ι +� +, +(B4) +where κ0 is independent of both the luminosity distance +(for fixed Mz) and the inclination angle. The sky- and +polarisation-averaged Fisher matrix (8) has then the +form11 +Γ0 +ij = +�ρκ0 +dL +�2 +ˆΓ0 +ij, +(B5) +11 Where we used ⟨F +(f)F ×∗(f)⟩ = 0 for the sky- and polarisation- +averaging of the cross terms [104]. +with i, j ∈ {log dL, ι}, where ρ2 +κ0 = (κ0|κ0) and the matrix +ˆΓ0 = +� +(1 + cos2 ι)2 + 4 cos2 ι (3 + cos2 ι) sin(2ι) +(3 + cos2 ι) sin(2ι) +4(1 + cos2 ι) sin2 ι +� +, +depends only on the inclination. This matrix is clearly sin- +gular for face-on/off binaries, and it is diagonal for edge-on +ones (implying that the two parameters are uncorrelated), +ˆΓ0(ι ∈ {0, π}) = +� +1 0 +0 0 +� +, +ˆΓ0(ι = π +2 ) = +� +1 0 +0 1 +� +. +It is easy to see that for inclination angles 0 < ι < π +2 +( π +2 < ι < π) the distance and the inclination are negatively +(positively) correlated. +This result shows that the degeneracy we focused in +in this work is driven by the dependence of the (leading) +quadrupole waveform on the inclination, and the particu- +lar combination of plus and cross polarisations measured +by the detector, which, in particular, lead to a singular +Fisher matrix for face-on/off configurations. This is a +well-known issue in the literature and special care must +be taken close to the singular points, where one should +use a beyond-Gaussian analysis [101, 102]. However, for +these face-on/off configurations the memory is almost van- +ishing, so that in this work our focus is on intermediate +inclination angles that are not too close to the singular +points. Despite that, in the main text our analysis in- +cludes higher modes, which break the complete degeneracy +(“regularising” the Fisher matrix) (c.f. Fig. 6). +Now we repeat the above computation, but for the +memory signal. Using the 0 PN waveform in Eq. (6) we +find that (in the frequency domain) the GW memory at +the detector is +� +δh = κmF + +dL +sin2 ι(17 + cos2 ι), +(B6) +with a factor κm independent of both the distance and +the inclination, and such that κ0/κm ∼ O(100). The sky- +and polarisation-averaged Fisher matrix of the memory is +Γm +ij = +�ρκm +dL +�2 +ˆΓm +ij, +(B7) +where ρ2 +κm = (κm|κm) and the matrix elements +ˆΓm +log dL,log dL = sin4 ι +2 +(17 + cos2 ι)2, +ˆΓm +log dL,ι = − sin(2ι) sin2 ι (8 + cos2 ι)(17 + cos2 ι), +ˆΓm +ι,ι = 2 sin2(2ι)(8 + cos2 ι)2, +depend only on the inclination. Because of the simple +structure of the memory signal (at leading order), its +Fisher matrix is singular for all inclination angles. This +is not an issue, since this singularity is cured through the +inclusion of (subleading) higher modes of the memory. +Contrarily to what happens with the primary signal, here +the distance and the inclination are positively (negatively) + +16 +correlated for inclination angles 0 < ι < +π +2 ( π +2 < ι < +π). This opposite behaviour is nicely illustrated by the +orthogonality of the two confidence ellipses in Fig. 3. +Note that although these results were derived for the 0 +PN waveforms, this picture still holds generically, since it +relies mostly on the leading dependence on the inclination. +The Fisher matrix for the total (primary + memory) +waveform Γtot includes additional cross-terms, +Γtot +ij = Γ0 +ij + Γm +ij + (∂i� +δh|∂j˜h0) + (∂i˜h0|∂j� +δh). +(B8) +Above we focused on Γ0 +ij and Γm +ij. This is because we veri- +fied that, due to the rapid oscillations of the integrands in +the cross-terms, the individual Fisher matrices dominate +with respect to those. +Appendix C: Numerical Fisher matrix +To calculate the Fisher matrix elements we need to +numerically compute derivatives of the waveform. 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Wongwathanarat, Modeling +core-collapse supernovae gravitational-wave memory in +laser interferometric data, Phys. Rev. D 105, 103008 +(2022), arXiv:2109.01582 [astro-ph.HE]. + diff --git a/y9FQT4oBgHgl3EQfBzWr/content/tmp_files/load_file.txt b/y9FQT4oBgHgl3EQfBzWr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2a82277dd52f0b89cbb740dd0cb7350db43d501 --- /dev/null +++ b/y9FQT4oBgHgl3EQfBzWr/content/tmp_files/load_file.txt @@ -0,0 +1,1796 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf,len=1795 +page_content='Can gravitational-wave memory help constrain binary black-hole parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' A LISA case study Silvia Gasparotto,1, ∗ Rodrigo Vicente,1 Diego Blas,1, 2 Alexander C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Jenkins,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3 and Enrico Barausse4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 5 1Institut de Fisica d’Altes Energies (IFAE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The Barcelona Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Campus UAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 08193 Bellaterra (Barcelona),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Spain 2Grup de Física Teòrica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Departament de Física,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Universitat Autònoma de Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 08193 Bellaterra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Spain 3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' London WC1E 6BT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' United Kingdom 4SISSA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Via Bonomea 265,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 34136 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Italy and INFN Sezione di Trieste 5IFPU - Institute for Fundamental Physics of the Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Via Beirut 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 34014 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Italy Besides the transient effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' the passage of a gravitational wave also causes a persistent displacement in the relative position of an interferometer’s test masses through the nonlinear memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This effect is generated by the gravitational backreaction of the waves themselves, and encodes additional information about the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In this work, we explore the implications of using this information for the parameter estimation of massive binary black holes with LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Based on a Fisher analysis, our results show that the memory can help to reduce the degeneracy between the luminosity distance and the inclination for binaries observed only for a short time (∼ few hours) before merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' To assess how many such short signals will be detected, we utilized state-of-the-art predictions for the population of massive black hole binaries and models for the gaps expected in the LISA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We forecast from tens to few hundreds of binaries with observable memory, but only ∼ O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1) events in 4 years for which the memory helps to reduce the degeneracy between distance and inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Based on this, we conclude that the new information from the non-linear memory, while promising for testing general relativity in the strong field regime, has probably a limited impact on further constraining the uncertainty on massive black hole binary parameters with LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' INTRODUCTION The direct detection of gravitational waves (GWs), pre- dicted by Einstein in 1916 [1], is one of the greatest accomplishments in modern physics, showing (at present) a spectacular agreement with the theory of general relativ- ity (GR) [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' By now, almost a hundred GW signals have been observed and interpreted as resulting from the coalescence of compact binaries by LIGO/Virgo [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As the sensitivity of current detectors improves and new detectors become available, it will be possible to estim- ate the binary parameters more accurately and to find GWs from new types of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This will allow us to not only better test GR in its strong-field regime, but also to probe astrophysics, cosmology and fundamental physics [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The future space-borne detector Laser Interferometer Space Antenna (LISA) [11] will play a key role in this quest, due to both its expected high signal-to- noise ratio (SNR) measurements and the rich population of sources expected to inhabit its frequency band (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1 mHz to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1 Hz) [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Progress may still be hindered by the fact that some binary parameters — such as the luminosity distance dL and inclination of the orbital plane with respect to the line of sight ι — may be highly correlated in GW sig- nals, limiting our ability to accurately estimate them [15– 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is especially important in the context of standard sirens [18, 19], where the precision with which ∗ sgasparotto@ifae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='es one can estimate the present-day Hubble parameter H0 depends primarily on how accurately one can meas- ure dL [17, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Indeed, this was the main con- tribution to the large uncertainty on the first estim- ate of H0 from GW170817 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The distance- inclination degeneracy can be simply understood by noting that, at leading (Newtonian) order, an inspiralling binary sources the two GW polarisations h+ ∝ (1 + cos2 ι)/dL and h× ∝ cos ι/dL [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' if the detector network is mostly sensitive to one particular combination of h+ and h×, the luminosity distance and inclination are therefore degener- ate [15, 17] (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The degree of this degeneracy depends on the sky location of the binary and the specific detector network [17], and can be greatly reduced by the observation of the afterglow light curve of an electromag- netic counterpart (which critically depends on ι) [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Interestingly, the degeneracy may also be mitigated by us- ing subleading effects in the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Examples include the effect of spins misaligned with the orbital angular momentum [26–28] (which lead to the precession of the orbital plane), of higher multipole modes (HMs) [29–34] (in particular, for unequal component masses), or using binary Love relations [35] (for neutron star binaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1 Another subleading effect in the GW signal with the potential to break the distance-inclination degeneracy is 1 Nevertheless, in the ∼ 100 GW signals observed to date there is only limited evidence for higher multipole content (with no evidence at all beyond ℓ = 3) [36] and only one measurement of strong-field precession has been claimed [37] (though some doubt has been cast on this claim due to data-quality issues [38, 39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='13228v1 [gr-qc] 30 Jan 2023 2 the nonlinear (Christodoulou) GW memory [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is a well-grounded prediction of GR which originates from a change in the radiative multipole moments of the gravita- tional field sourced by the flux of gravitational radiation itself, resulting in a permanent displacement of free-falling test masses upon the passage of GWs [40–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 While essentially any source of GWs will generate nonlinear memory,3 our focus here (and in much of the relevant literature) is on binary black holes (BBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The reason for this is twofold: firstly, binaries are the one source of detectable GWs we definitively know to exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' and secondly, the amplitude of the memory scales with the total GW en- ergy radiated, which for binaries is ∼ 1 − 10% of the total mass [52], favouring BBHs over lighter binaries containing neutron stars or white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The gravitational wave memory modifies the BBH wave- form by introducing a slowly-growing offset of the oscilla- tions that builds over the whole coalescence and whose time evolution follows that of the instantaneous orbital frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This shift rises over the radiation-reaction timescale during the inspiral [44, 45, 53] and accumulates rapidly during the merger before saturating to its final value during the ringdown [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Although the memory arises from a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 PN nonlinear interaction in a post- Newtonian (PN) expansion of Einstein equations, because it accumulates over the whole coalescence, it affects the gravitational waveform at leading (Newtonian) order [54], increasing the prospects of observing it in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Several searches for memory from BBHs have been per- formed using LIGO/Virgo data, returning only null results thus far [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is in agreement with forecasts for LIGO/Virgo, which show that the detection of memory from a single event would require a much more massive and nearby binary than any yet observed, and that to find collective evidence of memory in the total population of observed binaries one would need ∼ 5 yr of collected data [59–61] (or ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 yr taking into account the expected improvement of detector sensitivities [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The difficulty in detecting the memory with current ground-based inter- ferometers resides mostly in the fact that, besides being responsible for only a small amplitude offset, its power is larger at lower frequencies where the detectors sensitivity is limited by several sources of noise [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, the prospects for memory detection in single events are consid- erably better for third-generation ground-based detectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Einstein Telescope [64] and Cosmic Explorer [65]) and for the future space-based detectors LISA and Tian- Qin, due to their better sensitivity and low frequency coverage [54, 62, 66–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 2 A similar effect sourced by the flux of matter or non-gravitational radiation was actually discovered first and is called the linear GW memory [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Hereafter, we use “memory” to refer exclusively to Christodoulou’s GW memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3 See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [50, 51], which studied the nonlinear memory generated by cosmic string loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 4 Memory from the merger of supermassive BHs is also a target of In this work we investigate the impact of the nonlinear memory on parameter estimation via a Fisher matrix analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In particular, we focus on how the memory signal breaks the distance-inclination degeneracy, which is crucially important if these binaries are to be used as standard sirens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We found that the information of the memory can indeed reduce the uncertainty on the luminosity distance by reducing its correlation with the inclination angle, whereas it has almost no impact on the uncertainty of the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For LISA sources its greatest effect involves cases where (i) the constituent BHs are light enough [MBH ≲ 105 M⊙/(1 + z), with z the source’s cosmological redshift] that the merger takes place near the upper edge of the LISA band, and (ii) the information from the primary waveform is limited to a few cycles before the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The presence of gaps in the data stream and confusion noise from other sources will reduce the effective dura- tion of usable LISA data, and thus the observed number of cycles for BBHs [74], making the memory potentially useful for the distance estimation of some BBH events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Therefore, using the state-of-art astrophysical BBH popu- lation models described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [75, 76] (and based upon previous work presented in [77–79]), we assess quantitat- ively the impact of the memory in the distance estima- tion of LISA sources, taking into account the presence of gaps in the data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Considering the particular gap model used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [80] we did not find any significant enhancement in the distance estimation by the inclusion of memory on the BBH waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We do however find a greater number of events with detectable memory at LISA as compared to previous forecasts [67, 69], especially in our models with heavy BH seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II we re- view the computation of the memory and describe the phenomenology of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' III we describe our Fisher forecasting analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV we present our results for the distance-inclination inference of individual BBHs, and discuss the impact of the binary parameters and signal duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' V we perform population- level forecasts for LISA using synthetic BBH catalogues, and assess the impact of including the memory on the luminosity distance estimation in the presence of gaps in the data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Some technical material is discussed in the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We use geometric units throughout (c = G = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' GW MEMORY WAVEFORM II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Computation scheme: Thorne’s formula The most direct way to compute the memory contri- bution to waveforms would be to extract it directly from Pulsar Timing Arrays (PTAs) [70, 71], but searches in PTA data thus far have returned only null results [72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3 numerical relativity simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, most simula- tions to date have struggled to accurately capture this information for a number of reasons (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Some exceptions are Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [81], where the dominant memory mode (ℓ, m) = (2, 0) was first resolved, and the recent work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [82], which used a Cauchy-characteristic extraction (CCE) technique to extract the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Al- ternatively, the Bondi, van der Burg, Metzner, and Sachs (BMS) balance laws [83] have recently been used to add the memory to waveforms [84, 85] (see also [57, 69]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Instead, in this work we use a perturbative approach to evaluate the memory [44, 46], which we now briefly review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' A GW strain h0 (which we call the “primary” signal) sources an additional memory strain δh, which can be expressed in the transverse-traceless (TT) gauge using Thorne’s formula [86], δhTT ij (u) = 4 R � u −∞ du′ � R dΩ d2EGW du′dΩ � ninj 1 − nkN k �TT , (1) where the angular integral is over the solid angle dΩ of a (large) sphere of radius R surrounding the source, and ni and N i are the unit radial vectors pointing, respectively, to dΩ and the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The TT superscript represents a TT projection with respect to the direction of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The time integral is over the entire history of the source up to retarded time u, which shows that the memory is a hereditary effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The GW energy flux carried by the primary GW is [44] d2EGW dtdΩ = R2 16π (˙h2 0,+ + ˙h2 0,×), (2) where ˙h ≡ dh/dt and h+,× ≡ hTT ij eij +,×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We use the same choice of TT-polarisation tensors eij +,× as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' From the spin-weighted spherical harmonic decomposition h+ − ih× ≡ � ℓ≥2 � |m|≤ℓ hℓm(u, r) −2Yℓm(ι, φ), (3) it is possible to show that the sourced memory can be expressed as [56] δhℓm(u) = −R � ℓ′,ℓ′′≥2 � m′,m′′ � (ℓ − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (ℓ + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' × � dΩ Y ∗ ℓm −2Y ∗ ℓ′m′ −2Yℓ′′m′′ � u −∞ du′ ˙h∗ℓ′m′ 0 ˙hℓ′′m′′ 0 , (4) which allows us to straightforwardly compute the memory modes δhℓm from the primary waveform modes hℓm 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We can then reconstruct δh+ and δh× from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (3) to give the total strain h ≈ h0 + δh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In principle, this process should be iterated to give higher-order contributions (the “memory of the memory” [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In practice, these extra terms are subleading, and it is sufficient for our purposes to consider just the leading-order memory effect, δh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' To generate our primary waveforms, we use the surrog- ate NRHybSur3dq8 model [88], which includes all higher spherical harmonic modes up to (ℓ, |m|) = (4, 4) and is con- siderably more accurate than other often-used phenomen- ological models in modelling the merger stage of BBH coalescences [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We use the publicly available GWmemory package [89] to implement the calculation scheme de- scribed above for the corresponding memory signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Equation (4) is valid on a background Minkowski space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' It can be extended to a spatially flat Friedmann- Lemaître-Robertson-Walker (FLRW) spacetime using the fact that, for sources at the same luminosity distance dL, the memory amplitude in FLRW is enhanced over the Minkowski case by the redshift factor (1 + z) [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Additionally, we shall use the time at the detector t ≡ tpeak − (1 + z)(upeak − u), where tpeak is the instant when the primary strain reaches its peak amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Summar- izing, in this work we use δhℓm FLRW(t) = (1 + z)δhℓm Mink � u(t) � R→dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (5) This can be shown to be equivalent to using redshifted component masses Mi,z ≡ (1 + z)Mi, with i ∈ {1, 2}, and luminosity distance dL to generate the primary signal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', with NRHybSur3dq8), plugging this primary directly into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (4), and identifying (R, upeak − u) → (dL, tpeak − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We omit the subscript “FLRW” throughout, but a spatially flat FLRW is implicitly assumed in all our expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Phenomenology of the signal Equation (4) shows how the memory modes are sourced by pairs of primary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For a BBH coalescence oc- curring in the x-y plane, the primary modes have the form hℓm 0 ∝ e−imϕ(t) with ϕ(t) the orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' So, from the time integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (4) it is clear that the lead- ing contribution to the memory modes at low frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', u → +∞) comes from the non-oscillatory (DC) terms with m′−m′′ = 0 which accumulate in time [44, 45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' these source m = 0 memory modes (from the angular integ- ral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Note, however, that oscillatory m ̸= 0 memory modes do become dominant at high frequencies (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The memory sourced in the quasi-circular inspiral of non-spinning BBHs is known analytically up to 3 PN [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Due to the accumulation over the in- spiral, the non-oscillatory contribution to the memory enters at Newtonian (0 PN) order in the waveform, δh(0PN) + = ηMz 48 dL [Mω(t)] 2 3 sin2 ι (17 + cos2 ι), (6) and δh(0PN) × = 0, with the conventional choice of po- larization triad [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The orbital frequency is ω(t) ≡ ˙ϕ(t) and the symmetric reduced mass η ≡ M1M2/M 2, where M ≡ M1 + M2 is the total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The memory has the same scaling as the Newtonian primary waveform (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (B1)), but rather than the main time depend- ence coming from the oscillatory term, its time evolution 4 10-5 10-4 10-3 10-2 10-1 f[Hz] 10-23 10-22 10-21 10-20 10-19 hc(f) (2, 0) (4, 0) (2, 1) (3, 1) (4, 1) (2, 2) (3, 2) (4, 2) 1/60Mz fpeak 2fpeak QNM1 QNM2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Memory characteristic strain hℓm c (f) ≡ 2f| � δhℓm| of the most important modes computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We consider the non-spinning “heavy” BBH studied in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 2 and 3, with total mass M = 2 × 105 M⊙, mass ratio q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 and redshift z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The non-oscillatory (ℓ, m) = (2, 0) mode dominates at low frequencies, but is suppressed at f ≳ 1/60Mz, where oscillatory m ̸= 0 modes start becoming important (in particular, at their maxima f ∼ mfpeak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We can also see the presence of the ringdown in the memory spectrum in the form of high-frequency peaks [QNM1 is at f = (τ −1 221 + τ −1 222)/2πMz, and QNM2 at f = τ −1 222/πMz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (QNMs were computed using the Python package qnm [92], and the final mass and spin of the remnant BH via surfinBH [93], whose fitting procedure is described in [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=') is captured by the instantaneous orbital frequency ω(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This explains the typical step shape of the memory, which has a steep increase in the merger-plunge phase and a saturation during the ringdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Moreover, the memory is characterized by a different dependence on the inclination angle ι and an overall amplitude ∼ 20 times weaker than the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In particular, the two signals have oppos- ite monotonic dependence on the inclination angle, and while the primary signal is maximised for face-on binaries (ι = 0), the memory is instead maximised for edge-on bin- aries (ι = π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This behaviour is maintained when using primary waveforms generated by NRHybSur3dq8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The dif- ferent dependence on ι is what makes the memory helpful in reducing the (ι, dL) correlation (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Figure 1 shows the spectral shape of memory mode char- acteristic strains hℓm c (f) ≡ 2f|� δhℓm(f)|, with � δh(f) ≡ � dt e−i2πftδh(t) the Fourier transform (FT) of δh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' All modes exhibit a plateau at low frequencies, but the spec- tral content of the non-oscillatory (m = 0) modes is clearly distinct from the oscillatory (m ̸= 0) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The plateau of the m = 0 modes is easily understood from the approximation δhℓ0 ≈ ∆hℓ0H(t − tpeak), with H the Heaviside step function and where ∆hℓ0 scales with the fraction of radiated EGW that sources δhℓ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' this results in a constant hc ≈ ∆hℓ0/π [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Taking into account that the memory growth is not in- stantaneous, but happens in τ ∼ 60Mz (the timescale over which most of EGW is radiated [95]), one can understand the suppression at f ≳ 1/60Mz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For the m ̸= 0 modes the low-frequency plateau has a similar origin, but its value is always subdominant because, due to the oscil- lations in the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (4), the memory does not ac- cumulate, averaging out to a net small value that depends strongly on the value of the orbital phase at which the BHs merge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' this is also expected from PN results [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We note that for m ̸= 0 the maximum of the strain scales as mfpeak where the peak frequency fpeak ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1/πMz roughly corresponds to the moment in which most of the energy is radiated [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The oscillations at the left of these maxima are numerically stable (in particular, they are not artefacts of our FT) and come from inter- ference between different ℓ-modes in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' On the other hand, the high-frequency peaks seen in all modes are associated with the ringdown stage and are located at f ≈ (τ −1 ℓ′m′n′ + τ −1 ℓ′′m′′n′′)/2πMz, where the complex quasi- normal modes are σℓmn ≡ (ωℓmn+iτ −1 ℓmn)/Mz [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' It’s in- teresting to note that these peaks occur at a frequency set by the QNM decay rate τ −1 ℓmn (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' imaginary frequency), not the oscillatory part ωℓmn (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' real frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This can be confirmed analytically using Favata’s minimal waveform model (MWM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (14) of [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Figure 2 shows the characteristic strain hc(f, ι, ϕc) ≡ 2f|�h+ − i�h×| of the primary and memory – containing all modes up to (ℓ, |m|) = (4, 4) – for two binaries with different total masses and redshifts, seen from a fixed dir- ection ι = 40 deg and ϕc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' At this particular direction, the primary characteristic strain is O(102) greater than 5 10-5 10-4 10-3 10-2 10-1 100 f [Hz] 10-22 10-20 10-18 10-16 hc(f) 2 × 105M ⊙ 2 × 104M ⊙ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Characteristic strain hc(f, ι, ϕc) ≡ 2f|�h+ − i�h×| of the primary (solid curve) and memory (dashed curve) computed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (3) and (4), seen from a fixed direc- tion ι = 40 deg and ϕc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We consider two fiducial non- spinning BBHs, which will also be used in the following sections: a “light” binary (in violet) with total mass M = 2 × 104M⊙ at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5, and an “heavy” binary (in blue) with M = 2 × 105M⊙ at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Both BBH have mass ratio q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We consider the last 25 cycles before merger for both BBHs (which corresponds to ∼ 6 minutes for the “light” source, and ∼ 2 hours for the “heavy”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Thus, the memory adds information to para- meter estimation only if the number of cycles that can be observed during inspiral is limited (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' this may happen due to gaps in the data stream and/or confusion noise from other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Indeed, whereas truncating the primary waveform at some minimum fin (related to the time/cycles prior to merger) significantly reduces the SNR of the primary, the SNR of the memory is almost unchanged (as also noted in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [62, 69]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As long as the memory is observed in LISA for a period of at least 103 s ≈ 15 min after merger, its SNR is approxim- ately independent of the observation time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In this work we focus on quasi-circular and non-spinning BBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For the dependence of the memory on the spins, mass ratio and eccentricity, we refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [57, 68, 81, 89, 97, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' FISHER MATRIX AND PARAMETER ESTIMATION We use a Fisher matrix analysis (commonly used in GW astronomy [15, 99–103]) to quantify the impact of GW memory on the parameter estimation of the BBH source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We only consider events with merger occurring during the mission lifetime of LISA, since the memory is mainly generated during this phase of the binary evol- ution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The total signal h = h0 + δh is generated in time-domain and is given by the sum of the (primary) sur- rogate NRHybSur3dq8 waveform h0, and the memory δh computed through the GWMemory package [89] from the primary waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In this work we focus on non-spinning binaries, so the input parameters for the waveforms are the redshifted total mass Mz = (1+z)M, the mass ratio q, the luminosity distance dL, the inclination angle ι, and the coalescence phase ϕc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' thus, Θ = {ln Mz, q, ln dL, ι, ϕc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We only investigate the dependence on these parameters, meaning that we ignore the particular sky position of the source and the LISA response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Accounting for them should not affect our results, since the LISA orbital motion is a year time scale, whereas the longest inspirals where we see the effect of the memory are of the order of hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the strong-signal limit the probability distribution of the parameters is a multivariate Gaussian distribution centred on the true values Θ = ¯Θ, with the covariance matrix Σij described by the inverse of the Fisher inform- ation matrix Γij, up to corrections of order of the inverse signal-to-noise ratio, Σij = (Γ−1)ij[1 + O(SNR−1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (7) The SNR ρ and the Fisher matrix are computed as ρ2 = (˜h|˜h), Γij ≡ � ∂˜h ∂Θi ��� ∂˜h ∂Θj ���� Θ= ¯Θ, (8) where we have used the standard inner product (˜a|˜b) ≡ 4Re � fmax fmin df ˜a∗(f)˜b(f) Sn(f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (9) The Sn(f) is the sky-position- and polarisation-averaged, but not inclination-averaged, LISA power spectral density, as found in [104], with the confusion noise corresponding to Tobs = 4 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Computing the Fisher matrix thus allows us to find the variance of each parameter and correlation of each pair of parameters due to measurement errors, σ2 i = (Γ−1)ii, cij = (Γ−1)ij σiσj , (10) (with no summation implied over repeated indices here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The total frequency-domain signal ˜h = ˜h0 + δ˜h is given by the sum of the FT of the primary and the memory signals, which we compute numerically via the fast Fourier transform (FFT) implemented in NumPy [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' A we explain in detail our choices for manipulating the signal, such as the padding and the window function applied, while in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' C we elaborate on the numerical computation of the Fisher matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' RESULTS FOR THE DISTANCE-INCLINATION ESTIMATION The goal of this section is to study the impact of the memory on the parameter estimation of intermediate-mass 6 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The 1σ confidence ellipses computed for the primary waveform without memory (in magenta), with memory (in blue), and with only the memory (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For each of those we compare the effect of including higher modes (solid lines) and excluding them (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The left panel shows the result for the “light” binary, whereas the right panel shows it for the “heavy” binary (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' in both cases we consider the last 25 cycles before merger and a line of sight ι = 40 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For the “heavy” binary, the memory has negligible impact, since we cannot distinguish the purple and the blue lines (the green contours of the memory fall beyond the panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' to supermassive BBHs with LISA, with a particular fo- cus on breaking the distance-inclination degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Our motivation is twofold: (i) although nonlinear memory is expected to be observed in single events at LISA, no attention (to our knowledge) has been paid on its impact on the parameter estimation, and (ii) the memory signal has an opposite correlation between distance and inclina- tion cln dL,ι with respect to the primary signal, which can make it useful to break the distance-inclination degener- acy, thus improving the accuracy of distance estimations (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Firstly, using our Fisher analysis we noted that the primary signal alone already constrains the binary in- trinsic parameters quite well, and that the additional information provided by the memory does not constrain them further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' On the other hand, as already anticip- ated, we found that the dependence of the memory on the extrinsic parameters is complementary to that of the primary signal, and can mitigate the uncertainty on the luminosity distance dL and the inclination ι estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3 where we explicitly show the constraints coming from the primary signal and the memory, as well as the effect of including all higher modes (HMs) up to (ℓ, |m|) = (4, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' These results are also summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3, the ellipses computed from the memory signal and the primary waveform are ortho- gonal, this can be intuitively expected from the opposite monotonic dependence on the inclination of the two com- ponents, as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We analytically derive the opposite correlation for the two signals in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' B, where we compute the 2 × 2 Fisher matrix of this pair of parameters {ι, ln dL} for the dominant mode (DM) of the primary waveform and of the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This property leads to a lower correlation cι,ln dL of the overall Fisher matrix, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 4, which can mitigate the error on the luminosity distance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The estimation of the other parameters is less affected, thus justifying our approach to focus on this particular pair of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For signals with sufficiently large SNR or whose sources are at sufficiently high redshifts, the uncertainty on the luminosity distance estimation becomes dominated by weak lensing effects (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [21, 106]), and our Fisher analysis (which neglects lensing) ceases to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, as we show below, the memory is helpful to parameter estimation only for binaries whose primary signal is not very loud, and whose total redshifted mass Light h0,DM h0,HM δhDM δhHM SNR 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3 h0,DM h0,HM hDM hHM σdL/dL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='18 σι [rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='23 Heavy h0,DM h0,HM δhDM δhHM SNR 1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3 h0,DM h0,HM hDM hHM σdL/dL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0107 σι [rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0139 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Summary of the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The first two columns correspond to the results coming from the primary signal alone, whereas the last two correspond to the total (including the memory) waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' While the memory improves significantly the estimation of the distance-inclination of the “light” binary, it does not help much with the “heavy” binary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 1o, ho 1o, Sh 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='50 10,h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 [rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 1000 2000 3000 4000 5000 dL [Mpc]1o, ho 1o,h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='71 [rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='67 15600 15800 16000 dL [Mpc]7 q lnMz lndL ι ϕc q lnMz lndL ι ϕc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='06 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Correlation matrices cij for the “light” binary of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The upper panel shows the results for the primary signal without memory, while the lower panel is for the total waveform including memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The major effect of including the memory is decreasing the (dL, ι) correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' is in the range 104M⊙ ≲ Mz ≲ 105M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The memory of these sources will only be observed if they are sufficiently close z ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5, where lensing effects are not too strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Therefore, our results indicate that the memory has an impact on the distance estimation only in cases where its intrinsic uncertainty (even after adding the memory) is much larger than the contribution due to lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' It is natural to compare the effect of the memory with that of HMs, as the latter can also improve the BBH para- meter estimation and partially break the aforementioned degeneracy [29–32, 107–113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Their main contribution to the SNR comes from extending the signal to higher fre- quencies as compared to the dominant (2, 2)-mode, thus being more relevant when the merger falls well inside the sensitivity curve of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Importantly, HMs have a different dependence on the inclination angle than the DM, which is again why they may be useful in breaking the distance-inclination degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The memory signal, on the other hand, is always subdominant, but can play a significant role, as we will see, when the information from the primary at lower frequencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', from the in- spiral) is absent/degraded, and the memory becomes the only contribution at those frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the following we explain in detail the dependence of memory-assisted parameter estimation on the binary mass, duration of the signal, and line of sight, and we discuss the impact of HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Dependence on the binary mass By evaluating the SNR defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (8), we find that LISA can detect memory (more precisely, its SNR is > 1 [114, 115]) from binary mergers with total redshifted mass in the range [104, 108] M⊙, thus confirming previous results [54, 67, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For lower masses, the memory strain is too weak to be detected, whereas for larger masses the turnover frequency at which the memory drops off falls below the LISA band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Because of the dependence of the memory characteristic strain on the binary mass, LISA will be most sensitive to the low-frequency plateau part of the signal for light binaries, and to the subsequent high- frequency features associated with the merger/ringdown stage for more massive binaries (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 We found that nearly all the SNR of the memory accumulates during the merger and it is maximised for Mz ∼ 106M⊙, in which case the memory can be detectable up to redshift z ∼ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For short/degraded primary signals (with fixed number of cycles), we found that the memory is most helpful in parameter estimation for binaries with total redshifted mass Mz ≲ 105M⊙, in which case the memory falls in the most sensitive part of the LISA sensitivity band, whereas the merger covers only the high-frequency edge of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For this reason, the hierarchy of the SNR between the memory and the primary signals is less severe than for more massive binaries Mz ≳ 106M⊙, whose mergers occur in the middle/low-frequency part of the LISA band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The difference between “light” and “heavy” binaries is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3, which shows the effect of considering short signals (in this case, the last 25 cycles) for two differ- ent binary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the left panel, for the “light” binary, there is a manifest reduction in the parameter uncertain- ties, which is due to the intersection of the primary and memory confidence ellipses that results in the shrinking of the total signal’s confidence ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Such an effect is 5 See also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 8 10 15 20 25 30 35 40 45 50 Ncycles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 σdL, wm/σdL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='14 ρm/ρ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 σdL, wm/σdL 2 ×104M ⊙ 5 ×104M ⊙ 8 ×104M ⊙ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Effect of the memory on the luminosity distance estimation when the primary signal is truncated at increasing number of cycles prior to merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The memory becomes less important for more massive binaries, for which the merger occurs before or close to the peak of LISA sensitivity and the memory adds negligible contribution to the total SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The sources are kept at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 and line of sight ι = 40 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' not present in the right panel, for the “heavy” binary, because, due to the large SNR of the primary signal, its confidence ellipse is already entirely enclosed within the memory’s confidence ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The main factor controlling the relative sizes of the confidence ellipses, and thus the impact of the memory in parameter estimation, is the ratio between the memory and the primary signal SNRs (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Dependence on the signal duration As just discussed, our results show that the most reli- able indicator for how much the memory contributes to constrain the binary parameters (for a fixed line of sight) is the ratio between the SNR of the memory and of the primary signal, ρm/ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This ratio depends on the total mass of the binary (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' I), but it is also a function of the inclination and the time duration of the data taken before the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Decreasing the mass and the duration of the signal prior to merger tends to increase this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We quantify the improvement in parameter estimation in terms of the ratio between the standard deviation of the luminosity distance with and without the memory, σdL,wm/σdL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Our results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 5 for different masses in the range [104, 105]M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the upper panel, we show the σ-ratio as a function of the number of cycles Ncycles ob- served prior to merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We note that for any given number of observed cycles, the memory of more massive binaries leads to a smaller improvement in distance estimation as compared to lighter binaries (as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the lower panel, we show that there exists a mono- tonic, one-to-one relationship between the σ-ratio and the ρ-ratio, for a fixed inclination, which is approximately insensitive to the mass of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This observation will allows us to determine, in the next section, the “crit- ical” ρ-ratio needed to achieve a given improvement in the distance estimation as a function of the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Dependence on the inclination Fixing the other parameters, the impact of the memory depends strongly on the inclination of the binary, as we show in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In this plot we present the relative uncertainty in the luminosity distance at different inclination angles, considering the “light” binary (discussed previously) with mass-ratio q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 and fixing the observed signal to 25 cycles prior to merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We restrict the plot to values of ι ∈ [0, π/2], since its behavior is symmetric with respect to ι = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Note that, in the absence of the memory and the HMs, the uncertainty on the luminosity distance diverges for face-on ι → 0 (and face-off ι → π) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As we discuss in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' B, this is due to the fact that the Fisher matrix is singular 9 20 30 40 50 60 70 80 90 ι[deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 σdL/dL h0, HM hHM h0, DM hDM 30 40 50 60 70 80 90 ι[deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 ρm/ρ0 q = 3 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Upper panel: the effect of the memory on the estimation of the luminosity distance as a function of the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The parameters of the source are those of the “light” binary of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3 with q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2, and the number of cycles before merger is Ncycles = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Lower panel: the ρ- ratio required to achieve a 10% improvement in the luminosity distance estimation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', such that σdL,wm/σdL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='9) as a function of inclination for two different values of the mass- ratio q = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' at this inclination and, thus, the Fisher analysis for the primary DM ceases to be valid in a neighborhood of ι = 0 (and of ι = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, for unequal-mass binaries the HMs regularize the Fisher matrix, and even a slightly asymmetric binary as q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 has a relative uncertainty in the distance smaller than ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='8 for all inclination angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6 we compare the impact of HMs on the distance estimation with that of the memory for different inclination angles, keeping the mass-ratio q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, we also verified that HMs become more relevant the more asymmetric the binary is, whereas the impact of the memory is larger for smaller q ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='6 The large effect of the memory observed in the plot – improving the distance estimation by more than a factor of 4 for some inclinations – is partially related to the short duration of the signal considered (truncated at 6 minutes prior to merger);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2, for longer signals the information in the primary (which accumulates over the inspiral) eventually becomes dominant, with the memory contributing negligibly to parameter estimation (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6 we show the critical ρ- ratio needed to achieve a 10% reduction in σdL (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', such that σdL,wm/σdL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='9) as function of the inclination, for two different mass-ratios q = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' From the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2, these curves are approximately independent of the binary’s total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Moreover, we see from this plot that they are also only mildly dependent on the mass- ratio q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Another important observation is that closer to face-on a smaller ρ-ratio is needed to achieve a given improvement in the distance estimation compared to edge- on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' So, close to face-on the memory can be relevant even if the primary is observed for relatively long periods (∼ few hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For example, if our “light” binary (with q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2) is seen close to face-on, the inclusion of the memory information leads to a 10% reduction in σdL if the inspiral is observed over less than 6 hours prior to merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Impact of higher modes Here we discuss the contribution of HMs to the SNR of the primary and memory signals, and its consequent influence on the estimation of the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We consider the “light” and “heavy” binaries of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 2, both of them with mass ratio q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 and seen at an intermediate inclination angle ι = 40 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Our results indicate that HMs have a much greater impact on the loudness of the memory than on the primary signal, boosting the memory SNR by ∼ 19%, as opposed to just ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5% for the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is mainly caused by HMs increasing the total radiated EGW, which pushes the low- frequency plateau to larger amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The increase in the memory SNR with HMs has only a mild dependence on the inclination, but it depends strongly on the mass ratio, since for more asymmetric configurations more energy is released into HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6 The influence of the mass-ratio on the impact of the memory can be understood from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (6), δh(0PN) ∝ q/(1 + q)2, which results in the memory characteristic strain f � δh (0PN) ∝ q/(1 + q)2, whereas the primary characteristic strain is [15] f�h(0PN) 0 ∝ √q/(1 + q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 10 Regarding parameter estimation, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' I shows that HMs have a greater impact for the “heavy” binary than for the “light” one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This can be understood from the fact that the HMs add information mostly close to merger (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', at high frequencies), which is therefore masked for the “light” binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is opposed to the memory which, as we have seen, contributes the most to parameter estimation for the “light” binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As discussed in the last section, the impact of HMs increases with the mass-ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' indeed we verified that for q ≲ 8 the relative error in the distance estimation may be reduced by up to a factor of ∼ 2 − 3 with the inclusion of HMs, as compared to the values of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The information contained in the memory and in the HMs is complementary for parameter estimation, due to their different dependence on the mass-ratio and total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' POPULATION FORECASTS In this section we study the detectability of the non- linear memory for realistic population models of massive black holes, and assess its potential impact on parameter estimation considering the presence of gaps in the data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We update the previous forecasts of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [67, 69] for the measurability of the memory in single events with space-based detectors, by using the recent population models described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [75, 76] (and recently used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [80]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Framework In our analysis we consider all 8 models described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [75, 76], each of those corresponding to different as- sumptions about some of the main uncertainties in the cos- mological evolution of massive BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' These involve [75,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 76]: (i) the high-redshift mass function of the “seeds” of the massive black hole population [“light seeds” (LS) origin- ating from population III stars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' or or “heavy seeds” (HS) originating from direct collapse of protogalactic gaseous disks],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (ii) the time delay between the galaxy merger and the corresponding BBH mergers (realistic “delays”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' or “short delays” neglecting the contribution from scales of the order of hundreds of pc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' and (iii) the presence or not of supernova feedback (“SN” or “noSN”) on the accretion disk of massive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For each of the 8 models we compute the memory random realizations of mergers corresponding to 4 years of LISA mission, and present average results over many such realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We consider only the final 20 cycles prior to merger, which is enough to give us a reliable estimate of the SNR of the memory (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Due to the limited parameter space covered by the waveforms NRHybSur3dq8, we restrict the mass-ratio to q ≤ 8 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', we artificially fix q = 8 for every merger with higher values of q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This assumption is stronger for the LS case than for the HS one, since the two cases have quite different mass-ratio distributions, with the latter having a sharper peak close to equal mass (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [75]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Since the waveforms NRHybSur3dq8 do not cover precessing BBHs, we consider only the spin components orthogonal to the orbital plane and restrict them to |χi,ˆz| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='8, i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='7 We removed from the catalogues the sources with Mz ≥ 108 M⊙, because evalu- ating their SNR is computationally very expensive and, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1, they fall outside the parameter space of interest for memory observation with LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We take the inclination angle and the coalescence phase to be uniformly distributed in cos ι ∈ [−1, 1] and ϕc ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Detectability of memory in single events In the different 4-year realisations, we looked for events with SNR of the memory above the threshold value ρth ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Indeed, it has been shown that this is the condition to claim distinguishability of two waveforms (that dif- fer by δh) over the detector noise [114, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='8 Table II summarises our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For each population model we Astrophysical Catalogues Light seeds Heavy seeds SN-delays Ntot = 47 Ntot = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3 Nth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1) Nth = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 (10) ⟨ρ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='04 ⟨ρ⟩ = 6 ρmax = 7 ρmax = 97 noSN-delay Ntot = 191 Ntot = 10 Nth = 6 (1) Nth = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 (4) ⟨ρ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='17 ⟨ρ⟩ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='9 ρmax = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='64 ρmax = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='7 SN-short Ntot = 149 Ntot = 1245 Delays Nth = 1 (1) Nth = 418 (33) ⟨ρ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='04 ⟨ρ⟩ = 1 ρmax = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='01 ρmax = 43 noSN-short Ntot = 1203 Ntot = 1251 Delays Nth = 12 (2) Nth = 392 (29) ⟨ρ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='06 ⟨ρ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1 ρmax = 17 ρmax = 51 Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' SNR of the memory for the astrophysical models of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For each model we consider a random realisation of 4 years of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We denote the total number of events by Ntot and the number of those with SNR above the threshold value ρm > 1 (5) by Nth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The ⟨ρ⟩ and ρmax are, respectively, the average and the maximum ρm in the particular realisation of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For the SN-delays models and the NoSN-delay HS model, the reported values are the averages over 10 realisations of 4-year events, since for these models the total number of mergers are much smaller than for the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 7 Using the waveforms NrSur7dq2 which allow for precessing BBHs, we found that introducing the spin components along the orbital plane generally leads to an increase of the memory SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 8 This is equivalent to require that, if we parametrize the model waveform as h = h0 + λδh with the fudge parameter λ ∈ [0, 1], the statistical error of the fudge parameter is σλ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 11 0 2 4 6 8 10 12 14 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='6 PDF SN delays noSN delays SN short delays noSN short delays 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 log10(Mtot/M ⊙ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 PDF SN delays noSN delays SN short delays noSN short delays Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The redshift and total mass distributions of mergers with observable memory ρm > 1 for the various HS population models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The models with “delays” have mergers typically at lower redshift, explaining the corresponding higher fraction of events with observable memory (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The different redshift distribution translates into slightly different peak locations in the total mass distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='00 PDF SN delays noSN delays SN short delays noSN short delays 0 20 40 60 80 100 120 140 160 180 ι [deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0125 PDF memory events prior Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Left panel: the mass-ratio distribution for the events of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Right panel: the inclination angle distribution of the events with detectable memory of the “SN short delays” model (with Nth = 418) as compared to the prior isotropic (cosine) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' denote by Ntot the total number of events considered in the 4-year realisation, and by Nth the number of those with memory SNR above the threshold ρth = 1 (inside parentheses ρth = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We also indicate the average SNR of the memory ⟨ρ⟩ and its maximum value ρmax in the particular 4-year realisation of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The number of events with significant memory SNR depends strongly on the astrophysical population model, with clear differences between the LS and the HS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Our results are especially promising for the HS scenario, as it suggests that about 75−78% of events for the “delays” model and 31 − 33% for the “short-delays” model will have observable memory with ρm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The numbers inside the parenthesis can be directly compared with the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [69], where the “Q3d” and “Q3nod” models considered there (and presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [116], based on Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [77–79]) can be compared, respectively, with “delays” and “short-delays” HS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For those, we found that about 36 − 40% and 25%, respectively, of the total events have detectable memory with ρm > 5, as opposed to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='7% and 1% for the “Q3d” and “Q3nod” models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The main reason for this large mismatch is that the new population models of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [75, 76] that we used in this work have a different (more realistic) delay model, which shifts the mergers to lower redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Figure 7 shows the distribution of the redshift and the total mass of the events with ρm > 1 for the various HS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The distribution of models with “delays” peaks at lower redshift, while that of “short-delays” models extends up to z ∼ 13, which explains why the former have a bigger fraction of events with memory SNR above the threshold than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Interestingly, we found some events with particularly high SNR (ρm ≳ 50), as can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' these belong to the low redshift 12 tail of the distributions (z ≲ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The location of the peak of the total mass distribution changes slightly for the various models, but it is such that the total redshifted mass is about ∼ 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 8 we show the mass-ratio distribution for the same events of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Most of the events with detectable memory have a mass-ratio close to unity with a sharp suppression at higher values, so that the restriction to q < 8 turns out not to affect our results for the HS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The “noSN delays” model is the only one presenting a mild accumulation of events with q = 8 due to this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The situation is quite different for the LS models, which have a much broader distribution in the mass-ratio, res- ulting in a substantial (fictitious) accumulation of events at q = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' So, we repeated our analysis removing directly the binaries with q > 8 from the catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In this more conservative approach, for “noSN-short delays” we noted a reduction from 9 to 12 events with ρm > 1, but no change in the number of events with ρm > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For “SN- short delays” and “SN delays” models we found no events with detectable memory, and for “noSN-short delays” a reduction from 6 to 4 events with ρm > 1, and no events at all with ρm > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Therefore, for LS population models our results indicate that the prospects of observing the memory with LISA do not seem promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We repeated our analysis of LS population models also for the future generation ground-based detectors Cosmic Explorer (CE) [65] and Einstein Telescope (ET) [117], which have better sensitivity at higher frequencies, and thus to lower masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='9 We have found almost no events with observable memory ρm > 1 in 4 years of observation (there was just one event with ρm ≃ 2 for the “SN short delays” model with ET), and an average memory SNR within 10−1 − 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Impact of the memory on distance estimation Given that HS models predict such a large number of events with observable memory at LISA (which can be almost up to 80% of the total number of events, in the most optimistic scenario), we consider here the impact of the memory on the luminosity distance estimation for these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As we have shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='2 the impact of including the memory is highly dependent on the ratio of the SNR of the memory and the primary signals, with the memory helping substantially to constrain the distance when the information (or the duration) of the primary signal is somehow limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Among other possibilities, this could happen due to the presence of gaps in the data stream, which causes a partial loss of signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 9 We used the sensitivity curve of the configurations ET-D of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [118] and CE_40km_lf of https://cosmicexplorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='org/ sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='html, which have the best sensitivity at low fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Two kinds of gaps are expected at LISA: the sched- uled ones, related to the regular maintenance of the de- tector, and the unscheduled ones, due to unexpected prob- lems/events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [80] it was shown that the scheduled gaps have little or no impact at all on the parameter estim- ation of massive BBHs, but the unscheduled ones could degrade significantly the parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Thus, there is the intriguing possibility that, in the presence of unscheduled gaps, the memory may add useful informa- tion to constrain the binary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='10 To quantify this effect we consider the particular gap model used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [80], which is consistent with a 75% duty cycle, as expected for LISA [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We simulate the presence of gaps by windowing the signals as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [80], considering scheduled gaps with a typical duration of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='5 hours every week and unscheduled gaps with a duration of 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The time interval between two unscheduled gaps is treated as a random variable following an expo- nential probability distribution p(∆T) = λ exp (−λ∆T) with 1/λ = 9 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' With these choices, we simulate the effective data taking of the mission and we distribute the merger times uniformly over the 4-year mission duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' From our study in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IV, we expect the memory to be helpful in constraining the binary (extrinsic) parameters for a particular chunk of data if the merger happens within the first few hours from the last gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For concreteness, let us compute the (average) total number of mergers occurring within 6 hours from the last gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The number of unscheduled gaps can be estimated by Ngap ∼ Tmission/Tgap, where Tgap is the sum of the average time interval between gaps and the gap dura- tion, ⟨∆T⟩ + 3 ≃ 12 days, thus Ngap ∼ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We focus on the “SN-short delays” HS population model, the most optimistic scenario with the highest number of events with observable memory, Nth = 418;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' note that due to the presence of gaps this number is reduced by 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Thus, we can estimate the number of mergers by multiplying the probability of having at least one merger in 6 hours by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75NthNgap, which gives ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We checked this result numerically by simulating 50 times the distri- bution of mergers over the gap realisation and we found consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' To find the number of events for which the inclusion of the memory decreases by more than 5% the uncertainty on the luminosity distance, we computed the ρ−ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', ρm/ρ0) for each event occurring within 6 hours from the last gap in 50 numerical realisations, neglecting the information accumulated in the inspiral before the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Subsequently, we compared those ρ-ratios with the critical values needed to achieve a 5% improvement on the lumin- osity distance estimation, which depend on the particular binary inclination (as in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6, but for σdLwm/σdL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We found that, on average, only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='14 events of the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='4 occurring close after a gap have an 10 Note that the LISA data analysis will be further complicated by the presence of many overlapping signals [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 13 improvement of more than 5% on the distance estimation from including the memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' this corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='04% of the total number of events with observable memory in this population model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We believe that this low value is due to the fact that most of the BBH mergers with observable memory cor- respond to configurations relatively close to edge-on (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 8), where the critical ρ−ratio is much higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Another reason is that the majority of the BBHs with observable memory in the population considered have a redshifted total mass ≳ 106M⊙, whereas as dis- cussed in sec IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='1 the memory is more helpful for lighter binaries Mz ≲ 105M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The “noSN-short delays” HS population model, which is the second most optimistic in terms of number of events with detectable memory, suffers from these same issues and is, thus, expected to give a similarly small result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The other population models have far fewer events with observable memory, thus it is very unlikely that any of these mergers will happen sufficiently close to a gap to have a sufficiently large ρ−ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In summary, applying our Fisher analysis to state-of- the-art synthetic catalogues of massive BBHs indicates that the memory will not help constraining further the binary parameters at LISA, even in the presence of gaps in the data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, there is substantial uncertainty on the assumptions adopted in this analysis, in particular, regarding the population and gap models, and we cannot exclude the possibility that there may exist additional effects leading to a larger degradation of the primary signal than those considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' CONCLUSION In this work we have investigated the prospects of using the nonlinear GW memory to help infer the parameters of merging BBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In particular, we have focused on massive BBHs detections with the future space-based interfero- meter LISA, as these are the most promising individual sources of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Our motivation is to use the additional source of information provided by the memory signal to break the degeneracy between inclination ι and lumin- osity distance dL, which is present in the leading-order GW signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is especially important for attempts to use these BBHs as standard sirens (either via statistical identification of the host galaxy [120], or possibly using an electromagnetic counterpart due to the merger taking place in a gas-rich environment [21, 24, 25]), as the un- certainty on the Hubble constant H0 crucially depends on the uncertainty on dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We find that the memory can indeed play a significant role in breaking this inclination–luminosity distance de- generacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This occurs in cases where the redshifted total mass is relatively small (≲ 105 M⊙), the binary is seen not very close to edge-on, and the observation time is limited to a few hours prior to merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The limitation on the observation time could occur due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=', gaps in the data stream caused by interferometer downtime, or confu- sion noise from the presence of many other simultaneous signals in the LISA frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In order to understand the relevance of these results for the LISA mission, we started by performing a population study using new synthetic catalogues of massive BBHs to forecast the number of BBH events with observable memory (ρm > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' While there are currently large theor- etical uncertainties on the astrophysical processes leading to these mergers, we find a substantially larger number of events with significant memory as compared to previ- ous forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The prospects are particularly bright for the heavy seed model with “short delays” [75, 76], which presents about 400 memory events for a 4-year mission time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' On the other hand, most of the mergers coming from light seed models [75, 76] are undetectable by LISA (and so is their memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Finally, we considered a commonly used gap model, which includes both the scheduled and unscheduled types, to quantify the benefit of the memory in the estimation of the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For the most optimistic “short delays” heavy seed models [75, 76], we found that, out of the ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='75 × 400 observable memory events in a 4- year mission time, just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='14 events will produce a larger than 5% decrease in σdL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Thus, our analysis indicates that the information in the memory signal will not help constraining further the binary parameters at LISA, even in the presence of gaps in the data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is due to the fact that most of the events with observable memory are seen close to edge-on, in which case the luminosity distance and inclination are only slightly correlated in the primary signal and, thus, the information added by the memory is negligible for parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Our study, based on a Fisher matrix analysis, could be further improved by performing a full Bayesian analysis and by investigating the effect of priors on the lumin- osity distance estimation, which is especially important when the parameters are not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Another interesting extension of our work would be to consider the impact of the memory on parameter estimation for binaries with precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, we expect that our key finding — that the memory signal can only play an im- portant role in BBH parameter estimation when there is limited information from the inspiral — holds generically, due to the different orders of magnitude of the primary and memory signal characteristic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We also leave open the possibility that some currently unforeseen effects may lead to a much larger degradation of the primary signal than the one due to the presence of gaps, which could make the memory information more relevant to parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As a final remark, we note that even if the informa- tion in the memory turns out not to be very useful in constraining the binary parameters, the amount of events with detectable memory we found for LISA (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Table II) suggests that it may still play a significant role as a test of GR in the strong-gravity nonlinear regime, since most of the memory is generated close to merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We leave these questions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 14 ACKNOWLEDGMENTS The authors would like to thank Juan Calderon Bustillo, Xisco Jimenez Forteza, Giada Caneva and Marc An- drés for their technical help in the first stage of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We are also grateful to Neil Cornish for his valuable comments on a draft of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In this study we used the software packages matplotlib [121], numpy [105], scipy [122], LISA Sensitivity [104], gwmemory [59], GWsurrogate [123], surfinBH [93], and qnm [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' RV is supported by grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' FJC2021-046551-I funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='13039/501100011033 and by the European Union NextGenerationEU/PRTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' RV also acknowledges support by grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' CERN/FIS- PAR/0023/2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' DB is supported by a ‘Ayuda Beatriz Galindo Senior’ from the Spanish ‘Ministerio de Uni- versidades’, grant BG20/00228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The research leading to these results has received funding from the Spanish Ministry of Science and Innovation (PID2020-115845GB- I00/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='13039/501100011033).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This work was partly enabled by the UCL Cos- moparticle Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' EB acknowledges support from the European Union’s H2020 ERC Consolidator Grant “GRav- ity from Astrophysical to Microscopic Scales” (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' GRAMS-815673) and the EU Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 101007855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Appendix A: Signal processing In this section we provide more details about our choices in manipulating the BBH waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We first generate the primary signal with a sampling time ∆t = 1/4 s, and we subsequently generate its memory via the GWmemory package [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' III, we compute the total signal in frequency domain, summing the individual FFTs of the primary waveform and of the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' How- ever, we find that a spurious contribution of the primary waveform at frequencies f < fin generates cross-terms between the primary and the memory signal of order O(hc) which affect the computation of the SNR and the Fisher matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' To prevent these artefacts from affecting our results, we removed the contribution from f < fin of the primary signal before summing the individual FFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We follow a standard procedure to manipulate the primary waveform, namely, applying a window function, padding the signal, and taking the FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We apply the following window function to the primary signal: z(t) = 1 4 � 1 + tanh � t−t0 σ0/4 ��� 1 − tanh � t−th σh/4 �� , (A1) with Mt0 = 150 and Mth = 110, respectively, at the beginning and at the end of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The duration of the windowing is set by Mσ0 = 50 and Mσh = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For the memory signal we follow a different procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We first extend the generated δh(t) (evaluated numerically for t0 ≤ t ≤ tf) at the beginning and the end with the con- stant values δh(t0) and δh(tf), respectively, using the same padding length as for the primary signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Subsequently, we apply the following window (as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [124]): w(t) = � � � � � 1, t − td < 0 1 2 � 1 + cos[2πfd(t − td)] � , t − td ≥ 0 0, t − td ≥ 1 2fd (A2) and choose Mtd = 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The choice of the decay fre- quency of the window function fd greatly impacts the spectral shape of the memory at low frequencies, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 9, where we show the character- istic strain hc of the memory for different values of fd ∈ {10−2, 10−3, 10−4, 10−5} Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Note that higher values of fd inject spurious power at the frequencies for which LISA is most sensitive, thus leading to an artificial increase of the respective memory SNR ρm = {7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='37, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='35, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='13, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Thus, while taking a lower fd is more reliable, in the sense that it does not overestimate the memory SNR, it implies a corresponding longer observation time of the memory, which can become inconsistent with the max- imum observation time considered for the primary signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, this does not pose a real problem since most of our analysis applies to cases where the primary signal is observed for a few hours, whereas the SNR of the memory does not greatly change as long as the memory is observed for more than 15 minutes (fd ≲ 10−3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In computing the SNR and the Fisher matrix we take fmin = 1/T, where T is the total length of the signal, and fmax = min{1 Hz, f440}, since we find that the QNM f440 ≡ ω440/2πMz is a good measure of the maximum frequency present in the signal (note that our waveforms include modes up to ℓmax = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' For total masses between [104, 105]M⊙ we find that fixing the min- imum frequency at fmin = 10−4 Hz does not change our SNR and Fisher forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Appendix B: Analytic considerations on the distance-inclination Fisher matrix Here we review the Fisher matrix derivation of the (dL, ι) degeneracy by computing the relative 2 × 2 matrix analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Including other parameters have little effect close to the degenerate points, since the main source of error comes from this submatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Subsequently, we show that the results represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3 can be understood by taken into account simply the main angular dependence of the primary and the memory signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' In order to estimate the effect of the memory on this degeneracy it is enough to focus on the part of the Fisher matrix regarding the extrinsic parameters {dL, cos(ι), φ, ϕc} since, at linear order in SNR−1, it is decoupled from the one associated to the intrinsic parameters {Mz, q, tc, ψ, Spins} [15], where ψ is the polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Indeed, while the intrinsic parameters are mainly extracted from the phase evolution of the waveform, the extrinsic parameters depend on 15 10-5 10-4 10-3 10-2 10-1 100 f [Hz] 10-22 10-20 10-18 10-16 hc(f) fd= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='01 fd= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='001 fd= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='0001 fd= 1e-05 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Characteristic strain hc(f, ι, Φ) of the memory for dif- ferent choices of decay frequency of the window function (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The parameters of the binary are the same as those of the “light” binary in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 2, but with an inclination ι = 90 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' the amplitudes h+ and h× [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Here we ignore the dependence on the coalescence phase ϕc, since (at leading order) it does not affect the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' At Newtonian (0 PN) order the primary waveform is h+,0 = 2ηMz dL [Mω(t)] 2 3 (1 + cos2 ι) cos[2ϕ(t)], (B1) h×,0 = 4ηMz dL [Mω(t)] 2 3 cos ι sin[2ϕ(t)], (B2) using the polarisation conventions of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The GW amplitude in the detector can be written in the frequency domain as [104] ˜h(f) = F +(f)˜h+(f) + F ×(f)˜h×(f), (B3) where F +,×(ι, φ, ψ, f) are the frequency-dependent de- tector response functions, which also depend on the source sky-location and polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Substituting the primary waveform FTs in the last expression we find ˜h0 = κ0 dL � F +(1 + cos2 ι) − 2iF × cos ι � , (B4) where κ0 is independent of both the luminosity distance (for fixed Mz) and the inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The sky- and polarisation-averaged Fisher matrix (8) has then the form11 Γ0 ij = �ρκ0 dL �2 ˆΓ0 ij, (B5) 11 Where we used ⟨F +(f)F ×∗(f)⟩ = 0 for the sky- and polarisation- averaging of the cross terms [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' with i, j ∈ {log dL, ι}, where ρ2 κ0 = (κ0|κ0) and the matrix ˆΓ0 = � (1 + cos2 ι)2 + 4 cos2 ι (3 + cos2 ι) sin(2ι) (3 + cos2 ι) sin(2ι) 4(1 + cos2 ι) sin2 ι � , depends only on the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This matrix is clearly sin- gular for face-on/off binaries, and it is diagonal for edge-on ones (implying that the two parameters are uncorrelated), ˆΓ0(ι ∈ {0, π}) = � 1 0 0 0 � , ˆΓ0(ι = π 2 ) = � 1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' It is easy to see that for inclination angles 0 < ι < π 2 ( π 2 < ι < π) the distance and the inclination are negatively (positively) correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This result shows that the degeneracy we focused in in this work is driven by the dependence of the (leading) quadrupole waveform on the inclination, and the particu- lar combination of plus and cross polarisations measured by the detector, which, in particular, lead to a singular Fisher matrix for face-on/off configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is a well-known issue in the literature and special care must be taken close to the singular points, where one should use a beyond-Gaussian analysis [101, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' However, for these face-on/off configurations the memory is almost van- ishing, so that in this work our focus is on intermediate inclination angles that are not too close to the singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Despite that, in the main text our analysis in- cludes higher modes, which break the complete degeneracy (“regularising” the Fisher matrix) (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Now we repeat the above computation, but for the memory signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Using the 0 PN waveform in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (6) we find that (in the frequency domain) the GW memory at the detector is � δh = κmF + dL sin2 ι(17 + cos2 ι), (B6) with a factor κm independent of both the distance and the inclination, and such that κ0/κm ∼ O(100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The sky- and polarisation-averaged Fisher matrix of the memory is Γm ij = �ρκm dL �2 ˆΓm ij, (B7) where ρ2 κm = (κm|κm) and the matrix elements ˆΓm log dL,log dL = sin4 ι 2 (17 + cos2 ι)2, ˆΓm log dL,ι = − sin(2ι) sin2 ι (8 + cos2 ι)(17 + cos2 ι), ˆΓm ι,ι = 2 sin2(2ι)(8 + cos2 ι)2, depend only on the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Because of the simple structure of the memory signal (at leading order), its Fisher matrix is singular for all inclination angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is not an issue, since this singularity is cured through the inclusion of (subleading) higher modes of the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Contrarily to what happens with the primary signal, here the distance and the inclination are positively (negatively) 16 correlated for inclination angles 0 < ι < π 2 ( π 2 < ι < π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This opposite behaviour is nicely illustrated by the orthogonality of the two confidence ellipses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Note that although these results were derived for the 0 PN waveforms, this picture still holds generically, since it relies mostly on the leading dependence on the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' The Fisher matrix for the total (primary + memory) waveform Γtot includes additional cross-terms, Γtot ij = Γ0 ij + Γm ij + (∂i� δh|∂j˜h0) + (∂i˜h0|∂j� δh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' (B8) Above we focused on Γ0 ij and Γm ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' This is because we veri- fied that, due to the rapid oscillations of the integrands in the cross-terms, the individual Fisher matrices dominate with respect to those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Appendix C: Numerical Fisher matrix To calculate the Fisher matrix elements we need to numerically compute derivatives of the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We do so using a second-order finite differences, ∂˜h ∂Θi ≈ ∂˜h(Θi + ∆Θi) − ∂˜h(Θi − ∆Θi) 2∆Θi , (C1) except for the luminosity distance, for which we have the exact result ∂˜h ∂ log dL = −˜h, (C2) since ˜h ∝ 1/dL (keeping Mz fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We checked that our Fisher matrices are numerically stable in our region of interest in the parameter space for the increments: ∆ΘMz = 10−3M⊙, ∆Θq = 10−4, ∆Θι,ϕc = 10−6 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Varying by an order of magnitude the finite increments gives just a few per cent change in the final matrix ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We also checked that the Fisher matrices are stable by computing the derivatives with a higher-order finite differences method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' We verified the reliability of our Fisher matrix inversion by confirming that in all cases, max(|ΓijΓ−1 ij − Iij|) < 10−6, (C3) where Iij is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Einstein, Näherungsweise Integration der Feldgleichungen der Gravitation, Sitzber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FQT4oBgHgl3EQfBzWr/content/2301.13228v1.pdf'} 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