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HOLROYD∗, RADU CIMPEANU∗, AND SUSANA N. GOMES∗ +Abstract. +We propose and analyse a new methodology based on linear-quadratic regulation +(LQR) for stabilizing falling liquid films via blowing and suction at the base. LQR methods enable +rapidly responding feedback control by precomputing a gain matrix, but are only suitable for systems +of linear ordinary differential equations (ODEs). +By contrast, the Navier-Stokes equations that +describe the dynamics of a thin liquid film flowing down an inclined plane are too complex to +stabilise with standard control-theoretical techniques. +To bridge this gap we use reduced-order +models – the Benney equation and a weighted-residual integral boundary layer model – obtained +via asymptotic analysis to derive a multi-level control framework. This framework consists of an +LQR feedback control designed for a linearised and discretised system of ODEs approximating the +reduced-order system, which are then applied to the full Navier-Stokes system. The control scheme +is tested via direct numerical simulation (DNS), and compared to analytical predictions of linear +stability thresholds and minimum required actuator numbers. Comparing the strategy between the +two reduced-order models we show that in both cases we can successfully stabilise towards a uniform +flat film across their respective ranges of valid parameters, with the more accurate weighted-residual +model outperforming the Benney-derived controls. +The weighted-residual controls are also found +to work successfully far beyond their anticipated range of applicability. The proposed methodology +increases the feasibility of transferring robust control techniques towards real-world systems, and is +also generalisable to other forms of actuation. +Key words. Feedback control, Stabilisation, Falling liquid films, Asymptotic analysis, Reduced- +order modelling, Direct numerical simulation +1. Introduction. Modelling and stabilisation of falling liquid films is a funda- +mental problem at the intersection of fluid dynamics, asymptotic analysis, and control +theory. Manipulation of these multi-scale systems is key to a number of industrial +applications ranging from coating flows in liquid crystal display devices to microchip +manufacture. +Such systems have a high degree of complexity, which makes them +challenging to model, control and simulate accurately and efficiently. +Although the control of complex systems is a challenge found in a wide range of +applied sciences – from preventing ice buildup on aerofoils [34] to avoiding obstacles in +self-driving vehicles [33] and crowd management [8] – falling liquid films are a proto- +typical example of such a control problem. Governed by the two-phase Navier-Stokes +equations, these are multi-scale setups used in many applications as well as beautiful +day-to-day phenomena such as wavy films on a window on a rainy day. The resulting +flow becomes unstable above a critical Reynolds number (a parameter depending on +velocity, inclination angle, film thickness, and fluid density), exhibiting a rich set of +behaviours starting with two-dimensional (2D) waves and leading to 3D spatiotempo- +ral chaos. Since the 1960s the problem has attracted much analytical focus, resulting +in a number of periodic reduced-order models [6, 43, 47, 58] based on the assumption +that perturbations are ‘long-wave’, i.e. their wavelength is much larger than their +amplitude. Such models range from single equation models describing the dynam- +ics of the liquid film height, to systems of multiple equations describing the height, +downstream flux, and potentially additional independent quantities. A broad range +of thin-film models are covered extensively in two reviews by Craster and Matar [11] +and Kalliadasis et al. [22], and these models were recently coupled with validation +in experimental settings by Denner et al. [13]. +More recently, Richard et al [42], +∗Mathematics +Institute, +University +of +Warwick, +Coventry +CV4 +7AL, +UK +(o.holroyd@warwick.ac.uk, radu.cimpeanu@warwick.ac.uk, susana.gomes@warwick.ac.uk). +1 +arXiv:2301.11379v1 [math.OC] 26 Jan 2023 + +2 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +Usha, Chattopadhyay, and Tiwari [53], Mukhopadhyay, Ruyer-Quil, and Usha [28] +have provided new insights into how two-equation models behave, particularly when +attempting to apply them beyond their expected range of validity. +Thin liquid films have numerous industrial applications, most notably in coating +flows [57], heat and mass transfer [56], thin-film thermoelectric cooling [12], as well as +ice accretion prevention on aircraft surfaces [27]. These applications require control- +ling the interface to a specific shape, whether that be flat for coating flows or highly +corrugated for heat transfer. There are almost as many physical input mechanisms +as there are applications, and there is an extensive body of literature dealing with +the effects that these have on stability and the critical Reynolds number above which +unstable modes exist. These include heating and cooling of the fluid [5], electric [52] +or magnetic [2] fields, porous [31] or deformable [15] walls, and many more. Here, +as shown in Figure 1, we focus on blowing and suction through the base via dis- +crete actuators [50], since they act over faster (and therefore more computationally +accessible) timescales, and their effect on the overall flow control is greater. While +continuous controls, i.e., controls applied through the entire domain (flow base), are +more mathematically tractable, discrete controls applied through small holes or slots +are a necessary step towards real applications, since applying blowing and suction +controls throughout the whole domain is unfeasible. +In the last two decades, the field has progressed from studying the effects that +static, predetermined alterations to the system (such as fixed heating patterns or cor- +rugated baseplates) can produce towards feedback control, where information from +the interface is used to update control inputs as the film evolves in time. Armaou +and Christofides [3, 9] and more recently Gomes, Papageorgiou, and Pavliotis [18] and +Gomes et al. [17, 19], studied the simplest thin-film model, the Kuramoto-Sivashinsky +(KS) equation, from both analytic and computational perspectives, and successfully +applied methods from linear control theory based on [60]. Thompson et al [49] further +extended these results to the Benney [6] and weighted-residual [43] equations using a +family of linear-quadratic regulator (LQR) methods. However these long-wave models +remain one stage removed from the physical system that they approximate. Unfor- +tunately, the full two-phase Navier-Stokes system is too complex and nonlinear to +directly extend the previous work on long-wave models. Cimpeanu, Gomes, and Pa- +pageorgiou [10] first analysed aspects of model reduction applicability, and delineated +the discrepancies between thin-film modelling and direct numerical simulation (DNS) +approaches. They considered a scenario in which actuator input is simply propor- +tional to the observed interfacial deviation at its position, which previous numerical +evidence suggested could successfully stabilise the unperturbed (flat) state of the +Benney and weighted-residual systems [49], the so-called Nusselt solution. They then +showed how this model information can be interpreted and transferred into a more +accurate simulation framework to successfully stabilise the full Navier-Stokes system +via DNS. Nevertheless, a rigorous optimal control approach capable of surpassing +the limitations of proportional control setups remains a challenge yet to be addressed. +An alternative approach which incorporated model predictive control (MPC) was pro- +posed by Wray, Cimpeanu and Gomes [59] in the context of electrostatic actuation. +This includes designing optimal controls for reduced-order models and enhanced in- +teraction between model and simulation techniques, where any re-initialisation of the +optimal control problem uses readings from a direct numerical simulation of the full +problem. The computational cost of this framework may however prove prohibitive +in real-world contexts. +In this work we aim to close the gap between robust controls designed for long- + +FALLING LIQUID FILM CONTROL +3 +wave models and more physically relevant systems such as those governed by the +two-phase Navier-Stokes equations, ultimately bringing real-world applications closer +within reach. The paper is structured as follows. In section 2 we begin with a de- +scription of the models that make up the two-tier hierarchical structure of the control +framework: the two-phase Navier-Stokes equations as the target system and either the +Benney or weighted-residual model as the control system. In section 3 we outline the +control method: a set of discrete actuators injecting and removing fluid at the base of +the film. We further simplify these reduced-order models to a system of linear partial +differential equations (PDEs), and finally to a finite set of ODEs, which permits the +use of a linear-quadratic regulator, an established control-theoretical technique. We +then demonstrate for the first time that, by applying this control strategy to the full +Benney, weighted-residual, and Navier-Stokes systems, there is good agreement be- +tween the linear predictions and a series of nonlinear numerical experiments. Finally, +in section 4 we illustrate that this agreement spans a large region of the parameter +space corresponding to physically relevant fluids. Furthermore, our stability analysis +results indicate that the control performance significantly exceeds the range of validity +of the underpinning model assumptions in certain regions of the explored parameter +space. +2. Governing Equations. We consider a thin film of fluid flowing down a plane +tilted at an angle θ from the horizontal, as shown in Figure 1. We restrict ourselves to +2D flows, largely to make the problem more computationally tractable. Nevertheless, +this setup exhibits a highly nontrivial and physically rich behaviour, while also cap- +turing the initial wave development stages before cross-flow effects begin to appear in +3D contexts [22]. In many control scenarios manipulating the dynamics of these early +stages is the key objective (and only realisable strategy) of the control framework +before highly nonlinear and often undesirable flow features arise. +We use a coordinate system rotated with the plane, where x points downstream +and y is the perpendicular distance to the wall. There is a free surface at the upper +interface of the fluid at y = h(x, t) where the fluid and gas meet. We inject and remove +fluid through the rigid lower wall regions as dictated by our resulting control strat- +egy, with no-slip and impermeability conditions governing the remaining uncontrolled +boundary. Finally, we consider periodic boundary conditions in the x-direction; while +an experiment would be realised on an open domain with inflow and outflow, the +speed with which a wave fully develops after the inlet [13] means that for sufficiently +large domains – which we consider here – periodic boundary conditions provide a +reasonable approximation. Furthermore, periodicity allows us to perform the analysis +performed in section 4 below, which would be more challenging, if not impossible, to +undertake given the inability to compute eigenvalues of the problem explicitly with +open boundaries. +The problem is governed by the conservation of mass and momentum in both +the liquid film and the gas above it, coupled at the interface. Typically, the large +density and viscosity ratios between the two media mean that we can consider the gas +region to be hydrodynamically passive, and can ignore the flow in the gas, and model +the liquid film alone. The fluid flow is governed by the acceleration due to gravity +g, the inclination angle θ, and the physical properties of the liquid phase: constant +density ρ, viscosity µ and surface tension coefficient γ. A full list of physically-relevant +parameters and the values used in this investigation can be found in Appendix A. +For a liquid film with mean height hs, the uncontrolled system admits a uniform +solution known as the Nusselt solution [30], where h(x, t) = hs, which has a parabolic + +4 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +y +x +v +u +θ +h(x, t) +f(x, t) +1 +Fig. 1. Diagrammatic representation of a falling liquid film under gravity with basal forcing f. +Controls are applied through the wall at y = 0, and the behaviour of the interface h is governed by +the fluid parameters and the inclination angle θ. +velocity profile with surface velocity Us = ρgh2 +s sin θ +2µ +. We then nondimensionalise the +problem based on the length scale hs, velocity scale Us and pressure scale µUs +hs , defining +the Reynolds and capillary numbers +(2.1) +Re = ρUshs +µ +, +Ca = µUs +γ , +which measure the relative importance of inertial and viscous terms, and of gravity +and surface tension, respectively. +2.1. Navier-Stokes equations. The full liquid film flow is governed by the 2D +Navier-Stokes equations, which are solved for velocity u(x, y, t) = (u, v) and pressure +p(x, y, t) under the action of external forces. +The governing (nondimensionalised) +internal momentum equations are +Re(ut + uux + vuy) = −px + 2 + uxx + uyy, +(2.2) +Re(vt + uvx + vvy) = −py − 2 cot θ + vxx + vyy, +(2.3) +and the continuity equation reads +(2.4) +ux + vy = 0. +The system is completed by its boundary conditions: periodic boundaries in the +x-direction, no-slip and fluid injection/removal at the wall, +(2.5) +u = 0, +v = f(x, t), +the nonlinear dynamic stress balance (or momentum jump) at the interface, y = +h(x, t), +(vx + uy)(1 − h2 +x) + 2hx(vy − ux) = 0, +(2.6) +p − +2 +1 + h2x +(vy + uxh2 +x − hx(vx + uy)) = − 1 +Ca +hxx +(1 + h2x)3/2 , +(2.7) +and finally the kinematic boundary condition +(2.8) +ht = v − uhx. +In lieu of physical experiments, we perform computational analogues by simu- +lating the Navier-Stokes equations using a volume-of-fluid approach developed by +Popinet [37]. The methodology is well known, following more than two decades of + +FALLING LIQUID FILM CONTROL +5 +successful development and usage in the community [37, 38, 39], and therefore we +restrict our attention to details relevant to our particular setting in Appendix B. +Defining the down-slope flux q(x, t) by integrating over the height of the film +(2.9) +q(x, t) = +� h +0 +u(x, y, t) dy, +we combine (2.4), (2.5), and (2.8) to obtain the 1D mass conservation equation +(2.10) +ht + qx = f. +We can continue to use the Navier-Stokes equations to compute q, or we can use +one of a number of simplified models for the flux. Here we consider the Benney [6] +and weighted-residual [43] equations, which are valid in the long-wave limit. By using +a pair of reduced-order models we are better able to gauge the relative capabilities +of both, weighing model and computational complexity against control performance. +The following pair of reduced-order models are based on first-order asymptotic expan- +sions in the long-wave parameter ϵ = 1/L (where L is the aspect ratio of the domain). +In addition to the requirement that ϵ ≪ 1, we make the assumption that Re = O(1) +and Ca = O(ϵ2) to retain inertial and surface tension effects, and that f = O(ϵ) so +that the magnitude of the imposed control is comparable to the perturbed flow. +2.2. Benney equation. The first choice of model for the downstream flux is +the Benney system [6], which was extended to include the effects of O(ϵ) controls by +Thompson, Tseluiko, and Papageorgiou [50]. This enslaves the flux to the interfacial +height h via +(2.11) +q(x, t) = h3 +3 +� +2 − 2hx cot θ + hxxx +Ca +� ++ Re +�8h6hx +15 +− 2h4f +3 +� +, +resulting in a single equation for the evolution of the interface when coupled to (2.10). +The system above is a significant improvement over equations such as the KS equation +– the simplest nonlinear model of thin film flows [47]. While capturing some important +aspects of the chaotic behaviour of falling liquid films such as travelling waves, the +KS equation is more useful as a paradigmatic example in a dynamical systems sense +than as a predictive model outside of a very restricted region of the parameter space. +However, the Benney system exhibits undesirable behaviours such as unphysical finite- +time blow-up outside of a narrow range of parameters corresponding to low Reynolds +numbers [41], as is demonstrated in Figure 2. +2.3. Weighted-residual system. To overcome the unrealistic behaviour de- +scribed above, Ruyer-Quil and Manneville [43, 44] proposed an improved weighted- +residual methodology based on approximating u by a truncated sum of basis functions +satisfying no-slip boundary conditions at the wall (2.5) and zero tangential stress at +the interface (2.6). Here we use the first-order truncation, which matches well with +the second-order truncation up to Re ≈ 5 [43], a significant improvement over pre- +vious models [45]. When combined with the basal forcing this results in a separate +evolution equation for the flux [50] +(2.12) 2Re +5 h2qt+q = h3 +3 +� +2 − 2hx cot θ + hxxx +Ca +� ++Re +�18q2hx +35 +− 34hqqx +35 ++ hqf +5 +� +, +which together with (2.10) forms a system of two PDEs for the height h(x, t) and the +flux q(x, t). Equation (2.12) better captures many features of the full Navier-Stokes + +6 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +film, such as spontaneous back-flow [32], and Figure 2 illustrates how it provides a +good match for the interfacial shape even at moderate Reynolds and capillary num- +bers. However, it does overestimate the amplitude of the capillary ripples, as observed +by Ruyer-Quil and Manneville [43] for the weighted-residuals system and also for other +first-order models [1]. Unlike the Benney equation, it also does not exhibit unphys- +ical finite-time blow-up, although it too diverges from the Navier-Stokes model at +moderate Reynolds numbers, Re ≈ 10 [43]. +0 +1 +2 +t = 0 +h +0 +1 +2 +t = 2 +h +0 +1 +2 +t = 50 +h +0 +5 +10 +15 +20 +25 +30 +0 +1 +2 +t = 300 +x +h +4 +Fig. 2. Evolution of interfacial heights h for Navier-Stokes (black), weighted-residual (red), +and Benney (blue) systems, with peaks shifted to 3L/4. Here, the parameters used are Re = 10, +Ca = 0.05, θ = π/3. The Benney equation blows up shortly after t = 2, but the weighted-residual +and Navier-Stokes interfaces have very similar structures aside from some spurious oscillations in +the weighted-residual case, which are observable at t = 300 above. +3. Control methodology. We focus on controlling the interface towards the +Nusselt solution, which under our nondimensionalisation is the uniform film h(x, t) = +1. +All the controls we consider are a class of time-dependent controls known as +feedback controls, which we introduce here. Take a controlled quantity x governed by +the system +(3.1) +xt = Ax + Bu, +y = Cx, +where u is the control, y is some observation of the system, and A, B, and C are +arbitrary operators describing the uncontrolled system dynamics, the control actua- +tion mechanism, and the observations respectively. In the case of feedback controls, +we have the restriction that u = Ky for some operator K, so that the system can be +written in closed-loop form +(3.2) +xt = (A + BKC)x. +An overview of some important control theory definitions is provided in Appendix C. +In the case of falling liquid films, the full system (2.2)–(2.8) is too complex for +standard (linear) control-theoretical techniques to be tractable. Instead, we design +feedback controls for the reduced-order models presented in subsections 2.2 and 2.3 + +FALLING LIQUID FILM CONTROL +7 +and apply them to the full Navier-Stokes system by passing the Navier-Stokes in- +terfacial height to the feedback control scheme. The full framework is pictured in +Figure 3. +Initial condition +1 +Apply controls +4 +Time step +5 +Reduced order model +2 +Compute controls +3 +2 +Fig. 3. Multi-layer control methodology for the control of Navier-Stokes thin liquid films. From +the initial condition, we treat the interface as though it were described by the chosen reduced-order +model and generate the feedback control accordingly. We then apply this control to the full model +and time step forward to repeat the process. +Given the difficulties in observing both the height [24, 55] and flux [20] of falling +liquid films, it is understandable that in our case we might wish to express our control +f as a function of time only, known as offline control (as can be done when generating +controls for the KS equation for instance, see [14]). +Unfortunately, as shown by +Cimpeanu, Gomes, and Papageorgiou [10], although such hierarchical controls show +promising initial dampening, they invariably fail over longer timescales, as the Navier- +Stokes model eventually diverges from the chosen long-wave model, and the action +of the control no longer affects the intended state, with the Navier-Stokes system +ultimately converging back to its uncontrolled behaviour. +3.1. Control actuation mechanism. Although the control term f in (2.10)– +(2.12) is general, in this study we restrict ourselves to controls taking the form of +Dirac-delta distributions injecting or removing fluid at the wall at a finite number +of locations x1, . . . , xM – which is more experimentally achievable than continuous +controls, i.e. controls applied everywhere in the domain. Furthermore, both due to +realistic considerations and computational restrictions imposed by the direct numeri- +cal simulation setup, we must approximate these point sources by finite regions which +we select to be smooth, periodic functions (shown in Figure 4), of the type +(3.3) +d(x) = A exp +�cos(2πx/L) − 1 +ω2 +� +, +where ω controls the width of the function, and A is chosen so that +� L +0 d(x) dx = 1. +More refined discretisations support smaller values of ω, and d(x) → δ(x) as ω → 0. +The basal forcing term f is thus +(3.4) +f(x, t) = +M +� +i=1 +ui(t)d(x − xi), + +8 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +where ui(t) are the individual, time-dependent, control amplitudes. Despite the prac- +tical difficulties of obtaining full observations, the main goal of this investigation is +to test the feasibility of the control methodology, and so for the moment we assume +we are able to observe the full interface h(x). This means that C, the observation +operator from (3.2), is the identity. We will address the observability of the problem, +as well as issues introduced by noisy or partial observations of the interface, in future +work. +Note that, since the domain has periodic boundaries on the left and right, the +problem is translationally invariant, and so x1, . . . , xM should be evenly placed along +the base. For 3D and non-periodic flows, the optimal placement of the actuators is a +nontrivial problem [51]. +Finally, we introduce a cost functional to compare different control strategies, +taking into account the 2-norms of the deviation from the target state and penalising +the use of the controls. We thus define the total cost of a control by +(3.5) +κ = +� ∞ +0 +� L +0 +βˆh(x)2 + (1 − β)f 2 dx dt, +where ˆh(x) = h(x) − 1 is the deviation of the interface from the target uniform state, +and the parameter β controls the relative importance of the interfacial deviation and +the magnitude of the controls. +3.2. Linear-Quadratic Regulator (LQR). Despite the long-wave simplifi- +cation to one of the two reduced-order models, control methodologies for nonlinear +PDEs of the type considered here are still a rapidly developing area of active research, +with the most relevant efforts by Boujo and Sellier [7], Lunz [25], or the current au- +thors [10, 18, 49]. In spite of these recent advances, we cannot directly choose the +optimal control operator K with analytical methods. To make this problem tractable, +we assume that the perturbation away from the Nusselt solution (ˆh = h − 1 = 0, +ˆq = q −2/3 = 0, ˆf = 0) is small, so that we can linearise (2.11) to obtain the equation +(3.6) +ˆht = +� +−2∂x + +�2 cot θ +3 +− 8Re +15 +� +∂xx − +1 +3Ca ∂xxxx +� +ˆh + +� +1 + 2Re +3 ∂x +� +ˆf. +We then discretise to form a system of N ODEs, +(3.7) +dˆh +dt = Jˆh + Ψu, +u = KΦˆh. +Here, J ∈ RN×N captures the system dynamics, Ψ ∈ RN×M is the linearised actuator +matrix, and Φ ∈ RN×N is the linearised observation matrix (which we take to be the +identity). K ∈ RM×N is the gain matrix, which is chosen to minimise the discrete +cost +(3.8) +c = +� ∞ +0 +ˆhTUˆh + uTV u dt, +where U = βL +N I ∈ RN×N and V = (1 − β)I ∈ RM×M are matrices whose entries are +chosen so as to form the discrete analogue of the continuous cost (3.5). The process is +similar for the weighted-residual system, but with twice the system size at each stage, +since there are two unknowns, ˆh and ˆq. The linearisation of (2.12) results in +ˆht = −ˆqx + ˆf, +(3.9) +ˆqt = +� 5 +Re + +�4 +7 − 5 cot θ +3Re +� +∂x + +5 +6ReCa ∂xxx +� +ˆh − +� 5 +2Re + 34 +21∂x +� +ˆq + +�1 +3 +� +ˆf, +(3.10) + +FALLING LIQUID FILM CONTROL +9 +and the resulting discretised system has 2N equations rather than N. Finally, al- +though we are assuming full observations of the interfacial height, Thompson et +al. [49] showed that it is sufficient to use the leading order approximation ˆq = 2ˆh +to remove the need to directly observe the flux, incurring a small penalty in the size +of the largest eigenvalue but not fundamentally affecting stability. This is especially +important because the flux is challenging to measure in an application setting [20]. +This setup forms a classic problem in control theory: the linear-quadratic regu- +lator (LQR) problem, which is a subset of a broader class of static output feedback +(SOF) problems in which one can also have restricted observations (i.e. rank(Φ) < N). +Here, we provide an overview of how this class of problems is solved. For more details +see [21, 48]. +For the discretised linear control system (3.7), we write the cost as +(3.11) +c = +� ∞ +0 +ˆhTUˆh + uTV u dt = +� ∞ +0 +ˆhT(U + ΦTKTV KΦ)ˆh dt, +where U, V are assumed to be symmetric positive definite matrices. +If we suppose there exists a symmetric, positive semi-definite matrix P such that +(3.12) +d +dt(ˆhTPˆh) = −ˆhT(U + ΦTKTV KΦ)ˆh, +then, as long as the controlled system matrix A = J + ΨKΦ is asymptotically stable, +i.e., all its eigenvalues have negative real part, we can write (3.11) as +c = ˆh(0)TPˆh(0) − lim +t→∞ +ˆh(t)TPˆh(t) += ˆh(0)TPˆh(0). +(3.13) +By expanding out the left hand side of (3.12) and observing that this is true for all +initial conditions ˆh(0) ∈ RN, we have +(3.14) +ATP + PA + U + ΦTKTV KΦ = 0. +This further implies that the choice of P is independent of the initial condition ˆh(0), +and so +(3.15) +c = tr(PX), +where X = ˆh(0)ˆh(0)T. Since we wish to choose an optimal K for all initial condi- +tions, we set X = E[ˆh(0)ˆh(0)T] = I, the identity matrix, as we assume all initial +perturbations ˆh(0) are equally likely. +The problem thus becomes equivalent to selecting K to minimise (3.15) subject +to the constraint (3.14). This can be solved via Lagrange multipliers. Defining the +symmetric matrix of Lagrange multipliers S, we then have the resulting Hamiltonian +(3.16) +H = tr(PI) + tr((ATP + PA + U + ΦTKTV KΦ)S). +By setting ∂SH = ∂P H = ∂KH = 0 we have the conditions for the solution to the +SOF problem: +0 = ATP + PA + U + ΦTKTV KΦ, +(3.17) +0 = AS + SAT + I, +(3.18) +0 = V KΦSΦT + ΨTPSΦT. +(3.19) + +10 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +The final condition can be more usefully written as +(3.20) +K = −V −1ΨTPSΦT(ΦSΦT)−1. +Equations (3.17)–(3.19) cannot be solved directly, and so an iterative procedure +must be used. However, in the special case of the LQR problem where we have Φ = I, +we may discard (3.18) and rewrite (3.17) and (3.20) as +0 = JTP + PJ + U − PΨV −1ΨTP, +(3.21) +K = −V −1ΨTP. +(3.22) +Equation (3.21), which is known as the continuous algebraic Riccati equation +(CARE), can be solved for P directly, and then used to compute K. The structure +of the matrices J, U, V , and Ψ – with U and V diagonal, J periodic banded and Ψ +having translational symmetries – means that the specific CARE for this problem is +typically well-conditioned. Thus we can make use of the classical eigenvector approach +described by MacFarlane [26], Potter [40] and Vaughan [54]. Alternatively, Schur [23] +and generalised eigenvector [4] approaches may offer improved numerical stability +for larger systems (which would be encountered in 3D) and more unstable regimes +(where some of the interim matrices used in the classical method become singular or +near-singular). +We note that equations (3.15) and (3.22) illustrate why the single parameter β is +sufficient to fully explore the cost-space with regards to K: if we instead introduce a +pair of control parameters α and β so that +(3.23) +U ′ = αU = αβL +N I, +V ′ = αV = α(1 − β)I, +we can set the entries of U ′ and V ′ independently. The cost is then +(3.24) +c′ = αc = 1 +2 tr(αPX) = 1 +2 tr(P ′X). +Carrying the new cost matrices through to (3.22) we have +(3.25) +K′ = −(V ′)−1ΨTP ′ = −(αV )−1ΨTαP = K. +The above result indicates that scaling the cost makes no difference to the optimal K, +and so a single parameter describing the ratio of significance of the two components is +sufficient. Gibson [16] showed that, under certain conditions, the discretised feedback +operator K does converge to its infinite-dimensional counterpart K. +Once the optimal gain matrix K has been computed, we can calculate the mth +actuator amplitude as um = Km · ˆh(t), where Km is the mth row of K and · denotes +the inner product. This means that Km,i can be interpreted as describing the im- +portance of the ith entry of ˆh to um. As can be seen in Figure 4, this allows us to +examine the rows of K to develop an understanding of how the controls operate. The +weighted-residual gains are tightly clustered around the location of the actuator across +a wide range of Reynolds numbers, with minimal up- and down-stream contributions. +By contrast, the Benney gains are much broader and depend more strongly on the +interfacial shape away from the actuator location. They also are much more sensitive +to the Reynolds number (it is worth noting that, for Re = 30, the Benney-derived +controls fail to stabilise the Navier-Stokes system). +With a method to compute the gain matrix for the two reduced-order models we +are now well-positioned to deploy the methodology described in Figure 3 and direct +it towards the modelled physical system of interest. + +FALLING LIQUID FILM CONTROL +11 +0 +5 +10 +15 +20 +25 +30 +−1 +−0.5 +0 +0.5 +1 +1.5 +2 +x +Feedback gain +Actuator shape +B, Re = 0.5 +B, Re = 10 +B, Re = 30 +WR, Re = 0.5 +WR, Re = 10 +WR, Re = 30 +3 +Fig. 4. The second row of the gain matrix computed using either the Benney equation (in blue) +or weighted-residual system (in red) as the reduced-order model, as Re varies and Ca is fixed at +0.05. The gains are shown alongside the corresponding actuator (in black). Although the weighted- +residual gains remain clustered around the actuator, the Benney gains have significant nonlocal +contributions. +3.3. Preliminary results. Previous work by Thompson et al. [49] confirmed +that LQR controls with full observations are able to stabilise both the Benney and +weighted-residual systems. The same authors also found that the Benney controls +stabilise the weighted-residual model. In Figure 5 we can see that for a similar pa- +rameter regime (Re = 5, Ca = 0.05 – selected such that Re is not so high so as +to make numerical simulation difficult, and Ca is large enough that surface tension +alone cannot stabilise the liquid film with properties experimentally aligned with a +relatively thick and viscous oil flow), these controls can be extended to the Navier- +Stokes system, where we achieve similar results. Figure 5 shows how the interface is +allowed to develop from a small sinusoidal perturbation into a travelling wave, before +the application of controls at t = 0. Representative interfacial snapshots are pictured +in Figure 5. The interfacial deviation then decays exponentially, suggesting that the +use of linear models to design the gain matrix is appropriate in both cases. +4. Stability analysis. It is encouraging to see that we can control the film in +the specific setting of Figure 5, but a better aim is to predict the stabilisability of +the system given the flow parameters Re, Ca, θ and number of controls M. Since +we lack a closed-form expression for either the continuous control f(h), or its discrete +counterpart ΨKˆh, we cannot directly estimate the stability properties of the controlled +system. However, we can predict the damping rate by finding the largest eigenvalue +λ∗ of the controlled system matrix A = J + ΨKΦ and compare that to rates fitted to +the data produced in our numerical simulations. +From Figure 6 we observe that the Benney-derived controls directly stabilise the +Benney and weighted-residual systems (in a similar setup to that used by Thomp- +son et al. [49]) over a wide range of Reynolds numbers, and that their ability to +stabilise towards the uniform film extends to the hierarchical controls applied to the +Navier-Stokes film. We note that the weighted-residual and Navier-Stokes systems +are stabilised even above the stability threshold, after which the linearised weighted- +residual model predicts that five actuators are not sufficient to stabilise the uniform + +12 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +0 +1 +2 t = 0 +h +t = 0 +0 +1 +2 t = 0.1 +h +t = 0.1 +0 +1 +2 t = 1 +h +t = 1 +0 +5 +10 15 20 25 30 +0 +1 +2 t = 10 +x +h +0 +5 +10 15 20 25 30 +t = 10 +x +t = 0 +−1 +0 +1 +t = 0 +f +t = 0.1 +−1 +0 +1 +t = 0.1 +f +t = 1 +−1 +0 +1 +t = 1 +f +t = 10 +−1 +0 +1 +t = 10 +f +−100 +−50 +0 +50 +100 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +t +|h − 1| +8 +Fig. 5. +Interfacial shapes before and after the controls are switched on: Benney equation +derived controls in blue (left), weighted-residual derived controls in red (centre). A travelling wave +is allowed to develop until t = 0, when the controls are activated. Both controls successfully damp +out the perturbation, with the control amplitudes decreasing in proportion to |h − 1|. We note that, +although similar, the controls are not identical – see the second and fourth rows in particular. In +both cases (Benney in blue, weighted-residual in red) the 2-norm of the deviation of the interface +from the target state decays exponentially (right). After t ≈ 50 the deviation is small enough that +machine precision interferes with computing the deviation. In these simulations we used Re = 5, +Ca = 0.05. +0 +10 +20 +30 +40 +50 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0 +0.1 +Re +λ∗ +stability threshold +linearised Benney +linearised WR +Benney +WR +NS +5 +Fig. 6. Comparison of (fitted) damping rates for Benney-derived LQR control applied to Benney +(blue), weighted-residual (red), and Navier-Stokes (black) systems (all solid) to the predictions from +the linearised systems of ODEs. Here, we used M = 5 controls and Ca = 0.05. For all three systems, +the numerical models break down at sufficiently large Re. +state. +All three models display unphysical blow-up at sufficiently large Reynolds num- +bers. Although this is expected behaviour in the case of the Benney film [29], in +the case of the weighted-residual and Navier-Stokes models this is attributed to the +eventual breakdown of the controls as the Benney model finally loses the last of its + +FALLING LIQUID FILM CONTROL +13 +predictive capacity at larger values of Re. +0 +10 +20 +30 +40 +50 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0 +0.1 +Re +λ∗ +stability threshold +linearised Benney +linearised WR +Benney +WR +NS +6 +Fig. 7. Comparison of (fitted) damping rates for weighted-residual-derived LQR control ap- +plied to Benney (blue), weighted-residual (red), and Navier-Stokes (black) systems (all solid) to the +predictions from the linearised systems of ODEs. Here, we used M = 5 controls Ca = 0.05. +While the Benney-derived control rules stabilise all three models (at least for +small-to-moderate Reynolds numbers), the weighted-residual derived controls fail to +stabilise the Benney equation for Re > 7, in agreement with the linear predictions +given by the eigenvalues of A. The weighted-residual and Navier-Stokes models have +reasonable agreement with the linear damping rates but remain stabilisable even at +Re = 50, when the linear system is not. +In order to make analytical progress, we turn to an equivalent way to produce the +gain matrix K, where we first convert (3.7) to Fourier space (so ˜h = Fˆh, where F is +the Fourier transform). We can then reorder the wavenumbers to separate stable and +unstable modes: +(4.1) +d˜h +dt = ˜J˜h + ˜Ψ ˜K˜h = +� ˜Ju +0 +0 +˜Js +� +˜h + +�˜Ψu +˜Ψs +� +˜K˜h. +Concentrating on the unstable modes more explicitly, i.e., +d +dt +�˜hu +˜hs +� += +� ˜Ju +0 +0 +˜Js +� �˜hu +˜hs +� ++ +�˜Ψu +˜Ψs +� +˜K +�˜hu +˜hs +� += +� ˜Ju + ˜Ψu ˜Ku +0 +˜Ψs ˜Ks +˜Js +� �˜hu +˜hs +� +, +(4.2) +we find that since the matrix on the right-hand side of (4.2) is block lower triangular, +the controls leave the eigenvalues of the stable modes unchanged, and so they remain +stable. We thus reduce the control problem to +(4.3) +d˜hu +dt = ˜Ju˜hu + ˜Ψu ˜Ku˜hu. +By solving the problem in Fourier space it is clear that – for the purely linear case +at least – we should expect that M actuators would be sufficient to control any + +14 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +system satisfying M ≥ rank( ˜Ju). This would amount to one control per unstable +mode plus one more to satisfy conservation of mass, as pointed out by Armaou and +Christofides [9]. +The rank of the unstable Jacobian ˜Ju corresponds to the number of unstable +modes of the linearised system ((3.6) or (3.9) and (3.10)). We compute this rank for a +perturbation with wavenumber k, where the linearised Benney equation has a single +eigenvalue +(4.4) +λ = −2ik + +�8Re +15 − 2 +3 cot θ − +1 +3Ca k2 +� +k2, +and the weighted-residual system has a pair of eigenvalues that solve the quadratic +equation +(4.5) +λ2 + +� 5 +2Re + 34 +21ik +� +λ + +� 5 +Re ik − +�4 +7 − 5 cot θ +3Re +� +k2 + +5 +6ReCa k4 +� += 0. +Setting the real part ℜ(λ) = 0 we can solve for the critical wavenumber k0 (the +boundary between stable and unstable unimodal systems). For both (4.4) and (4.5), +this is +(4.6) +k0 = ± +� +Ca +�8 +5Re − 2 cot θ +� +. +After rescaling to account for L ̸= 2π, this expression admits a single zero eigenmode +and pairs of positive and negative modes with k < k0, resulting in the number of +unstable modes being +(4.7) +nu = 1 + 2 +� L +2π k0 +� += 1 + 2 +� +L +2π +� +Ca +�8 +5Re − 2 cot θ +�� +. +In Figure 8 we compare our predictions for nu from expression (4.7) to the min- +imum number of controls required to stabilise the film in our numerical experiments +of the Navier-Stokes system as Re and Ca vary. We see that, as expected, the system +is stabilisable at M ≥ nu in all cases. In fact, in the majority of the parameter space, +the minimum number of actuators required to stabilise the uniform state is lower than +the number predicted by the linear analysis, particularly at lower Reynolds numbers. +As previous work by Salamon, Armstrong, and Brown [45] and Ruyer-Quil and Man- +neville [43] shows that important physical characteristics such as travelling wave speed +begin to diverge from DNS results at Re ≈ 5, the fact that controls based on a lin- +earisation of these equations match (or even exceed) the expected performance up to +Re ≈ 100 is remarkable. However, after this point it becomes clear that we are reach- +ing the limit of the model’s validity, and the ability of the controls to stabilise the +uniform state becomes less predictable. We note that at larger Reynolds and capillary +numbers the film takes much longer to respond to the effects of the controls, making it +more challenging to assess whether the uniform state is stabilisable. By dynamically +estimating the sign of the fitted damping rate we can avoid running simulations over +unfeasibly long times. +5. Conclusion. The research presented herein has demonstrated new and sig- +nificant capabilities in terms of design and analysis of optimal feedback controls for + +FALLING LIQUID FILM CONTROL +15 +100 +101 +102 +10−4 +10−3 +10−2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +3 +3 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +3 +3 +3 +3 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +3 +3 +3 +3 +3 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +3 +3 +5 +5 +5 +5 +7 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +3 +5 +5 +5 +5 +5 +7 +7 +9 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +5 +5 +5 +5 +5 +5 +7 +7 +9 +11 +Re +Ca +1 +3 +5 +7 +9 +11 +7 +Fig. 8. The minimum number of actuators required to stabilise the Navier-Stokes film compared +to the number of unstable modes of the linearised weighted-residual system (red). The number of +controls needed to stabilise the uniform film never exceeds the number of unstable modes of the linear +system nu as given in (4.7). The ranges for the two parameters cover a broad range of different +fluids, select examples are listed in Appendix A. Videos of selected instances of film evolution and +control are available as supplementary material. +complex physical systems. The stabilisation of the canonical multi-scale framework of +a thin liquid film falling down an inclined plane by employing reduced-order models +such as the Benney and first-order weighted-residual equations has been used as the +physical setup for our proposed methodology. We developed an LQR approach via +blowing and suction controls which has been shown to outperform the predictions +of linear stability theory, and can successfully function beyond the region of model +validity for either the Benney- or the weighted-residual-derived controls. +We have shown that even the crude controls used here far exceed their expected +performance, and this opens up numerous avenues for future work. It remains to be +seen whether higher-order models such as the second-order weighted-residual integral +boundary layer model proposed by Ruyer-Quil and Manneville [43] can be used to fur- +ther improve the type of control demonstrated here. In addition, it would be desirable +to remove the control dependence on discretisation by developing infinite-dimensional +controls, which might also allow for an improved analysis of control performance. +Although here we have performed numerical experiments to showcase the control +efficacy, physical experiments on real fluids are an obvious next step that we hope our +work will inspire. In order to achieve this in practice there are a number of useful +assumptions that must be relaxed, namely the 2D nature of the flow and periodic +boundary condition formulation. The additional dimension will allow for cross-flow +instabilities (an interaction which needs to be further quantified), and the boundaries +can also affect the stability of the film [35]. The blowing and suction controls used in +the present work offer a valuable theoretical foundation permitting a comprehensive +examination of control performance for this system. We envision realistic embodi- +ments thereof to require further analysis. Nevertheless, the developed methodological +platform offers a promising springboard for both mathematical progress and trans- +fer towards other forms of actuation within related control mechanisms. Finally, we +recognise that the assumption that full observations of the interfacial height are avail- + +16 +O. A. HOLROYD, R. CIMPEANU, AND S. N. GOMES +able is often unrealistic. In these scenarios, adaptations of the LQR method such as +static and dynamic output feedback controls have been used to stabilise long-wave +models [49], and so we are hopeful that future methods underpinned by the present +work will generalise to the full Navier-Stokes system, and further to physical experi- +ments. +Appendix A. Parameter values. +Although the majority of the results in this +paper are applied to the dimensionless systems governed by the dimensionless numbers +L, θ, Re, and Ca, it is important not to forget the physical roots of these systems. For +all of the numerical simulations in this work, we have fixed the aspect ratio L = 30, +the inclination angle θ = π/3, gravitational acceleration g = 9.807ms−2, and control +width ω = 0.1. A range of values for the dimensional parameters (and the resulting +dimensionless numbers) is provided in Table 1. A wide range of physical configurations +of interest are thus described by a parametric envelope given by 100 < Re < 102 and +10−4 < Ca < 10−2. +Fluid +ρ ( kgm−3) +µ ( kgm−1 s−1) +γ ( Nm−1) +Re +Ca +Water +999.8 +8.91 × 10−4 +0.072 +28.2 +0.0018 +Ethanol +789.5 +1.06 × 10−3 +0.022 +12.6 +0.0047 +Pentane +626.0 +2.24 × 10−4 +0.018 +178 +0.0045 +Nitrogen +3.44 +6.88 × 10−6 +0.0085 +5.69 +5.26 × 10−5 +Table 1 +Parameters (and resulting dimensionless numbers) for a range of physical fluids with a Nusselt +film height of 175 × 10−6 m. +Appendix B. Numerical simulations. +The Navier-Stokes equations ((2.2), +(2.3), and (2.5)–(2.7)) are solved on a finite domain Ω = [0, L] × [0, 8] (the permissive +height setup has been designed to prevent spurious pressure waves in the gas affect- +ing the film) using the volume-of-fluid (VOF) method [46]. The computations were +performed using Basilisk [36], a free extension to the C language designed to simplify +writing code to numerically solve PDEs. It solves the incompressible Navier-Stokes +equations on an adaptive quadtree grid [39] using the Bell-Collela-Glaz advection +scheme with a CFL-limited time step, and an implicit viscosity solver (as did its pre- +decessor, Gerris [37, 38]). The grid spacing ranges from L × 2−8 (covering the liquid +film) to L × 2−6 (smoothing out spurious pressure waves in the gas at the top of the +finite computational domain). The time step is capped at 0.05 to prevent sudden +jumps in the actuator inputs. +Since the control strategy is fundamentally agnostic to the specifics of the PDE +system being controlled aside from the entries of the linearised matrices J and Ψ, the +control code can be largely separated from the fluid simulation code. It would thus +be relatively easy to transfer the same framework to a different problem. +The Benney and weighted-residual equations are solved using second-order finite- +difference stencils for the spacial grid and a second-order backward finite-difference +scheme (BDF2) in time as, in Thompson, Tseluiko, and Papageorgiou [50]. +The +resulting problem is fully implicit and is solved via direct Newton iteration. All the +computations in this paper were performed on a grid with a spacing of L × 2−8 to +match the resolution of the Basilisk grid. +Appendix C. Control theory fundamentals. +Here we provide a brief over- +view of some important definitions in control theory relevant to our study. For more + +FALLING LIQUID FILM CONTROL +17 +detailed aspects we refer the interested reader to the seminal work of Zabczyk [60]. +Suppose we have the linear control system +(C.1) +ˆht = Jˆh + Ψu, +u = Ky, +y = Φˆh, +which can be written ˆht = (J + ΨKΦ)ˆh. The pair (J, Ψ) is controllable if, for any +pair of states ˆh0, ˆh1 ∈ RN there exists a control u that takes ˆh from ˆh0 to ˆh1 in +finite time. The pair (J, Φ) is observable if for all initial conditions ˆh0 ∈ RN there +exists a time T > 0 after which ˆh0 is uniquely determined from the observations +{y(t)|t ∈ [0, T]}. +Controllability and observability are duals, that is, if (J, Ψ) is +controllable then (J∗, Ψ∗) (where ·∗ is the conjugate transpose) is observable and +conversely if (J, Φ) is observable then (J∗, Φ∗) is controllable. +We can check if a pair (J, Ψ) is controllable with the Kalman rank condition: +(J, Ψ) is controllable if rank([J|Ψ]) = N, where +(C.2) +[J|Ψ] = [Ψ JΨ J2Ψ . . . JN−1Ψ] +is known as the controllability matrix. +In this paper, we are concerned with controlling towards the state ˆh = 0 rather +than an arbitrary interface (see Thompson et al. [49]), and so we require a weaker +form of controllability. For this we require (J, Ψ) to be stabilisable, which means that +there exists a gain matrix K such that J +ΨK is stable (i.e. has strictly negative real +parts to all its eigenvalues). Similarly, in this case (J, Φ) is detectable if we can choose +an L such that J + LΦ is stable, corresponding to being able to observe all of the +unstable modes of the system. As for controllability and observability, stabilisability +and detectability are dual properties (simply set L = K∗ and vice versa). +Supplementary Material. +Supplementary material showing the evolution of +the interface before and after the application of controls alongside the corresponding +2-norm deviations across a range of Reynolds and capillary numbers will be available +upon publication. +The version of the code used for this paper, along with installation instructions +and documentation, can be found on GitHub. +On a single core a full simulation +(for instance the one shown in Figure 5) takes ∼ 10 hours for the Navier-Stokes and +weighted-residual systems (the Benney system is considerably faster). +Acknowledgements. +Oscar Holroyd is grateful for the computing resources +supplied by the University of Warwick Scientific Computing Research Technology +Platform (SCRTP) and funding from the UK Engineering and Physical Sciences Re- +search Council (EPSRC) grant EP/S022848/1 for the University of Warwick Centre +for Doctoral Training in Modelling of Heterogeneous Systems (HetSys). 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Zabczyk, Mathematical control theory: an introduction, Birkenh¨auser, 1992. + diff --git a/0dFIT4oBgHgl3EQf3CsM/content/tmp_files/load_file.txt b/0dFIT4oBgHgl3EQf3CsM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88bc58b4993266fae1c17bbad25fe66afffac2f0 --- /dev/null +++ b/0dFIT4oBgHgl3EQf3CsM/content/tmp_files/load_file.txt @@ -0,0 +1,1395 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf,len=1394 +page_content='LINEAR QUADRATIC REGULATION CONTROL FOR FALLING LIQUID FILMS OSCAR A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD∗, RADU CIMPEANU∗, AND SUSANA N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We propose and analyse a new methodology based on linear-quadratic regulation (LQR) for stabilizing falling liquid films via blowing and suction at the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' LQR methods enable rapidly responding feedback control by precomputing a gain matrix, but are only suitable for systems of linear ordinary differential equations (ODEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' By contrast, the Navier-Stokes equations that describe the dynamics of a thin liquid film flowing down an inclined plane are too complex to stabilise with standard control-theoretical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' To bridge this gap we use reduced-order models – the Benney equation and a weighted-residual integral boundary layer model – obtained via asymptotic analysis to derive a multi-level control framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This framework consists of an LQR feedback control designed for a linearised and discretised system of ODEs approximating the reduced-order system, which are then applied to the full Navier-Stokes system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The control scheme is tested via direct numerical simulation (DNS), and compared to analytical predictions of linear stability thresholds and minimum required actuator numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Comparing the strategy between the two reduced-order models we show that in both cases we can successfully stabilise towards a uniform flat film across their respective ranges of valid parameters, with the more accurate weighted-residual model outperforming the Benney-derived controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The weighted-residual controls are also found to work successfully far beyond their anticipated range of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The proposed methodology increases the feasibility of transferring robust control techniques towards real-world systems, and is also generalisable to other forms of actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Feedback control, Stabilisation, Falling liquid films, Asymptotic analysis, Reduced- order modelling, Direct numerical simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Modelling and stabilisation of falling liquid films is a funda- mental problem at the intersection of fluid dynamics, asymptotic analysis, and control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Manipulation of these multi-scale systems is key to a number of industrial applications ranging from coating flows in liquid crystal display devices to microchip manufacture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Such systems have a high degree of complexity, which makes them challenging to model, control and simulate accurately and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Although the control of complex systems is a challenge found in a wide range of applied sciences – from preventing ice buildup on aerofoils [34] to avoiding obstacles in self-driving vehicles [33] and crowd management [8] – falling liquid films are a proto- typical example of such a control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Governed by the two-phase Navier-Stokes equations, these are multi-scale setups used in many applications as well as beautiful day-to-day phenomena such as wavy films on a window on a rainy day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The resulting flow becomes unstable above a critical Reynolds number (a parameter depending on velocity, inclination angle, film thickness, and fluid density), exhibiting a rich set of behaviours starting with two-dimensional (2D) waves and leading to 3D spatiotempo- ral chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Since the 1960s the problem has attracted much analytical focus, resulting in a number of periodic reduced-order models [6, 43, 47, 58] based on the assumption that perturbations are ‘long-wave’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' their wavelength is much larger than their amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Such models range from single equation models describing the dynam- ics of the liquid film height, to systems of multiple equations describing the height, downstream flux, and potentially additional independent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A broad range of thin-film models are covered extensively in two reviews by Craster and Matar [11] and Kalliadasis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [22], and these models were recently coupled with validation in experimental settings by Denner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' More recently, Richard et al [42], ∗Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='holroyd@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='uk, radu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='cimpeanu@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='uk, susana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='gomes@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='11379v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='OC] 26 Jan 2023 2 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES Usha, Chattopadhyay, and Tiwari [53], Mukhopadhyay, Ruyer-Quil, and Usha [28] have provided new insights into how two-equation models behave, particularly when attempting to apply them beyond their expected range of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Thin liquid films have numerous industrial applications, most notably in coating flows [57], heat and mass transfer [56], thin-film thermoelectric cooling [12], as well as ice accretion prevention on aircraft surfaces [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' These applications require control- ling the interface to a specific shape, whether that be flat for coating flows or highly corrugated for heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' There are almost as many physical input mechanisms as there are applications, and there is an extensive body of literature dealing with the effects that these have on stability and the critical Reynolds number above which unstable modes exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' These include heating and cooling of the fluid [5], electric [52] or magnetic [2] fields, porous [31] or deformable [15] walls, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here, as shown in Figure 1, we focus on blowing and suction through the base via dis- crete actuators [50], since they act over faster (and therefore more computationally accessible) timescales, and their effect on the overall flow control is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' While continuous controls, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=', controls applied through the entire domain (flow base), are more mathematically tractable, discrete controls applied through small holes or slots are a necessary step towards real applications, since applying blowing and suction controls throughout the whole domain is unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In the last two decades, the field has progressed from studying the effects that static, predetermined alterations to the system (such as fixed heating patterns or cor- rugated baseplates) can produce towards feedback control, where information from the interface is used to update control inputs as the film evolves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Armaou and Christofides [3, 9] and more recently Gomes, Papageorgiou, and Pavliotis [18] and Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [17, 19], studied the simplest thin-film model, the Kuramoto-Sivashinsky (KS) equation, from both analytic and computational perspectives, and successfully applied methods from linear control theory based on [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Thompson et al [49] further extended these results to the Benney [6] and weighted-residual [43] equations using a family of linear-quadratic regulator (LQR) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' However these long-wave models remain one stage removed from the physical system that they approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Unfor- tunately, the full two-phase Navier-Stokes system is too complex and nonlinear to directly extend the previous work on long-wave models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Cimpeanu, Gomes, and Pa- pageorgiou [10] first analysed aspects of model reduction applicability, and delineated the discrepancies between thin-film modelling and direct numerical simulation (DNS) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' They considered a scenario in which actuator input is simply propor- tional to the observed interfacial deviation at its position, which previous numerical evidence suggested could successfully stabilise the unperturbed (flat) state of the Benney and weighted-residual systems [49], the so-called Nusselt solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' They then showed how this model information can be interpreted and transferred into a more accurate simulation framework to successfully stabilise the full Navier-Stokes system via DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Nevertheless, a rigorous optimal control approach capable of surpassing the limitations of proportional control setups remains a challenge yet to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' An alternative approach which incorporated model predictive control (MPC) was pro- posed by Wray, Cimpeanu and Gomes [59] in the context of electrostatic actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This includes designing optimal controls for reduced-order models and enhanced in- teraction between model and simulation techniques, where any re-initialisation of the optimal control problem uses readings from a direct numerical simulation of the full problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The computational cost of this framework may however prove prohibitive in real-world contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In this work we aim to close the gap between robust controls designed for long- FALLING LIQUID FILM CONTROL 3 wave models and more physically relevant systems such as those governed by the two-phase Navier-Stokes equations, ultimately bringing real-world applications closer within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In section 2 we begin with a de- scription of the models that make up the two-tier hierarchical structure of the control framework: the two-phase Navier-Stokes equations as the target system and either the Benney or weighted-residual model as the control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In section 3 we outline the control method: a set of discrete actuators injecting and removing fluid at the base of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We further simplify these reduced-order models to a system of linear partial differential equations (PDEs), and finally to a finite set of ODEs, which permits the use of a linear-quadratic regulator, an established control-theoretical technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We then demonstrate for the first time that, by applying this control strategy to the full Benney, weighted-residual, and Navier-Stokes systems, there is good agreement be- tween the linear predictions and a series of nonlinear numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Finally, in section 4 we illustrate that this agreement spans a large region of the parameter space corresponding to physically relevant fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Furthermore, our stability analysis results indicate that the control performance significantly exceeds the range of validity of the underpinning model assumptions in certain regions of the explored parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Governing Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We consider a thin film of fluid flowing down a plane tilted at an angle θ from the horizontal, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We restrict ourselves to 2D flows, largely to make the problem more computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Nevertheless, this setup exhibits a highly nontrivial and physically rich behaviour, while also cap- turing the initial wave development stages before cross-flow effects begin to appear in 3D contexts [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In many control scenarios manipulating the dynamics of these early stages is the key objective (and only realisable strategy) of the control framework before highly nonlinear and often undesirable flow features arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We use a coordinate system rotated with the plane, where x points downstream and y is the perpendicular distance to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' There is a free surface at the upper interface of the fluid at y = h(x, t) where the fluid and gas meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We inject and remove fluid through the rigid lower wall regions as dictated by our resulting control strat- egy, with no-slip and impermeability conditions governing the remaining uncontrolled boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Finally, we consider periodic boundary conditions in the x-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' while an experiment would be realised on an open domain with inflow and outflow, the speed with which a wave fully develops after the inlet [13] means that for sufficiently large domains – which we consider here – periodic boundary conditions provide a reasonable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Furthermore, periodicity allows us to perform the analysis performed in section 4 below, which would be more challenging, if not impossible, to undertake given the inability to compute eigenvalues of the problem explicitly with open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The problem is governed by the conservation of mass and momentum in both the liquid film and the gas above it, coupled at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Typically, the large density and viscosity ratios between the two media mean that we can consider the gas region to be hydrodynamically passive, and can ignore the flow in the gas, and model the liquid film alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The fluid flow is governed by the acceleration due to gravity g, the inclination angle θ, and the physical properties of the liquid phase: constant density ρ, viscosity µ and surface tension coefficient γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A full list of physically-relevant parameters and the values used in this investigation can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For a liquid film with mean height hs, the uncontrolled system admits a uniform solution known as the Nusselt solution [30], where h(x, t) = hs, which has a parabolic 4 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES y x v u θ h(x, t) f(x, t) 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Diagrammatic representation of a falling liquid film under gravity with basal forcing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Controls are applied through the wall at y = 0, and the behaviour of the interface h is governed by the fluid parameters and the inclination angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' velocity profile with surface velocity Us = ρgh2 s sin θ 2µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We then nondimensionalise the problem based on the length scale hs, velocity scale Us and pressure scale µUs hs , defining the Reynolds and capillary numbers (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1) Re = ρUshs µ , Ca = µUs γ , which measure the relative importance of inertial and viscous terms, and of gravity and surface tension, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The full liquid film flow is governed by the 2D Navier-Stokes equations, which are solved for velocity u(x, y, t) = (u, v) and pressure p(x, y, t) under the action of external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The governing (nondimensionalised) internal momentum equations are Re(ut + uux + vuy) = −px + 2 + uxx + uyy, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2) Re(vt + uvx + vvy) = −py − 2 cot θ + vxx + vyy, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3) and the continuity equation reads (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4) ux + vy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The system is completed by its boundary conditions: periodic boundaries in the x-direction, no-slip and fluid injection/removal at the wall, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5) u = 0, v = f(x, t), the nonlinear dynamic stress balance (or momentum jump) at the interface, y = h(x, t), (vx + uy)(1 − h2 x) + 2hx(vy − ux) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='6) p − 2 1 + h2x (vy + uxh2 x − hx(vx + uy)) = − 1 Ca hxx (1 + h2x)3/2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7) and finally the kinematic boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='8) ht = v − uhx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In lieu of physical experiments, we perform computational analogues by simu- lating the Navier-Stokes equations using a volume-of-fluid approach developed by Popinet [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The methodology is well known, following more than two decades of FALLING LIQUID FILM CONTROL 5 successful development and usage in the community [37, 38, 39], and therefore we restrict our attention to details relevant to our particular setting in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Defining the down-slope flux q(x, t) by integrating over the height of the film (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='9) q(x, t) = � h 0 u(x, y, t) dy, we combine (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='8) to obtain the 1D mass conservation equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10) ht + qx = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We can continue to use the Navier-Stokes equations to compute q, or we can use one of a number of simplified models for the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here we consider the Benney [6] and weighted-residual [43] equations, which are valid in the long-wave limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' By using a pair of reduced-order models we are better able to gauge the relative capabilities of both, weighing model and computational complexity against control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The following pair of reduced-order models are based on first-order asymptotic expan- sions in the long-wave parameter ϵ = 1/L (where L is the aspect ratio of the domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In addition to the requirement that ϵ ≪ 1, we make the assumption that Re = O(1) and Ca = O(ϵ2) to retain inertial and surface tension effects, and that f = O(ϵ) so that the magnitude of the imposed control is comparable to the perturbed flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Benney equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The first choice of model for the downstream flux is the Benney system [6], which was extended to include the effects of O(ϵ) controls by Thompson, Tseluiko, and Papageorgiou [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This enslaves the flux to the interfacial height h via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='11) q(x, t) = h3 3 � 2 − 2hx cot θ + hxxx Ca � + Re �8h6hx 15 − 2h4f 3 � , resulting in a single equation for the evolution of the interface when coupled to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The system above is a significant improvement over equations such as the KS equation – the simplest nonlinear model of thin film flows [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' While capturing some important aspects of the chaotic behaviour of falling liquid films such as travelling waves, the KS equation is more useful as a paradigmatic example in a dynamical systems sense than as a predictive model outside of a very restricted region of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' However, the Benney system exhibits undesirable behaviours such as unphysical finite- time blow-up outside of a narrow range of parameters corresponding to low Reynolds numbers [41], as is demonstrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Weighted-residual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' To overcome the unrealistic behaviour de- scribed above, Ruyer-Quil and Manneville [43, 44] proposed an improved weighted- residual methodology based on approximating u by a truncated sum of basis functions satisfying no-slip boundary conditions at the wall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5) and zero tangential stress at the interface (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here we use the first-order truncation, which matches well with the second-order truncation up to Re ≈ 5 [43], a significant improvement over pre- vious models [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' When combined with the basal forcing this results in a separate evolution equation for the flux [50] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='12) 2Re 5 h2qt+q = h3 3 � 2 − 2hx cot θ + hxxx Ca � +Re �18q2hx 35 − 34hqqx 35 + hqf 5 � , which together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10) forms a system of two PDEs for the height h(x, t) and the flux q(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='12) better captures many features of the full Navier-Stokes 6 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES film, such as spontaneous back-flow [32], and Figure 2 illustrates how it provides a good match for the interfacial shape even at moderate Reynolds and capillary num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' However, it does overestimate the amplitude of the capillary ripples, as observed by Ruyer-Quil and Manneville [43] for the weighted-residuals system and also for other first-order models [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Unlike the Benney equation, it also does not exhibit unphys- ical finite-time blow-up, although it too diverges from the Navier-Stokes model at moderate Reynolds numbers, Re ≈ 10 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 0 1 2 t = 0 h 0 1 2 t = 2 h 0 1 2 t = 50 h 0 5 10 15 20 25 30 0 1 2 t = 300 x h 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Evolution of interfacial heights h for Navier-Stokes (black), weighted-residual (red), and Benney (blue) systems, with peaks shifted to 3L/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here, the parameters used are Re = 10, Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05, θ = π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The Benney equation blows up shortly after t = 2, but the weighted-residual and Navier-Stokes interfaces have very similar structures aside from some spurious oscillations in the weighted-residual case, which are observable at t = 300 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Control methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We focus on controlling the interface towards the Nusselt solution, which under our nondimensionalisation is the uniform film h(x, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' All the controls we consider are a class of time-dependent controls known as feedback controls, which we introduce here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Take a controlled quantity x governed by the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1) xt = Ax + Bu, y = Cx, where u is the control, y is some observation of the system, and A, B, and C are arbitrary operators describing the uncontrolled system dynamics, the control actua- tion mechanism, and the observations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In the case of feedback controls, we have the restriction that u = Ky for some operator K, so that the system can be written in closed-loop form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2) xt = (A + BKC)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' An overview of some important control theory definitions is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In the case of falling liquid films, the full system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='8) is too complex for standard (linear) control-theoretical techniques to be tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Instead, we design feedback controls for the reduced-order models presented in subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3 FALLING LIQUID FILM CONTROL 7 and apply them to the full Navier-Stokes system by passing the Navier-Stokes in- terfacial height to the feedback control scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The full framework is pictured in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Initial condition 1 Apply controls 4 Time step 5 Reduced order model 2 Compute controls 3 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Multi-layer control methodology for the control of Navier-Stokes thin liquid films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' From the initial condition, we treat the interface as though it were described by the chosen reduced-order model and generate the feedback control accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We then apply this control to the full model and time step forward to repeat the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Given the difficulties in observing both the height [24, 55] and flux [20] of falling liquid films, it is understandable that in our case we might wish to express our control f as a function of time only, known as offline control (as can be done when generating controls for the KS equation for instance, see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Unfortunately, as shown by Cimpeanu, Gomes, and Papageorgiou [10], although such hierarchical controls show promising initial dampening, they invariably fail over longer timescales, as the Navier- Stokes model eventually diverges from the chosen long-wave model, and the action of the control no longer affects the intended state, with the Navier-Stokes system ultimately converging back to its uncontrolled behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Control actuation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Although the control term f in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10)– (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='12) is general, in this study we restrict ourselves to controls taking the form of Dirac-delta distributions injecting or removing fluid at the wall at a finite number of locations x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' , xM – which is more experimentally achievable than continuous controls, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' controls applied everywhere in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Furthermore, both due to realistic considerations and computational restrictions imposed by the direct numeri- cal simulation setup, we must approximate these point sources by finite regions which we select to be smooth, periodic functions (shown in Figure 4), of the type (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3) d(x) = A exp �cos(2πx/L) − 1 ω2 � , where ω controls the width of the function, and A is chosen so that � L 0 d(x) dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' More refined discretisations support smaller values of ω, and d(x) → δ(x) as ω → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The basal forcing term f is thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4) f(x, t) = M � i=1 ui(t)d(x − xi), 8 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES where ui(t) are the individual, time-dependent, control amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Despite the prac- tical difficulties of obtaining full observations, the main goal of this investigation is to test the feasibility of the control methodology, and so for the moment we assume we are able to observe the full interface h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This means that C, the observation operator from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2), is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We will address the observability of the problem, as well as issues introduced by noisy or partial observations of the interface, in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Note that, since the domain has periodic boundaries on the left and right, the problem is translationally invariant, and so x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' , xM should be evenly placed along the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For 3D and non-periodic flows, the optimal placement of the actuators is a nontrivial problem [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Finally, we introduce a cost functional to compare different control strategies, taking into account the 2-norms of the deviation from the target state and penalising the use of the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We thus define the total cost of a control by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5) κ = � ∞ 0 � L 0 βˆh(x)2 + (1 − β)f 2 dx dt, where ˆh(x) = h(x) − 1 is the deviation of the interface from the target uniform state, and the parameter β controls the relative importance of the interfacial deviation and the magnitude of the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Linear-Quadratic Regulator (LQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Despite the long-wave simplifi- cation to one of the two reduced-order models, control methodologies for nonlinear PDEs of the type considered here are still a rapidly developing area of active research, with the most relevant efforts by Boujo and Sellier [7], Lunz [25], or the current au- thors [10, 18, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In spite of these recent advances, we cannot directly choose the optimal control operator K with analytical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' To make this problem tractable, we assume that the perturbation away from the Nusselt solution (ˆh = h − 1 = 0, ˆq = q −2/3 = 0, ˆf = 0) is small, so that we can linearise (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='11) to obtain the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='6) ˆht = � −2∂x + �2 cot θ 3 − 8Re 15 � ∂xx − 1 3Ca ∂xxxx � ˆh + � 1 + 2Re 3 ∂x � ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We then discretise to form a system of N ODEs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7) dˆh dt = Jˆh + Ψu, u = KΦˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here, J ∈ RN×N captures the system dynamics, Ψ ∈ RN×M is the linearised actuator matrix, and Φ ∈ RN×N is the linearised observation matrix (which we take to be the identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' K ∈ RM×N is the gain matrix, which is chosen to minimise the discrete cost (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='8) c = � ∞ 0 ˆhTUˆh + uTV u dt, where U = βL N I ∈ RN×N and V = (1 − β)I ∈ RM×M are matrices whose entries are chosen so as to form the discrete analogue of the continuous cost (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The process is similar for the weighted-residual system, but with twice the system size at each stage, since there are two unknowns, ˆh and ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The linearisation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='12) results in ˆht = −ˆqx + ˆf, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='9) ˆqt = � 5 Re + �4 7 − 5 cot θ 3Re � ∂x + 5 6ReCa ∂xxx � ˆh − � 5 2Re + 34 21∂x � ˆq + �1 3 � ˆf, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10) FALLING LIQUID FILM CONTROL 9 and the resulting discretised system has 2N equations rather than N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Finally, al- though we are assuming full observations of the interfacial height, Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [49] showed that it is sufficient to use the leading order approximation ˆq = 2ˆh to remove the need to directly observe the flux, incurring a small penalty in the size of the largest eigenvalue but not fundamentally affecting stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This is especially important because the flux is challenging to measure in an application setting [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This setup forms a classic problem in control theory: the linear-quadratic regu- lator (LQR) problem, which is a subset of a broader class of static output feedback (SOF) problems in which one can also have restricted observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' rank(Φ) < N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here, we provide an overview of how this class of problems is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For more details see [21, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For the discretised linear control system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7), we write the cost as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='11) c = � ∞ 0 ˆhTUˆh + uTV u dt = � ∞ 0 ˆhT(U + ΦTKTV KΦ)ˆh dt, where U, V are assumed to be symmetric positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' If we suppose there exists a symmetric, positive semi-definite matrix P such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='12) d dt(ˆhTPˆh) = −ˆhT(U + ΦTKTV KΦ)ˆh, then, as long as the controlled system matrix A = J + ΨKΦ is asymptotically stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=', all its eigenvalues have negative real part, we can write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='11) as c = ˆh(0)TPˆh(0) − lim t→∞ ˆh(t)TPˆh(t) = ˆh(0)TPˆh(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='13) By expanding out the left hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='12) and observing that this is true for all initial conditions ˆh(0) ∈ RN, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='14) ATP + PA + U + ΦTKTV KΦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This further implies that the choice of P is independent of the initial condition ˆh(0), and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='15) c = tr(PX), where X = ˆh(0)ˆh(0)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Since we wish to choose an optimal K for all initial condi- tions, we set X = E[ˆh(0)ˆh(0)T] = I, the identity matrix, as we assume all initial perturbations ˆh(0) are equally likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The problem thus becomes equivalent to selecting K to minimise (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='15) subject to the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This can be solved via Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Defining the symmetric matrix of Lagrange multipliers S, we then have the resulting Hamiltonian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='16) H = tr(PI) + tr((ATP + PA + U + ΦTKTV KΦ)S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' By setting ∂SH = ∂P H = ∂KH = 0 we have the conditions for the solution to the SOF problem: 0 = ATP + PA + U + ΦTKTV KΦ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='17) 0 = AS + SAT + I, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='18) 0 = V KΦSΦT + ΨTPSΦT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='19) 10 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES The final condition can be more usefully written as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='20) K = −V −1ΨTPSΦT(ΦSΦT)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='17)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='19) cannot be solved directly, and so an iterative procedure must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' However, in the special case of the LQR problem where we have Φ = I, we may discard (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='18) and rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='20) as 0 = JTP + PJ + U − PΨV −1ΨTP, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='21) K = −V −1ΨTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='22) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='21), which is known as the continuous algebraic Riccati equation (CARE), can be solved for P directly, and then used to compute K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The structure of the matrices J, U, V , and Ψ – with U and V diagonal, J periodic banded and Ψ having translational symmetries – means that the specific CARE for this problem is typically well-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Thus we can make use of the classical eigenvector approach described by MacFarlane [26], Potter [40] and Vaughan [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Alternatively, Schur [23] and generalised eigenvector [4] approaches may offer improved numerical stability for larger systems (which would be encountered in 3D) and more unstable regimes (where some of the interim matrices used in the classical method become singular or near-singular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We note that equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='22) illustrate why the single parameter β is sufficient to fully explore the cost-space with regards to K: if we instead introduce a pair of control parameters α and β so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='23) U ′ = αU = αβL N I, V ′ = αV = α(1 − β)I, we can set the entries of U ′ and V ′ independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The cost is then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='24) c′ = αc = 1 2 tr(αPX) = 1 2 tr(P ′X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Carrying the new cost matrices through to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='22) we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='25) K′ = −(V ′)−1ΨTP ′ = −(αV )−1ΨTαP = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The above result indicates that scaling the cost makes no difference to the optimal K, and so a single parameter describing the ratio of significance of the two components is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Gibson [16] showed that, under certain conditions, the discretised feedback operator K does converge to its infinite-dimensional counterpart K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Once the optimal gain matrix K has been computed, we can calculate the mth actuator amplitude as um = Km · ˆh(t), where Km is the mth row of K and · denotes the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This means that Km,i can be interpreted as describing the im- portance of the ith entry of ˆh to um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' As can be seen in Figure 4, this allows us to examine the rows of K to develop an understanding of how the controls operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The weighted-residual gains are tightly clustered around the location of the actuator across a wide range of Reynolds numbers, with minimal up- and down-stream contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' By contrast, the Benney gains are much broader and depend more strongly on the interfacial shape away from the actuator location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' They also are much more sensitive to the Reynolds number (it is worth noting that, for Re = 30, the Benney-derived controls fail to stabilise the Navier-Stokes system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' With a method to compute the gain matrix for the two reduced-order models we are now well-positioned to deploy the methodology described in Figure 3 and direct it towards the modelled physical system of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' FALLING LIQUID FILM CONTROL 11 0 5 10 15 20 25 30 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 2 x Feedback gain Actuator shape B, Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 B, Re = 10 B, Re = 30 WR, Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 WR, Re = 10 WR, Re = 30 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The second row of the gain matrix computed using either the Benney equation (in blue) or weighted-residual system (in red) as the reduced-order model, as Re varies and Ca is fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The gains are shown alongside the corresponding actuator (in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Although the weighted- residual gains remain clustered around the actuator, the Benney gains have significant nonlocal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Previous work by Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [49] confirmed that LQR controls with full observations are able to stabilise both the Benney and weighted-residual systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The same authors also found that the Benney controls stabilise the weighted-residual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In Figure 5 we can see that for a similar pa- rameter regime (Re = 5, Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05 – selected such that Re is not so high so as to make numerical simulation difficult, and Ca is large enough that surface tension alone cannot stabilise the liquid film with properties experimentally aligned with a relatively thick and viscous oil flow), these controls can be extended to the Navier- Stokes system, where we achieve similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Figure 5 shows how the interface is allowed to develop from a small sinusoidal perturbation into a travelling wave, before the application of controls at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Representative interfacial snapshots are pictured in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The interfacial deviation then decays exponentially, suggesting that the use of linear models to design the gain matrix is appropriate in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' It is encouraging to see that we can control the film in the specific setting of Figure 5, but a better aim is to predict the stabilisability of the system given the flow parameters Re, Ca, θ and number of controls M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Since we lack a closed-form expression for either the continuous control f(h), or its discrete counterpart ΨKˆh, we cannot directly estimate the stability properties of the controlled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' However, we can predict the damping rate by finding the largest eigenvalue λ∗ of the controlled system matrix A = J + ΨKΦ and compare that to rates fitted to the data produced in our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' From Figure 6 we observe that the Benney-derived controls directly stabilise the Benney and weighted-residual systems (in a similar setup to that used by Thomp- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [49]) over a wide range of Reynolds numbers, and that their ability to stabilise towards the uniform film extends to the hierarchical controls applied to the Navier-Stokes film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We note that the weighted-residual and Navier-Stokes systems are stabilised even above the stability threshold, after which the linearised weighted- residual model predicts that five actuators are not sufficient to stabilise the uniform 12 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES 0 1 2 t = 0 h t = 0 0 1 2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 h t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 0 1 2 t = 1 h t = 1 0 5 10 15 20 25 30 0 1 2 t = 10 x h 0 5 10 15 20 25 30 t = 10 x t = 0 −1 0 1 t = 0 f t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 −1 0 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 f t = 1 −1 0 1 t = 1 f t = 10 −1 0 1 t = 10 f −100 −50 0 50 100 10−5 10−4 10−3 10−2 10−1 100 t |h − 1| 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Interfacial shapes before and after the controls are switched on: Benney equation derived controls in blue (left), weighted-residual derived controls in red (centre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A travelling wave is allowed to develop until t = 0, when the controls are activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Both controls successfully damp out the perturbation, with the control amplitudes decreasing in proportion to |h − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We note that, although similar, the controls are not identical – see the second and fourth rows in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In both cases (Benney in blue, weighted-residual in red) the 2-norm of the deviation of the interface from the target state decays exponentially (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' After t ≈ 50 the deviation is small enough that machine precision interferes with computing the deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In these simulations we used Re = 5, Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 0 10 20 30 40 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 Re λ∗ stability threshold linearised Benney linearised WR Benney WR NS 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Comparison of (fitted) damping rates for Benney-derived LQR control applied to Benney (blue), weighted-residual (red), and Navier-Stokes (black) systems (all solid) to the predictions from the linearised systems of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here, we used M = 5 controls and Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For all three systems, the numerical models break down at sufficiently large Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' All three models display unphysical blow-up at sufficiently large Reynolds num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Although this is expected behaviour in the case of the Benney film [29], in the case of the weighted-residual and Navier-Stokes models this is attributed to the eventual breakdown of the controls as the Benney model finally loses the last of its FALLING LIQUID FILM CONTROL 13 predictive capacity at larger values of Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 0 10 20 30 40 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 Re λ∗ stability threshold linearised Benney linearised WR Benney WR NS 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Comparison of (fitted) damping rates for weighted-residual-derived LQR control ap- plied to Benney (blue), weighted-residual (red), and Navier-Stokes (black) systems (all solid) to the predictions from the linearised systems of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here, we used M = 5 controls Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' While the Benney-derived control rules stabilise all three models (at least for small-to-moderate Reynolds numbers), the weighted-residual derived controls fail to stabilise the Benney equation for Re > 7, in agreement with the linear predictions given by the eigenvalues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The weighted-residual and Navier-Stokes models have reasonable agreement with the linear damping rates but remain stabilisable even at Re = 50, when the linear system is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In order to make analytical progress, we turn to an equivalent way to produce the gain matrix K, where we first convert (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7) to Fourier space (so ˜h = Fˆh, where F is the Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We can then reorder the wavenumbers to separate stable and unstable modes: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1) d˜h dt = ˜J˜h + ˜Ψ ˜K˜h = � ˜Ju 0 0 ˜Js � ˜h + �˜Ψu ˜Ψs � ˜K˜h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Concentrating on the unstable modes more explicitly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=', d dt �˜hu ˜hs � = � ˜Ju 0 0 ˜Js � �˜hu ˜hs � + �˜Ψu ˜Ψs � ˜K �˜hu ˜hs � = � ˜Ju + ˜Ψu ˜Ku 0 ˜Ψs ˜Ks ˜Js � �˜hu ˜hs � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2) we find that since the matrix on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2) is block lower triangular, the controls leave the eigenvalues of the stable modes unchanged, and so they remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We thus reduce the control problem to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3) d˜hu dt = ˜Ju˜hu + ˜Ψu ˜Ku˜hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' By solving the problem in Fourier space it is clear that – for the purely linear case at least – we should expect that M actuators would be sufficient to control any 14 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES system satisfying M ≥ rank( ˜Ju).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' This would amount to one control per unstable mode plus one more to satisfy conservation of mass, as pointed out by Armaou and Christofides [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The rank of the unstable Jacobian ˜Ju corresponds to the number of unstable modes of the linearised system ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='6) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We compute this rank for a perturbation with wavenumber k, where the linearised Benney equation has a single eigenvalue (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4) λ = −2ik + �8Re 15 − 2 3 cot θ − 1 3Ca k2 � k2, and the weighted-residual system has a pair of eigenvalues that solve the quadratic equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5) λ2 + � 5 2Re + 34 21ik � λ + � 5 Re ik − �4 7 − 5 cot θ 3Re � k2 + 5 6ReCa k4 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Setting the real part ℜ(λ) = 0 we can solve for the critical wavenumber k0 (the boundary between stable and unstable unimodal systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5), this is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='6) k0 = ± � Ca �8 5Re − 2 cot θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' After rescaling to account for L ̸= 2π, this expression admits a single zero eigenmode and pairs of positive and negative modes with k < k0, resulting in the number of unstable modes being (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7) nu = 1 + 2 � L 2π k0 � = 1 + 2 � L 2π � Ca �8 5Re − 2 cot θ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In Figure 8 we compare our predictions for nu from expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7) to the min- imum number of controls required to stabilise the film in our numerical experiments of the Navier-Stokes system as Re and Ca vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We see that, as expected, the system is stabilisable at M ≥ nu in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In fact, in the majority of the parameter space, the minimum number of actuators required to stabilise the uniform state is lower than the number predicted by the linear analysis, particularly at lower Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' As previous work by Salamon, Armstrong, and Brown [45] and Ruyer-Quil and Man- neville [43] shows that important physical characteristics such as travelling wave speed begin to diverge from DNS results at Re ≈ 5, the fact that controls based on a lin- earisation of these equations match (or even exceed) the expected performance up to Re ≈ 100 is remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' However, after this point it becomes clear that we are reach- ing the limit of the model’s validity, and the ability of the controls to stabilise the uniform state becomes less predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We note that at larger Reynolds and capillary numbers the film takes much longer to respond to the effects of the controls, making it more challenging to assess whether the uniform state is stabilisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' By dynamically estimating the sign of the fitted damping rate we can avoid running simulations over unfeasibly long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The research presented herein has demonstrated new and sig- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='nificant capabilities in terms of design and analysis of optimal feedback controls for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='FALLING LIQUID FILM CONTROL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The minimum number of actuators required to stabilise the Navier-Stokes film compared to the number of unstable modes of the linearised weighted-residual system (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The number of controls needed to stabilise the uniform film never exceeds the number of unstable modes of the linear system nu as given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The ranges for the two parameters cover a broad range of different fluids, select examples are listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Videos of selected instances of film evolution and control are available as supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' complex physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The stabilisation of the canonical multi-scale framework of a thin liquid film falling down an inclined plane by employing reduced-order models such as the Benney and first-order weighted-residual equations has been used as the physical setup for our proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We developed an LQR approach via blowing and suction controls which has been shown to outperform the predictions of linear stability theory, and can successfully function beyond the region of model validity for either the Benney- or the weighted-residual-derived controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We have shown that even the crude controls used here far exceed their expected performance, and this opens up numerous avenues for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' It remains to be seen whether higher-order models such as the second-order weighted-residual integral boundary layer model proposed by Ruyer-Quil and Manneville [43] can be used to fur- ther improve the type of control demonstrated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In addition, it would be desirable to remove the control dependence on discretisation by developing infinite-dimensional controls, which might also allow for an improved analysis of control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Although here we have performed numerical experiments to showcase the control efficacy, physical experiments on real fluids are an obvious next step that we hope our work will inspire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In order to achieve this in practice there are a number of useful assumptions that must be relaxed, namely the 2D nature of the flow and periodic boundary condition formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The additional dimension will allow for cross-flow instabilities (an interaction which needs to be further quantified), and the boundaries can also affect the stability of the film [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The blowing and suction controls used in the present work offer a valuable theoretical foundation permitting a comprehensive examination of control performance for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We envision realistic embodi- ments thereof to require further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Nevertheless, the developed methodological platform offers a promising springboard for both mathematical progress and trans- fer towards other forms of actuation within related control mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Finally, we recognise that the assumption that full observations of the interfacial height are avail- 16 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' HOLROYD, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' CIMPEANU, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' GOMES able is often unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In these scenarios, adaptations of the LQR method such as static and dynamic output feedback controls have been used to stabilise long-wave models [49], and so we are hopeful that future methods underpinned by the present work will generalise to the full Navier-Stokes system, and further to physical experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Although the majority of the results in this paper are applied to the dimensionless systems governed by the dimensionless numbers L, θ, Re, and Ca, it is important not to forget the physical roots of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For all of the numerical simulations in this work, we have fixed the aspect ratio L = 30, the inclination angle θ = π/3, gravitational acceleration g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='807ms−2, and control width ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A range of values for the dimensional parameters (and the resulting dimensionless numbers) is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' A wide range of physical configurations of interest are thus described by a parametric envelope given by 100 < Re < 102 and 10−4 < Ca < 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Fluid ρ ( kgm−3) µ ( kgm−1 s−1) γ ( Nm−1) Re Ca Water 999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='91 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='072 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='0018 Ethanol 789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='06 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='022 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='0047 Pentane 626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='24 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='018 178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='0045 Nitrogen 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='88 × 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='0085 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='69 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='26 × 10−5 Table 1 Parameters (and resulting dimensionless numbers) for a range of physical fluids with a Nusselt film height of 175 × 10−6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The Navier-Stokes equations ((2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='3), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='7)) are solved on a finite domain Ω = [0, L] × [0, 8] (the permissive height setup has been designed to prevent spurious pressure waves in the gas affect- ing the film) using the volume-of-fluid (VOF) method [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The computations were performed using Basilisk [36], a free extension to the C language designed to simplify writing code to numerically solve PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' It solves the incompressible Navier-Stokes equations on an adaptive quadtree grid [39] using the Bell-Collela-Glaz advection scheme with a CFL-limited time step, and an implicit viscosity solver (as did its pre- decessor, Gerris [37, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The grid spacing ranges from L × 2−8 (covering the liquid film) to L × 2−6 (smoothing out spurious pressure waves in the gas at the top of the finite computational domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The time step is capped at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='05 to prevent sudden jumps in the actuator inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Since the control strategy is fundamentally agnostic to the specifics of the PDE system being controlled aside from the entries of the linearised matrices J and Ψ, the control code can be largely separated from the fluid simulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' It would thus be relatively easy to transfer the same framework to a different problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The Benney and weighted-residual equations are solved using second-order finite- difference stencils for the spacial grid and a second-order backward finite-difference scheme (BDF2) in time as, in Thompson, Tseluiko, and Papageorgiou [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The resulting problem is fully implicit and is solved via direct Newton iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' All the computations in this paper were performed on a grid with a spacing of L × 2−8 to match the resolution of the Basilisk grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Control theory fundamentals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Here we provide a brief over- view of some important definitions in control theory relevant to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For more FALLING LIQUID FILM CONTROL 17 detailed aspects we refer the interested reader to the seminal work of Zabczyk [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Suppose we have the linear control system (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='1) ˆht = Jˆh + Ψu, u = Ky, y = Φˆh, which can be written ˆht = (J + ΨKΦ)ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The pair (J, Ψ) is controllable if, for any pair of states ˆh0, ˆh1 ∈ RN there exists a control u that takes ˆh from ˆh0 to ˆh1 in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The pair (J, Φ) is observable if for all initial conditions ˆh0 ∈ RN there exists a time T > 0 after which ˆh0 is uniquely determined from the observations {y(t)|t ∈ [0, T]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Controllability and observability are duals, that is, if (J, Ψ) is controllable then (J∗, Ψ∗) (where ·∗ is the conjugate transpose) is observable and conversely if (J, Φ) is observable then (J∗, Φ∗) is controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' We can check if a pair (J, Ψ) is controllable with the Kalman rank condition: (J, Ψ) is controllable if rank([J|Ψ]) = N, where (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='2) [J|Ψ] = [Ψ JΨ J2Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' JN−1Ψ] is known as the controllability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' In this paper, we are concerned with controlling towards the state ˆh = 0 rather than an arbitrary interface (see Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' [49]), and so we require a weaker form of controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For this we require (J, Ψ) to be stabilisable, which means that there exists a gain matrix K such that J +ΨK is stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' has strictly negative real parts to all its eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Similarly, in this case (J, Φ) is detectable if we can choose an L such that J + LΦ is stable, corresponding to being able to observe all of the unstable modes of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' As for controllability and observability, stabilisability and detectability are dual properties (simply set L = K∗ and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Supplementary material showing the evolution of the interface before and after the application of controls alongside the corresponding 2-norm deviations across a range of Reynolds and capillary numbers will be available upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' The version of the code used for this paper, along with installation instructions and documentation, can be found on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' On a single core a full simulation (for instance the one shown in Figure 5) takes ∼ 10 hours for the Navier-Stokes and weighted-residual systems (the Benney system is considerably faster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Oscar Holroyd is grateful for the computing resources supplied by the University of Warwick Scientific Computing Research Technology Platform (SCRTP) and funding from the UK Engineering and Physical Sciences Re- search Council (EPSRC) grant EP/S022848/1 for the University of Warwick Centre for Doctoral Training in Modelling of Heterogeneous Systems (HetSys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Radu Cim- peanu and Susana Gomes also acknowledge EPSRC support via grant EP/V051385/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' For the purpose of open access, the authors have applied a Creative Commons Attri- bution (CC BY) licence to any arising Author Accepted Manuscript version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Alekseenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' Nakoryakov, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf'} +page_content=' G.' 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PREPRINT VERSION. ACCEPTED DECEMBER, 2022 +1 +SCENE: Reasoning about Traffic Scenes using +Heterogeneous Graph Neural Networks +Thomas Monninger†,1, Julian Schmidt†,2,3, +Jan Rupprecht2, David Raba2, Julian Jordan2, +Daniel Frank4, Steffen Staab4,5 and Klaus Dietmayer3, Senior Member, IEEE +Abstract—Understanding traffic scenes requires considering +heterogeneous information about dynamic agents and the static +infrastructure. In this work we propose SCENE, a methodology +to encode diverse traffic scenes in heterogeneous graphs and to +reason about these graphs using a heterogeneous Graph Neural +Network encoder and task-specific decoders. The heterogeneous +graphs, whose structures are defined by an ontology, consist of +different nodes with type-specific node features and different +relations with type-specific edge features. In order to exploit +all the information given by these graphs, we propose to use +cascaded layers of graph convolution. The result is an encoding +of the scene. Task-specific decoders can be applied to predict +desired attributes of the scene. Extensive evaluation on two +diverse binary node classification tasks show the main strength +of this methodology: despite being generic, it even manages to +outperform task-specific baselines. The further application of +our methodology to the task of node classification in various +knowledge graphs shows its transferability to other domains. +Index Terms—Semantic Scene Understanding, AI-Based Meth- +ods, Behavior-Based Systems +I. INTRODUCTION +U +NDERSTANDING traffic scenes is important for an +autonomous vehicle such that it may develop a safe, +effective and efficient plan of how to move forward. For +instance, whether a stationary car is parked or just temporarily +stopped determines whether the autonomous vehicle should +wait or overtake. Understanding of traffic scenes requires +reasoning about dynamic agents and static infrastructure in +order to predict the intents of nearby dynamic agents (e.g., +Manuscript received: August 26, 2022; Revised: November 21, 2022; +Accepted: December 20, 2022. +This paper was recommended for publication by Editor Markus Vincze +upon evaluation of the Associate Editor and Reviewers’ comments. This +work was supported by the BMWK within the project ”KI Delta Learning” +(F¨orderkennzeichen 19A19013A) and the Deutsche Forschungsgemeinschaft +(DFG, German Research Foundation) under Germany’s Excellence Strategy - +EXC 2075 – 390740016. (Corresponding author: Julian Schmidt) +†Thomas Monninger and Julian Schmidt are co-first authors. The order was +determined alphabetically. +1Thomas Monninger is with Mercedes-Benz R&D North America, Sunny- +vale, CA, USA (e-mail: thomas.monninger@mercedes-benz.com) +2Julian Schmidt, Jan Rupprecht, David Raba and Julian Jordan are with +Mercedes-Benz AG, R&D, Stuttgart, Germany (e-mail: {julian.sj.schmidt, +jan.rupprecht, david.raba, julian.jordan}@mercedes-benz.com) +3Julian Schmidt and Klaus Dietmayer are with Ulm University, Institute +of Measurement, Control and Microtechnology, Ulm, Germany (e-mail: +klaus.dietmayer@uni-ulm.de) +4Daniel Frank and Steffen Staab are with University of Stuttgart, Institute +of Parallel and Distributed Systems, Stuttgart, Germany (e-mail: {daniel.frank, +steffen.staab}@ipvs.uni-stuttgart.de) +5Steffen Staab is with University of Southampton, Electronics and Com- +puter Science, Southampton, United Kingdom +Dynamic +Agents +Static +Infrastructure +GNN +GNN +Heterogeneous +Scene Graph +Task-Speci�c +Decoder +GNN Encoder +Fig. 1. Overview of SCENE: The traffic scene is modeled in a heterogeneous +scene graph with different node types and different relation types between +these nodes. The combination of a generic GNN architecture, making use of +cascaded layers of graph convolution, and a task-specific decoder is used to +predict relevant information about the given scene. +parked or temporarily stopped). To this end, the vehicle needs +to correctly estimate which sensory information is reliable +and it must reason about the relative positions, features and +trajectories of dynamic agents. +© 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or +reuse of any copyrighted component of this work in other works. +Information about dynamic +agents is conveyed by the perception systems of autonomous +vehicles. We raise the hypothesis that considering additional +heterogeneous entities in a traffic scene might add valu- +able information. In particular, reasoning should also involve +knowledge about static infrastructure, which may either be +perceived or in our case is provided by a High Definition +(HD) map. Thus, the problem of understanding traffic scenes +boils down to integrating a plenitude of heterogeneous data, +which may change over time, and reasoning about it in order to +predict intents of nearby traffic agents. This is difficult because +data from heterogeneous sources may be structured in a myriad +of ways and reasoning may require deriving complex relations +and patterns across this heterogeneous data. +Related +work +has +tackled +the +problem +of +scene +understanding from heterogeneous data by using machine +learning approaches to reason about the scene. Existing +machine learning approaches that jointly leverage information +about dynamic agents and static infrastructure, so far, have +been based on rasterized representations (e.g., [1]), have been +handcrafted and task-specific (e.g., [2]) or have been limited +in their ability to consider heterogeneous data (e.g., [3]). +Shortcomings of rasterized representations lie in the loss of +information and task-specific approaches lack the ability to +generalize to further tasks. +arXiv:2301.03512v1 [cs.CV] 9 Jan 2023 + +2 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2022 +We propose SCENE (SCene Encoding NEtwork), a graph- +based methodology to encode and perform reasoning about +a traffic scene. An overview of SCENE is given in Fig. 1. +Inputs to SCENE are features provided by upstream perception +components, which represent dynamic agents over a dura- +tion of 3 s, as well as the abstract representation of static +infrastructure, in our case given in an HD map. This input is +encoded in a heterogeneous scene graph with different node +types and a set of typed relations between nodes. In addition +to this expressive representation, we provide the means for +versatile reasoning and predictions in traffic scene graphs +using a novel architecture based on Graph Neural Networks +(GNNs). With this work we contribute to the research on +heterogeneous GNNs, defined by a recent survey as a core +field of future work [4]. Our GNN model learns from examples +how information about dynamic agents and static infrastructure +of a traffic scene may be integrated and reasoned about, +such that it can correctly predict unknown characteristics of +entities or relations. In order to show that this methodology +is not task-specific, we evaluate it on two different binary +node-classification tasks that correspond to the predictions (i) +whether a car is parked or temporarily stopped and (ii) whether +perceived information is reliable or not. +Our main contributions are: +• We propose a novel way to model information about +dynamic agents and static infrastructure of traffic scenes +in one heterogeneous graph structure with edge features, +allowing for an extensible and generic representation. +• We propose a novel GNN architecture that is able to +perform reasoning on this heterogeneous graph. +• We extensively evaluate our proposed methodology on +two diverse learning tasks. +• We quantify the effect of including heterogeneous data +about additional scene entities and relations for those +learning tasks in detailed ablation studies. +• We show that our GNN architecture transfers to applica- +tions beyond scene understanding, by applying it to the +task of node classification in knowledge graphs. +II. RELATED WORK +In this section, related work regarding reasoning about +traffic scenes is discussed. Existing approaches focus on the +prediction of intents and trajectories of agents. +A. Grid-based Approaches +Grid-based approaches rasterize information in a bird’s-eye +view grid with multiple channels and use Convolutional Neural +Networks (CNN) to learn from patterns in the given data in +order to perform reasoning about the traffic scene. One option +is to use raw sensor data as input and project it into a bird’s- +eye view grid. [5]. Most recent approaches receive dynamic +agents, extracted and processed from an upstream perception +component, and information about the static infrastructure as +input and render both into different channels of a grid [1], [6], +[7], [8]. Different to the heterogeneous graph from our work, +a grid cannot represent complex relationships in an abstract +form, e.g., the right of way between lanes [3], [9]. +B. Hybrid Approaches +Hybrid approaches introduce a graph-based representation +of the provided dynamic agents, but keep the grid-based +representation for the static infrastructure. The graph-based +representation of agents allows for an agent-wise encoding, +considering semantic attributes and temporal information. Rea- +soning on these encodings is done via GNNs [10], [11], [12], +which can consider edges in the graph to derive complex +interaction patterns between agents. In contrast to our work, +these hybrid approaches still come with the aforementioned +limitations of not representing complex relationships that +involve static infrastructure. +C. Graph-based Approaches +Graph-based approaches model both, dynamic agents and +static infrastructure via graph structures. This idea of holisti- +cally modeling scenes in a graph structure and reasoning on +it originated from the field of image retrieval [13]. +Since graph-based approaches work on sparse graphs in- +stead of dense grids, these approaches tend to be more memory +efficient [14]. Analogously to the hybrid approaches, GNNs +are used to model interactions between dynamic agents. +Early work of Ulbrich et al. [15] proposes an ontology for +representing a scene graph for autonomous driving, but do +not provide means for reasoning on the graph. Tian et al. [16] +propose a simplified approach by not modeling lanes explicitly. +This is a limitation compared to our work because their ap- +proach cannot capture topological nor regulatory relationships +between lanes. +Gao et al. [17] propose VectorNet, which shares the concept +of creating one global graph and serves as a baseline in our +evaluations. In contrast, their graph is homogeneous and fully- +connected. The homogeneous representation is obtained by +learning a node embedding for each of the heterogeneous +entities in the scene, including dynamic agents and static in- +frastructure (e.g., crosswalks and lanes). Their fully-connected +graph has only one type of edges, which requires the network +to implicitly learn different semantic relations between nodes +based on their embeddings. As a drawback, their representa- +tion does not capture edge features. However, edge features +enable the inclusion of additional relational information in the +graph, which we evaluate as advantageous in our ablation +study. The use of edge features in general is very limited +in recent publications. Approaches either only use spatial +relations as edge features (e.g., distances or headings between +dynamic agents) [18], [19], or intermediate representations by +combining information of two connected nodes [20], [21]. +Li et al. [22] use one graph to model the interactions +between an ego vehicle and its nearby vehicles (ego-thing +graph) and one graph to model interactions between the ego +vehicle and its static infrastructure (ego-stuff graph). For the +ego-stuff graph, only graph edges between the node of the +ego vehicle and stuff nodes are allowed. This modeling limits +their reasoning to the ego vehicle only, while our approach +is capable of also reasoning over patterns between non-ego +vehicles. Kumar et al. [23] lift that restriction and create a +heterogeneous graph in which all agents are fully connected + +MONNINGER AND SCHMIDT et al.: SCENE +3 +to each other as well as to nodes of the static infrastructure +within a fixed radius. Still, no relations between the static +infrastructure are explicitly modeled in the graph. Both ap- +proaches suffer from the aforementioned limitation that the +graph contains no explicitly modeled edge features between +entities of the static infrastructure. +Other approaches [3], [9], [14] explicitly model the lane +topology in a graph in order to incorporate knowledge about +the static infrastructure. These approaches utilize specialized +mechanisms in the inference process to include lane informa- +tion, allowing a transductive exchange of information between +agents via the underlying lanes. However, they do not cover +other entities of the static infrastructure, such as crosswalks or +traffic lights. In contrast, our methodology is generic and freely +extensible in a sense that it can capture various information in +a heterogeneous graph by using typed nodes. Furthermore, our +methodology allows for modeling relations with arbitrary type +and edge features between these nodes, which we demonstrate +to be a valuable addition. The result is one heterogeneous +graph that explicitly models all aspects of a given traffic scene +without limitations to specific use cases. +III. METHODOLOGY +This section describes our proposed methodology. Firstly, +we define a graph ontology to model the given dynamic +agents and static infrastructure in a heterogeneous scene graph. +Secondly, we use a learning-based approach to predict relevant +information from this scene graph. +A. Heterogeneous Scene Graph Ontology +We represent a scene by a directed heterogeneous graph +G = (V, E, T , R, φ). Every node vi ∈ V has a feature vector +vi. The edge ej,r,i = (vj, r, vi) ∈ E between the source node +vj and the destination node vi with the relation type r ∈ R has +a feature vector ej,r,i. The type of node v is defined by the type +operator φ : V → T , with T being the set of allowed node +types. We define the domain type operator dom : R → T +and range type operator ran : R → T to map a relation +type r to the source and target node types, respectively. Each +relation type r has a fixed source and destination node type: +∀(vj, r, vi) +φ(vj) ∈ dom(r) and φ(vi) ∈ ran(r). +Fig. 2 illustrates our used node types and our used relation +types between these node types. Each node and edge has +a corresponding feature vector. The relation types belong to +three groups. Relations between agents are of type interacts. +They are dynamically generated for each pair based on the +assumption that all agents can interact with each other. Re- +lations between agents and map entities are of the types on, +under and crosses. All valid relations are dynamically derived +from the geometric constellation. The remaining relation types +link map entities and are given by the HD map. +B. Reasoning on the Heterogeneous Scene Graph +Reasoning on the generated heterogeneous scene graph is +done with the encoder-decoder architecture presented below. +The encoder first aggregates information of a traffic scene into +crosswalk +light +lane +agent +stop +interacts +on +under +conflict +connection +precedence +overlaps +controls +crosses +signals +stops +Fig. 2. +Node types and allowed relation types of the proposed ontology. +Different colors are used to indicate the order of our proposed flow of +information. +embeddings of the agent nodes. From these embeddings, task- +specific decoders can directly predict agent-specific attributes +(e.g., intents or trajectories). Encoder and decoder are jointly +trained with task-specific data in a supervised manner. The +focus of this work is the generic encoder. +1) Encoder: Multiple layers of graph convolution are cas- +caded to aggregate information regarding the heterogeneous +scene. Thanks to their invariance properties [24], graph con- +volutional layers can learn general, abstract patterns from con- +crete scenes. We show that this principle works for different +classification tasks. +For graph convolution, a variety of operators is applicable. +We follow the principle described in [25], allowing to incorpo- +rate edge features in the established Graph Attention Network +(GAT) operator [26]. For better error propagation and to avoid +over-smoothing, we add a residual connection for Θs,r · vi to +the operator. The update of node vi under consideration of +neighboring nodes connected via the relation type r is given +by +v′ +i,r = EdgeGATr(vi) = +Θs,r · vi+ +��� +K +k=1 +� +� +� +j∈Nr(vi) +αk +j,r,i +� +Θk +n,r · vj + Θk +e,r · ej,r,i +� +� +� . +(1) +Θ is used to denote learnable weight matrices for the trans- +formation of features of the node to update (s=self), neigh- +boring nodes (n=neighbor) and edge features (e=edge). K +corresponds to the number of attention heads and ∥ denotes +the concatenation operator. Attention weights are obtained by +αk +j,r,i = softmaxr,i +� +LeakyReLU +� +ak +r +T [Θk +n,r · vi||Θk +n,r · vj||Θk +e,r · ej,r,i] +�� +, +(2) +with a corresponding to a learnable vector. softmaxr,i stands +for the normalization by all incoming edges of node i con- +nected via relation type r. + +4 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2022 +While EdgeGAT is able to aggregate information of one +specific relation type, reasoning on a heterogeneous graph +requires aggregating information of neighboring nodes that are +possibly connected via different relation types. Adapted from +Schlichtkrull et al. [27], we define the node update of one +heterogeneous GNN layer as +v′ +i = ReLU +�� +r∈R +v′ +i,r +� +. +(3) +We denote the combination of EdgeGAT and the aggregation +of the resulting embeddings over multiple relation types as +HetEdgeGAT. +We propose to use cascaded layers of HetEdgeGAT in order +to aggregate information of the scene into agent nodes. The +flow of information towards the agent nodes is represented by +the color of the relation types in Fig. 2: The first layer aggre- +gates information into crosswalk nodes (green). Subsequent +layers aggregate information into lane nodes (blue) and agent +nodes (red). The last layer of graph convolution then considers +social interaction between agents and updates the agent nodes +again (pink). +2) Decoder: The decoder is task-specific. We use a Multi- +layer Perceptron (MLP) and apply it to the encodings of agent +nodes for the two binary node classification tasks considered +in the experiments section. +IV. EXPERIMENTS +In this section we describe the extensive evaluation of our +proposed methodology. +A. Learning Tasks +We evaluate our methodology on two diverse binary node +classification tasks: +1) Classification whether an agent is parked or not. We +consider one prior publication introducing and address- +ing this task as a baseline [2]. +2) Classification whether an agent is a ghost or not. Ghosts +are unreliable detections of agents by upstream per- +ception components that do not exist in real world, +i.e., false positive detections. We are not aware of +prior publications that consider static infrastructure for +this task. Comparison is done with an approach that +considers only dynamic information [28]. +B. Dataset +Experiments are carried out on a large-scale, in-house +dataset with over 22 400 sequences, each coming with 3 s of +temporal history. They are extracted from in-vehicle recordings +from different areas in Germany and the U.S. with a sampling +rate of 10 Hz. Camera, LiDAR and Radar detections are fused +together in order to detect surrounding dynamic agents. The +dataset includes diverse environments (e.g., urban, rural and +highway) as well as diverse scenarios (e.g., driving and yield- +ing). To the best of our knowledge, there is no publicly avail- +able dataset for scene understanding that similarly provides +manually annotated semantic attributes for dynamic agents in +TABLE I +NODE TYPE-SPECIFIC FEATURES +Node type +Features +agent +• State vector (position, velocity, acceleration, yaw, yaw rate) with +corresponding covariances +• Tracking properties (e.g., max. velocity, tracked time) +• Bounding box dimensions +• Estimate of agent type (e.g., car, truck, two-wheeler) +• Sensor specific detection and existence probabilities +• Existence confidence calculated according to [28] +• Trajectory of the past three seconds as a series of positional and +angular differences +lane +• Type (car, bike, shoulder, parking) +• Geometric properties (length, min. and max. width, max. curvature) +• Maximum legal speed +• Left and right boundary types +• Turn type +crosswalk +• Is signaled +stop +• Type (e.g., stop, crosswalk, yield) +light +• Type (e.g., car, pedestrian) +• State (e.g., red, yellow) +• Is deactivatable +TABLE II +RELATION TYPE-SPECIFIC EDGE FEATURES +Relation type +Features +interacts +• Geometric differences (position, velocity, angle) +under +• Assignment probability +• Frenet state (position, velocity) at agent position +• Lane properties at agent position +• Gap to lane boundaries at agent position +• Behavior primitive of agent in lane (e.g., following, crossing) +connection +• Type (e.g., precede, left neighbor) +conflict +• Type (e.g., cross, merge) +precedence +• Type (e.g., higher, lower) +stops +• Longitudinal position in lane +combination with an extensively attributed heterogeneous HD +map. +By deriving correspondences between manually annotated +agents and agent detections from upstream perception com- +ponents, more than 430 000 labels per training tasks are +generated. For our experiments, we use a dataset split of 60% +(training), 30% (validation) and 10% (testing). +C. Model Implementation Details +We use an extensive set of features for nodes and edges in +the scene graph to explicitly model all available knowledge of +the scene. Features are provided by the upstream perception +components and the HD map. Table I and Table II list the +used feature sets for each node type and each relation type. +To propagate uncertainty of the perception component to the +model, the feature vectors of agent nodes contain covariances +and confidence values. All features that express a category +type are one-hot encoded. While this set of features is given +by our perception and HD map, the proposed methodology +can use any arbitrary set of features. +Details of the final implementation are shown in Fig. 3, +including the dimensions of all feature vectors. The features +of all nodes and edges are type-specifically encoded with a +single linear layer and ReLU. Static and temporal aspects of +agents are encoded separately and concatenated thereafter. The +static encoding uses a linear layer with ReLU. The temporal +encoding uses a Gated Recurrent Unit (GRU) and ReLU to + +MONNINGER AND SCHMIDT et al.: SCENE +5 +1 +16 +50 +Agent +Features +30x3 +Agent +Trajectory +1 +Crosswalk +Features +12 +Light +Features +5 +Stop +Features +21 +Lane +Features +5 +Connection +Features +5 +Con�ict +Features +6 +Precedence +Features +15 +Under +Features +3 +Interacts +Features +64 +Linear +64 +GRU +32 +Linear +16 +Linear +16 +Linear +Linear +64 +64 +Linear +Predicted +Binary Label +64 +128 +Linear +HetEdgeGAT +Encoder +Decoder +64 +1 +Stops +Features +16 +Linear +16 +Linear +16 +Linear +16 +Linear +16 +Linear +16 +Linear +16 +Linear + + + + +Fig. 3. Implementation details of the SCENE encoder and the task-specific MLP-decoder: Four cascaded layers of HetEdgeGAT (green, blue, red, pink) are +used to combine information of nodes of different types and their relations in order to update the feature vector of agent nodes. Residual connections (orange) +prevent over-smoothing. For the two binary node classification tasks, an MLP is then used to generate a classification score. +process the trajectory feature. Following the idea of [3], the +trajectory feature contains a fixed-length series of positional +and angular differences of the last 30 timesteps (3 s). We add +a binary flag that indicates whether an entry contains a valid +measurement for each timestep. +In order to avoid over-smoothing, which is one of the main +issues of multilayer GNNs [29], we exploit two concatenated +residual connections (orange). These allow the decoder to +combine high and low-level features of agent nodes. +Binary cross-entropy is used as a loss function. The model +is trained with Adam optimizer [30] with a learning rate of +10−5 and a batch size of 32. Dropout with a rate of 0.3 is +used for the two linear layers of the decoder. +D. Baselines +For both tasks, we compare our generic methodology to +multiple task-specific and generic baselines. +1) Task-specific Baselines for the Parked Attribute: The +velocity baseline evaluates the velocity of each car in a given +scene. Stationary cars (zero velocity) are labeled as parked and +vice versa. +The logistic regression baseline uses a handcrafted set of +features based on the inputs provided by upstream perception +components and the HD map. This baseline has been specifi- +cally designed for classifying the parked attribute. +We also consider two approaches of one prior publication +addressing parked car classification [2], namely the heuristic +approach and the MLP approach, which operates on only three +features for each agent. Both approaches utilize features that +contain information about the agent and one underlying lane. +We call these baselines heuristic and MoveMLP3. +2) Task-specific Baseline for the Ghost Attribute: The ex- +istence confidence baseline gives an estimate about the exis- +tence of an agent. This baseline approach [28] uses a method +based on Dempster-Shafer evidence theory to estimate a fused +existence confidence about an agent based on detections from +multiple sensor modalities. Note that the resulting existence +confidence is also part of the input features of agent nodes. +A comparison to this baseline therefore shows the benefit of +additionally considering social context and map context. +3) Generic Baselines: The MLP baseline contains four +linear layers with ReLU between these layers. It operates +directly on the features of agent nodes and does not process +the graph structure. In comparison to our proposed model, this +baseline shows the effect of neglecting relational information +about social context, defined by nearby dynamic agents, and +map context, defined by the static infrastructure. +The R-GCN baseline applies the Relational Graph Convo- +lutional Network [27], typically used as a common approach to +reason about knowledge graphs, to our scene graph. We extend +the original R-GCN approach by introducing edge features, +which allows our implementation of R-GCN to use the same +input features as SCENE. After four layers of R-GCN, an MLP +decoder is used to predict the labels. Feature vector sizes of +nodes and edges are similar to the ones used in SCENE. +To compare our methodology to the current-state-of-the- +art in scene encoding, we adapt VectorNet [17] to our input +representation. The VectorNet-like baselines therefore rely +on learning from a fully-connected, homogeneous graph. In +contrast to SCENE, the features of all heterogeneous nodes +are type-specificially encoded into a joint feature space with +size 64 to get a homogeneous graph. The vanilla VectorNet- +like baseline does not use edge features. In order to allow a fair +comparison to SCENE, we also extend the original VectorNet +approach by the introduction of edge features. All existing +edges are type-specifically encoded to a size of 16. To obtain +the required fully-connected graph, edges are instantiated with +a zero vector of size 16 between all nodes that are not yet +connected. A binary flag concatenated to the edge feature +vector is used to indicate whether an edge is valid (1) or invalid +(0). Despite this leading to a fully-connected, homogeneous +graph, connectivity information of our initial scene graph is +still conserved in the edge features. One layer of EdgeGAT +is used on the resulting fully-connected graph. Similar to +SCENE, the labels are then predicted with an MLP decoder. + +6 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2022 +TABLE III +RESULTS ON THE TEST SET +Method +Parked +Ghost +F1 (%) +Acc (%) +F1 (%) +Acc (%) +Velocity +76.51 +81.41 +- +- +Logistic regression +89.79 +93.21 +- +- +Heuristic [2] +86.75 +90.47 +- +- +MoveMLP3 [2] +88.56±0.15 92.78±0.07 - +- +Existence confidence [28] +- +- +53.48 +66.01 +Naive prior +0.00 +67.33 +66.88 +50.24 +MLP +75.79±0.89 83.23±0.40 79.93±0.67 80.73±0.44 +R-GCN [27] +89.68±0.81 93.11±0.58 78.76±0.36 79.87±0.24 +VectorNet-like [17] +73.90±1.43 82.71±0.74 74.63±4.17 75.84±1.68 +VectorNet-like (w/ edge feat) [17] +89.18±1.28 93.08±0.75 80.93±1.26 81.40±0.98 +Ours (using HetEdgeGAT)1 +91.17±0.71 94.29±0.46 80.56±0.77 81.44±0.71 +Ours (using HetEdgeGatedGCN) +90.09±0.29 93.54±0.15 82.42±1.33 82.83±1.07 +Ours (using HetEdgeSAGE) +91.11±0.43 94.19±0.34 80.72±1.23 81.68±0.71 +Ours (using HetEdgeGAT)∗ +90.16±1.42 93.49±1.10 81.05±1.26 81.93±0.80 +1Selected for all further experiments. +∗Multi-task training. +E. Metrics +F-Score (F1) and accuracy (Acc) are used for evaluation. +F. Quantitative Results +The models were trained over five random seeds to min- +imize stochasticity in the results. The resulting average and +standard deviation of the performance metrics on the test +split are shown in Table III. Besides GAT, we evaluated +our approach using different operators for graph convolution, +including variants of Gated Graph Convolutional Neural Net- +works (EdgeGatedGCN) [31] and GraphSAGE (EdgeSAGE) +[32]. They all consistently perform well, which suggests +that the graph convolution operator is interchangeable, also +with regards to the architecture. Therefore, our methodology +can benefit from upcoming advances in the field of GNNs. +HetEdgeGAT was selected for all further experiments because +it uses the least number of parameters. Also, Schmidt et al. +[33] show that the resulting attention weights of interacts +relations offer additional interpretability, as they are a direct +measure for interactions. +Comparing the results of our methodology with the MLP +baseline supports our initial hypothesis that our proposed way +of modeling a scene in a heterogeneous scene graph adds +valuable information. +Our generic methodology outperforms all task-specific and +generic baselines on both tasks. The last row in Table III +shows the result of the model simultaneously trained on both +learning tasks. The performance is on par with the single- +task setup, which supports the indication that our method +works as a generic scene encoder. The VectorNet-like baseline +extended with edge features performs much better than the +vanilla VectorNet-like baseline, which supports the intuition +that adding relational attributes provides valuable information +for scene understanding. Comparing the average number of +Floating-Point Operations (FLOPs) of the VectorNet-like base- +line with edge features (6.24·108 FLOPs) and our SCENE ap- +proach (4.57 · 107 FLOPs) shows that our approach has more +than an order of magnitude less computational complexity. The +higher complexity of the VectorNet-like baselines comes from +TABLE IV +CONTEXT ABLATION STUDY ON THE TEST SET +Context +#Params +Parked +Ghost +Agent Lane Remaining +F1 (%) +Acc (%) +F1 (%) +Acc (%) +50k +75.35±0.70 82.96±0.54 80.13±1.14 80.70±0.59 +✓ +59k +81.85±2.80 88.31±2.00 79.85±0.51 80.69±0.43 +✓ +✓ +99k +91.03±1.46 94.28±0.89 80.63±0.63 81.40±0.47 +✓ +✓ +✓ +118k +91.17±0.71 94.29±0.46 80.56±0.77 81.44±0.71 +TABLE V +ARCHITECTURAL ABLATION STUDY ON THE TEST SET +Architecture +#Params +Parked +Ghost +Temp Res +Edge feat2 +F1 (%) +Acc (%) +F1 (%) +Acc (%) +77k +89.47±1.19 93.28±0.63 77.42±0.96 78.74±0.85 +✓ +✓ +101k +90.47±0.76 93.92±0.45 80.26±0.55 80.92±0.55 +✓ +✓ +102k +90.81±1.17 94.06±0.72 80.12±1.47 81.02±1.26 +✓ +✓ +111k +89.06±0.72 92.92±0.60 77.13±0.46 78.71±0.23 +✓ +✓ +✓ +118k +91.17±0.71 94.29±0.46 80.56±0.77 81.44±0.71 +2In contrast to the temporal and residual architectural measures, edge features +introduce additional information into the graph. +applying convolution over the fully-connected graph, which +also results in significantly higher GPU memory requirements +and training time than our approach. +G. Ablation Studies +In two ablation studies we analyze how well our approach +can leverage the various sources of information and the +effectiveness of our architectural measures. +Table IV ablates the value of various sources of information +coming from the dynamic agents and static infrastructure and +the ability of our approach to leverage this information. With- +out any context at all, the model performs the classification +tasks with the features of agent nodes only. Our experiments +show that considering social interactions to nearby agents +(agent context), lane information (lane context) and informa- +tion given by crosswalks, stops and lights (remaining context) +can have a strong positive effect on model performance. The +relevance of each contextual aspect differs between tasks. +The remaining context has only a small effect, since much +information is implicitly present in the lane model already. +Overall, the results show that incrementally adding further +context information improves model performance. This con- +firms the value of the additional information as hypothesized +in the introduction as well as the capability of the generic +model to exploit the provided information. +Table V ablates the performance of our approach in terms +of applied architectural measures. These measures are the +temporal encoding of each agent’s trajectory, the residual +connections and the inclusion of edge features in the layers of +graph convolution. The results indicate that omitting individual +architectural measures decreases model performance compared +to applying the full set of measures. This is particularly +noteworthy for the edge features, suggesting a benefit of +adding relational information to the graph. The architecture +that combines all measures either excels or comes very close to +the best results. This suggests that the individual architectural +measures benefit from each other. + +MONNINGER AND SCHMIDT et al.: SCENE +7 +Parked classification +Parked classification +Ghost classification +Fig. 4. Qualitative results of SCENE for the classification of the parked attribute (left and center) and the classification of the ghost attribute (right). The +upper row shows the front camera frame, which is one of multiple sensors used by upstream perception, and the lower row renders the corresponding scene +with lanes in gray. Bounding boxes of agents are drawn as rectangles, with the past trajectory visualized by an orange line. The color of the rectangle outline +indicates the ground-truth label and the fill color indicates the label predicted by our model. Green corresponds to an agent being labeled as non-parked or +non-ghost. Red corresponds to an agent being labeled as parked or ghost. All agents are correctly classified as indicated by matching fill and outline colors. +A purple outline indicates a missing label. The autonomous vehicle is colored in blue. +H. Qualitative Results +Fig. 4 shows qualitative results for both tasks. Color codes +are described in the caption of the figure. In the three examples +all agents with available ground-truth label are correctly clas- +sified, which is represented by consistent coloring of outline +and fill. +The figure on the left shows an urban scenario with vehicles +parked on the road side (red) and vehicles driving in the center +(green). Interestingly, the vehicle with white paint inside the +paved intersection is parked, which is correctly predicted by +our model. +The figure displayed in the center is a rare case where two +cars are parked on the left lane of a highway on-ramp. Again, +those are correctly classified by our model. The prediction is +likely supported by the humans nearby, which are detected by +the system (purple outline due to no parked label for agents of +type human) and provide social context to the parked vehicles. +The figure on the right shows a highway scenario, where +all nearby agents besides one are correctly classified as non- +ghost (green outline and fill). The prediction of the one ghost +agent (red fill) can be confirmed by its trajectory showing a +wrong direction of travel. This specific detection is probably +caused by sensor reflections of a bridge. The corresponding +ground-truth label also classifies it as ghost (red outline). +V. TRANSFERABILITY TO OTHER APPLICATIONS +As an extension to evaluating the prediction of unknown +characteristics of traffic agents, in this section we show that, +without any modifications, the use of cascaded layers of graph +convolution can be transferred to applications that go beyond +the domain of scene understanding. We therefore apply our +methodology for the task of node classification to multiple +knowledge graphs of different sizes. The source code of +these experiments, including our graph convolution operator, +is publicly available3. +A. Datasets +Evaluation is done on four publicly available heterogeneous +knowledge graph datasets, namely AIFB, MUTAG, BGS and +3Source code: https://github.com/schmidt-ju/scene +TABLE VI +ACCURACY (%) ON THE MASKED NODES OF KNOWLEDGE GRAPHS +Dataset +WL [36] +RDF2Vec [37] Walk Tree [35] R-GCN [27] Ours +AIFB +80.55±0.00 +88.88±0.00 +89.44±2.08 +95.83±0.62 95.83±1.96 +MUTAG +80.88±0.00 67.20±1.24 +73.82±5.61 +73.23±0.48 +75.44±2.50 +BGS +86.20±0.00 +87.24±0.89 +86.90±1.38 +83.10±0.80 +92.41±2.72 +AM +87.37±0.00 +88.33±0.61 +86.77±0.59 +89.29±0.35 +90.05±1.07 +AM [34]. The datasets cover varying graph sizes, ranging from +small (AIFB, 8 285 nodes) to large (AM, 1 666 764 nodes) +[27]. Given classes for some nodes of a target node type, the +goal is to correctly classify the classes of masked target nodes. +B. Model and Results +We remove low-degree nodes and initialize the features +of each node with a learnable bias vector. Four layers of +HetEdgeGAT are arranged in cascaded form. The first two +layers sequentially update all nodes not of type target based on +neighboring nodes of the same and of other types. The second +two layers sequentially aggregate information into nodes of +type target by considering neighboring nodes of other types +and of type target. The model is trained full-batch with cross- +entropy loss. Average accuracy and standard deviation for ten +runs is reported in Table VI. Results of the compared methods +are taken from prior publications [27], [35], [36], [37]. +Despite the task being different from predicting unknown +characteristics of traffic agents, the results show that our +methodology manages to yield state-of-the-art performance for +the task of node classification in knowledge graphs. +VI. CONCLUSION +This paper proposes a method using cascaded layers of +graph convolution in order to predict relevant information from +heterogeneous graphs and examines it on the task of reasoning +about traffic scenes. Combining the cascaded layers of graph +convolution with our novel way for modeling traffic scenes +in heterogeneous graphs results in a generic and extensible +method to reason about traffic scenes. The heterogeneous +graph ontology can be extended with additional types or +features of nodes and edges. Our methodology outperforms all + +8 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2022 +task-specific baselines on two diverse tasks. Furthermore, we +compared it to multiple generic state-of-the-art encoders and +demonstrated that our method has significant advantages with +regard to performance metrics and computational complexity. +The application of our methodology to the task of node +classification in knowledge graphs indicates another key prop- +erty of our methodology: it is, without any modifications, +applicable to areas that go beyond the domain of scene +understanding. By making source code and GNN operator +publicly available, we contribute to the progress in this field. +REFERENCES +[1] J. Hong, B. Sapp, and J. Philbin, “Rules of the road: Predicting driving +behavior with a convolutional model of semantic interactions,” in 2019 +IEEE/CVF Conference on Computer Vision and Pattern Recognition +(CVPR), 2019, pp. 8446–8454. +[2] K. Behrendt, O. Mangin, N. Bhakta, and S. Lefevre, “Is this car going to +move? parked car classification for automated vehicles,” in 2019 IEEE +Intelligent Vehicles Symposium (IV), 2019, pp. 541–548. +[3] M. Liang, B. Yang, R. Hu, Y. Chen, R. Liao, S. Feng, and R. Urta- +sun, “Learning lane graph representations for motion forecasting,” in +Computer Vision – ECCV 2020, 2020, pp. 541–556. +[4] Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu, “A +Comprehensive Survey on Graph Neural Networks,” IEEE Transactions +on Neural Networks and Learning Systems, vol. 32, no. 1, pp. 4–24, +2021. +[5] S. Casas, W. Luo, and R. 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Paulheim, +“Rdf2vec: Rdf graph embeddings and their applications,” The Semantic +Web, vol. 10, pp. 721–752, 2019. + diff --git a/2dE1T4oBgHgl3EQf5gVO/content/tmp_files/load_file.txt b/2dE1T4oBgHgl3EQf5gVO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ce60cce09c19030765ee6fcc5dc020c716eab47 --- /dev/null +++ b/2dE1T4oBgHgl3EQf5gVO/content/tmp_files/load_file.txt @@ -0,0 +1,921 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf,len=920 +page_content='IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' ACCEPTED DECEMBER, 2022 1 SCENE: Reasoning about Traffic Scenes using Heterogeneous Graph Neural Networks Thomas Monninger†,1, Julian Schmidt†,2,3, Jan Rupprecht2, David Raba2, Julian Jordan2, Daniel Frank4, Steffen Staab4,5 and Klaus Dietmayer3, Senior Member, IEEE Abstract—Understanding traffic scenes requires considering heterogeneous information about dynamic agents and the static infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In this work we propose SCENE, a methodology to encode diverse traffic scenes in heterogeneous graphs and to reason about these graphs using a heterogeneous Graph Neural Network encoder and task-specific decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The heterogeneous graphs, whose structures are defined by an ontology, consist of different nodes with type-specific node features and different relations with type-specific edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In order to exploit all the information given by these graphs, we propose to use cascaded layers of graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The result is an encoding of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Task-specific decoders can be applied to predict desired attributes of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Extensive evaluation on two diverse binary node classification tasks show the main strength of this methodology: despite being generic, it even manages to outperform task-specific baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The further application of our methodology to the task of node classification in various knowledge graphs shows its transferability to other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Index Terms—Semantic Scene Understanding, AI-Based Meth- ods, Behavior-Based Systems I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' INTRODUCTION U NDERSTANDING traffic scenes is important for an autonomous vehicle such that it may develop a safe, effective and efficient plan of how to move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' For instance, whether a stationary car is parked or just temporarily stopped determines whether the autonomous vehicle should wait or overtake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Understanding of traffic scenes requires reasoning about dynamic agents and static infrastructure in order to predict the intents of nearby dynamic agents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', Manuscript received: August 26, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Revised: November 21, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Accepted: December 20, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This paper was recommended for publication by Editor Markus Vincze upon evaluation of the Associate Editor and Reviewers’ comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This work was supported by the BMWK within the project ”KI Delta Learning” (F¨orderkennzeichen 19A19013A) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2075 – 390740016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' (Corresponding author: Julian Schmidt) †Thomas Monninger and Julian Schmidt are co-first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The order was determined alphabetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 1Thomas Monninger is with Mercedes-Benz R&D North America, Sunny- vale, CA, USA (e-mail: thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='monninger@mercedes-benz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='com) 2Julian Schmidt, Jan Rupprecht, David Raba and Julian Jordan are with Mercedes-Benz AG, R&D, Stuttgart, Germany (e-mail: {julian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='schmidt, jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='rupprecht, david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='raba, julian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='jordan}@mercedes-benz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='com) 3Julian Schmidt and Klaus Dietmayer are with Ulm University, Institute of Measurement, Control and Microtechnology, Ulm, Germany (e-mail: klaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='dietmayer@uni-ulm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='de) 4Daniel Frank and Steffen Staab are with University of Stuttgart, Institute of Parallel and Distributed Systems, Stuttgart, Germany (e-mail: {daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='frank, steffen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='staab}@ipvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='de) 5Steffen Staab is with University of Southampton, Electronics and Com- puter Science, Southampton, United Kingdom Dynamic Agents Static Infrastructure GNN GNN Heterogeneous Scene Graph Task-Speci�c Decoder GNN Encoder Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Overview of SCENE: The traffic scene is modeled in a heterogeneous scene graph with different node types and different relation types between these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The combination of a generic GNN architecture, making use of cascaded layers of graph convolution, and a task-specific decoder is used to predict relevant information about the given scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' parked or temporarily stopped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' To this end, the vehicle needs to correctly estimate which sensory information is reliable and it must reason about the relative positions, features and trajectories of dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Information about dynamic agents is conveyed by the perception systems of autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We raise the hypothesis that considering additional heterogeneous entities in a traffic scene might add valu- able information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In particular, reasoning should also involve knowledge about static infrastructure, which may either be perceived or in our case is provided by a High Definition (HD) map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Thus, the problem of understanding traffic scenes boils down to integrating a plenitude of heterogeneous data, which may change over time, and reasoning about it in order to predict intents of nearby traffic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This is difficult because data from heterogeneous sources may be structured in a myriad of ways and reasoning may require deriving complex relations and patterns across this heterogeneous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Related work has tackled the problem of scene understanding from heterogeneous data by using machine learning approaches to reason about the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Existing machine learning approaches that jointly leverage information about dynamic agents and static infrastructure, so far, have been based on rasterized representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', [1]), have been handcrafted and task-specific (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', [2]) or have been limited in their ability to consider heterogeneous data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Shortcomings of rasterized representations lie in the loss of information and task-specific approaches lack the ability to generalize to further tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='03512v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='CV] 9 Jan 2023 2 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' ACCEPTED DECEMBER, 2022 We propose SCENE (SCene Encoding NEtwork), a graph- based methodology to encode and perform reasoning about a traffic scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' An overview of SCENE is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Inputs to SCENE are features provided by upstream perception components, which represent dynamic agents over a dura- tion of 3 s, as well as the abstract representation of static infrastructure, in our case given in an HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This input is encoded in a heterogeneous scene graph with different node types and a set of typed relations between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In addition to this expressive representation, we provide the means for versatile reasoning and predictions in traffic scene graphs using a novel architecture based on Graph Neural Networks (GNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' With this work we contribute to the research on heterogeneous GNNs, defined by a recent survey as a core field of future work [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Our GNN model learns from examples how information about dynamic agents and static infrastructure of a traffic scene may be integrated and reasoned about, such that it can correctly predict unknown characteristics of entities or relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In order to show that this methodology is not task-specific, we evaluate it on two different binary node-classification tasks that correspond to the predictions (i) whether a car is parked or temporarily stopped and (ii) whether perceived information is reliable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Our main contributions are: We propose a novel way to model information about dynamic agents and static infrastructure of traffic scenes in one heterogeneous graph structure with edge features, allowing for an extensible and generic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We propose a novel GNN architecture that is able to perform reasoning on this heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We extensively evaluate our proposed methodology on two diverse learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We quantify the effect of including heterogeneous data about additional scene entities and relations for those learning tasks in detailed ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We show that our GNN architecture transfers to applica- tions beyond scene understanding, by applying it to the task of node classification in knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' RELATED WORK In this section, related work regarding reasoning about traffic scenes is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Existing approaches focus on the prediction of intents and trajectories of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Grid-based Approaches Grid-based approaches rasterize information in a bird’s-eye view grid with multiple channels and use Convolutional Neural Networks (CNN) to learn from patterns in the given data in order to perform reasoning about the traffic scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' One option is to use raw sensor data as input and project it into a bird’s- eye view grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Most recent approaches receive dynamic agents, extracted and processed from an upstream perception component, and information about the static infrastructure as input and render both into different channels of a grid [1], [6], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Different to the heterogeneous graph from our work, a grid cannot represent complex relationships in an abstract form, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', the right of way between lanes [3], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Hybrid Approaches Hybrid approaches introduce a graph-based representation of the provided dynamic agents, but keep the grid-based representation for the static infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The graph-based representation of agents allows for an agent-wise encoding, considering semantic attributes and temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Rea- soning on these encodings is done via GNNs [10], [11], [12], which can consider edges in the graph to derive complex interaction patterns between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In contrast to our work, these hybrid approaches still come with the aforementioned limitations of not representing complex relationships that involve static infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Graph-based Approaches Graph-based approaches model both, dynamic agents and static infrastructure via graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This idea of holisti- cally modeling scenes in a graph structure and reasoning on it originated from the field of image retrieval [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Since graph-based approaches work on sparse graphs in- stead of dense grids, these approaches tend to be more memory efficient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Analogously to the hybrid approaches, GNNs are used to model interactions between dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Early work of Ulbrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [15] proposes an ontology for representing a scene graph for autonomous driving, but do not provide means for reasoning on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [16] propose a simplified approach by not modeling lanes explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This is a limitation compared to our work because their ap- proach cannot capture topological nor regulatory relationships between lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [17] propose VectorNet, which shares the concept of creating one global graph and serves as a baseline in our evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In contrast, their graph is homogeneous and fully- connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The homogeneous representation is obtained by learning a node embedding for each of the heterogeneous entities in the scene, including dynamic agents and static in- frastructure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', crosswalks and lanes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Their fully-connected graph has only one type of edges, which requires the network to implicitly learn different semantic relations between nodes based on their embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' As a drawback, their representa- tion does not capture edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' However, edge features enable the inclusion of additional relational information in the graph, which we evaluate as advantageous in our ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The use of edge features in general is very limited in recent publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Approaches either only use spatial relations as edge features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', distances or headings between dynamic agents) [18], [19], or intermediate representations by combining information of two connected nodes [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [22] use one graph to model the interactions between an ego vehicle and its nearby vehicles (ego-thing graph) and one graph to model interactions between the ego vehicle and its static infrastructure (ego-stuff graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' For the ego-stuff graph, only graph edges between the node of the ego vehicle and stuff nodes are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This modeling limits their reasoning to the ego vehicle only, while our approach is capable of also reasoning over patterns between non-ego vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [23] lift that restriction and create a heterogeneous graph in which all agents are fully connected MONNINGER AND SCHMIDT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' : SCENE 3 to each other as well as to nodes of the static infrastructure within a fixed radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Still, no relations between the static infrastructure are explicitly modeled in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Both ap- proaches suffer from the aforementioned limitation that the graph contains no explicitly modeled edge features between entities of the static infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Other approaches [3], [9], [14] explicitly model the lane topology in a graph in order to incorporate knowledge about the static infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' These approaches utilize specialized mechanisms in the inference process to include lane informa- tion, allowing a transductive exchange of information between agents via the underlying lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' However, they do not cover other entities of the static infrastructure, such as crosswalks or traffic lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In contrast, our methodology is generic and freely extensible in a sense that it can capture various information in a heterogeneous graph by using typed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Furthermore, our methodology allows for modeling relations with arbitrary type and edge features between these nodes, which we demonstrate to be a valuable addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The result is one heterogeneous graph that explicitly models all aspects of a given traffic scene without limitations to specific use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' METHODOLOGY This section describes our proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Firstly, we define a graph ontology to model the given dynamic agents and static infrastructure in a heterogeneous scene graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Secondly, we use a learning-based approach to predict relevant information from this scene graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Heterogeneous Scene Graph Ontology We represent a scene by a directed heterogeneous graph G = (V, E, T , R, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Every node vi ∈ V has a feature vector vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The edge ej,r,i = (vj, r, vi) ∈ E between the source node vj and the destination node vi with the relation type r ∈ R has a feature vector ej,r,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The type of node v is defined by the type operator φ : V → T , with T being the set of allowed node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We define the domain type operator dom : R → T and range type operator ran : R → T to map a relation type r to the source and target node types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Each relation type r has a fixed source and destination node type: ∀(vj, r, vi) φ(vj) ∈ dom(r) and φ(vi) ∈ ran(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 2 illustrates our used node types and our used relation types between these node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Each node and edge has a corresponding feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The relation types belong to three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Relations between agents are of type interacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' They are dynamically generated for each pair based on the assumption that all agents can interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Re- lations between agents and map entities are of the types on, under and crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' All valid relations are dynamically derived from the geometric constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The remaining relation types link map entities and are given by the HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Reasoning on the Heterogeneous Scene Graph Reasoning on the generated heterogeneous scene graph is done with the encoder-decoder architecture presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The encoder first aggregates information of a traffic scene into crosswalk light lane agent stop interacts on under conflict connection precedence overlaps controls crosses signals stops Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Node types and allowed relation types of the proposed ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Different colors are used to indicate the order of our proposed flow of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' embeddings of the agent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' From these embeddings, task- specific decoders can directly predict agent-specific attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', intents or trajectories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Encoder and decoder are jointly trained with task-specific data in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The focus of this work is the generic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 1) Encoder: Multiple layers of graph convolution are cas- caded to aggregate information regarding the heterogeneous scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Thanks to their invariance properties [24], graph con- volutional layers can learn general, abstract patterns from con- crete scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We show that this principle works for different classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' For graph convolution, a variety of operators is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We follow the principle described in [25], allowing to incorpo- rate edge features in the established Graph Attention Network (GAT) operator [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' For better error propagation and to avoid over-smoothing, we add a residual connection for Θs,r · vi to the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The update of node vi under consideration of neighboring nodes connected via the relation type r is given by v′ i,r = EdgeGATr(vi) = Θs,r · vi+ ��� K k=1 � � � j∈Nr(vi) αk j,r,i � Θk n,r · vj + Θk e,r · ej,r,i � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' (1) Θ is used to denote learnable weight matrices for the trans- formation of features of the node to update (s=self), neigh- boring nodes (n=neighbor) and edge features (e=edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' K corresponds to the number of attention heads and ∥ denotes the concatenation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Attention weights are obtained by αk j,r,i = softmaxr,i � LeakyReLU � ak r T [Θk n,r · vi||Θk n,r · vj||Θk e,r · ej,r,i] �� , (2) with a corresponding to a learnable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' softmaxr,i stands for the normalization by all incoming edges of node i con- nected via relation type r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 4 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' ACCEPTED DECEMBER, 2022 While EdgeGAT is able to aggregate information of one specific relation type, reasoning on a heterogeneous graph requires aggregating information of neighboring nodes that are possibly connected via different relation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Adapted from Schlichtkrull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [27], we define the node update of one heterogeneous GNN layer as v′ i = ReLU �� r∈R v′ i,r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' (3) We denote the combination of EdgeGAT and the aggregation of the resulting embeddings over multiple relation types as HetEdgeGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We propose to use cascaded layers of HetEdgeGAT in order to aggregate information of the scene into agent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The flow of information towards the agent nodes is represented by the color of the relation types in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 2: The first layer aggre- gates information into crosswalk nodes (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Subsequent layers aggregate information into lane nodes (blue) and agent nodes (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The last layer of graph convolution then considers social interaction between agents and updates the agent nodes again (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 2) Decoder: The decoder is task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We use a Multi- layer Perceptron (MLP) and apply it to the encodings of agent nodes for the two binary node classification tasks considered in the experiments section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' EXPERIMENTS In this section we describe the extensive evaluation of our proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Learning Tasks We evaluate our methodology on two diverse binary node classification tasks: 1) Classification whether an agent is parked or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We consider one prior publication introducing and address- ing this task as a baseline [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 2) Classification whether an agent is a ghost or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Ghosts are unreliable detections of agents by upstream per- ception components that do not exist in real world, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', false positive detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We are not aware of prior publications that consider static infrastructure for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Comparison is done with an approach that considers only dynamic information [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Dataset Experiments are carried out on a large-scale, in-house dataset with over 22 400 sequences, each coming with 3 s of temporal history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' They are extracted from in-vehicle recordings from different areas in Germany and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' with a sampling rate of 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Camera, LiDAR and Radar detections are fused together in order to detect surrounding dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The dataset includes diverse environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', urban, rural and highway) as well as diverse scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', driving and yield- ing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' To the best of our knowledge, there is no publicly avail- able dataset for scene understanding that similarly provides manually annotated semantic attributes for dynamic agents in TABLE I NODE TYPE-SPECIFIC FEATURES Node type Features agent State vector (position, velocity, acceleration, yaw, yaw rate) with corresponding covariances Tracking properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' velocity, tracked time) Bounding box dimensions Estimate of agent type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', car, truck, two-wheeler) Sensor specific detection and existence probabilities Existence confidence calculated according to [28] Trajectory of the past three seconds as a series of positional and angular differences lane Type (car, bike, shoulder, parking) Geometric properties (length, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' and max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' width, max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' curvature) Maximum legal speed Left and right boundary types Turn type crosswalk Is signaled stop Type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', stop, crosswalk, yield) light Type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', car, pedestrian) State (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', red, yellow) Is deactivatable TABLE II RELATION TYPE-SPECIFIC EDGE FEATURES Relation type Features interacts Geometric differences (position, velocity, angle) under Assignment probability Frenet state (position, velocity) at agent position Lane properties at agent position Gap to lane boundaries at agent position Behavior primitive of agent in lane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', following, crossing) connection Type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', precede, left neighbor) conflict Type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', cross, merge) precedence Type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=', higher, lower) stops Longitudinal position in lane combination with an extensively attributed heterogeneous HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' By deriving correspondences between manually annotated agents and agent detections from upstream perception com- ponents, more than 430 000 labels per training tasks are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' For our experiments, we use a dataset split of 60% (training), 30% (validation) and 10% (testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Model Implementation Details We use an extensive set of features for nodes and edges in the scene graph to explicitly model all available knowledge of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Features are provided by the upstream perception components and the HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Table I and Table II list the used feature sets for each node type and each relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' To propagate uncertainty of the perception component to the model, the feature vectors of agent nodes contain covariances and confidence values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' All features that express a category type are one-hot encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' While this set of features is given by our perception and HD map, the proposed methodology can use any arbitrary set of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Details of the final implementation are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 3, including the dimensions of all feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The features of all nodes and edges are type-specifically encoded with a single linear layer and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Static and temporal aspects of agents are encoded separately and concatenated thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The static encoding uses a linear layer with ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The temporal encoding uses a Gated Recurrent Unit (GRU) and ReLU to MONNINGER AND SCHMIDT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' : SCENE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Agent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='30x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Agent ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Predicted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Binary Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='HetEdgeGAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='Encoder ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Implementation details of the SCENE encoder and the task-specific MLP-decoder: Four cascaded layers of HetEdgeGAT (green, blue, red, pink) are used to combine information of nodes of different types and their relations in order to update the feature vector of agent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Residual connections (orange) prevent over-smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' For the two binary node classification tasks, an MLP is then used to generate a classification score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' process the trajectory feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Following the idea of [3], the trajectory feature contains a fixed-length series of positional and angular differences of the last 30 timesteps (3 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We add a binary flag that indicates whether an entry contains a valid measurement for each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In order to avoid over-smoothing, which is one of the main issues of multilayer GNNs [29], we exploit two concatenated residual connections (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' These allow the decoder to combine high and low-level features of agent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Binary cross-entropy is used as a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The model is trained with Adam optimizer [30] with a learning rate of 10−5 and a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Dropout with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='3 is used for the two linear layers of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Baselines For both tasks, we compare our generic methodology to multiple task-specific and generic baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 1) Task-specific Baselines for the Parked Attribute: The velocity baseline evaluates the velocity of each car in a given scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Stationary cars (zero velocity) are labeled as parked and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The logistic regression baseline uses a handcrafted set of features based on the inputs provided by upstream perception components and the HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This baseline has been specifi- cally designed for classifying the parked attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We also consider two approaches of one prior publication addressing parked car classification [2], namely the heuristic approach and the MLP approach, which operates on only three features for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Both approaches utilize features that contain information about the agent and one underlying lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We call these baselines heuristic and MoveMLP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 2) Task-specific Baseline for the Ghost Attribute: The ex- istence confidence baseline gives an estimate about the exis- tence of an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This baseline approach [28] uses a method based on Dempster-Shafer evidence theory to estimate a fused existence confidence about an agent based on detections from multiple sensor modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Note that the resulting existence confidence is also part of the input features of agent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A comparison to this baseline therefore shows the benefit of additionally considering social context and map context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 3) Generic Baselines: The MLP baseline contains four linear layers with ReLU between these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' It operates directly on the features of agent nodes and does not process the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In comparison to our proposed model, this baseline shows the effect of neglecting relational information about social context, defined by nearby dynamic agents, and map context, defined by the static infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The R-GCN baseline applies the Relational Graph Convo- lutional Network [27], typically used as a common approach to reason about knowledge graphs, to our scene graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We extend the original R-GCN approach by introducing edge features, which allows our implementation of R-GCN to use the same input features as SCENE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' After four layers of R-GCN, an MLP decoder is used to predict the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Feature vector sizes of nodes and edges are similar to the ones used in SCENE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' To compare our methodology to the current-state-of-the- art in scene encoding, we adapt VectorNet [17] to our input representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The VectorNet-like baselines therefore rely on learning from a fully-connected, homogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In contrast to SCENE, the features of all heterogeneous nodes are type-specificially encoded into a joint feature space with size 64 to get a homogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The vanilla VectorNet- like baseline does not use edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In order to allow a fair comparison to SCENE, we also extend the original VectorNet approach by the introduction of edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' All existing edges are type-specifically encoded to a size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' To obtain the required fully-connected graph, edges are instantiated with a zero vector of size 16 between all nodes that are not yet connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A binary flag concatenated to the edge feature vector is used to indicate whether an edge is valid (1) or invalid (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Despite this leading to a fully-connected, homogeneous graph, connectivity information of our initial scene graph is still conserved in the edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' One layer of EdgeGAT is used on the resulting fully-connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Similar to SCENE, the labels are then predicted with an MLP decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 6 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' ACCEPTED DECEMBER, 2022 TABLE III RESULTS ON THE TEST SET Method Parked Ghost F1 (%) Acc (%) F1 (%) Acc (%) Velocity 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='51 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='41 Logistic regression 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='79 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='21 Heuristic [2] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='75 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='47 MoveMLP3 [2] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='15 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='07 - Existence confidence [28] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='48 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='01 Naive prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='00 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='33 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='88 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='24 MLP 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='89 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='40 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='67 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='44 R-GCN [27] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='81 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='58 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='36 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='24 VectorNet-like [17] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='90±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='43 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='74 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='63±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='17 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='84±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='68 VectorNet-like (w/ edge feat) [17] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='18±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='28 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='75 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='93±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='26 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='98 Ours (using HetEdgeGAT)1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='71 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='46 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='77 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='71 Ours (using HetEdgeGatedGCN) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='29 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='15 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='42±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='33 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='83±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='07 Ours (using HetEdgeSAGE) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='43 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='34 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='72±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='23 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='71 Ours (using HetEdgeGAT)∗ 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='16±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='42 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='49±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='10 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='05±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='26 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='80 1Selected for all further experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' ∗Multi-task training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Metrics F-Score (F1) and accuracy (Acc) are used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Quantitative Results The models were trained over five random seeds to min- imize stochasticity in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The resulting average and standard deviation of the performance metrics on the test split are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Besides GAT, we evaluated our approach using different operators for graph convolution, including variants of Gated Graph Convolutional Neural Net- works (EdgeGatedGCN) [31] and GraphSAGE (EdgeSAGE) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' They all consistently perform well, which suggests that the graph convolution operator is interchangeable, also with regards to the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Therefore, our methodology can benefit from upcoming advances in the field of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' HetEdgeGAT was selected for all further experiments because it uses the least number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Also, Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' [33] show that the resulting attention weights of interacts relations offer additional interpretability, as they are a direct measure for interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Comparing the results of our methodology with the MLP baseline supports our initial hypothesis that our proposed way of modeling a scene in a heterogeneous scene graph adds valuable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Our generic methodology outperforms all task-specific and generic baselines on both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The last row in Table III shows the result of the model simultaneously trained on both learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The performance is on par with the single- task setup, which supports the indication that our method works as a generic scene encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The VectorNet-like baseline extended with edge features performs much better than the vanilla VectorNet-like baseline, which supports the intuition that adding relational attributes provides valuable information for scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Comparing the average number of Floating-Point Operations (FLOPs) of the VectorNet-like base- line with edge features (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='24·108 FLOPs) and our SCENE ap- proach (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='57 · 107 FLOPs) shows that our approach has more than an order of magnitude less computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The higher complexity of the VectorNet-like baselines comes from TABLE IV CONTEXT ABLATION STUDY ON THE TEST SET Context #Params Parked Ghost Agent Lane Remaining F1 (%) Acc (%) F1 (%) Acc (%) 50k 75.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='77 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='71 2In contrast to the temporal and residual architectural measures, edge features introduce additional information into the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' applying convolution over the fully-connected graph, which also results in significantly higher GPU memory requirements and training time than our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Ablation Studies In two ablation studies we analyze how well our approach can leverage the various sources of information and the effectiveness of our architectural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Table IV ablates the value of various sources of information coming from the dynamic agents and static infrastructure and the ability of our approach to leverage this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' With- out any context at all, the model performs the classification tasks with the features of agent nodes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Our experiments show that considering social interactions to nearby agents (agent context), lane information (lane context) and informa- tion given by crosswalks, stops and lights (remaining context) can have a strong positive effect on model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The relevance of each contextual aspect differs between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The remaining context has only a small effect, since much information is implicitly present in the lane model already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Overall, the results show that incrementally adding further context information improves model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This con- firms the value of the additional information as hypothesized in the introduction as well as the capability of the generic model to exploit the provided information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Table V ablates the performance of our approach in terms of applied architectural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' These measures are the temporal encoding of each agent’s trajectory, the residual connections and the inclusion of edge features in the layers of graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The results indicate that omitting individual architectural measures decreases model performance compared to applying the full set of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This is particularly noteworthy for the edge features, suggesting a benefit of adding relational information to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The architecture that combines all measures either excels or comes very close to the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This suggests that the individual architectural measures benefit from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' MONNINGER AND SCHMIDT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' : SCENE 7 Parked classification Parked classification Ghost classification Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Qualitative results of SCENE for the classification of the parked attribute (left and center) and the classification of the ghost attribute (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The upper row shows the front camera frame, which is one of multiple sensors used by upstream perception, and the lower row renders the corresponding scene with lanes in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Bounding boxes of agents are drawn as rectangles, with the past trajectory visualized by an orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The color of the rectangle outline indicates the ground-truth label and the fill color indicates the label predicted by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Green corresponds to an agent being labeled as non-parked or non-ghost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Red corresponds to an agent being labeled as parked or ghost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' All agents are correctly classified as indicated by matching fill and outline colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A purple outline indicates a missing label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The autonomous vehicle is colored in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Qualitative Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' 4 shows qualitative results for both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Color codes are described in the caption of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' In the three examples all agents with available ground-truth label are correctly clas- sified, which is represented by consistent coloring of outline and fill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The figure on the left shows an urban scenario with vehicles parked on the road side (red) and vehicles driving in the center (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Interestingly, the vehicle with white paint inside the paved intersection is parked, which is correctly predicted by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The figure displayed in the center is a rare case where two cars are parked on the left lane of a highway on-ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Again, those are correctly classified by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The prediction is likely supported by the humans nearby, which are detected by the system (purple outline due to no parked label for agents of type human) and provide social context to the parked vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The figure on the right shows a highway scenario, where all nearby agents besides one are correctly classified as non- ghost (green outline and fill).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The prediction of the one ghost agent (red fill) can be confirmed by its trajectory showing a wrong direction of travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' This specific detection is probably caused by sensor reflections of a bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The corresponding ground-truth label also classifies it as ghost (red outline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' TRANSFERABILITY TO OTHER APPLICATIONS As an extension to evaluating the prediction of unknown characteristics of traffic agents, in this section we show that, without any modifications, the use of cascaded layers of graph convolution can be transferred to applications that go beyond the domain of scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' We therefore apply our methodology for the task of node classification to multiple knowledge graphs of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The source code of these experiments, including our graph convolution operator, is publicly available3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Datasets Evaluation is done on four publicly available heterogeneous knowledge graph datasets, namely AIFB, MUTAG, BGS and 3Source code: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='com/schmidt-ju/scene TABLE VI ACCURACY (%) ON THE MASKED NODES OF KNOWLEDGE GRAPHS Dataset WL [36] RDF2Vec [37] Walk Tree [35] R-GCN [27] Ours AIFB 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='00 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='00 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='44±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='08 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='62 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='83±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='96 MUTAG 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='00 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='24 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='82±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='61 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='48 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='44±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='50 BGS 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='00 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='89 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='90±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='38 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='80 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='41±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='72 AM 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='00 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='61 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='59 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='35 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='05±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content='07 AM [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The datasets cover varying graph sizes, ranging from small (AIFB, 8 285 nodes) to large (AM, 1 666 764 nodes) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Given classes for some nodes of a target node type, the goal is to correctly classify the classes of masked target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Model and Results We remove low-degree nodes and initialize the features of each node with a learnable bias vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Four layers of HetEdgeGAT are arranged in cascaded form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The first two layers sequentially update all nodes not of type target based on neighboring nodes of the same and of other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The second two layers sequentially aggregate information into nodes of type target by considering neighboring nodes of other types and of type target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The model is trained full-batch with cross- entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Average accuracy and standard deviation for ten runs is reported in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Results of the compared methods are taken from prior publications [27], [35], [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Despite the task being different from predicting unknown characteristics of traffic agents, the results show that our methodology manages to yield state-of-the-art performance for the task of node classification in knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' CONCLUSION This paper proposes a method using cascaded layers of graph convolution in order to predict relevant information from heterogeneous graphs and examines it on the task of reasoning about traffic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Combining the cascaded layers of graph convolution with our novel way for modeling traffic scenes in heterogeneous graphs results in a generic and extensible method to reason about traffic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The heterogeneous graph ontology can be extended with additional types or features of nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Our methodology outperforms all 8 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' ACCEPTED DECEMBER, 2022 task-specific baselines on two diverse tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' Furthermore, we compared it to multiple generic state-of-the-art encoders and demonstrated that our method has significant advantages with regard to performance metrics and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' The application of our methodology to the task of node classification in knowledge graphs indicates another key prop- erty of our methodology: it is, without any modifications, applicable to areas that go beyond the domain of scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' By making source code and GNN operator publicly available, we contribute to the progress in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQf5gVO/content/2301.03512v1.pdf'} diff --git a/6NE1T4oBgHgl3EQf6wWy/content/tmp_files/2301.03527v1.pdf.txt b/6NE1T4oBgHgl3EQf6wWy/content/tmp_files/2301.03527v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbe903f00dde3db1ce506a237d4b9e3cf182a952 --- /dev/null +++ b/6NE1T4oBgHgl3EQf6wWy/content/tmp_files/2301.03527v1.pdf.txt @@ -0,0 +1,690 @@ +MNRAS 000, 1–7 (2022) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Covariance Matrix of Fast Radio Bursts Dispersion +Robert Reischke⋆1 and Steffen Hagstotz†2,3 +1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), +German Centre for Cosmological Lensing, 44780 Bochum, Germany +2 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, +Scheinerstraße 1, D-81679 München, Germany +3 Excellence Cluster ORIGINS, Boltzmannstraße 2, D-85748 Garching, Germany +10 January 2023 +ABSTRACT +The dispersion of fast radio bursts (FRBs) is a measure of the large-scale electron distribution. +It enables measurements of cosmological parameters, especially of the expansion rate and the +cosmic baryon fraction. The number of events is expected to increase dramatically over the +coming years, and of particular interest are bursts with identified host galaxy and therefore +redshift information. In this paper, we explore the covariance matrix of the dispersion mea- +sure (DM) of FRBs induced by the large-scale structure, as bursts from a similar direction on +the sky are correlated by long wavelength modes of the electron distribution. We derive ana- +lytical expressions for the covariance matrix and examine the impact on parameter estimation +from the FRB dispersion measure - redshift relation. The covariance also contains additional +information that is missed by analysing the events individually. For future samples containing +over ∼ 300 FRBs with host identification over the full sky, the covariance needs to be taken +into account for unbiased inference, and the effect increases dramatically for smaller patches +of the sky. +Key words: cosmology: theory, large-scale structure of Universe, radio continuum: transients +1 +INTRODUCTION +Fast radio bursts (FRBs) are very short transients lasting usually only a few milliseconds, with a frequency range from ∼ 100 MHz to several +GHz. The original pulse gets dispersed due to free electrons in the ionised intergalactic medium. This leads to a delayed arrival time of the +pulse frequencies ∆t(ν) ∝ ν−2, where the proportionality constant is called dispersion measure (DM) (e.g. Thornton et al. 2013; Petroff et al. +2015; Connor et al. 2016; Champion et al. 2016; Chatterjee et al. 2017) and is related the column density of electrons along the line-of-sight +to the FRB. +While the mechanism for the radio emission is still under debate, their isotropic occurrence and large observed DM suggest an extra- +galactic origin for the vast majority of events (even though some might also be galactic, see Andersen et al. 2020), so that the DM can be +used to test the distribution of diffuse electrons in the large-scale structure (LSS). Several authors therefore proposed to use the DM inferred +from FRBs as a cosmological probe, using either the average dispersion measure up to a given redshift (Zhou et al. 2014; Walters et al. 2018; +Hagstotz et al. 2022; Macquart et al. 2020; Wu et al. 2022; James et al. 2022) or the statistics of DM fluctuations (Masui & Sigurdson 2015; +Shirasaki et al. 2017; Rafiei-Ravandi et al. 2020; Reischke et al. 2021; Bhattacharya et al. 2021; Takahashi et al. 2021; Rafiei-Ravandi et al. +2021; Reischke et al. 2022). While the former requires host identification to acquire an independent redshift estimate, the latter can be done +without it, as the homogeneous component can serve as a (noisy) estimate for the redshift. Angular statistics of the DM are formally very +similar to cosmic shear since one is dealing with projections of cosmic fields. In this paper, however, we will focus on the homogeneous +component of the DM, the so-called DM−z relation, which can be employed in similar ways as supernovae Ia (SN Ia) measurements (see e.g. +Riess et al. 2022; Brout et al. 2022, for the most recent results). The dispersion is used as a distance estimate and consequently as a probe +of the geometry of the Universe. The total amplitude of the dispersion is also sensitive to the overall baryon content, the ionisation fraction +and the Hubble constant. These are perfectly degenerate at the background level, so additional information about some of these quantities +have to be considered to constrain the remaining one. A common choice is to adapt a prior on the baryon density coming from big bang +nucleosynthesis as described in Hagstotz et al. (2022) in order to measure the Hubble parameter at late times. +Studies that employ FRBs to measure either the cosmic baryon density Macquart et al. (2020) or the Hubble constant Hagstotz et al. +⋆ E-mail: reischke@astro.ruhr-uni-bochum.de +† E-mail: steffen.hagstotz@lmu.de +© 2022 The Authors +arXiv:2301.03527v1 [astro-ph.CO] 9 Jan 2023 + +2 +Reischke & Hagstotz +(2022) treat the individual bursts and their DM as independent. However, since the signal from an FRB travels through the large-scale +structure (LSS), events within angular proximity on the sky become correlated. In this paper, we intend to fill this gap in current analyses +and are concerned with deriving the covariance and its consequences for using the mean FRB dispersion for the inference of astrophysical +and cosmological parameters. We emphasise that even though the observed signal does only depend on the cosmological background, the +covariance itself is sensitive to fluctuations and therefore to perturbations charaterised by the 2-point correlation function of the electron +distribution. +The paper is structured as follows: In Section 2 we summarise the theory of FRBs, the DM and derive the expression for the covariance +matrix. Section 3 presents and discusses the results for a current sample of FRBs (Petroff et al. 2016) and the prospects for future analysis +with FRBs. Finally, we summarise our findings in Section 4. Throughout the paper we fix the cosmological parameters to a ΛCDM model +with the best-fit values from the Planck mission Aghanim et al. (2020a) and vary only one parameter for illustration, usually chosen to be the +Hubble constant H0. +2 +TESTING THE COSMOLOGICAL BACKGROUND WITH FAST RADIO BURSTS +In this section we will review the basic theoretical framework of FRBs and how it is related to properties of the LSS. We will then derive +main result of this paper, the covariance matrix for FRBs with host identification induced by the correlated LSS along nearby lines of sight. +2.1 +Dispersion Measure +Cosmological tests using FRBs with host identification, that is with an independent redshift estimate, aim to fit the DM-z diagram. The DM +itself is estimated from the pulse’s dispersion +∆t ∝ DMtot(ˆx, z) ν−2 , +(1) +defining the estimated DM of an FRB at the sky position ˆx and redshift z. Dispersion itself is caused by scattering with the free electrons +along the line of sight. These electrons are either associated with the host halo, with the Milky Way, or with the large-scale structure (LSS). +Therefore, the average total contribution can be split into three parts: +DMtot(ˆx, z) = DMhost(z) + DMMW(ˆx) + DMLSS(z, ˆx) . +(2) +Here the contribution from the Milky Way does not depend on redshift, since it is a local effect. Likewise the contribution from the host does +not depend on the direction. The LSS contribution, however, depends both on redshift and direction, which will become important later on. +Note that each of these contributions takes the form of a PDF with scatter around the mean values. +For this work, we will focus on the contribution from the LSS. We write explicitly +DMLSS(ˆx, z) = +� z +0 +ne(ˆx, z′) fIGM(z′) 1 + z′ +H(z′) dz′ , +(3) +where ne(ˆx, z) is the comoving cosmic free electron density, H(z) = H0E(z) is the Hubble function with the expansion function E(z) and the +Hubble constant H0. The overall DM is usually multiplied with the fraction fIGM(z) of electrons in the IGM that are not bound in structures. +For redshifts z < 3 almost all baryons are ionised, it is thus useful to express the electron density by the number of baryons in the Universe: +ne(ˆx, z) = χe +ρb(ˆx, z) +mp += χe +¯ρb +mp +�1 + δe(ˆx, z)) , +(4) +with the baryon density ρb, the proton mass mp and the electron fraction +χe = YH + 1 +2YHe +(5) +≈ 1 − 1 +2YHe , +(6) +calculated from the primordial hydrogen and helium abundances YH and YHe. Here, we assume YH ≈ 1 − YHe and YHe = 0.24, found to high +precision both by CMB measurements (Aghanim et al. 2020a) and by spectroscopic observations of metal-poor gas clouds (Aver et al. 2015). +The baryon number density in Equation (4) is commonly expanded around its background value ¯ρb/mp with the electron density contrast +δe, whose mean vanishes by definition. Hence the DM is in principle a probe of the LSS by measuring DM statistics. This, however, requires +a larger sample of FRBs than currently available. +The electron fraction in the IGM in Equation (3) is calculated by subtracting the fraction bound in stars, compact objects and the dense +interstellar medium (ISM) +fIGM(z) = 1 − f⋆(z) − fISM(z) . +(7) +We compute1 f⋆ and fISM using the estimates of star formation rate and ISM mass fraction from Fukugita & Peebles (2004); Madau & +1 The code for the calculations is publicly available at https://github.com/FRBs/FRB, provided by Macquart et al. (2020). +MNRAS 000, 1–7 (2022) + +Covariance matrix for located FRBs +3 +10−2 +10−1 +100 +zi +10−2 +10−1 +100 +zj +26 +31 +36 +41 +46 +51 +56 +61 +66 +71 +Cij(ℓ = 2) +10−2 +10−1 +100 +zi +10−2 +10−1 +100 +zj +0.000 +0.125 +0.250 +0.375 +0.500 +0.625 +0.750 +0.875 +1.000 +1.125 +Cij(ℓ = 128) +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +zi +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +zj +0.000 +0.003 +0.006 +0.009 +0.012 +0.015 +0.018 +0.021 +0.024 +Cij(ℓ = 1090) +Figure 1. Angular power Cij(ℓ) for different multipoles in the (zi, zj)-plane as defined in Equation (18). Note that the colour scale changes and as well as the +axis scaling of the rightmost plot. +Dickinson (2014). We keep fIGM = 0.84 constant for the purposes of this paper. Putting everything together, we write the DM – redshift +relation in Equation (3) as +DMLSS(ˆx, z) = 3Ωb0H0 +8πGmp +χe fIGM +� z +0 +1 + z′ +E(z′) +�1 + δe(ˆx, z′)�dz′ , +(8) +with the dimensionless baryon density parameter Ωb0 and the dimensionless expansion function E(z) = H(z)/H0. Averaging Equation (8) +provides the well known mean DM-redshift relation: +DMLSS(z) � ⟨DMLSS(ˆx, z)⟩ = 3Ωb0H0 +8πGmp +χe fIGM +� z +0 +1 + z′ +E(z′) dz′ . +(9) +The measurement of FRBs together with a host redshift yields pairs {DMi, zi} and can be used to constrain any parameter from Equation (9) +in addition to the cosmic expansion history. +2.2 +Covariance of the LSS component +Observations of FRBs with host identification consist of a set of NFRB measurements �DMi, ˆxi, zi +�, i = 1, ..., NFRB, with the observed DM, the +direction of the burst ˆxi and its redshift. We are interested in the contribution to the covariance induced by the LSS between events labelled +i, j: +covi j � +� +DMLSS(ˆxi, zi)DMLSS(ˆx j, z j) +� +− DMLSS(zi)DMLSS(z j). +(10) +Using Equation (8) and Equation (9) one finds +covi j = +� zi +0 +dz′ +iWDM(z′ +i) +� zj +0 +dz′ +j WDM(z′ +j) +� +δe(ˆxi, z′ +i)δe(ˆxj, z′ +j) +� +, +(11) +with the DM weight function: +WDM(z) = 3Ωb0H0 +8πGmp +χe fIGM +1 + z +E(z) . +(12) +What is left is to do is to work out the correlator in the integrand: +� +δe(ˆxi, zi)δe(ˆx j, zj) +� += +� +d3k +(2π)3 eik·(xi−x j)Pe(k, zi, z j) , +(13) +where we introduced the electron power spectrum and carried out the k′-integration. Expanding the exponential into plane waves yields: +� +δe(ˆxi, zi)δe(ˆx j, zj) +� += 2 +π +� +k2dk +� +dΩkPe(k, zi, zj) +� +ℓ,ℓ′ +� +m,m′ +iℓ(−i)ℓ′Yℓm(ˆk)Y∗ +ℓm( ˆxi) jℓ(kχi)Y∗ +ℓ′m′(ˆk)Yℓ′m′( ˆxi) jℓ′(kχi) +(14) += 2 +π +� +k2dkPe(k, zi, z j) +� +ℓ +� +m +Y∗ +ℓm( ˆxi)jℓ(kχi)Yℓm( ˆxi)jℓ(kχi) +(15) += +1 +2π2 +� +ℓ +(2ℓ + 1) +� +k2dkPe(k, zi, zj) jℓ(kχi) jℓ(kχj)Pℓ(cos θ) . +(16) +In the last step, we made use of the isotropy of cosmological fields and used +� +m +Yℓm( ˆxi)Y∗ +ℓm( ˆxi) = 2ℓ + 1 +4π +Pℓ(cos θ) , +(17) +MNRAS 000, 1–7 (2022) + +4 +Reischke & Hagstotz +FRB190523 +FRB190711 +FRB181112 +FRB190611 +FRB180924 +FRB190102 +FRB121102 +FRB190608 +FRB180916 +FRB190523 +FRB190711 +FRB181112 +FRB190611 +FRB180924 +FRB190102 +FRB121102 +FRB190608 +FRB180916 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +log10(covij) +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +h +posterior +full covij +diagonal covii +Figure 2. Left: Covariance matrix, Equation (18), for the FRB catalogue (Petroff et al. 2016) with host identification. Right: Posterior distribution of the Hubble +constant (or other any amplitude of the DM), similar to the analysis carried out in Hagstotz et al. (2022). The solid blue lines use the accurate covariance matrix, +while the dashed orange lines only use the diagonal elements, i.e. the events are uncorrelated. Parameter dependence of the covariance does not change the +results for this sample. +with the Legendre polynomials Pℓ(x) and we denote the angular separation between pairs of FRBs as ˆxi· ˆxj = cos θ. Furthermore, x = (ˆxχ, χ), +where χ = ∥x∥, with the comoving distance χ(z). Thus, altogether, by using Pe(k, zi, z j) = �Pe(k, zi)Pe(k, z j), we arrive at +covi j(cos θ, zi, z j) = +1 +2π2 +� +ℓ +(2ℓ + 1)Pℓ(ˆxi · ˆx j) +� +k2dk +� zi +0 +dz′ +iWDM(z′ +i) +� +Pe(k, z′ +i) jℓ(kχi) +� zj +0 +dz′ +j WDM(z′ +j) +� +Pe(k, z′ +j) jℓ(kχj) += +� +ℓ +2ℓ + 1 +4π +Pℓ(cos θ)Cij(ℓ) , +(18) +which defines the angular power spectrum Cij(ℓ) between the two fields i and j. To calculate the electron power spectrum, we use HMX (Mead +et al. 2015, 2020; Tröster et al. 2022). In order to carry out the sum over ℓ, we collect multipoles up to ℓ = 5 × 104 on the diagonal and for +the other entries take up to ℓ = 100/|ˆxi − ˆx j| into account. +2.3 +Remarks on parameter dependence of the covariance +Since the covariance in Equation (22) depends on cosmological parameters, it contains additional information. There has been a long debate in +the cosmological community whether it is necessary to account for this dependence or not. Current LSS (e.g. Asgari et al. 2021; Abbott et al. +2022) or CMB measurements (Aghanim et al. 2020b) adjust the covariance interatively, that is they chose a fiducial cosmology, perform the +inference for preliminary model parameters, update the covariance matrix to the preliminary best-fit model and start the inference again. This +process is repeated until convergence is reached. Carron (2013) discussed the assumption of a parameter (in)dependent covariance matrices +when two-point statistics are used as the model and data, showing that the (Gaussian) covariance matrix never carries any independent +information (as it is again just a product of two-point functions) and is rather a sign of non-Gaussian information. In Reischke et al. (2017) the +overall parameter dependence of the cosmic shear two-point covariance was investigated with analytic methods and ray-tracing simulations. +This work was followed up by Kodwani et al. (2019), where the effect of a parameter dependent covariance matrix on the inference process +with future LSS surveys was investigated and found to be negligible. However, one should keep in mind that these papers worked with +averaged data and not simulated realisations of the data. The situation studied in this paper is different since the average DM - redshift +relation only contains information about the cosmological background, while the correlations in the data are induced by the perturbations +characterised by the electron power spectrum. Therefore the covariance matrix contains additional information without any double-counting. +3 +RESULTS AND DISCUSSION +In this section we present the results for the covariance matrix. We start by discussing some intermediate results for the angular power spectra +in the (zi, zj)-plane. Figure 1 shows the corresponding covariance for three different multipoles, ℓ = 2, 128, 1090, from left to right. The +colour bar encodes the covariance in redshift at these fixed angular scales ℓ ∼ θ−1. All covariances have a clear rectangular structure which +stems from the integration bounds in Equation (18) reflecting the fact that the DM of two FRBs is only correlated for redshifts z ≤ min(zi, z j). +Furthermore, the structure of the covariance also shows that on larger angular scales the correlation is stronger at lower redshifts. This can be +understood by the fact that the Bessel function jℓ(kχ) peaks around kχ = ℓ + 0.5, thus small ℓ require small χ and hence z to reach the peak +MNRAS 000, 1–7 (2022) + +Covariance matrix for located FRBs +5 +of the power spectrum at k ≈ 0.01 h−1Mpc. Lastly, we also note that the variance obtained from Equation (18), i.e. +σ2 +i = +� +ℓ +(2ℓ + 1)Cii(ℓ)/(4π), +(19) +agrees well with the results from the empirical formula presented in (McQuinn 2014; Zhang et al. 2021): +p(∆) ∝ ∆−β exp +�(∆−α − C0)2 +2α2σ2 +� +, +(20) +with α = β = 3, ∆ = DMLSS/⟨DMLSS⟩ and the fitting values from N-body simulations presented in table 1 of Zhang et al. (2021). At redshift +z = 0.1 we find 10 per-cent agreement with our analytical approach. +3.1 +Current Data +We now turn to current data using all FRBs from the FRB catalogue (Petroff et al. 2016) with host identification. For illustrative purposes, +we use them to fit the Hubble constant by putting a tight prior on the baryon density parameter Ωb0. There are more events available at +the time of writing (James et al. 2022), but including a slightly larger sample does not affect the role of the covariance. In Hagstotz et al. +(2022) the value of the physical density parameter, ωb = Ωb0h2, as measured by big bang nucleosynthesis (Cooke et al. 2018) was used, +changing the overall scaling with h slightly. With the approach followed here, one could think of the constraints just by looking at any linear +amplitude parameter of the DM, Equation (9). In Figure 2 we show the covariance matrix on the left for the 9 host-identified FRBs from +the FRBCAT. Clearly the variance is largest for the highest redshifts, the cross-covariance, however, is largest between FRB190102 and +FRB190611 which are in close proximity on the sky. Withal, the correlation coefficient is below 0.2. The right panel shows the fit to the +Hubble constant H0 = 100 h kms−1Mpc−1 for these 9 FRBs. We assume a Gaussian likelihood +χ2(θ) = log det C(θ) + (d − µ(θ))T C−1(θ) (d − µ(θ)) , +(21) +where we made the dependence on the parameters θ explicit. The covariance consists out of three contributions +C = CLSS + CMW + Chost , +(22) +and the components of CLSS are given by Equation (18), while we assume for the Milky Way CMW = σ2 +MWI, with σMW = 30 and the host +Chost = σ2 +hostI, with σhost = 50/(1 + z) +The results are shown on the right side of Figure 2, where the solid blue line denotes the posterior using the full covariance matrix, +while the assumption of independent events (taking only the diagonal of the covariance into account) leads to the dashed orange result. For +the small sample size available right now, both approaches agree very well. In this case, the parameter dependence of the covariance is also +still negligible. As we explain in the next section, this changes once the samples grow larger. +3.2 +Future Data +In order to illustrate when the proper treatment of correlated errors in the FRB dispersion becomes important, we now generate synthetic +samples containing a total number of NFRB FRBs distributed over redshift. For the redshift distribution, we assume a standard magnitude +limited sample (e.g. Reischke et al. 2021): +n(z) ∝ z2 exp(−zα) , +(23) +with α = 5. Next, we draw random positions for each FRB uniformly over patches in the sky with sky fractions fsky = 1, 10−2 and 10−3, +so that the effective number density is n = f −1 +skyNFRB/(4π). For this sample, we calculate the covariance matrix Equation (18) of the LSS +component which in turn yields the final covariance via Equation (22). We used this full covariance matrix to sample the NFRB DM values +for the generated events, completing the triples �DMi, ˆxi, zi +� in our synthetic catalog. +In Figure 3 we show the correlation coefficient rij = covij/(coviicov jj)1/2 for 500 FRBs distributed over different parts of the sky. While +the covariance for a few hundred events distributed over the full sphere is dominated by the diagonal elements, the same number of FRBs +distributed on a small fraction of the sky leads to a tight correlation due to the small angular separation of events. +The full covariance modelling is crucial for parameter estimation from larger FRB catalogs. In Figure 4 we show the posterior of h +from several synthetic catalogues of 500 events distributed over various fractions of the sky. The catalogue is always generated using the +true covariance matrix, and analysed using either the full covariance (blue solid) or only the diagonal (assuming uncorrelated events, orange +dashed). Thick lines are showing the average over many realisations, while single realisations of the data and the corresponding inference are +shown with shaded lines. The assumption of uncorrelated DMs leads to a severe underestimation of the error by 40%, 60% and up to 85% for +events covering either the full sky, or fsky = 10−2 and fsky = 10−3 respectively. While a linear parameter cannot be biased on average, single +realisations using the diagonal correlation matrix can easily show more than 3σ deviation from the true value used to generate the samples. +In the lower panels of Figure 4 we show the effect of the additional cosmological information contained in the covariance of the samples. +We compare again inference using the full, parameter-dependent covariance matrix (solid blue) with the case of a fixed covariance matrix +(dashed red). The width of the posterior shrinks by 30%, 45% and up to 70% depending on the sky fraction. +In Figure 5 we demonstrate the influence of the covariance as a function of the number of observed FRBs, again for the same sky +fractions. Note that the synthetic data used in Figure 4 is not necessarily the same as in Figure 5, but both are compatible with the full +covariance. The solid line shows the maximum posterior values while the shaded areas correspond to the 95% confidence interval. From the +plots it is noticeable that the uncertainty on h is severely underestimated for NFRB ≥ 300 even for a full sky sample when using a diagonal +MNRAS 000, 1–7 (2022) + +6 +Reischke & Hagstotz +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +fsky = 1 +fsky = 10−3 +−5 +−4 +−3 +−2 +−1 +0 +log10(rij) +Figure 3. Correlation coefficient, rij = covij/(coviicov jj)1/2, for 500 FRBs with host identification for a full sky (lower half) compared to the same sample +only on a small subset fsky = 10−3 of the sky (upper half), where the correlation of the data points becomes much stronger. The number of events corresponds +to n ≈ 5 × 10−3 deg−2 for the full sky sample, and n ≈ 5 deg−2 for the case of a small sky fraction. +posterior +fsky = 1 +full covij +diagonal covii +fsky = 10−2 +fsky = 10−3 +−0.10 +−0.05 +0.00 +0.05 +0.10 +∆h/h +posterior +fsky = 1 +covij(θ) +covij(θ0) +−0.10 +−0.05 +0.00 +0.05 +0.10 +∆h/h +fsky = 10−2 +−0.10 +−0.05 +0.00 +0.05 +0.10 +∆h/h +fsky = 10−3 +Figure 4. Upper panels: Posterior distribution for various samples of 500 FRBs drawn with the respective covariance plotted in Figure 3. Solid blue lines use +the full covariance, while dashed orange lines treat the FRBs as independent and just use the diagonal of the covariance matrix and underestimate the true error +severely by 40%, 60% and 85% for the respective panels. The x-axis shows the relative deviation from the fiducial value (used to generate the synthetic data). +Thick lines denote the average effect over many realisations, and shaded lines show different realisations of the noisy data. Single realisations analysed using +diagonal covariance can lead to false parameter estimations. Lower panels: Posterior distributions either using a parameter-dependent covariance (solid blue) +or a fixed covariance (dashed red). The cosmological dependence of the covariance matrix contains additional information, shrinking the error bars by 30%, +45% and 70% for the respective sky fractions compared to a covariance calculated at fixed parameters. +covariance. Although significant biases are unlikely to arise in this scenario, 3σ deviations from the true underlying value are possible if the +covariance between events is neglected. For fsky = 10−3 these effects are already present for smaller NFRB and the error can be misestimated +by up to 50 per-cent for NFRB as low as 40. While the last case is mostly of academical nature, selecting subsets of close by FRBs which are +close by and ignoring there covariance might be dangerous. +We close this section with a short comparison with the approach used in e.g. Macquart et al. (2020),Wu et al. (2022) or James et al. +(2022). These works use a likelihood derived from the one-point probability distribution function of DMLSS and take into account the full +non-Gaussianity of the DM distribution since it is measured directly from numerical simulations. While this captures the high DM tail of the +distribution, the final likelihood is still dominated by the variance rather than its skewness. On the other hand, it then is generally difficult to +take the covariance between different FRBs into account since in principle an NFRB-point function is needed to obtain the accurate shape of +the likelihood. Measuring all necessary moments from numerical simulations is inherently difficult due to the high noise in these estimates. +Furthermore, it is challenging to include the parameter dependence in these approaches, since the numerical simulations are only evaluated +at a single cosmology, although some of it has already been taken care of by looking at the relative DM, i.e. compared to the background +MNRAS 000, 1–7 (2022) + +Covariance matrix for located FRBs +7 +200 +400 +600 +800 +1000 +NFRB +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +∆h/h +fsky = 1 +fsky = 1 +fsky = 1 +full covij +diagonal covii +200 +400 +600 +800 +1000 +NFRB +fsky = 10−2 +fsky = 10−2 +fsky = 10−2 +200 +400 +600 +800 +1000 +NFRB +fsky = 10−3 +fsky = 10−3 +fsky = 10−3 +Figure 5. Best fit values and 95% confidence interval (shaded bands) against the number of FRBs with host identification generated with a known redshift +distribution for a full sky sample, fsky = 1 and fsky = 10−2, fsky = 10−3 from left to right. Results from synthetic data analysed either using only the diagonal +covariance (orange) or the full covariance including off-diagonal elements (blue). The diagonal covariance underestimates the true error, so the inferred value +of h is offset from the fiducial value. +cosmology. It would be interesting which effect is more important: the correlation or the high DM tail of the distribution. We refer this +investigation to future work. +4 +CONCLUSIONS +In this paper we have investigated the impact of the LSS induced correlation between FRBs with host identification. We have derived the +covariance matrix in harmonic and real space for FRBs observed at redshift z and ˆxi position. This new covariance matrix was then used to +reanalyse the FRBs from the FRB catalogue (Petroff et al. 2016) and to explore the influence on a single parameter, the Hubble constant h, +measured from current and future samples. Our main findings can be summarised as follows: +(i) The number of current FRBs with host identification does not require the inclusion of the covariance between them as the statistical +significance of the measurement is too low. Here we find similar results as Hagstotz et al. (2022). +(ii) For a full sky sample we find that the Hubble constant h or any other linear model parameter picks up an underestimated error of +roughly 50 per-cent for 500 FRBs in the best case. In the worst case there can be significant biases for any single realisation of the data. This +situation becomes even more serious if the number of FRBs increases. +(iii) If the parameter dependence of the covariance is not accounted for, biases can arise already for smaller numbers of FRBs in the case +of partial sky fraction. We generally advise to take the dependence on the model parameters of the covariance (diagonal or not) into account, +as it contains complementary information to the background dispersion measure. +(iv) When small patches of the sky are observed (fsky = 10−3 or smaller) the influence of the full covariance can be seen already for +NFRB = 40, leading to underestimated errors. +We therefore conclude that the LSS covariance matrix of the DM of FRBs with host identification can become important in the future when +more such FRBs (∼ 102) have been observed. Here we only investigated isotropically distributed FRB samples over sky patches of different +sizes. In case of a more complex selection function the results found here might become more severe, but we leave this for future work. +Another issue is the inclusion of the non-Gaussian structure of the likelihood which in principle is naturally included in approaches using +a formula fitted to numerical simulations (Macquart et al. 2020; Wu et al. 2022; James et al. 2022). These, however, lack the possibility to +account for the correlations between the different FRBs. This approach is feasible at the moment, but will lead to errorneous conclusions in +the future. Lastly, there are studies investigating the possibility to constrain reionization with FRBs (Heimersheim et al. 2022). These studies, +due to their high redshift FRBs would be much stringer affected by the covariance matrix due to the longer integration path. +Data Availability: The data and code underlying this article will be shared on request to the corresponding author. +ACKNOWLEDGMENTS +RR is supported by the European Research Council (Grant No. 770935). SH was supported by the Excellence Cluster ORIGINS which is +funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - +390783311. SH and RR acknowledge support by Institut Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium +2022, with the support of the P2IO Laboratory of Excellence (program “Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and +ANR-10-LABX-0038), the P2I axis of the Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA, APPEC, IAS, +OSUPS, and the IN2P3 master project UCMN. +MNRAS 000, 1–7 (2022) + +8 +Reischke & Hagstotz +REFERENCES +Abbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520 +Aghanim N., et al., 2020a, Astron. Astrophys., 641, A6 +Aghanim N., et al., 2020b, Astron. Astrophys., 641, A6 +Andersen B., et al., 2020, Nature, 587, 54 +Asgari M., et al., 2021, Astron. Astrophys., 645, A104 +Aver E., Olive K. A., Skillman E. D., 2015, JCAP, 07, 011 +Bhattacharya M., Kumar P., Linder E. V., 2021, Phys. Rev. D, 103, 103526 +Brout D., et al., 2022, Astrophys. 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J., 906, 49 +Zhou B., Li X., Wang T., Fan Y.-Z., Wei D.-M., 2014, Physical Review D, 89, 107303 +MNRAS 000, 1–7 (2022) + diff --git a/6NE1T4oBgHgl3EQf6wWy/content/tmp_files/load_file.txt b/6NE1T4oBgHgl3EQf6wWy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af3fec4eab53fe92411a270b7735cede152e932e --- /dev/null +++ b/6NE1T4oBgHgl3EQf6wWy/content/tmp_files/load_file.txt @@ -0,0 +1,573 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf,len=572 +page_content='MNRAS 000, 1–7 (2022) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='0 Covariance Matrix of Fast Radio Bursts Dispersion Robert Reischke⋆1 and Steffen Hagstotz†2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='3 1 Ruhr University Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Faculty of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Astronomical Institute (AIRUB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' German Centre for Cosmological Lensing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 44780 Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Germany 2 Universitäts-Sternwarte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Fakultät für Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Ludwig-Maximilians Universität München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Scheinerstraße 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' D-81679 München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Germany 3 Excellence Cluster ORIGINS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Boltzmannstraße 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Germany 10 January 2023 ABSTRACT The dispersion of fast radio bursts (FRBs) is a measure of the large-scale electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' It enables measurements of cosmological parameters, especially of the expansion rate and the cosmic baryon fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The number of events is expected to increase dramatically over the coming years, and of particular interest are bursts with identified host galaxy and therefore redshift information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In this paper, we explore the covariance matrix of the dispersion mea- sure (DM) of FRBs induced by the large-scale structure, as bursts from a similar direction on the sky are correlated by long wavelength modes of the electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We derive ana- lytical expressions for the covariance matrix and examine the impact on parameter estimation from the FRB dispersion measure - redshift relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The covariance also contains additional information that is missed by analysing the events individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For future samples containing over ∼ 300 FRBs with host identification over the full sky, the covariance needs to be taken into account for unbiased inference, and the effect increases dramatically for smaller patches of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Key words: cosmology: theory, large-scale structure of Universe, radio continuum: transients 1 INTRODUCTION Fast radio bursts (FRBs) are very short transients lasting usually only a few milliseconds, with a frequency range from ∼ 100 MHz to several GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The original pulse gets dispersed due to free electrons in the ionised intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This leads to a delayed arrival time of the pulse frequencies ∆t(ν) ∝ ν−2, where the proportionality constant is called dispersion measure (DM) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Thornton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Petroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Champion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Chatterjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2017) and is related the column density of electrons along the line-of-sight to the FRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' While the mechanism for the radio emission is still under debate, their isotropic occurrence and large observed DM suggest an extra- galactic origin for the vast majority of events (even though some might also be galactic, see Andersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2020), so that the DM can be used to test the distribution of diffuse electrons in the large-scale structure (LSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Several authors therefore proposed to use the DM inferred from FRBs as a cosmological probe, using either the average dispersion measure up to a given redshift (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Walters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Hagstotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022) or the statistics of DM fluctuations (Masui & Sigurdson 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Shirasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Rafiei-Ravandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Reischke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Rafiei-Ravandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Reischke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' While the former requires host identification to acquire an independent redshift estimate, the latter can be done without it, as the homogeneous component can serve as a (noisy) estimate for the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Angular statistics of the DM are formally very similar to cosmic shear since one is dealing with projections of cosmic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In this paper, however, we will focus on the homogeneous component of the DM, the so-called DM−z relation, which can be employed in similar ways as supernovae Ia (SN Ia) measurements (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Brout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022, for the most recent results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The dispersion is used as a distance estimate and consequently as a probe of the geometry of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The total amplitude of the dispersion is also sensitive to the overall baryon content, the ionisation fraction and the Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' These are perfectly degenerate at the background level, so additional information about some of these quantities have to be considered to constrain the remaining one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' A common choice is to adapt a prior on the baryon density coming from big bang nucleosynthesis as described in Hagstotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2022) in order to measure the Hubble parameter at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Studies that employ FRBs to measure either the cosmic baryon density Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2020) or the Hubble constant Hagstotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' ⋆ E-mail: reischke@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='ruhr-uni-bochum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='de † E-mail: steffen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='hagstotz@lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='de © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='03527v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='CO] 9 Jan 2023 2 Reischke & Hagstotz (2022) treat the individual bursts and their DM as independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' However, since the signal from an FRB travels through the large-scale structure (LSS), events within angular proximity on the sky become correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In this paper, we intend to fill this gap in current analyses and are concerned with deriving the covariance and its consequences for using the mean FRB dispersion for the inference of astrophysical and cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We emphasise that even though the observed signal does only depend on the cosmological background, the covariance itself is sensitive to fluctuations and therefore to perturbations charaterised by the 2-point correlation function of the electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The paper is structured as follows: In Section 2 we summarise the theory of FRBs, the DM and derive the expression for the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Section 3 presents and discusses the results for a current sample of FRBs (Petroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2016) and the prospects for future analysis with FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Finally, we summarise our findings in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Throughout the paper we fix the cosmological parameters to a ΛCDM model with the best-fit values from the Planck mission Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2020a) and vary only one parameter for illustration, usually chosen to be the Hubble constant H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2 TESTING THE COSMOLOGICAL BACKGROUND WITH FAST RADIO BURSTS In this section we will review the basic theoretical framework of FRBs and how it is related to properties of the LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We will then derive main result of this paper, the covariance matrix for FRBs with host identification induced by the correlated LSS along nearby lines of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='1 Dispersion Measure Cosmological tests using FRBs with host identification, that is with an independent redshift estimate, aim to fit the DM-z diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The DM itself is estimated from the pulse’s dispersion ∆t ∝ DMtot(ˆx, z) ν−2 , (1) defining the estimated DM of an FRB at the sky position ˆx and redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Dispersion itself is caused by scattering with the free electrons along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' These electrons are either associated with the host halo, with the Milky Way, or with the large-scale structure (LSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Therefore, the average total contribution can be split into three parts: DMtot(ˆx, z) = DMhost(z) + DMMW(ˆx) + DMLSS(z, ˆx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2) Here the contribution from the Milky Way does not depend on redshift, since it is a local effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Likewise the contribution from the host does not depend on the direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The LSS contribution, however, depends both on redshift and direction, which will become important later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Note that each of these contributions takes the form of a PDF with scatter around the mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For this work, we will focus on the contribution from the LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We write explicitly DMLSS(ˆx, z) = � z 0 ne(ˆx, z′) fIGM(z′) 1 + z′ H(z′) dz′ , (3) where ne(ˆx, z) is the comoving cosmic free electron density, H(z) = H0E(z) is the Hubble function with the expansion function E(z) and the Hubble constant H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The overall DM is usually multiplied with the fraction fIGM(z) of electrons in the IGM that are not bound in structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For redshifts z < 3 almost all baryons are ionised, it is thus useful to express the electron density by the number of baryons in the Universe: ne(ˆx, z) = χe ρb(ˆx, z) mp = χe ¯ρb mp �1 + δe(ˆx, z)) , (4) with the baryon density ρb, the proton mass mp and the electron fraction χe = YH + 1 2YHe (5) ≈ 1 − 1 2YHe , (6) calculated from the primordial hydrogen and helium abundances YH and YHe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Here, we assume YH ≈ 1 − YHe and YHe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='24, found to high precision both by CMB measurements (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2020a) and by spectroscopic observations of metal-poor gas clouds (Aver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The baryon number density in Equation (4) is commonly expanded around its background value ¯ρb/mp with the electron density contrast δe, whose mean vanishes by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Hence the DM is in principle a probe of the LSS by measuring DM statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This, however, requires a larger sample of FRBs than currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The electron fraction in the IGM in Equation (3) is calculated by subtracting the fraction bound in stars, compact objects and the dense interstellar medium (ISM) fIGM(z) = 1 − f⋆(z) − fISM(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (7) We compute1 f⋆ and fISM using the estimates of star formation rate and ISM mass fraction from Fukugita & Peebles (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Madau & 1 The code for the calculations is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='com/FRBs/FRB, provided by Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' MNRAS 000, 1–7 (2022) Covariance matrix for located FRBs 3 10−2 10−1 100 zi 10−2 10−1 100 zj 26 31 36 41 46 51 56 61 66 71 Cij(ℓ = 2) 10−2 10−1 100 zi 10−2 10−1 100 zj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='875 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='125 Cij(ℓ = 128) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 zi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 zj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='024 Cij(ℓ = 1090) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Angular power Cij(ℓ) for different multipoles in the (zi, zj)-plane as defined in Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Note that the colour scale changes and as well as the axis scaling of the rightmost plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Dickinson (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We keep fIGM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='84 constant for the purposes of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Putting everything together, we write the DM – redshift relation in Equation (3) as DMLSS(ˆx, z) = 3Ωb0H0 8πGmp χe fIGM � z 0 1 + z′ E(z′) �1 + δe(ˆx, z′)�dz′ , (8) with the dimensionless baryon density parameter Ωb0 and the dimensionless expansion function E(z) = H(z)/H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Averaging Equation (8) provides the well known mean DM-redshift relation: DMLSS(z) � ⟨DMLSS(ˆx, z)⟩ = 3Ωb0H0 8πGmp χe fIGM � z 0 1 + z′ E(z′) dz′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (9) The measurement of FRBs together with a host redshift yields pairs {DMi, zi} and can be used to constrain any parameter from Equation (9) in addition to the cosmic expansion history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='2 Covariance of the LSS component Observations of FRBs with host identification consist of a set of NFRB measurements �DMi, ˆxi, zi �, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=', NFRB, with the observed DM, the direction of the burst ˆxi and its redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We are interested in the contribution to the covariance induced by the LSS between events labelled i, j: covi j � � DMLSS(ˆxi, zi)DMLSS(ˆx j, z j) � − DMLSS(zi)DMLSS(z j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (10) Using Equation (8) and Equation (9) one finds covi j = � zi 0 dz′ iWDM(z′ i) � zj 0 dz′ j WDM(z′ j) � δe(ˆxi, z′ i)δe(ˆxj, z′ j) � , (11) with the DM weight function: WDM(z) = 3Ωb0H0 8πGmp χe fIGM 1 + z E(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (12) What is left is to do is to work out the correlator in the integrand: � δe(ˆxi, zi)δe(ˆx j, zj) � = � d3k (2π)3 eik·(xi−x j)Pe(k, zi, z j) , (13) where we introduced the electron power spectrum and carried out the k′-integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Expanding the exponential into plane waves yields: � δe(ˆxi, zi)δe(ˆx j, zj) � = 2 π � k2dk � dΩkPe(k, zi, zj) � ℓ,ℓ′ � m,m′ iℓ(−i)ℓ′Yℓm(ˆk)Y∗ ℓm( ˆxi) jℓ(kχi)Y∗ ℓ′m′(ˆk)Yℓ′m′( ˆxi) jℓ′(kχi) (14) = 2 π � k2dkPe(k, zi, z j) � ℓ � m Y∗ ℓm( ˆxi)jℓ(kχi)Yℓm( ˆxi)jℓ(kχi) (15) = 1 2π2 � ℓ (2ℓ + 1) � k2dkPe(k, zi, zj) jℓ(kχi) jℓ(kχj)Pℓ(cos θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (16) In the last step, we made use of the isotropy of cosmological fields and used � m Yℓm( ˆxi)Y∗ ℓm( ˆxi) = 2ℓ + 1 4π Pℓ(cos θ) , (17) MNRAS 000, 1–7 (2022) 4 Reischke & Hagstotz FRB190523 FRB190711 FRB181112 FRB190611 FRB180924 FRB190102 FRB121102 FRB190608 FRB180916 FRB190523 FRB190711 FRB181112 FRB190611 FRB180924 FRB190102 FRB121102 FRB190608 FRB180916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='5 log10(covij) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='1 h posterior full covij diagonal covii Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Left: Covariance matrix, Equation (18), for the FRB catalogue (Petroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2016) with host identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Right: Posterior distribution of the Hubble constant (or other any amplitude of the DM), similar to the analysis carried out in Hagstotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The solid blue lines use the accurate covariance matrix, while the dashed orange lines only use the diagonal elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' the events are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Parameter dependence of the covariance does not change the results for this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' with the Legendre polynomials Pℓ(x) and we denote the angular separation between pairs of FRBs as ˆxi· ˆxj = cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Furthermore, x = (ˆxχ, χ), where χ = ∥x∥, with the comoving distance χ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Thus, altogether, by using Pe(k, zi, z j) = �Pe(k, zi)Pe(k, z j), we arrive at covi j(cos θ, zi, z j) = 1 2π2 � ℓ (2ℓ + 1)Pℓ(ˆxi · ˆx j) � k2dk � zi 0 dz′ iWDM(z′ i) � Pe(k, z′ i) jℓ(kχi) � zj 0 dz′ j WDM(z′ j) � Pe(k, z′ j) jℓ(kχj) = � ℓ 2ℓ + 1 4π Pℓ(cos θ)Cij(ℓ) , (18) which defines the angular power spectrum Cij(ℓ) between the two fields i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' To calculate the electron power spectrum, we use HMX (Mead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2015, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Tröster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In order to carry out the sum over ℓ, we collect multipoles up to ℓ = 5 × 104 on the diagonal and for the other entries take up to ℓ = 100/|ˆxi − ˆx j| into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='3 Remarks on parameter dependence of the covariance Since the covariance in Equation (22) depends on cosmological parameters, it contains additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' There has been a long debate in the cosmological community whether it is necessary to account for this dependence or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Current LSS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022) or CMB measurements (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2020b) adjust the covariance interatively, that is they chose a fiducial cosmology, perform the inference for preliminary model parameters, update the covariance matrix to the preliminary best-fit model and start the inference again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This process is repeated until convergence is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Carron (2013) discussed the assumption of a parameter (in)dependent covariance matrices when two-point statistics are used as the model and data, showing that the (Gaussian) covariance matrix never carries any independent information (as it is again just a product of two-point functions) and is rather a sign of non-Gaussian information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In Reischke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2017) the overall parameter dependence of the cosmic shear two-point covariance was investigated with analytic methods and ray-tracing simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This work was followed up by Kodwani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2019), where the effect of a parameter dependent covariance matrix on the inference process with future LSS surveys was investigated and found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' However, one should keep in mind that these papers worked with averaged data and not simulated realisations of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The situation studied in this paper is different since the average DM - redshift relation only contains information about the cosmological background, while the correlations in the data are induced by the perturbations characterised by the electron power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Therefore the covariance matrix contains additional information without any double-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 3 RESULTS AND DISCUSSION In this section we present the results for the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We start by discussing some intermediate results for the angular power spectra in the (zi, zj)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Figure 1 shows the corresponding covariance for three different multipoles, ℓ = 2, 128, 1090, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The colour bar encodes the covariance in redshift at these fixed angular scales ℓ ∼ θ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' All covariances have a clear rectangular structure which stems from the integration bounds in Equation (18) reflecting the fact that the DM of two FRBs is only correlated for redshifts z ≤ min(zi, z j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Furthermore, the structure of the covariance also shows that on larger angular scales the correlation is stronger at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This can be understood by the fact that the Bessel function jℓ(kχ) peaks around kχ = ℓ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='5, thus small ℓ require small χ and hence z to reach the peak MNRAS 000, 1–7 (2022) Covariance matrix for located FRBs 5 of the power spectrum at k ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='01 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Lastly, we also note that the variance obtained from Equation (18), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' σ2 i = � ℓ (2ℓ + 1)Cii(ℓ)/(4π), (19) agrees well with the results from the empirical formula presented in (McQuinn 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021): p(∆) ∝ ∆−β exp �(∆−α − C0)2 2α2σ2 � , (20) with α = β = 3, ∆ = DMLSS/⟨DMLSS⟩ and the fitting values from N-body simulations presented in table 1 of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' At redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='1 we find 10 per-cent agreement with our analytical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='1 Current Data We now turn to current data using all FRBs from the FRB catalogue (Petroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2016) with host identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For illustrative purposes, we use them to fit the Hubble constant by putting a tight prior on the baryon density parameter Ωb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' There are more events available at the time of writing (James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022), but including a slightly larger sample does not affect the role of the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In Hagstotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2022) the value of the physical density parameter, ωb = Ωb0h2, as measured by big bang nucleosynthesis (Cooke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2018) was used, changing the overall scaling with h slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' With the approach followed here, one could think of the constraints just by looking at any linear amplitude parameter of the DM, Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In Figure 2 we show the covariance matrix on the left for the 9 host-identified FRBs from the FRBCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Clearly the variance is largest for the highest redshifts, the cross-covariance, however, is largest between FRB190102 and FRB190611 which are in close proximity on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Withal, the correlation coefficient is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The right panel shows the fit to the Hubble constant H0 = 100 h kms−1Mpc−1 for these 9 FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We assume a Gaussian likelihood χ2(θ) = log det C(θ) + (d − µ(θ))T C−1(θ) (d − µ(θ)) , (21) where we made the dependence on the parameters θ explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The covariance consists out of three contributions C = CLSS + CMW + Chost ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (22) and the components of CLSS are given by Equation (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' while we assume for the Milky Way CMW = σ2 MWI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' with σMW = 30 and the host Chost = σ2 hostI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' with σhost = 50/(1 + z) The results are shown on the right side of Figure 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' where the solid blue line denotes the posterior using the full covariance matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' while the assumption of independent events (taking only the diagonal of the covariance into account) leads to the dashed orange result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For the small sample size available right now, both approaches agree very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In this case, the parameter dependence of the covariance is also still negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' As we explain in the next section, this changes once the samples grow larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='2 Future Data In order to illustrate when the proper treatment of correlated errors in the FRB dispersion becomes important, we now generate synthetic samples containing a total number of NFRB FRBs distributed over redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For the redshift distribution, we assume a standard magnitude limited sample (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Reischke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2021): n(z) ∝ z2 exp(−zα) , (23) with α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Next, we draw random positions for each FRB uniformly over patches in the sky with sky fractions fsky = 1, 10−2 and 10−3, so that the effective number density is n = f −1 skyNFRB/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For this sample, we calculate the covariance matrix Equation (18) of the LSS component which in turn yields the final covariance via Equation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We used this full covariance matrix to sample the NFRB DM values for the generated events, completing the triples �DMi, ˆxi, zi � in our synthetic catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In Figure 3 we show the correlation coefficient rij = covij/(coviicov jj)1/2 for 500 FRBs distributed over different parts of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' While the covariance for a few hundred events distributed over the full sphere is dominated by the diagonal elements, the same number of FRBs distributed on a small fraction of the sky leads to a tight correlation due to the small angular separation of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The full covariance modelling is crucial for parameter estimation from larger FRB catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In Figure 4 we show the posterior of h from several synthetic catalogues of 500 events distributed over various fractions of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The catalogue is always generated using the true covariance matrix, and analysed using either the full covariance (blue solid) or only the diagonal (assuming uncorrelated events, orange dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Thick lines are showing the average over many realisations, while single realisations of the data and the corresponding inference are shown with shaded lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The assumption of uncorrelated DMs leads to a severe underestimation of the error by 40%, 60% and up to 85% for events covering either the full sky, or fsky = 10−2 and fsky = 10−3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' While a linear parameter cannot be biased on average, single realisations using the diagonal correlation matrix can easily show more than 3σ deviation from the true value used to generate the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In the lower panels of Figure 4 we show the effect of the additional cosmological information contained in the covariance of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We compare again inference using the full, parameter-dependent covariance matrix (solid blue) with the case of a fixed covariance matrix (dashed red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The width of the posterior shrinks by 30%, 45% and up to 70% depending on the sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In Figure 5 we demonstrate the influence of the covariance as a function of the number of observed FRBs, again for the same sky fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Note that the synthetic data used in Figure 4 is not necessarily the same as in Figure 5, but both are compatible with the full covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The solid line shows the maximum posterior values while the shaded areas correspond to the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' From the plots it is noticeable that the uncertainty on h is severely underestimated for NFRB ≥ 300 even for a full sky sample when using a diagonal MNRAS 000, 1–7 (2022) 6 Reischke & Hagstotz 0 100 200 300 400 0 100 200 300 400 fsky = 1 fsky = 10−3 −5 −4 −3 −2 −1 0 log10(rij) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Correlation coefficient, rij = covij/(coviicov jj)1/2, for 500 FRBs with host identification for a full sky (lower half) compared to the same sample only on a small subset fsky = 10−3 of the sky (upper half), where the correlation of the data points becomes much stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The number of events corresponds to n ≈ 5 × 10−3 deg−2 for the full sky sample, and n ≈ 5 deg−2 for the case of a small sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' posterior fsky = 1 full covij diagonal covii fsky = 10−2 fsky = 10−3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 ∆h/h posterior fsky = 1 covij(θ) covij(θ0) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 ∆h/h fsky = 10−2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 ∆h/h fsky = 10−3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Upper panels: Posterior distribution for various samples of 500 FRBs drawn with the respective covariance plotted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Solid blue lines use the full covariance, while dashed orange lines treat the FRBs as independent and just use the diagonal of the covariance matrix and underestimate the true error severely by 40%, 60% and 85% for the respective panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The x-axis shows the relative deviation from the fiducial value (used to generate the synthetic data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Thick lines denote the average effect over many realisations, and shaded lines show different realisations of the noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Single realisations analysed using diagonal covariance can lead to false parameter estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Lower panels: Posterior distributions either using a parameter-dependent covariance (solid blue) or a fixed covariance (dashed red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The cosmological dependence of the covariance matrix contains additional information, shrinking the error bars by 30%, 45% and 70% for the respective sky fractions compared to a covariance calculated at fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Although significant biases are unlikely to arise in this scenario, 3σ deviations from the true underlying value are possible if the covariance between events is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' For fsky = 10−3 these effects are already present for smaller NFRB and the error can be misestimated by up to 50 per-cent for NFRB as low as 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' While the last case is mostly of academical nature, selecting subsets of close by FRBs which are close by and ignoring there covariance might be dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We close this section with a short comparison with the approach used in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2020),Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2022) or James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' These works use a likelihood derived from the one-point probability distribution function of DMLSS and take into account the full non-Gaussianity of the DM distribution since it is measured directly from numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' While this captures the high DM tail of the distribution, the final likelihood is still dominated by the variance rather than its skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' On the other hand, it then is generally difficult to take the covariance between different FRBs into account since in principle an NFRB-point function is needed to obtain the accurate shape of the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Measuring all necessary moments from numerical simulations is inherently difficult due to the high noise in these estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Furthermore, it is challenging to include the parameter dependence in these approaches, since the numerical simulations are only evaluated at a single cosmology, although some of it has already been taken care of by looking at the relative DM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' compared to the background MNRAS 000, 1–7 (2022) Covariance matrix for located FRBs 7 200 400 600 800 1000 NFRB −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='15 ∆h/h fsky = 1 fsky = 1 fsky = 1 full covij diagonal covii 200 400 600 800 1000 NFRB fsky = 10−2 fsky = 10−2 fsky = 10−2 200 400 600 800 1000 NFRB fsky = 10−3 fsky = 10−3 fsky = 10−3 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Best fit values and 95% confidence interval (shaded bands) against the number of FRBs with host identification generated with a known redshift distribution for a full sky sample, fsky = 1 and fsky = 10−2, fsky = 10−3 from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Results from synthetic data analysed either using only the diagonal covariance (orange) or the full covariance including off-diagonal elements (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' The diagonal covariance underestimates the true error, so the inferred value of h is offset from the fiducial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' It would be interesting which effect is more important: the correlation or the high DM tail of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We refer this investigation to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 4 CONCLUSIONS In this paper we have investigated the impact of the LSS induced correlation between FRBs with host identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We have derived the covariance matrix in harmonic and real space for FRBs observed at redshift z and ˆxi position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This new covariance matrix was then used to reanalyse the FRBs from the FRB catalogue (Petroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2016) and to explore the influence on a single parameter, the Hubble constant h, measured from current and future samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Our main findings can be summarised as follows: (i) The number of current FRBs with host identification does not require the inclusion of the covariance between them as the statistical significance of the measurement is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Here we find similar results as Hagstotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (ii) For a full sky sample we find that the Hubble constant h or any other linear model parameter picks up an underestimated error of roughly 50 per-cent for 500 FRBs in the best case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In the worst case there can be significant biases for any single realisation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This situation becomes even more serious if the number of FRBs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (iii) If the parameter dependence of the covariance is not accounted for, biases can arise already for smaller numbers of FRBs in the case of partial sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We generally advise to take the dependence on the model parameters of the covariance (diagonal or not) into account, as it contains complementary information to the background dispersion measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' (iv) When small patches of the sky are observed (fsky = 10−3 or smaller) the influence of the full covariance can be seen already for NFRB = 40, leading to underestimated errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' We therefore conclude that the LSS covariance matrix of the DM of FRBs with host identification can become important in the future when more such FRBs (∼ 102) have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Here we only investigated isotropically distributed FRB samples over sky patches of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' In case of a more complex selection function the results found here might become more severe, but we leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Another issue is the inclusion of the non-Gaussian structure of the likelihood which in principle is naturally included in approaches using a formula fitted to numerical simulations (Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' These, however, lack the possibility to account for the correlations between the different FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' This approach is feasible at the moment, but will lead to errorneous conclusions in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Lastly, there are studies investigating the possibility to constrain reionization with FRBs (Heimersheim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' These studies, due to their high redshift FRBs would be much stringer affected by the covariance matrix due to the longer integration path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' Data Availability: The data and code underlying this article will be shared on request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' ACKNOWLEDGMENTS RR is supported by the European Research Council (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' 770935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=' SH was supported by the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} +page_content=', 2014, Physical Review D, 89, 107303 MNRAS 000, 1–7 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQf6wWy/content/2301.03527v1.pdf'} diff --git a/8dA0T4oBgHgl3EQfOv8t/content/tmp_files/2301.02164v1.pdf.txt b/8dA0T4oBgHgl3EQfOv8t/content/tmp_files/2301.02164v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5924d9321b424515e7273126626e34b2d7c54c2 --- /dev/null +++ b/8dA0T4oBgHgl3EQfOv8t/content/tmp_files/2301.02164v1.pdf.txt @@ -0,0 +1,763 @@ +Emergence of anyonic correlations from spin and charge dynamics in one dimension +Oleksandr Gamayun,1 Eoin Quinn,2 Kemal Bidzhiev,3 and Mikhail B. Zvonarev2 +1Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland +2Universit´e Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France +3PASQAL, 7 rue L´eonard de Vinci, 91300 Massy, France +(Dated: January 6, 2023) +We propose a transformation for spin and charge degrees of freedom in one-dimensional lattice +systems, constrained to have no doubly occupied sites, that allows direct access to the dynamical +correlations of the system. The transformation delivers particle creation and annihilation operators +in a form of a spinless particle and a non-local operator acting on the space of states of a spin- +1/2 chain. This permits a decomposition of dynamical correlation functions as a convolution of +those for impenetrable anyons together with those of a spin chain. Further analysis can be done by +methods tailored for each part of the convolution, greatly increasing the impact and flexibility of +the approach. +The physics of many-body quantum systems incorpo- +rates effects from interaction and statistics of bare par- +ticles. +The emerging quasi-particles could inherit the +statistics of their non-interacting peers, free fermions +turning into a Fermi liquid, and free bosons into a Bose- +Einstein condensate. Reducing a system’s dimensionality +enhances interaction effects and masks out signatures of +the statistics of the constitutent particles. +In one di- +mension, arbitrarily weak repulsion precludes a macro- +scopic occupation of a single state with the zero momen- +tum, that is, destroys the Bose-Einstein condensate [1]. +Furthermore, interactions may transform bosonic excita- +tion spectrum into a fermionic one, an example being the +bosons repelling each other through a δ-function poten- +tial of infinite strength, the system known as the Tonks- +Girardeau gas, whose excitation spectrum is identical to +that of a free Fermi gas [2]. +The interplay of spin and charge degrees of freedom +could be particularly intricate in one dimension. Systems +having linear excitation spectrum at low energies fall into +a Luttinger liquid (LL) universality class regardless of the +statistics of the bare particles. Spin and charge degrees +of freedom of the microscopic theory are represented by +commuting terms in the LL Hamiltonian and factor out +in the dynamical correlation functions, the phenomenon +referred to as spin-charge separation [3, 4]. Accounting +for non-linearities of the excitation spectrum within the +effective field theory approach requires proper modifica- +tion of the LL description, the cases studied recently +being spin and charge dynamics above the highly de- +generate ground state (spin-incoherent regime, Refs. [5– +7]), in presence of the quadratic branch of the excita- +tion spectrum (itinerant ferromagnetic regime, Refs. [8– +12]), and in the vicinity of the edge of excitation spec- +trum, Ref. [13]. +Whether and how the concept of the +spin-charge separation may be extended beyond the LL +effective field theory description is a challenging open +question, relevant, in particular, for ultracold gas exper- +iments [14]. +Studying systems with no double occupancy (NDO) +constraint (any two particles cannot occupy the same +lattice site) is a must for understanding how spin and +charge degrees of freedom are coupled at all energy scales. +Disregarding the unoccupied sites (“squeezing” the lat- +tice) reduces the space of states of the original system +containing N spin-1/2 particles to the space of states +of the spin-1/2 chain of length N. +The state of indi- +vidual spins on the squeezed lattice could be controlled +and manipulated directly by ultracold quantum gas mi- +croscopy [15–17]. On the theory side, some dynamical +correlation functions have been evaluated by making use +of the coordinate representation for the many body wave +functions, whose structure is very special due to the NDO +constraint [18–21]. The formalism of the second quanti- +zation, expressing basic microscopic fields of the system +in terms of the collective spin and charge variables, could +serve as a systemic approach revealing contributions from +spin and charge dynamics into any correlation function. +However, such a formalism has not been developed so far. +In this Letter we present a transformation from the +spin-1/2 fermions subjected to the NDO constraint to the +collective charge (spinless fermions on a lattice) and spin +(spin-1/2 operators on another lattice) variables. These +collective charge and spin variables commute with each +other, and enter into the transformation in a highly non- +local way, as shown in Eqs. (9)–(12). Being used for cor- +relation functions, the transformation leads to the charge +dynamics of the impenetrable anyons, whose statistical +angle is averaged out with the weight function defined by +spin configurations. +Transformation to spin and charge variables.— We +consider spin-1/2 fermions on an infinite one-dimensional +lattice. There, ˆψ† +jα, ˆψjα, and ˆnjα = ˆψ† +jα ˆψjα are the cre- +ation, annihilation, and the particle number operators for +a site j (−∞ ≤ j ≤ ∞), and α =↑, ↓ is the spin index. +The local spin vector ˆs(j) = (ˆsx(j), ˆsy(j), ˆsz(j)) can be +represented as +ˆs(j) = 1 +2 +� +ˆψ† +j↑ +ˆψ† +j↓ +� +σ +� ˆψj↑ +ˆψj↓ +� +, +(1) +arXiv:2301.02164v1 [cond-mat.quant-gas] 5 Jan 2023 + +2 +where σ = (σx, σy, σz) is the vector composed of the +three Pauli matrices. The spin-ladder operators ˆs±(j) = +ˆsx(j) ± iˆsy(j) read ˆs+(j) = ˆψ† +j↑ ˆψj↓ and ˆs−(j) = ˆψ† +j↓ ˆψj↑, +respectively. +We require the total number of fermions +in the system, ˆN = � +j ˆnj, to be a conserved quantity. +There could only be either zero or one fermion on each +site, +ˆnj ≡ ˆnj↑ + ˆnj↓ = {0, 1}, +(2) +due to the NDO constraint. The projection operator +ˆ +X = +∞ +� +j=−∞ +(1 − ˆnj↑ˆnj↓) +(3) +applied to the basis state |Ψ⟩ = ˆψ† +j1α1 · · · ˆψ† +jNαN |0⟩ elimi- +nates those with any number of double occupancies. The +remaining ones can be uniquely identified as a product +of the states |f⟩ and |ℓ⟩: +|Ψ⟩ = |f⟩ ⊗ |ℓ⟩. +(4) +Here, |f⟩ = ˆc† +j1 · · · ˆc† +jN |0⟩ is defined by spinless fermions +on an infitite lattice placed at the positions of the origi- +nal spin-1/2 fermions. The vacuum |0⟩ for the states |Ψ⟩ +and |f⟩ contains no fermions, ˆψj|0⟩ = 0, and ˆcj|0⟩ = 0, +respectively. +The state |ℓ⟩ = |α1 · · · αN⟩ of a spin- +1/2 chain of length N can be represented as |ℓ⟩ = +ˆℓ−(m1) · · · ˆℓ−(mM)| ⇑⟩. The set {m1, . . . , mM} indicates +the positions of the down-spins among {α1, . . . , αN}, M +being the total number of the down-spins. +For exam- +ple, | ↑↓↑↓↓⟩ gives {m1, m2, m3} = {2, 4, 5}. The vac- +uum | ⇑⟩ is the spin-up polarized state. The operator +ˆℓ(m) = σ(m)/2 acts on the spin state of the mth parti- +cle, and ˆℓ± = ˆℓx ± iˆℓy. +We now express spin-1/2 fermion fields via operators +acting into the spaces formed by |f⟩ and |ℓ⟩. The number +of particles to the left from the jth site is +ˆ +Nj = +j +� +a=−∞ +ˆna. +(5) +Here, ˆnj = ˆc† +jˆcj acting onto |f⟩ corresponds to ˆnj defined +by Eq. (2), acting onto |Ψ⟩. Note that the spectrum of the +operator ˆ +Nj is integer-valued. Any operator ˆO depending +on ˆ +Nj can be understood by the following formula: +ˆO( ˆ +Nj) = +∞ +� +m=−∞ +ˆO(m)δm, ˆ +Nj. +(6) +The operator ˆO(m) characterizes the state of mth parti- +cle, and the Kronecker delta +δm, ˆ +Nj = +� 2π +0 +dλ +2π eiλ( ˆ +Nj−m) +(7) +1 +N + 1 +m′ +Pm′,N+1 +N + 1 +m +PN+1,m +N + 1 +m′ − 1 +PN+1,mPm′,N+1 +FIG. 1. +Shown is the action of the operator P onto the states +of the spin chain. The arrows indicate the directions of the +transfer of the local states. The outcome of the action of the +composition PN+1,mPm′,N+1 is illustrated for m′ > m. +is equal to one for the lattice site at which the mth par- +ticle is located, and is equal to zero otherwise. The com- +position law +ˆO1( ˆ +Nj) ˆO2( ˆ +Nj) = +∞ +� +m=−∞ +ˆO1(m) ˆO2(m)δm, ˆ +Nj +(8) +stems directly from Eqs. (6) and (7). +We propose the following expressions for the fermion +creation operators +ˆψ† +j↑ =P ˆ +Nj, ˆ +Nˆc† +j, +(9) +ˆψ† +j↓ =P ˆ +Nj, ˆ +N ˆℓ−( ˆN)ˆc† +j. +(10) +and the corresponding annihilation operators +ˆψj↑ =ˆcj ˆη( ˆN)P† +ˆ +Nj, ˆ +N, +(11) +ˆψj↓ =ˆcj ˆℓ+( ˆN)P† +ˆ +Nj, ˆ +N. +(12) +The operator ˆη = ˆℓ+ˆℓ− = | ↑⟩⟨↑ | in Eq. (11) acts on the +site of the spin chain defined by the value of the number +operator ˆN. A way to interpret the dependence on ˆ +Nj is +explained by Eqs. (6) and (7). The cyclic shift operator +Pm,m′ on a lattice encompassing the sites from m to m′ +is +Pm,m′ = Πm,m+1Πm+1,m+2 · · · Πm′−1,m′. +(13) +The permutation operator Πm,m′ interchanges the states +on the sites m and m′, in case of spin-1/2 particles it +reads +Πm,m′ = 1 +2[σ(m) ⊗ σ(m′) + I ⊗ I]. +(14) +Here, I is the identity matrix. Evidently, Π is its own +inverse, (Πm,m′)2 = I, Hermitian, Π† +m,m′ = Πm,m′, +and unitary, Π† +m,m′Πm,m′ = I. +This implies Pm′,m = +P−1 +m,m′ = P† +m,m′. The action of the operator (13) onto the +states of the spin chain is illustrated in Fig. 1. Note that + +3 +the local spin operator (1) consists of the pairs ˆψ† +jα ˆψjα′ +where ˆψ† and ˆψ are taken at the same site j. As a con- +sequence, the permutation operators cancels out when +using Eqs. (9)–(12), leading to the representation +ˆs(j) = ˆnjˆℓ( ˆ +Nj) +(15) +already known in the literature [12]. +We demonstrate +how efficacious are Eqs. (9)–(12) in revealing the contri- +butions from the spin and charge degrees of freedom into +the dynamical correlation functions in the remaining part +of the Letter. +Hamiltonian.— We apply the transformations (9)–(12) +to the Hamiltonian +ˆH = ˆHf + ˆHℓ, +(16) +where +ˆHf = ˆ +X +� +���−th +∞ +� +j=−∞ +α=↑,↓ +( ˆψ† +jα ˆψj+1α + H.c.) − h ˆN ++1 +2 +∞ +� +jj′=−∞ +: ˆnjUj−j′ ˆnj′ : +� +��� ˆ +X +(17) +is SU(2)-invariant, and the term +ˆHℓ = 2B ˆ +X ˆSz ˆ +X, +ˆSz = +∞ +� +j=−∞ +ˆsz(j) +(18) +breaks this symmetry due to the magnetic field B applied +along the z-projection of the total spin. The symbols H.c. +and : · · · : in Eq. (17) stand for the Hermitian conjugate +and the normal ordering, respectively. The projection op- +erator ˆ +X, given by Eq. (3), imposes the NDO constraint. +Note that the on-site interaction term : ˆn2 +j : U0/2 im- +plies an infinite energy cost for having two particles on +any site in the U0 → ∞ limit. This way, the use of ˆ +X +is equivalent to letting U0 → ∞ in the Hamiltonian (16) +with no ˆ +X. The actual value of U0 is irrelevant when ˆ +X +is used, since ˆ +X : ˆn2 +j : ˆ +X = 0. +Using the transformation (9)–(12) we get Eq. (17) writ- +ten in terms of the spinless fermions exclusively, +ˆHf = −th +∞ +� +j=−∞ +(ˆc† +jˆcj+1 + H.c.) − h ˆN ++ 1 +2 +∞ +� +j,j′=−∞ +: ˆnjUj−j′ ˆnj′ : +(19) +and Eq. (18) containing the spinless fermions as well as +the spin operators, +ˆHℓ = 2B +∞ +� +j=−∞ +ˆnj ˆℓz( ˆ +Nj). +(20) +Amazingly, the action of ˆHf ( ˆHℓ) onto the state (4) is +non-trivial for the |f⟩ (|ℓ⟩) part only: +ˆHf|Ψ⟩ = Ef|f⟩ ⊗ |ℓ⟩, +ˆHℓ|Ψ⟩ = |f⟩ ⊗ Eℓ|ℓ⟩. +(21) +The energy Eℓ = 2BLz, where Lz is the eigenvalue of the +operator ˆLz = �N +m=1 ˆℓz(m), measuring the z-projection +of the total spin for the state |ℓ⟩ of the spin chain. Hence, +the spin degeneracy of the Hamiltonian (16) takes place +for any Lz ̸= ±N/2. Furthermore, ˆHℓ = 0 for B = 0, +implying 2N-fold degeneracy as long as the system is not +put into a finite volume with some boundary conditions. +Field-field correlation functions in thermal state.—We +consider the one-body correlation functions, describing +the particle propagation, +Gα +p (j − j′, t) = 1 +Z ⟨ ˆψjα(t) ˆψ† +j′α(0)⟩T , +α =↑, ↓, +(22) +and the hole propagation, +Gα +h(j − j′, t) = 1 +Z ⟨ ˆψ† +jα(t) ˆψj′α(0)⟩T , +α =↑, ↓, +(23) +evaluated at temperature T, chemical potential h, and +magnetic field B, on a thermals state +⟨· · · ⟩T = +∞ +� +N=0 +� +f,ℓ +⟨Ψ|e−β ˆ +H · · · |Ψ⟩, +(24) +where |Ψ⟩ is given by Eq. (4). +The sum over f runs +through all possible values of the free-particle momenta +characterizing the N-fermion state |f⟩. +The sum over +ℓ runs through all possible configurations of the z- +projection of the spins, Z is the grand partition function, +and β = T −1. The symmetry +G↑ +p(h)(j − j′, t; h, B) = G↓ +p(h)(j − j′, t; h, −B) +(25) +makes it sufficient to evaluate G↑ only. +Using Eqs. (6)–(12) we factorize the matrix element +from Eq. (22) into two parts, +⟨Ψ| ˆψj↑(t) ˆψ† +j′↑(0)|Ψ⟩ = +∞ +� +m,m′=−∞ +� 2π +0 +dλ +2π +dλ′ +2π +e−iλm+iλ′m′e−β(Ef +Eℓ)Cp(λ, λ′; j − j′; t)S(m, m′). +(26) +The first one encompasses the contributions from the +state |f⟩ of spinless fermions, +Cp(λ, λ′; j − j′; t) = ⟨f|ˆcj(t)eiλ ˆ +Nj(t)e−iλ′ ˆ +Nj′(0)ˆc† +j′|f⟩. +(27) +Its non-trivial time evolution is governed by the Hamil- +tonian (19). The second one involves the state |ℓ⟩ of the +spin chain, and the existence of the free fermions is only +noticed through their total number N, which defines the + +4 +length of the chain, +S(m, m′) = ⟨ℓ|PN+1,mPm′,N+1|ℓ⟩ += ⟨ℓ| +max{m,m′}−1 +� +j=min{m,m′} +[1 +2I + ˆℓz(j)]|ℓ⟩. +(28) +This part is time-independent, since the cyclic shift op- +erator, Eq. (13) does not change the value of the z- +projection of the total spin, Lz. The action of the oper- +ator PN+1,mPm′,N+1, illustrated in Fig. 1, leads to van- +ishing S if any spin between the sites m and m′ is pointed +down. This way we get the right hand side of Eq. (28). +We proceed further by substituting Eq. (28) into +Eq. (22) and taking the sum over the spin configurations, +� +ℓ +e−βEℓS(m, m′) = [2 cosh(βB)]N +ν|m−m′| +, +(29) +where ν = 1 + e2βB. We get +G↑ +p(j − j′, t) = 1 +Z +� +{N} +e−β ˜ +Ef +� 2π +0 +dλ +2π +dλ′ +2π +× Cp(λ, λ′; j − j′; t) +∞ +� +m,m′=−∞ +e−iλm+iλ′m′ +ν|m−m′| +, +(30) +where +˜Ef = Ef − 1 +β N ln[2 cosh(βB)], +(31) +and the sum over {N} encompasses the ones over N +and f. +The partition function Z can be taken over +the fermion configurations f with the energies given by +Eq. (31). We have +∞ +� +m,m′=−∞ +e−iλm+iλ′m′ +ν|m−m′| += 2πδ(λ − λ′)F(λ; T), +(32) +where +F(λ; ν) = 1 + +∞ +� +m=1 +ν−m(eimλ + e−imλ). +(33) +Therefore, +G↑ +p(j − j′, t) = +� 2π +0 +dλ +2π F(λ; ν)Cp(λ; j − j′; t; T), +(34) +where +Cp(λ; j − j′; t; T) = 1 +Z +� +{N} +e−β ˜ +Ef Cp(λ; j − j′; t), +(35) +and we write Cp(λ) in place of Cp(λ, λ) in order to lighten +notations. +The summation on the right hand side of +Eq. (35) represents the definition of the thermal state +for the spinless fermions with the spectum given by ˜Ef. +The hole correlation function (23) is treated the same +way as the particle one. The result is given by Eqs. (34) +and (35) with Cp replaced by +Ch(λ; j − j′; t) = ⟨f|eiλ ˆ +Nj(t)ˆc† +j(t)ˆcj′e−iλ ˆ +Nj′(0)|f⟩. +(36) +Emergence of impenetrable anyons.— The operator +ˆaj = ˆcje−iλ ˆ +Nj satisfies the commutation relations +ˆajˆa† +j′ + e−iλϵ(j−j′)ˆa† +j′ˆaj = δjj′, +(37) +ˆajˆaj′ + eiλϵ(j−j′)ˆaj′ˆaj = 0, +(38) +where ϵ(x) = |x|/x, and ϵ(0) = 0. This is the fermion- +anyon mapping discussed in Ref. [22]. The function Cp(λ) +turns into +Cp(−λ; j − j′; t) = ⟨f|ˆaj(t)ˆa† +j′(0)|f⟩, +(39) +which is a correlation function of the impenetrable +anyons on a lattice, the variable λ being the statistical +angle. +The emergence of the anyon correlation function and +its subsequent integration over λ with the function F in +Eq. (34) could be understood as follows. Let us consider +a system with M spin-up and N − M spin-down par- +ticles. Pick one spin-up particle among them, and pull +it through the whole system, subsequently interchanging +its coordinate with those of the other particles. The in- +terchanges with the spin-down particles are non-trivial: +the spin part of the wave function could give any phase +factor since its symmetry is not restricted by the fermion +symmetry of the total wave function. We stress that for- +malizing our a posteriori explanation of the structure of +Eq. (34) by examining exact finite-N wave functions in +the coordinate representations (given, for example, in the +Refs. [21, 23]) goes beyond the scope of the Letter. +Place +among +other +approaches.— +The +Hamilto- +nian (16) with Uj−j′ = 0 represents the exactly solvable +t − 0 model, also known as the Hubbard model in the +limit of infinitely strong repulsion [24]. There, Eq. (34) +has been obtained in the form of a Fredholm determinant +with the use of the exact wave functions in the coordi- +nate representation [21, 23, 25]. The transformation (9)– +(12) leading to Eq. (34), combined with the ones given +in Ref. [26] for the function (27) bring us the same Fred- +holm determinant representation through much shorter +calculations. +Note that the model (16) is also exactly +solvable when Uj−j′ = Uδj,j′±1. In this case, the Hamil- +tonian (19) can be mapped onto the one of the XXZ +Heisenberg magnet, and the function (27) can, in princi- +ple, be calculated by the Bethe Ansatz method. +Special attention has been paid in the literature to +the model in the T → 0 limit. Its ground state is non- +degenerate and spin-up (-down) polarized for B negative + +5 +(positive). In the former case, Eq. (34) describes a spin- +up fermion propagating through a gas of the other spin- +up fermions. We have F = 2πδ(λ) in Eq. (33), hence +G↑ +p = ⟨ˆcj(t)ˆc† +j′⟩. In the latter case, Eq. (34) describes +a spin-up fermion (an impurity particle) propagating +through a gas of spin-down fermions. We have F = 1, +and the long time and distance asymptotic behaviour of +G↑ +p reveals the logarithmic diffusion phenomenon [8, 9]. +The non-degeneracy of the ground state at B ̸= 0 stands +in a sharp contrast to the high degeneracy at B = 0, +where F is given by Eq. (33) with ν = 2. This regime is +known as the spin-incoherent one [5–7]. A challenge put +forward in the aforementioned works was to find a low- +energy effective field theory, since the low-enegry spec- +trum of spin excitations cannot be linearized for B > 0 +and B = 0, and the LL theory is inapplicable. The repre- +sentation (34) resolves this problem in the following way: +the LL theory in applicable to the function Cp; the spin +excitations are accounted for by the integral over λ with +the weight function F without any approximation, which +is equivalent to counting the number of worldlines within +the first-quantized path integral approach implemented +in Refs. [6, 8]. +ACKNOWLEDGEMENTS +We thank V. Cheianov and K. Seetharam for fruitful dis- +cussions. O.G. acknowledges support from the Polish Na- +tional Agency for Academic Exchange (NAWA) through +the Grant No. PPN/ULM/2020/1/00247. O.G. is grate- +ful to Galileo Galilei Institute for hospitality and support +during the scientific program on “Randomness, Integra- +bility, and Universality”, where part of this work was +done. The work of E. Q. is supported by Grant No. ANR- +16-CE91-0009-01. +K.B. thanks S. Bocini, V. Mari´c, +L. Zadnik and M. Fagotti for useful discussions. +The +work of K.B. was partially supported by the European +Research Council under the Starting Grant No. 805252 +LoCoMacro. The work of M. B. Z. is supported by Grant +No. ANR-16-CE91-0009-01 and CNRS grant PICS06738. +M. B. Z. acknowledges Russian Quantum Center and +Prof. A. Fedorov for their hospitality during the work. +[1] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa- +tion (Oxford University Press, Oxford, 2003). +[2] M. Girardeau, Relationship between systems of impene- +trable bosons and fermions in one dimension, J. Math. +Phys. 1, 516 (1960). +[3] A. O. Gogolin, A. A. Nersesyan, and A. M. 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Stat. +Mech. 2009, P07035 (2009), arXiv:0812.4059. + diff --git a/8dA0T4oBgHgl3EQfOv8t/content/tmp_files/load_file.txt b/8dA0T4oBgHgl3EQfOv8t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f87072a194a3c790aaea700707e24407825af52 --- /dev/null +++ b/8dA0T4oBgHgl3EQfOv8t/content/tmp_files/load_file.txt @@ -0,0 +1,439 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf,len=438 +page_content='Emergence of anyonic correlations from spin and charge dynamics in one dimension Oleksandr Gamayun,1 Eoin Quinn,2 Kemal Bidzhiev,3 and Mikhail B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Zvonarev2 1Faculty of Physics, University of Warsaw, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Pasteura 5, 02-093 Warsaw, Poland 2Universit´e Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France 3PASQAL, 7 rue L´eonard de Vinci, 91300 Massy, France (Dated: January 6, 2023) We propose a transformation for spin and charge degrees of freedom in one-dimensional lattice systems, constrained to have no doubly occupied sites, that allows direct access to the dynamical correlations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The transformation delivers particle creation and annihilation operators in a form of a spinless particle and a non-local operator acting on the space of states of a spin- 1/2 chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' This permits a decomposition of dynamical correlation functions as a convolution of those for impenetrable anyons together with those of a spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Further analysis can be done by methods tailored for each part of the convolution, greatly increasing the impact and flexibility of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The physics of many-body quantum systems incorpo- rates effects from interaction and statistics of bare par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The emerging quasi-particles could inherit the statistics of their non-interacting peers, free fermions turning into a Fermi liquid, and free bosons into a Bose- Einstein condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Reducing a system’s dimensionality enhances interaction effects and masks out signatures of the statistics of the constitutent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' In one di- mension, arbitrarily weak repulsion precludes a macro- scopic occupation of a single state with the zero momen- tum, that is, destroys the Bose-Einstein condensate [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Furthermore, interactions may transform bosonic excita- tion spectrum into a fermionic one, an example being the bosons repelling each other through a δ-function poten- tial of infinite strength, the system known as the Tonks- Girardeau gas, whose excitation spectrum is identical to that of a free Fermi gas [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The interplay of spin and charge degrees of freedom could be particularly intricate in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Systems having linear excitation spectrum at low energies fall into a Luttinger liquid (LL) universality class regardless of the statistics of the bare particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Spin and charge degrees of freedom of the microscopic theory are represented by commuting terms in the LL Hamiltonian and factor out in the dynamical correlation functions, the phenomenon referred to as spin-charge separation [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Accounting for non-linearities of the excitation spectrum within the effective field theory approach requires proper modifica- tion of the LL description, the cases studied recently being spin and charge dynamics above the highly de- generate ground state (spin-incoherent regime, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [5– 7]), in presence of the quadratic branch of the excita- tion spectrum (itinerant ferromagnetic regime, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [8– 12]), and in the vicinity of the edge of excitation spec- trum, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Whether and how the concept of the spin-charge separation may be extended beyond the LL effective field theory description is a challenging open question, relevant, in particular, for ultracold gas exper- iments [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Studying systems with no double occupancy (NDO) constraint (any two particles cannot occupy the same lattice site) is a must for understanding how spin and charge degrees of freedom are coupled at all energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Disregarding the unoccupied sites (“squeezing” the lat- tice) reduces the space of states of the original system containing N spin-1/2 particles to the space of states of the spin-1/2 chain of length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The state of indi- vidual spins on the squeezed lattice could be controlled and manipulated directly by ultracold quantum gas mi- croscopy [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' On the theory side, some dynamical correlation functions have been evaluated by making use of the coordinate representation for the many body wave functions, whose structure is very special due to the NDO constraint [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The formalism of the second quanti- zation, expressing basic microscopic fields of the system in terms of the collective spin and charge variables, could serve as a systemic approach revealing contributions from spin and charge dynamics into any correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' However, such a formalism has not been developed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' In this Letter we present a transformation from the spin-1/2 fermions subjected to the NDO constraint to the collective charge (spinless fermions on a lattice) and spin (spin-1/2 operators on another lattice) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' These collective charge and spin variables commute with each other, and enter into the transformation in a highly non- local way, as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (9)–(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Being used for cor- relation functions, the transformation leads to the charge dynamics of the impenetrable anyons, whose statistical angle is averaged out with the weight function defined by spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Transformation to spin and charge variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='— We consider spin-1/2 fermions on an infinite one-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' There, ˆψ† jα, ˆψjα, and ˆnjα = ˆψ† jα ˆψjα are the cre- ation, annihilation, and the particle number operators for a site j (−∞ ≤ j ≤ ∞), and α =↑, ↓ is the spin index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The local spin vector ˆs(j) = (ˆsx(j), ˆsy(j), ˆsz(j)) can be represented as ˆs(j) = 1 2 � ˆψ† j↑ ˆψ† j↓ � σ � ˆψj↑ ˆψj↓ � , (1) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='02164v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='quant-gas] 5 Jan 2023 2 where σ = (σx, σy, σz) is the vector composed of the three Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The spin-ladder operators ˆs±(j) = ˆsx(j) ± iˆsy(j) read ˆs+(j) = ˆψ† j↑ ˆψj↓ and ˆs−(j) = ˆψ† j↓ ˆψj↑, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We require the total number of fermions in the system, ˆN = � j ˆnj, to be a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' There could only be either zero or one fermion on each site, ˆnj ≡ ˆnj↑ + ˆnj↓ = {0, 1}, (2) due to the NDO constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The projection operator ˆ X = ∞ � j=−∞ (1 − ˆnj↑ˆnj↓) (3) applied to the basis state |Ψ⟩ = ˆψ† j1α1 · · · ˆψ† jNαN |0⟩ elimi- nates those with any number of double occupancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The remaining ones can be uniquely identified as a product of the states |f⟩ and |ℓ⟩: |Ψ⟩ = |f⟩ ⊗ |ℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (4) Here, |f⟩ = ˆc† j1 · · · ˆc† jN |0⟩ is defined by spinless fermions on an infitite lattice placed at the positions of the origi- nal spin-1/2 fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The vacuum |0⟩ for the states |Ψ⟩ and |f⟩ contains no fermions, ˆψj|0⟩ = 0, and ˆcj|0⟩ = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The state |ℓ⟩ = |α1 · · · αN⟩ of a spin- 1/2 chain of length N can be represented as |ℓ⟩ = ˆℓ−(m1) · · · ˆℓ−(mM)| ⇑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The set {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' , mM} indicates the positions of the down-spins among {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' , αN}, M being the total number of the down-spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' For exam- ple, | ↑↓↑↓↓⟩ gives {m1, m2, m3} = {2, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The vac- uum | ⇑⟩ is the spin-up polarized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The operator ˆℓ(m) = σ(m)/2 acts on the spin state of the mth parti- cle, and ˆℓ± = ˆℓx ± iˆℓy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We now express spin-1/2 fermion fields via operators acting into the spaces formed by |f⟩ and |ℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The number of particles to the left from the jth site is ˆ Nj = j � a=−∞ ˆna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (5) Here, ˆnj = ˆc† jˆcj acting onto |f⟩ corresponds to ˆnj defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (2), acting onto |Ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Note that the spectrum of the operator ˆ Nj is integer-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Any operator ˆO depending on ˆ Nj can be understood by the following formula: ˆO( ˆ Nj) = ∞ � m=−∞ ˆO(m)δm, ˆ Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (6) The operator ˆO(m) characterizes the state of mth parti- cle, and the Kronecker delta δm, ˆ Nj = � 2π 0 dλ 2π eiλ( ˆ Nj−m) (7) 1 N + 1 m′ Pm′,N+1 N + 1 m PN+1,m N + 1 m′ − 1 PN+1,mPm′,N+1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Shown is the action of the operator P onto the states of the spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The arrows indicate the directions of the transfer of the local states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The outcome of the action of the composition PN+1,mPm′,N+1 is illustrated for m′ > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' is equal to one for the lattice site at which the mth par- ticle is located, and is equal to zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The com- position law ˆO1( ˆ Nj) ˆO2( ˆ Nj) = ∞ � m=−∞ ˆO1(m) ˆO2(m)δm, ˆ Nj (8) stems directly from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We propose the following expressions for the fermion creation operators ˆψ† j↑ =P ˆ Nj, ˆ Nˆc† j, (9) ˆψ† j↓ =P ˆ Nj, ˆ N ˆℓ−( ˆN)ˆc† j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (10) and the corresponding annihilation operators ˆψj↑ =ˆcj ˆη( ˆN)P† ˆ Nj, ˆ N, (11) ˆψj↓ =ˆcj ˆℓ+( ˆN)P† ˆ Nj, ˆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (12) The operator ˆη = ˆℓ+ˆℓ− = | ↑⟩⟨↑ | in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (11) acts on the site of the spin chain defined by the value of the number operator ˆN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' A way to interpret the dependence on ˆ Nj is explained by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The cyclic shift operator Pm,m′ on a lattice encompassing the sites from m to m′ is Pm,m′ = Πm,m+1Πm+1,m+2 · · · Πm′−1,m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (13) The permutation operator Πm,m′ interchanges the states on the sites m and m′, in case of spin-1/2 particles it reads Πm,m′ = 1 2[σ(m) ⊗ σ(m′) + I ⊗ I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (14) Here, I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Evidently, Π is its own inverse, (Πm,m′)2 = I, Hermitian, Π† m,m′ = Πm,m′, and unitary, Π† m,m′Πm,m′ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' This implies Pm′,m = P−1 m,m′ = P† m,m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The action of the operator (13) onto the states of the spin chain is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Note that 3 the local spin operator (1) consists of the pairs ˆψ† jα ˆψjα′ where ˆψ† and ˆψ are taken at the same site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' As a con- sequence, the permutation operators cancels out when using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (9)–(12), leading to the representation ˆs(j) = ˆnjˆℓ( ˆ Nj) (15) already known in the literature [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We demonstrate how efficacious are Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (9)–(12) in revealing the contri- butions from the spin and charge degrees of freedom into the dynamical correlation functions in the remaining part of the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='— We apply the transformations (9)–(12) to the Hamiltonian ˆH = ˆHf + ˆHℓ, (16) where ˆHf = ˆ X � ���−th ∞ � j=−∞ α=↑,↓ ( ˆψ† jα ˆψj+1α + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=') − h ˆN +1 2 ∞ � jj′=−∞ : ˆnjUj−j′ ˆnj′ : � ��� ˆ X (17) is SU(2)-invariant, and the term ˆHℓ = 2B ˆ X ˆSz ˆ X, ˆSz = ∞ � j=−∞ ˆsz(j) (18) breaks this symmetry due to the magnetic field B applied along the z-projection of the total spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The symbols H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' and : · · · : in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (17) stand for the Hermitian conjugate and the normal ordering, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The projection op- erator ˆ X, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (3), imposes the NDO constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Note that the on-site interaction term : ˆn2 j : U0/2 im- plies an infinite energy cost for having two particles on any site in the U0 → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' This way, the use of ˆ X is equivalent to letting U0 → ∞ in the Hamiltonian (16) with no ˆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The actual value of U0 is irrelevant when ˆ X is used, since ˆ X : ˆn2 j : ˆ X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Using the transformation (9)–(12) we get Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (17) writ- ten in terms of the spinless fermions exclusively, ˆHf = −th ∞ � j=−∞ (ˆc† jˆcj+1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=') − h ˆN + 1 2 ∞ � j,j′=−∞ : ˆnjUj−j′ ˆnj′ : (19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (18) containing the spinless fermions as well as the spin operators, ˆHℓ = 2B ∞ � j=−∞ ˆnj ˆℓz( ˆ Nj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (20) Amazingly, the action of ˆHf ( ˆHℓ) onto the state (4) is non-trivial for the |f⟩ (|ℓ⟩) part only: ˆHf|Ψ⟩ = Ef|f⟩ ⊗ |ℓ⟩, ˆHℓ|Ψ⟩ = |f⟩ ⊗ Eℓ|ℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (21) The energy Eℓ = 2BLz, where Lz is the eigenvalue of the operator ˆLz = �N m=1 ˆℓz(m), measuring the z-projection of the total spin for the state |ℓ⟩ of the spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Hence, the spin degeneracy of the Hamiltonian (16) takes place for any Lz ̸= ±N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Furthermore, ˆHℓ = 0 for B = 0, implying 2N-fold degeneracy as long as the system is not put into a finite volume with some boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Field-field correlation functions in thermal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='—We consider the one-body correlation functions, describing the particle propagation, Gα p (j − j′, t) = 1 Z ⟨ ˆψjα(t) ˆψ† j′α(0)⟩T , α =↑, ↓, (22) and the hole propagation, Gα h(j − j′, t) = 1 Z ⟨ ˆψ† jα(t) ˆψj′α(0)⟩T , α =↑, ↓, (23) evaluated at temperature T, chemical potential h, and magnetic field B, on a thermals state ⟨· · · ⟩T = ∞ � N=0 � f,ℓ ⟨Ψ|e−β ˆ H · · · |Ψ⟩, (24) where |Ψ⟩ is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The sum over f runs through all possible values of the free-particle momenta characterizing the N-fermion state |f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The sum over ℓ runs through all possible configurations of the z- projection of the spins, Z is the grand partition function, and β = T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The symmetry G↑ p(h)(j − j′, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' h, B) = G↓ p(h)(j − j′, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' h, −B) (25) makes it sufficient to evaluate G↑ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (6)–(12) we factorize the matrix element from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (22) into two parts, ⟨Ψ| ˆψj↑(t) ˆψ† j′↑(0)|Ψ⟩ = ∞ � m,m′=−∞ � 2π 0 dλ 2π dλ′ 2π e−iλm+iλ′m′e−β(Ef +Eℓ)Cp(λ, λ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t)S(m, m′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (26) The first one encompasses the contributions from the state |f⟩ of spinless fermions, Cp(λ, λ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t) = ⟨f|ˆcj(t)eiλ ˆ Nj(t)e−iλ′ ˆ Nj′(0)ˆc† j′|f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (27) Its non-trivial time evolution is governed by the Hamil- tonian (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The second one involves the state |ℓ⟩ of the spin chain, and the existence of the free fermions is only noticed through their total number N, which defines the 4 length of the chain, S(m, m′) = ⟨ℓ|PN+1,mPm′,N+1|ℓ⟩ = ⟨ℓ| max{m,m′}−1 � j=min{m,m′} [1 2I + ˆℓz(j)]|ℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (28) This part is time-independent, since the cyclic shift op- erator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (13) does not change the value of the z- projection of the total spin, Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The action of the oper- ator PN+1,mPm′,N+1, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' 1, leads to van- ishing S if any spin between the sites m and m′ is pointed down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' This way we get the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We proceed further by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (28) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (22) and taking the sum over the spin configurations, � ℓ e−βEℓS(m, m′) = [2 cosh(βB)]N ν|m−m′| , (29) where ν = 1 + e2βB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We get G↑ p(j − j′, t) = 1 Z � {N} e−β ˜ Ef � 2π 0 dλ 2π dλ′ 2π × Cp(λ, λ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t) ∞ � m,m′=−∞ e−iλm+iλ′m′ ν|m−m′| , (30) where ˜Ef = Ef − 1 β N ln[2 cosh(βB)], (31) and the sum over {N} encompasses the ones over N and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The partition function Z can be taken over the fermion configurations f with the energies given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We have ∞ � m,m′=−∞ e−iλm+iλ′m′ ν|m−m′| = 2πδ(λ − λ′)F(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' T), (32) where F(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' ν) = 1 + ∞ � m=1 ν−m(eimλ + e−imλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (33) Therefore, G↑ p(j − j′, t) = � 2π 0 dλ 2π F(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' ν)Cp(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' T), (34) where Cp(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' T) = 1 Z � {N} e−β ˜ Ef Cp(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t), (35) and we write Cp(λ) in place of Cp(λ, λ) in order to lighten notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The summation on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (35) represents the definition of the thermal state for the spinless fermions with the spectum given by ˜Ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The hole correlation function (23) is treated the same way as the particle one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The result is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34) and (35) with Cp replaced by Ch(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t) = ⟨f|eiλ ˆ Nj(t)ˆc† j(t)ˆcj′e−iλ ˆ Nj′(0)|f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (36) Emergence of impenetrable anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='— The operator ˆaj = ˆcje−iλ ˆ Nj satisfies the commutation relations ˆajˆa† j′ + e−iλϵ(j−j′)ˆa† j′ˆaj = δjj′, (37) ˆajˆaj′ + eiλϵ(j−j′)ˆaj′ˆaj = 0, (38) where ϵ(x) = |x|/x, and ϵ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' This is the fermion- anyon mapping discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The function Cp(λ) turns into Cp(−λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' j − j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' t) = ⟨f|ˆaj(t)ˆa† j′(0)|f⟩, (39) which is a correlation function of the impenetrable anyons on a lattice, the variable λ being the statistical angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The emergence of the anyon correlation function and its subsequent integration over λ with the function F in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34) could be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Let us consider a system with M spin-up and N − M spin-down par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Pick one spin-up particle among them, and pull it through the whole system, subsequently interchanging its coordinate with those of the other particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The in- terchanges with the spin-down particles are non-trivial: the spin part of the wave function could give any phase factor since its symmetry is not restricted by the fermion symmetry of the total wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We stress that for- malizing our a posteriori explanation of the structure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34) by examining exact finite-N wave functions in the coordinate representations (given, for example, in the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [21, 23]) goes beyond the scope of the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Place among other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='— The Hamilto- nian (16) with Uj−j′ = 0 represents the exactly solvable t − 0 model, also known as the Hubbard model in the limit of infinitely strong repulsion [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' There, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34) has been obtained in the form of a Fredholm determinant with the use of the exact wave functions in the coordi- nate representation [21, 23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The transformation (9)– (12) leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34), combined with the ones given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [26] for the function (27) bring us the same Fred- holm determinant representation through much shorter calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Note that the model (16) is also exactly solvable when Uj−j′ = Uδj,j′±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' In this case, the Hamil- tonian (19) can be mapped onto the one of the XXZ Heisenberg magnet, and the function (27) can, in princi- ple, be calculated by the Bethe Ansatz method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Special attention has been paid in the literature to the model in the T → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Its ground state is non- degenerate and spin-up (-down) polarized for B negative 5 (positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' In the former case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34) describes a spin- up fermion propagating through a gas of the other spin- up fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We have F = 2πδ(λ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (33), hence G↑ p = ⟨ˆcj(t)ˆc† j′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' In the latter case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (34) describes a spin-up fermion (an impurity particle) propagating through a gas of spin-down fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' We have F = 1, and the long time and distance asymptotic behaviour of G↑ p reveals the logarithmic diffusion phenomenon [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The non-degeneracy of the ground state at B ̸= 0 stands in a sharp contrast to the high degeneracy at B = 0, where F is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' (33) with ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' This regime is known as the spin-incoherent one [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' A challenge put forward in the aforementioned works was to find a low- energy effective field theory, since the low-enegry spec- trum of spin excitations cannot be linearized for B > 0 and B = 0, and the LL theory is inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The repre- sentation (34) resolves this problem in the following way: the LL theory in applicable to the function Cp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' the spin excitations are accounted for by the integral over λ with the weight function F without any approximation, which is equivalent to counting the number of worldlines within the first-quantized path integral approach implemented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Cheianov and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Seetharam for fruitful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' acknowledges support from the Polish Na- tional Agency for Academic Exchange (NAWA) through the Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' PPN/ULM/2020/1/00247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' is grate- ful to Galileo Galilei Institute for hospitality and support during the scientific program on “Randomness, Integra- bility, and Universality”, where part of this work was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The work of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' is supported by Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' ANR- 16-CE91-0009-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' thanks S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Bocini, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Mari´c, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Zadnik and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Fagotti for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The work of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' was partially supported by the European Research Council under the Starting Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' 805252 LoCoMacro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' The work of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' is supported by Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' ANR-16-CE91-0009-01 and CNRS grant PICS06738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' acknowledges Russian Quantum Center and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Fedorov for their hospitality during the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Pitaevskii and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'} +page_content=' Stringari, Bose-Einstein Condensa- tion (Oxford University Press, 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some light on these difficulties, we study a class of toy models for which +one-dimensional diffeomorphism invariance, namely time-reparametrization in- +variance, emerges at the classical level from energy conservation. An attempt +to quantize the models while taking the invariance seriously leads to toy ver- +sions of the problem of time in quantum gravity, of the cosmological constant +problem, and of the black hole firewall problem. Nevertheless, all these prob- +lems are easily resolved by taking into account that the invariance emerges only +at the classical level, while the fundamental theory that needs to be quantized +is not diffeomorphism invariant. +Keywords: diffeomorphism invariance; time in quantum gravity; cosmological con- +stant; black hole firewall +1 +Introduction +Classical general relativity [1, 2, 3] is one of the most elegant theories in physics. +Its most distinguished feature is diffeomorphism invariance, or invariance under ac- +tive general transformations of spacetime coordinates, which implies that spacetime +metric is a dynamical quantity. +But this elegance is a blessing and a curse. +It’s +a blessing in classical physics, but a curse in quantum physics because we still do +not fully understand how to quantize gravity [4, 5, 6], that is, how to implement +diffeomorphism invariance at the quantum level. The problems appear not only in +fully quantum gravity, but also in the semiclassical approximation [7, 8] where only +matter is quantized while gravity is treated classically. The problems that appear +are not only technical, but also conceptual. +The three conceptual problems that +1 + +stand out are the problem of time in quantum gravity [9, 10, 11, 12], the cosmolog- +ical constant problem [13, 14, 15, 16, 17], and the black hole information paradox +[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. +One possibility that potentially could help to resolve these conceptual problems is +the idea that general relativity and its diffeomorphism invariance is emergent, rather +than fundamental, while the underlying more fundamental theory rests on entirely +different principles. This idea can be realized in condensed-matter inspired theories +such as induced gravity [33], as well as in string theory [6]. However, there is no +any direct experimental evidence for such a more fundamental theory. +Moreover, +promising theoretical candidates such as string theory are still poorly understood in +their most fundamental terms. Consequently, it is very difficult to study the idea of +emergent diffeomorphism invariance in realistic models. In this paper, therefore, we +study this idea in toy models, similar to the toy models in [11, 9, 34] studied before +in the context of the problem of time in quantum gravity. In these models, the 4- +dimensional spacetime diffeomorphism invariance of general relativity is replaced with +a 1-dimensional diffeomorphism invariance realized as time-reparametrization invari- +ance. Even though such models cannot solve the problems of realistic 4-dimensional +systems with gravity, it is hoped that such simple models can at least serve as a +conceptual inspiration for dealing with more difficult realistic theories. +The paper is organized as follows. +In Sec. 2 we first introduce a class of toy +models without diffeomorphism invariance and then explain how 1-dimensional dif- +feomorphism invariance emerges from conservation of energy, namely, as a way to +implement the constraint that the classical system has definite energy. In Sec. 3 we +explain how the 1-dimensional diffeomorphism invariance leads to a toy version of +the problem of time in quantum gravity, and how the problem resolves when one +recalls that the diffeomorphism invariance is not fundamental. Similarly, in Sec. 4 +we explain how the 1-dimensional diffeomorphism invariance leads to a toy version of +the cosmological constant problem, and how the problem resolves when one recalls +that the diffeomorphism invariance is not fundamental. Likewise, in Sec. 5 we find +a solution of the constraint that in some aspects resembles the behavior in a black +hole exterior, explain how the diffeomorphism invariance can be used to extend the +solution to a region resembling the behavior in a black hole interior, and point out +that the interior is actually unphysical because the diffeomorphism invariance is not +fundamental. The non-existence of the interior can be understood as a toy version +of the black hole firewall [35, 36], which plays a key role in some approaches to solv- +ing the black hole information paradox. In Sec. 6 we briefly speculate how these toy +models could perhaps be generalized to real 4-dimensional diffeomorphism invariance. +Finally, in Sec. 7 we present a qualitative discussion of our results. +2 + +2 +The model and emergent diffeomorphism invari- +ance +2.1 +The model +We study a system with N dynamical degrees of freedom described by the collective +configuration variable q(t) = {q1(t), . . . , qN(t)}, the dynamics of which is described +by the action +A = +� +dt L(q, ˙q), +(1) +where the dot denotes the derivative with respect to time t and +L(q, ˙q) = +N +� +a=1 +ma ˙q2 +a +2 +− V (q). +(2) +The canonical momenta are well defined +pa = ∂L +∂ ˙qa += ma ˙qa, +(3) +so the Hamiltonian is +H(q, p) = +N +� +a=1 +pa ˙qa − L = +N +� +a=1 +p2 +a +2ma ++ V (q) +(4) +and can be interpreted as the energy of the system. +The system can be treated +either classically of quantum mechanically, in a straightforward manner. In particu- +lar, quantization can be performed via canonical quantization and dynamics can be +described by the Schr¨odinger equation +H|ψ(t)⟩ = i¯h∂t|ψ(t)⟩ +(5) +as usual, where H is the operator. Since the action does not have any a priori gauge +or diffeomorphism invariance, the quantization is straightforward. +2.2 +Emergent diffeomorphism invariance +Since the Hamiltonian H does not have an explicit time dependence, it is conserved. +In classical physics, this means that H has some definite constant value E of energy, +so we can write it as H(q, p) = E, or +H(q, p) = 0, +(6) +where +H(q, p) ≡ H(q, p) − E. +(7) +3 + +In the configuration space, the fact that the Hamiltonian has the value E can be +written as +N +� +a=1 +ma ˙q2 +a +2 ++ V (q) − E = 0. +(8) +If we imagine that (2) describes a whole Universe, then E is the energy of that +Universe. +The inhabitants of this Universe observe only one value of E, but the +theory cannot say which one. For the inhabitants of this Universe, the constant E is +a fundamental constant the value of which can be determined from experiments. +Since E appears as a fundamental constant, it seems natural to incorporate the +value of this constant into an effective action. One possibility is to incorporate the con- +straint (8) into the action by adding the Lagrange multiplier term λ [� +a ma ˙q2 +a/2 + V (q) − E]. +However, there is a much more interesting way to incorporate the constraint (8) into +the action. We do that not by introducing a Lagrange multiplier λ, but by introducing +a new configuration variable g(t) > 0 and replacing the action (1) with +˜A = +� +dt√g +� N +� +a=1 +ma ˙q2 +a +2g +− V (q) + E +� +. +(9) +Since this action does not depend on time derivatives of g(t), the g(t) is not a dy- +namical variable and the equation of motion for this variable is a constraint equation. +More precisely, the equation of motion δ ˜A/δg = 0 gives +− +1 +2√g +� N +� +a=1 +ma ˙q2 +a +2g ++ V (q) − E +� += 0, +(10) +which reduces to the constraint (8) if g = 1. But what is the rational for taking g = 1? +The answer is that the action (9) has the property of diffeomorphism invariance which +allows us to choose for g(t) any positive function we want, so g(t) = 1 is nothing but +a convenient choice of “gauge”. Since this diffeomorphism invariance is crucial, let us +explain it in more detail. +The g in (9) appears in two terms, which are proportional to +dt√g, +˙q2 +a +g = dq2 +a +g dt2. +(11) +Thus g appears either in the combination √gdt = +� +g dt2 or g dt2 = (√gdt)2. This +implies that the action is invariant under arbitrary transformations that keep +dτ 2 ≡ g(t)dt2 +(12) +invariant. +The dτ 2 is very much analogous to the spacetime line element ds2 = +gµν(x)dxµdxν in general relativity, so we see that g in (12) corresponds to g00 in +general relativity. Likewise, 1/g corresponds to g00. Just like general relativity is +invariant under arbitrary 4-dimensional spacetime diffeomorphisms xµ → x′µ = f µ(x) +4 + +which keep ds2 = gµν(x)dxµdxν invariant, the action (9) is invariant under arbitrary +1-dimensional time diffeomorphisms +t → t′ = f(t) +(13) +which keep (12) invariant. The invariance g dt2 = g′dt′2 implies that g transforms as +g → g′ = +� dt +dt′ +�2 +g. +(14) +This 1-dimensional diffeomorphism invariance is also known in literature under the +name time-reparametrization invariance [5, 12, 10]. +To summarize, we have started from the action (1) without diffeomorphism invari- +ance and, from the fact that energy has some constant value E in classical mechanics, +derived the corresponding action (9) with 1-dimensional diffeomorphism invariance. +In this way, the 1-dimensional diffeomorphism invariance is emergent from classical +energy conservation. +2.3 +The constraint in the canonical form +Now we want to develop some formal tools that will be used in further sections. The +action (9) can also be written as +˜A = +� +dt ˜L(q, ˙q, g) = +� +dt√g L(q, ˙q, g), +(15) +where +L(q, ˙q, g) = +N +� +a=1 +ma ˙q2 +a +2g +− V (q) + E, +˜L(q, ˙q, g) = √gL(q, ˙q, g). +(16) +The corresponding canonical momenta are +˜pa = ∂ ˜L +∂ ˙qa += ma ˙qa +√g , +pg = ∂ ˜L +∂ ˙g = 0, +(17) +so the Hamiltonian is +˜H(q, ˜p, g) = +N +� +a=1 +˜pa ˙qa − ˜L = √g H(q, ˜p), +(18) +where +H(q, ˜p) = +N +� +a=1 +˜p2 +a +2ma ++ V (q) − E. +(19) +5 + +The canonical equation of motion for pg is +˙pg = −∂ ˜H +∂g = − 1 +2√gH. +(20) +However, in (17) we have seen that pg = 0, which implies ˙pg = 0, so (20) implies +− +1 +2√gH = 0, +(21) +which is identical to the constraint (10). Thus, since g > 0, we see that the constraint +(10), or (21), can also be written as the Hamiltonian constraint +H(q, ˜p) = 0, +(22) +or equivalently +˜H(q, ˜p, g) = 0. +(23) +In the gauge g = 1, this reduces to the constraint (6). +3 +The problem of time in quantum gravity +Seduced by the beauty and elegance of the action with 1-dimensional diffeomorphism +invariance, one may be tempted to quantize it. The problem is, how to implement +the Hamiltonian constraint (22) in the quantum theory? The most natural approach +is to implement it as the constraint on physical states +H(q, ˜p)|ψ⟩ = 0, +(24) +where H(q, ˜p) is the quantum operator obtained via standard canonical quantization. +This constraint implies also +˜H(q, ˜p, g)|ψ⟩ = 0, +(25) +which is the quantum version of (23). However, the time evolution of the state should +be described by the corresponding Schr¨odinger equation +˜H(q, ˜p, g)|ψ(t)⟩ = i¯h∂t|ψ(t)⟩, +(26) +so compatibility with (25) implies +∂t|ψ(t)⟩ = 0. +(27) +Hence the state does not depend on time. But we know that the real world, or even +the toy world described by the toy model in Sec. 2.1, depends on time. Where does +the dependence on time come from, if the quantum state |ψ(t)⟩ does not depend on +time? This is the toy version of the problem of time in quantum gravity [9, 10, 11, 12]. +Within our model, it is not difficult to understand where the problem comes +from and how it should be resolved. In general, whenever a quantum system has a +6 + +well defined energy E, its wave function has trivial time dependence proportional to +e−iEt/¯h, which is just a time-dependent phase without any physical consequences. To +have a genuine time-dependent state in quantum mechanics, the state must not have +a well defined energy. Instead, the state must be in a superposition of two or more +different energies. +So what is wrong with (25)? This quantum constraint originates from the classical +action (9) in which the energy E is fixed. In fact, the whole diffeomorphism invariance +of (9) emerged from a desire to implement the classical value E of energy into the +action. +There is nothing wrong with it in classical physics, where energy indeed +has a well defined value. However, requiring that the quantum system should also +have a definite value of energy is wrong, because the energy of a quantum system +is, in general, uncertain. In other words, it is wrong to quantize the diffeomorphism +invariant effective action (9). What needs to be quantized is the original action (1), +which is not diffeomorphism invariant and leads to the proper Schr¨odinger equation +(5) without the problem of time. The emergent diffeomorphism invariance is only +valid at the classical level, where energy is well defined. At the quantum level, where +energy is uncertain, there is no diffeomorphism invariance. +To conclude, the problem of time in the toy version of quantum gravity originates +from taking the diffeomorphism invariance too seriously. When one takes into account +that this invariance is only emergent at the classical level, while fundamental quantum +theory does not have this invariance, the problem of time disappears in an obvious +way. +4 +The cosmological constant problem +Among the N degrees of freedom, let us suppose that Nheavy of them are “heavy” +and the rest Nlight = N − Nheavy are “light”. +We call them “heavy” and “light” +degrees because we assume that one can use a semiclassical approximation in which +the Nheavy degrees are treated classically, while the rest Nlight of them are quantized. +For simplicity, we also assume that V (q) can be split as +V (q) = Vheavy(qheavy) + Vlight(qlight), +(28) +where qheavy = {qb | b = 1, . . . , Nheavy} are heavy degrees, and qlight = {qa | a = +1, . . . , Nlight} are light degrees. +Thus the classical constraint (10) can be written +as +− +Nheavy +� +b=1 +mb ˙q2 +b +2g +− Vheavy(qheavy) = +Nlight +� +a=1 +ma ˙q2 +a +2g ++ Vlight(qlight) − E, +(29) +or more concisely +− Hheavy = Hlight − E, +(30) +with a self-explaining notation. This is a classical equation, but as we said, the idea +is to treat it semi-classically, so that the light degrees are quantized while the heavy +degrees are left classical. Thus one replaces (30) with a semiclassical equation +− Hheavy = ⟨ψ|Hlight|ψ⟩ − E, +(31) +7 + +where ⟨ψ|Hlight|ψ⟩ is the mean value of the operator Hlight in the quantum state |ψ⟩. +Next suppose that Vlight(qlight) is the potential of Nlight harmonic oscillators +Vlight(qlight) = +Nlight +� +a=1 +kaq2 +a +2 . +(32) +Then the operator Hlight can be written in the usual quantum harmonic oscillator +form +Hlight = +Nlight +� +a=1 +¯hωa +� +A† +aAa + 1 +2 +� +, +(33) +where ωa = +� +ka/ma, while A† +a and Aa are the raising and lowering operators, respec- +tively. In particular, in the quantum ground state defined by Aa|0⟩ = 0 we have +⟨0|Hlight|0⟩ = +Nlight +� +a=1 +¯hωa +2 , +(34) +so the semiclassical equation (31) becomes +− Hheavy = +Nlight +� +a=1 +¯hωa +2 +− E. +(35) +By contrast, the ground state energy of the classical harmonic oscillator is zero, so +the classical version of (35) is +− Hheavy = −E. +(36) +But Nlight is supposed to be very large, after all this is the number of light degrees +in the whole toy Universe. Thus, there is a large discrepancy between the classical +equation (36) and the semiclassical equation (35). The semiclassical equation (35) +can also be written as +− Hheavy = −Eeff, +(37) +where +− Eeff = −E + +Nlight +� +a=1 +¯hωa +2 . +(38) +The effective energy Eeff contains a very large contribution from the quantum zero- +point energy. +Finally, suppose that the inhabitants of the toy Universe measure Eeff and find a +value +− Eeff ≪ +Nlight +� +a=1 +¯hωa +2 . +(39) +Then it is the problem to explain why −Eeff is so small; why is it much smaller than +its natural value given by the right-hand side of (39)? +8 + +Clearly, this problem is analogous to the cosmological constant problem in semi- +classical gravity [13, 14, 15, 16, 17]. Eq. (30) multiplied with g +− Hheavyg = Hlightg − Eg +(40) +is analogous to the 00-component of the Einstein equation which, in appropriate units, +can be written as +Gµν = Tµν + Λgµν, +(41) +where Gµν is the Einstein tensor depending only on gravitational degrees, Tµν is the +energy-momentum tensor of matter, and Λ is the cosmological constant. In this anal- +ogy, “heavy” degrees are analogous to the gravitational degrees, “light” degrees are +analogous to the matter degrees, and the constant −E is analogous to the cosmo- +logical constant. In the semiclassical approximation one performs a quantization of +matter while keeping gravity classical, so (41) is replaced with +Gµν = ⟨Ψ|Tµν|Ψ⟩ + Λgµν, +(42) +the 00-component of which is analogous to (31) multiplied with g +− Hheavyg = ⟨ψ|Hlight|ψ⟩g − Eg. +(43) +In particular, in the matter ground state |Ψ⟩ = |0⟩ one finds a very large quantum +contribution analogous to (34), so there is a large discrepancy between the value of +cosmological constant defined by the quantum ground state and the small value of +cosmological constant found from cosmological observations [13, 14, 15, 16, 17]. +Within our model, it is not difficult to understand where the problem comes +from and how it should be resolved. In the diffeomorphism invariant action (9), the +constant energy −E has physical consequences because it is coupled to g via the +term proportional to √gE. This is analogous to the cosmological constant coupled +to gravity via the term proportional to +� +| det gµν|Λ. On the other hand, the action +(1) with (2) is not diffeomorphism invariant and hence does not contain √g. As a +consequence, adding a constant E to the Lagrangian (2) does not have any physical +consequences. In the corresponding quantum theory described by the Schr¨odinger +equation (5), the Hamiltonian is shifted by a constant value −E, which changes the +phase of the quantum state by an additional phase factor eiEt/¯h, which does not have +any physical consequences. The quantum ground state energy further shifts this value +from E to Eeff as given by (38), but the new phase factor eiEefft/¯h still does not have +any physical consequences. +Hence the conclusion is very similar to that in Sec. 3. The toy version of the +cosmological constant problem originates from taking the diffeomorphism invariance +too seriously. When one takes into account that this invariance is only emergent at +the classical level, while fundamental quantum theory does not have this invariance, +the toy cosmological constant problem disappears in an obvious way. +9 + +5 +Black hole and firewall +5.1 +The model +Consider a subsystem described by only two degrees of freedom q(t) = {x(t), y(t)}, +and suppose that the subsystem is invariant under rotations in the x-y plane. Suppose +also that E = 0. Under these conditions, the action (9) reduces to +˜A = +� +dt√g +�m( ˙x2 + ˙y2) +2g +− V (x, y) +� +, +(44) +where V (x, y) = V (x2 +y2). Due to the rotational symmetry, it is convenient to work +in polar coordinates +z = +� +x2 + y2, +ϕ = arctgy +x, +(45) +with ranges +z ∈ [0, ∞), +ϕ ∈ [0, 2π), +(46) +where the values ϕ = 0 and ϕ = 2π are identified. Note that z is the usual radial +coordinate, but we denote it with z, rather than with r, for the reasons that will +become clear later. Thus the action (44) can be written as +˜A = +� +dt√g +�m( ˙z2 + z2 ˙ϕ2) +2g +− V (z2) +� +, +(47) +and the corresponding constraint (10) reduces to +m( ˙z2 + z2 ˙ϕ2) +2g ++ V (z2) = 0. +(48) +To get an interesting solution of the constraint, let us suppose that the potential +V (z2) for small z has a form of an inverted harmonic oscillator +V (z2) = −kz2 +2 , +(49) +with k > 0. Thus, assuming in addition that ϕ(t) = 0 and choosing the gauge +g(t) = 1, +(50) +the constraint (48) finally reduces to +m ˙z2 +2 +− kz2 +2 += 0, +(51) +which is a differential equation for z(t) +�dz(t) +dt +�2 += γ2z2(t), +(52) +where γ = +� +k/m. We will see that (52) describes a motion analogous to the radial +motion of a particle around a black hole with a horizon at z = 0. +10 + +5.2 +Analogy with a black hole +The solution of the differential equation (52) is +z(t) = z(0)e±γt. +(53) +The solution z(t) = z(0)e−γt can be visualized as radial infalling towards z = 0. The +infalling exponentially slows down as z = 0 is approached, and it takes an infinite +time t to reach z = 0. Likewise, the solution z(t) = z(0)eγt is a time inversion of the +infalling, it describes an escaping from small z towards z → ∞. However, if it starts +from z(0) = 0, then it can never escape; it remains trapped at z(t) = 0 forever. This +behavior is very much analogous to infalling towards the black hole, or escaping from +it. In particular, it takes an infinite time to reach the black hole horizon, from the +point of view of observer staying at a fixed non-zero distance from the horizon. Also, +an object initially at the horizon can never escape from it. We see that the point +z = 0 is analogous to the black hole horizon. +Moreover, the analogy with black holes does not stop here. The solution (53) is +obtained in the gauge (50), but the theory is diffeomorphism invariant under time +reparametrizations (13). Thus we can introduce a new time variable t′ defined im- +plicitly by +e−γt = 1 − γt′, +(54) +so the infalling solution z(t) = z(0)e−γt can be written as +z(t(t′)) = z(0)[1 − γt′]. +(55) +Now the point z = 0 is reached after a finite time t′ = 1/γ. Furthermore, the solution +(55) can be extended to negative values of z (this is the reason why we denote it with +z, rather than with r), reached at times t′ > 1/γ. This is analogous to the Kruskal +extension (see e.g. [1, 2, 3]) of the Schwarzschild solution in general relativity, where +in appropriate spacetime coordinates a freely falling object reaches the horizon after +a finite time and the Schwarzschild solution is extended beyond the horizon, thus +describing not only the black hole exterior, but also its interior. Hence, the region +of negative z in the toy model is analogous to the black hole interior behind the +Schwarzschild horizon. +5.3 +Effective spacetime +The analogy above can also be made more explicit by introducing an effective space- +time metric. The constraint (52) can be written as γ2z2dt2 − dz2 = 0, which can +be interpreted as motion of a relativistic massless particle in a spacetime with the +effective metric +ds2 +eff = Ω(t, z)[γ2z2dt2 − dz2], +(56) +where Ω(t, z) > 0 is an arbitrary conformal factor. This effective metric has a horizon +at z = 0. In particular, the metric in the square bracket has the same form as the +Rindler metric [37, 1] +ds2 +Rindler = a2z2dt2 − dz2, +(57) +11 + +associated with an observer at z = 1/a accelerating with proper acceleration a. The +Rindler horizon at z = 0 is known to have many similarities with the black hole +horizon [37, 7, 8]. +Since (56) has a coordinate singularity at z = 0, we want to see what happens +with this singularity after the coordinate transformation (54). By applying (54) to +(56), we get +ds2 +eff = Ω +� γ2z2dt′2 +(1 − γt′)2 − dz2 +� +, +(58) +which is still singular at z = 0. However, the singular quantity +g′ +00 = +Ωγ2z2 +(1 − γt′)2 +(59) +is in fact regular along the infalling trajectory (55), i.e. +g′ +00 +traj += Ωγ2z2(0) +(60) +is regular provided that the initial position obeys z(0) ̸= 0. +A standard way to completely remove the coordinate singularity at the horizon +z = 0 is to introduce the new spacetime coordinates +T = z shγt, +Z = z chγt. +(61) +Indeed, an elementary calculus shows that dT 2 − dZ2 = γ2z2dt2 − dz2, so (56) can be +written as +ds2 +eff = Ω[dT 2 − dZ2]. +(62) +In these coordinates the relativistic massless particle obeys dT 2 − dZ2 = 0, so the +infalling solution is +Z(T) = Z(0) − T, +(63) +which corresponds to (55). +Now we want to express the position of the horizon z = 0 in the T, Z coordinates. +Inserting z = 0 into (61) gives (T, Z) = (0, 0), if t is finite. But what about the limit +t → ±∞? In this limit (61) gives Z/T = ±1 for any z, including the limit z → 0, +so the two lines Z = ±T are also consistent with z = 0. Thus the horizon is the +union of the point (T, Z) = (0, 0) (corresponding to finite t) and the lines Z = ±T +(corresponding to t → ±∞). But this union is simply the two lines Z = ±T, so we +conclude that the horizon is the two lines Z = ±T. The line Z = T is the future +horizon, which is characteristic for a black hole, while the line Z = −T is the past +horizon, which is characteristic for a white hole. +Thus we see that the infalling solution (63) crosses the future horizon Z = T and +extends beyond the future horizon, which corresponds to the extension beyond the +analogue horizon z = 0 in (55). +Finally note that the effective spacetime metric can be introduced not only for the +potential (49), but also for any potential V (x, y) in (44), provided that it is negative. +The constraint resulting from (44) is +m( ˙x2 + ˙y2) +2g ++ V (x, y) = 0, +(64) +12 + +which in the gauge g = 1 can be written as +− 2V (x, y) +m +dt2 − dx2 − dy2 = 0. +(65) +This can be interpreted as motion of a relativistic massless particle in a spacetime +with the effective metric +ds2 +eff = Ω(t, x, y) +� +−2V (x, y) +m +dt2 − dx2 − dy2 +� +, +(66) +where Ω(t, x, y) > 0 is an arbitrary conformal factor. This metric has the relativis- +tic signature (+ − −), provided that V (x, y) < 0. Taking Ω = 1 for convenience +and defining the effective “Newtonian” gravitational potential φgrav(x, y) through the +standard relation [3] +g00(x, y) = 1 + 2φgrav(x, y), +(67) +we see that the potentials V and φgrav are related as +φgrav(x, y) = −V (x, y) +m +− 1 +2. +(68) +The important message of (68) is that φgrav corresponds to −V , rather than to V as +one might naively expect. In particular, we see that a repulsive potential V such as +(49) corresponds to an attractive gravitational potential φgrav. +5.4 +The firewall +We have seen that the solution (55) can be extended to negative values of z, and +that this extension is analogous to the extension of black hole behind the horizon. +However, in the toy model, the extension is conceptually problematic. How can the +extension to negative values of z be compatible with the fact that the z-coordinate +was restricted to non-negative values by definition, in Eq. (46)? The answer is that it +cannot! Only non-negative values of z are physical. The region of space with negative +z does not exist. The motivation for extension to negative values of z has arisen from +(55), which, in turn, has arisen from a new time coordinate introduced in (54). But +the original model (1) with (2) is not diffeomorphism invariant, i.e. it does not allow +arbitrary redefinitions of the time coordinate. From this point of view, the gauge (50) +is not merely an arbitrary choice, but the correct physical value of g. The negative +values of z have arisen from taking the diffeomorphism invariance too seriously, while +this invariance is just an emergent feature resulting from a formalism that encoded +the classical value of energy E into the action, as described in Sec. 2.2. +The conclusion above that there is no region behind z = 0 is completely classi- +cal, it does not involve any quantum physics. Nevertheless, a semiclassical version +resembling Hawking radiation can also be constructed. Suppose that two entangled +particles are created at z > 0, one infalling and the other escaping, thus mimicking +13 + +the Hawking pair. Suppose also that the potential V (z2), given by (49) for small z, +is defined for all z ≥ 0 as +V (z2) = +� +−kz2/2 +for z ≤ z0 +−V0 +for z ≥ z0, +(69) +where +z0 = +� +2V0 +k . +(70) +This potential can be visualized as a flat valley at the constant potential −V0 for +z > z0, with a hill of height V0, radius z0, and the top at z = 0. It mimics a stationary +black hole approximated with flat geometry for r ≥ r0, which is justified if r0 is much +larger than the Schwarzschild radius. To mimic a non-stationary evaporating black +hole, we modify (69) and (70) to +V (z2, t) += +� −k(t)z2/2 +for z < z0(t) +−V0 +for z ≥ z0(t), +(71) +z0(t) += +� +2V0 +k(t), +(72) +where k(t) is an increasing function that, after a large but finite time t∗, becomes +infinite k(t∗) = ∞. Thus the radius z0(t) shrinks and becomes zero at time t∗, which +mimics the shrinking of the evaporating black hole. The information paradox can now +be formulated as follows. The peak of the infalling wave packet follows approximately +the classical trajectory (55), thus entering the region behind z = 0, i.e. behind the +top of the hill. But at late times t > t∗ the potential is V (z2) = −V0, so there is no +hill and hence no region behind the top of the hill. It looks as if the infalling particle +disappears at late times, so the remaining escaping particle in the mixed state seems +to contradict unitarity of quantum mechanics. This is the toy version of the black +hole information paradox. The solution of the paradox is that the region behind z = 0 +never existed in the first place. As we said, the motivation for extension to negative +values of z originated from (55), which, in turn, originated from introducing a new +time coordinate in (54), which, however, is not allowed in the fundamental theory +without diffeomorphism invariance. +Remarkably, the non-existence of the region behind z = 0 in the toy model has an +analogy in black hole physics. With a motivation to resolve the black hole information +paradox [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] in semiclassical +gravity, it has been proposed that the black hole interior does not exist; the black +hole horizon represents a physical boundary called firewall [35, 36, 27]. The problem +with the firewall is to reconcile it with standard classical general relativity, which +predicts that the black hole interior exists, and that the horizon is not a physical +boundary. But such a standard view of classical general relativity is a consequence +of the 4-dimensional diffeomorphism invariance. Alternatively, if the 4-dimensional +diffeomorphism invariance in general relativity is emergent in a way similar to the +emergence of the 1-dimensional diffeomorphism invariance in our toy model, then the +14 + +4-dimensional diffeomorphism invariance should not be taken too seriously even in +the classical theory. If so, then the existence of the black hole interior resulting from +the Kruskal extension should not be trusted. Such an alternative view of classical +gravity, if correct, makes the firewall perfectly compatible with classical physics, which +resolves the firewall problem. +Hence the conclusion is similar to that in Secs. 3 and 4. The toy version of the +firewall problem originates from taking the diffeomorphism invariance too seriously. +When one takes into account that this invariance is only emergent, while the funda- +mental theory does not have this invariance, the toy firewall problem disappears in +an obvious way. +6 +Towards emergent 4-dimensional diffeomorphism +invariance +The motivation for studying the toy models with 1-dimensional diffeomorphism invari- +ance is to teach us something about the real 4-dimensional diffeomorphism invariance, +namely, about real classical, semiclassical and quantum gravity. So the question is, +how the ideas of the toy models can be generalized to 4-dimensional diffeomorphism +invariance? Unfortunately, we do not have a full answer to that question. A full +answer would be tantamount to having a full theory of quantum gravity, which, of +course, we do not have. Nevertheless, inspired by the toy models, we sketch an idea +how such a generalization might look like. What we present here can be thought of +as a gist of a research program based on a series of educated guesses1, which at the +current level is very far from a fully developed theory. +Our starting point of view is that the spacetime curvature emerges from a massless +spin-2 field [38, 39, 40, 41, 42], and not the other way around. Roughly, this means +that in the formula +gµν(x) = ηµν + φspin-2 +µν +(x), +(73) +relating the curved spacetime metric gµν(x) to the flat Minkowski metric ηµν and +the massless spin-2 field φspin-2 +µν +(x), the quantities on the right-hand side are more +fundamental than that on the left-hand side. Philosophically, such a view complies +much better with string theory than with loop quantum gravity. In the fundamental +theory, the formula (73) is expected to be valid only in some approximative sense. +We assume that there is some fundamental action A[φ] without diffeomorphism +invariance, where φ = φ(x) is a collective symbol for all fundamental dynamical fields +φ = {φmatt, φspin-2, . . .}. +(74) +Here φmatt are the usual “matter” fields of spins 0, 1 +2 and 1, the field φspin-2 is the +massless spin-2 field, and the ellipses are possible other fields beyond the Standard +1“Educated guess” is (supposed to be) a well balanced term, between the over-pretentious “con- +jecture” and over-cynical “wishful thinking”. +15 + +Model of particle physics. The x denotes a spacetime position in 4 or more dimen- +sions. From the action A[φ] one can derive the symmetrized energy-momentum tensor +Tµν[φ; x], which is conserved when the equations of motion +δA/δφ(x) = 0 +(75) +are satisfied. In classical physics the fields φ(x) attain some definite values Φ(x), +where Φ(x) is a definite solution of (75). Thus we can define +Eµν(x) ≡ Tµν[Φ; x], +(76) +which is a generalization of the definite energy E appearing in (8). For example, in +a classical vacuum in Minkowski spacetime, the Eµν(x) may take the form +Eµν(x) = −Ληµν, +(77) +where Λ is a constant. But whatever the Eµν(x) is, in classical physics we can always +write +Tµν[φ; x] − Eµν(x) = 0, +(78) +which is a generalization of (8). +In some limit one expects that Tµν[φ; x] can be +decomposed as +Tµν[φ; x] = T matt +µν +[φ; x] + T spin-2 +µν +[φ; x] + . . . . +(79) +With this decomposition, (78) looks very much like the Einstein equation (41) written +in the non-geometric spin-2 language. +Now the idea is to think of (78) as a constraint derived from a new action ˜A[φ, g], +where g(x) = {gµν(x)} is a symmetric tensor field. By analogy with (9), one expects +that the new action ˜A[φ, g] is diffeomorphism invariant, so that the diffeomorphism- +covariant equation +δ ˜A/δgµν(x) = 0 +(80) +reduces to (78) when the gauge for gµν is chosen appropriately. One also expects +that, in a certain limit, the action ˜A[φ, g] reduces to the usual gravitational action +with the matter term, the Einstein-Hilbert term, and the cosmological term. This is, +roughly, how the 4-dimensional diffeomorphism is expected to emerge at the classical +level. However, the fundamental action that needs to be quantized in this scheme is +A[φ], not ˜A[φ, g]. +With this approach, it it easy to see that there is no problem of time in quantum +gravity, simply because the fundamental action A[φ] does not have a Hamiltonian +constraint. The Hamiltonian H derived from A[φ] does not need to vanish on-shell. +Likewise, there is no cosmological constant problem, in the sense that energy (asso- +ciated with H) of the quantum ground state does not have physical consequences. +Finally, the quantum time evolution defined by e−iHt/¯h is unitary, so all quantum pro- +cesses, including Hawking radiation, are compatible with unitarity. Nevertheless, at +this level, it is not clear how exactly the information paradox associated with Hawk- +ing radiation resolves. Since the quantum theory lacks diffeomorphism invariance, +the firewall scenario discussed in Sec. 5.4 scenario seems plausible. In the same spirit, +16 + +since quantum gravity is not fundamentally geometrical in this picture, inherently +geometrical proposals involving wormholes, such as ER=EPR [43] and black hole is- +lands [44], seem less plausible. Nevertheless, at the current level of understanding of +the ideas sketched above, it is impossible to make definite precise claims about the +quantum nature of black holes. +7 +Discussion and conclusion +In this paper we have constructed toy versions of the problem of time in quantum +gravity, of the cosmological constant problem, and of the black hole firewall problem. +Within the models, the problems originate from taking the 1-dimensional diffeomor- +phism invariance too seriously. This 1-dimensional diffeomorphism invariance, real- +ized as time-reparametrization invariance, is emergent, rather than fundamental, and +when one takes it into account the problems disappear in a rather natural way. The +problem of time disappears because quantum energy is uncertain in the absence of +fundamental time-reparametrization invariance. The cosmological constant problem +disappears because a shift of energy by a constant does not have physical conse- +quences in the absence of fundamental time-reparametrization invariance. The black +hole firewall problem disappears because a firewall at the horizon may be completely +compatible with classical physics when the diffeomorphism invariance is interpreted +as emergent, rather than fundamental. +Note also that the physical irrelevance of vacuum energy in the context of the cos- +mological constant problem is compatible with the Casimir effect. The description of +Casimir effect in terms of vacuum energy is just an effective macroscopic description, +while the fundamental microscopic origin of Casimir effect lies in van der Waals forces +[45, 46, 47]. In particular, it can be understood in terms of a toy model [47] similar +to that of the present paper. +In our toy models, the solutions of the problems of time and of the cosmological +constant are rather generic; the solutions do not depend on details of the models. In +particular, even though the cosmological constant problem is discussed for quantum +harmonic oscillators, the solution of the problem works in essentially the same way +for any other interaction V (q) that leads to a non-zero quantum ground state energy. +By contrast, our solution of the toy black hole firewall problem is not so generic, +it depends on details of the model. Perhaps different models could suggest totally +different solutions of the black hole information paradox, without any hints for the +existence of firewalls. Or perhaps some models would describe classical states resem- +bling black holes, but without any hints how to solve the information paradox. More +research is needed to better understand how the lack of fundamental diffeomorphism +invariance may, or may not, help to solve the information paradox. +More importantly, it is not at all clear whether such toy 1-dimensional ideas can, +and should, be generalized to the real 4-dimensional diffeomorphism invariance of +general relativity. In Sec. 6 we have sketched how such a generalization might look +like, but it is far from a fully developed theory. Nevertheless, the conceptual simplicity +of solutions of the toy problems seems suggestive, so we believe that this conceptual +17 + +simplicity could at least serve as a source of inspiration for further research. +In any case, we believe that our analysis of the toy models with emergent diffeo- +morphism invariance may influence how physicists think about general relativity at +an intuitive level. 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Phys. 383, 181 (2017); arXiv:1702.03291. +20 + diff --git a/B9E3T4oBgHgl3EQfUAr7/content/tmp_files/load_file.txt b/B9E3T4oBgHgl3EQfUAr7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a13cfb68931ddba4ecb5ef7f9bc4f3ce20b53699 --- /dev/null +++ b/B9E3T4oBgHgl3EQfUAr7/content/tmp_files/load_file.txt @@ -0,0 +1,580 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf,len=579 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='04448v1 [gr-qc] 11 Jan 2023 Emergent diffeomorphism invariance in toy models Hrvoje Nikoli´c Theoretical Physics Division, Rudjer Boˇskovi´c Institute, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 180, HR-10002 Zagreb, Croatia e-mail: hnikolic@irb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='hr January 12, 2023 Abstract Conceptual difficulties in semiclassical and quantum gravity arise from dif- feomorphism invariance of classical general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' With a motivation to shed some light on these difficulties, we study a class of toy models for which one-dimensional diffeomorphism invariance, namely time-reparametrization in- variance, emerges at the classical level from energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' An attempt to quantize the models while taking the invariance seriously leads to toy ver- sions of the problem of time in quantum gravity, of the cosmological constant problem, and of the black hole firewall problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Nevertheless, all these prob- lems are easily resolved by taking into account that the invariance emerges only at the classical level, while the fundamental theory that needs to be quantized is not diffeomorphism invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Keywords: diffeomorphism invariance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' time in quantum gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' cosmological con- stant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' black hole firewall 1 Introduction Classical general relativity [1, 2, 3] is one of the most elegant theories in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Its most distinguished feature is diffeomorphism invariance, or invariance under ac- tive general transformations of spacetime coordinates, which implies that spacetime metric is a dynamical quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But this elegance is a blessing and a curse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' It’s a blessing in classical physics, but a curse in quantum physics because we still do not fully understand how to quantize gravity [4, 5, 6], that is, how to implement diffeomorphism invariance at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The problems appear not only in fully quantum gravity, but also in the semiclassical approximation [7, 8] where only matter is quantized while gravity is treated classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The problems that appear are not only technical, but also conceptual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The three conceptual problems that 1 stand out are the problem of time in quantum gravity [9, 10, 11, 12], the cosmolog- ical constant problem [13, 14, 15, 16, 17], and the black hole information paradox [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' One possibility that potentially could help to resolve these conceptual problems is the idea that general relativity and its diffeomorphism invariance is emergent, rather than fundamental, while the underlying more fundamental theory rests on entirely different principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This idea can be realized in condensed-matter inspired theories such as induced gravity [33], as well as in string theory [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, there is no any direct experimental evidence for such a more fundamental theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Moreover, promising theoretical candidates such as string theory are still poorly understood in their most fundamental terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Consequently, it is very difficult to study the idea of emergent diffeomorphism invariance in realistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In this paper, therefore, we study this idea in toy models, similar to the toy models in [11, 9, 34] studied before in the context of the problem of time in quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In these models, the 4- dimensional spacetime diffeomorphism invariance of general relativity is replaced with a 1-dimensional diffeomorphism invariance realized as time-reparametrization invari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Even though such models cannot solve the problems of realistic 4-dimensional systems with gravity, it is hoped that such simple models can at least serve as a conceptual inspiration for dealing with more difficult realistic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 2 we first introduce a class of toy models without diffeomorphism invariance and then explain how 1-dimensional dif- feomorphism invariance emerges from conservation of energy, namely, as a way to implement the constraint that the classical system has definite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 3 we explain how the 1-dimensional diffeomorphism invariance leads to a toy version of the problem of time in quantum gravity, and how the problem resolves when one recalls that the diffeomorphism invariance is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Similarly, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 4 we explain how the 1-dimensional diffeomorphism invariance leads to a toy version of the cosmological constant problem, and how the problem resolves when one recalls that the diffeomorphism invariance is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Likewise, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 5 we find a solution of the constraint that in some aspects resembles the behavior in a black hole exterior, explain how the diffeomorphism invariance can be used to extend the solution to a region resembling the behavior in a black hole interior, and point out that the interior is actually unphysical because the diffeomorphism invariance is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The non-existence of the interior can be understood as a toy version of the black hole firewall [35, 36], which plays a key role in some approaches to solv- ing the black hole information paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 6 we briefly speculate how these toy models could perhaps be generalized to real 4-dimensional diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 7 we present a qualitative discussion of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 2 2 The model and emergent diffeomorphism invari- ance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='1 The model We study a system with N dynamical degrees of freedom described by the collective configuration variable q(t) = {q1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' , qN(t)}, the dynamics of which is described by the action A = � dt L(q, ˙q), (1) where the dot denotes the derivative with respect to time t and L(q, ˙q) = N � a=1 ma ˙q2 a 2 − V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (2) The canonical momenta are well defined pa = ∂L ∂ ˙qa = ma ˙qa, (3) so the Hamiltonian is H(q, p) = N � a=1 pa ˙qa − L = N � a=1 p2 a 2ma + V (q) (4) and can be interpreted as the energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The system can be treated either classically of quantum mechanically, in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particu- lar, quantization can be performed via canonical quantization and dynamics can be described by the Schr¨odinger equation H|ψ(t)⟩ = i¯h∂t|ψ(t)⟩ (5) as usual, where H is the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Since the action does not have any a priori gauge or diffeomorphism invariance, the quantization is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='2 Emergent diffeomorphism invariance Since the Hamiltonian H does not have an explicit time dependence, it is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In classical physics, this means that H has some definite constant value E of energy, so we can write it as H(q, p) = E, or H(q, p) = 0, (6) where H(q, p) ≡ H(q, p) − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (7) 3 In the configuration space, the fact that the Hamiltonian has the value E can be written as N � a=1 ma ˙q2 a 2 + V (q) − E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (8) If we imagine that (2) describes a whole Universe, then E is the energy of that Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The inhabitants of this Universe observe only one value of E, but the theory cannot say which one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' For the inhabitants of this Universe, the constant E is a fundamental constant the value of which can be determined from experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Since E appears as a fundamental constant, it seems natural to incorporate the value of this constant into an effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' One possibility is to incorporate the con- straint (8) into the action by adding the Lagrange multiplier term λ [� a ma ˙q2 a/2 + V (q) − E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, there is a much more interesting way to incorporate the constraint (8) into the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' We do that not by introducing a Lagrange multiplier λ, but by introducing a new configuration variable g(t) > 0 and replacing the action (1) with ˜A = � dt√g � N � a=1 ma ˙q2 a 2g − V (q) + E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (9) Since this action does not depend on time derivatives of g(t), the g(t) is not a dy- namical variable and the equation of motion for this variable is a constraint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' More precisely, the equation of motion δ ˜A/δg = 0 gives − 1 2√g � N � a=1 ma ˙q2 a 2g + V (q) − E � = 0, (10) which reduces to the constraint (8) if g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But what is the rational for taking g = 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The answer is that the action (9) has the property of diffeomorphism invariance which allows us to choose for g(t) any positive function we want, so g(t) = 1 is nothing but a convenient choice of “gauge”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Since this diffeomorphism invariance is crucial, let us explain it in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The g in (9) appears in two terms, which are proportional to dt√g, ˙q2 a g = dq2 a g dt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (11) Thus g appears either in the combination √gdt = � g dt2 or g dt2 = (√gdt)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This implies that the action is invariant under arbitrary transformations that keep dτ 2 ≡ g(t)dt2 (12) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The dτ 2 is very much analogous to the spacetime line element ds2 = gµν(x)dxµdxν in general relativity, so we see that g in (12) corresponds to g00 in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Likewise, 1/g corresponds to g00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Just like general relativity is invariant under arbitrary 4-dimensional spacetime diffeomorphisms xµ → x′µ = f µ(x) 4 which keep ds2 = gµν(x)dxµdxν invariant, the action (9) is invariant under arbitrary 1-dimensional time diffeomorphisms t → t′ = f(t) (13) which keep (12) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The invariance g dt2 = g′dt′2 implies that g transforms as g → g′ = � dt dt′ �2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (14) This 1-dimensional diffeomorphism invariance is also known in literature under the name time-reparametrization invariance [5, 12, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' To summarize, we have started from the action (1) without diffeomorphism invari- ance and, from the fact that energy has some constant value E in classical mechanics, derived the corresponding action (9) with 1-dimensional diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In this way, the 1-dimensional diffeomorphism invariance is emergent from classical energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='3 The constraint in the canonical form Now we want to develop some formal tools that will be used in further sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The action (9) can also be written as ˜A = � dt ˜L(q, ˙q, g) = � dt√g L(q, ˙q, g), (15) where L(q, ˙q, g) = N � a=1 ma ˙q2 a 2g − V (q) + E, ˜L(q, ˙q, g) = √gL(q, ˙q, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (16) The corresponding canonical momenta are ˜pa = ∂ ˜L ∂ ˙qa = ma ˙qa √g , pg = ∂ ˜L ∂ ˙g = 0, (17) so the Hamiltonian is ˜H(q, ˜p, g) = N � a=1 ˜pa ˙qa − ˜L = √g H(q, ˜p), (18) where H(q, ˜p) = N � a=1 ˜p2 a 2ma + V (q) − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (19) 5 The canonical equation of motion for pg is ˙pg = −∂ ˜H ∂g = − 1 2√gH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (20) However, in (17) we have seen that pg = 0, which implies ˙pg = 0, so (20) implies − 1 2√gH = 0, (21) which is identical to the constraint (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus, since g > 0, we see that the constraint (10), or (21), can also be written as the Hamiltonian constraint H(q, ˜p) = 0, (22) or equivalently ˜H(q, ˜p, g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (23) In the gauge g = 1, this reduces to the constraint (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 3 The problem of time in quantum gravity Seduced by the beauty and elegance of the action with 1-dimensional diffeomorphism invariance, one may be tempted to quantize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The problem is, how to implement the Hamiltonian constraint (22) in the quantum theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The most natural approach is to implement it as the constraint on physical states H(q, ˜p)|ψ⟩ = 0, (24) where H(q, ˜p) is the quantum operator obtained via standard canonical quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This constraint implies also ˜H(q, ˜p, g)|ψ⟩ = 0, (25) which is the quantum version of (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, the time evolution of the state should be described by the corresponding Schr¨odinger equation ˜H(q, ˜p, g)|ψ(t)⟩ = i¯h∂t|ψ(t)⟩, (26) so compatibility with (25) implies ∂t|ψ(t)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (27) Hence the state does not depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But we know that the real world, or even the toy world described by the toy model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='1, depends on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Where does the dependence on time come from, if the quantum state |ψ(t)⟩ does not depend on time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This is the toy version of the problem of time in quantum gravity [9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Within our model, it is not difficult to understand where the problem comes from and how it should be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In general, whenever a quantum system has a 6 well defined energy E, its wave function has trivial time dependence proportional to e−iEt/¯h, which is just a time-dependent phase without any physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' To have a genuine time-dependent state in quantum mechanics, the state must not have a well defined energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Instead, the state must be in a superposition of two or more different energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' So what is wrong with (25)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This quantum constraint originates from the classical action (9) in which the energy E is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In fact, the whole diffeomorphism invariance of (9) emerged from a desire to implement the classical value E of energy into the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' There is nothing wrong with it in classical physics, where energy indeed has a well defined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, requiring that the quantum system should also have a definite value of energy is wrong, because the energy of a quantum system is, in general, uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In other words, it is wrong to quantize the diffeomorphism invariant effective action (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' What needs to be quantized is the original action (1), which is not diffeomorphism invariant and leads to the proper Schr¨odinger equation (5) without the problem of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The emergent diffeomorphism invariance is only valid at the classical level, where energy is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' At the quantum level, where energy is uncertain, there is no diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' To conclude, the problem of time in the toy version of quantum gravity originates from taking the diffeomorphism invariance too seriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' When one takes into account that this invariance is only emergent at the classical level, while fundamental quantum theory does not have this invariance, the problem of time disappears in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 4 The cosmological constant problem Among the N degrees of freedom, let us suppose that Nheavy of them are “heavy” and the rest Nlight = N − Nheavy are “light”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' We call them “heavy” and “light” degrees because we assume that one can use a semiclassical approximation in which the Nheavy degrees are treated classically, while the rest Nlight of them are quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' For simplicity, we also assume that V (q) can be split as V (q) = Vheavy(qheavy) + Vlight(qlight), (28) where qheavy = {qb | b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' , Nheavy} are heavy degrees, and qlight = {qa | a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' , Nlight} are light degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus the classical constraint (10) can be written as − Nheavy � b=1 mb ˙q2 b 2g − Vheavy(qheavy) = Nlight � a=1 ma ˙q2 a 2g + Vlight(qlight) − E, (29) or more concisely − Hheavy = Hlight − E, (30) with a self-explaining notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This is a classical equation, but as we said, the idea is to treat it semi-classically, so that the light degrees are quantized while the heavy degrees are left classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus one replaces (30) with a semiclassical equation − Hheavy = ⟨ψ|Hlight|ψ⟩ − E, (31) 7 where ⟨ψ|Hlight|ψ⟩ is the mean value of the operator Hlight in the quantum state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Next suppose that Vlight(qlight) is the potential of Nlight harmonic oscillators Vlight(qlight) = Nlight � a=1 kaq2 a 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (32) Then the operator Hlight can be written in the usual quantum harmonic oscillator form Hlight = Nlight � a=1 ¯hωa � A† aAa + 1 2 � , (33) where ωa = � ka/ma, while A† a and Aa are the raising and lowering operators, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particular, in the quantum ground state defined by Aa|0⟩ = 0 we have ⟨0|Hlight|0⟩ = Nlight � a=1 ¯hωa 2 , (34) so the semiclassical equation (31) becomes − Hheavy = Nlight � a=1 ¯hωa 2 − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (35) By contrast, the ground state energy of the classical harmonic oscillator is zero, so the classical version of (35) is − Hheavy = −E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (36) But Nlight is supposed to be very large, after all this is the number of light degrees in the whole toy Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus, there is a large discrepancy between the classical equation (36) and the semiclassical equation (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The semiclassical equation (35) can also be written as − Hheavy = −Eeff, (37) where − Eeff = −E + Nlight � a=1 ¯hωa 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (38) The effective energy Eeff contains a very large contribution from the quantum zero- point energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Finally, suppose that the inhabitants of the toy Universe measure Eeff and find a value − Eeff ≪ Nlight � a=1 ¯hωa 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (39) Then it is the problem to explain why −Eeff is so small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' why is it much smaller than its natural value given by the right-hand side of (39)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 8 Clearly, this problem is analogous to the cosmological constant problem in semi- classical gravity [13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (30) multiplied with g − Hheavyg = Hlightg − Eg (40) is analogous to the 00-component of the Einstein equation which, in appropriate units, can be written as Gµν = Tµν + Λgµν, (41) where Gµν is the Einstein tensor depending only on gravitational degrees, Tµν is the energy-momentum tensor of matter, and Λ is the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In this anal- ogy, “heavy” degrees are analogous to the gravitational degrees, “light” degrees are analogous to the matter degrees, and the constant −E is analogous to the cosmo- logical constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In the semiclassical approximation one performs a quantization of matter while keeping gravity classical, so (41) is replaced with Gµν = ⟨Ψ|Tµν|Ψ⟩ + Λgµν, (42) the 00-component of which is analogous to (31) multiplied with g − Hheavyg = ⟨ψ|Hlight|ψ⟩g − Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (43) In particular, in the matter ground state |Ψ⟩ = |0⟩ one finds a very large quantum contribution analogous to (34), so there is a large discrepancy between the value of cosmological constant defined by the quantum ground state and the small value of cosmological constant found from cosmological observations [13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Within our model, it is not difficult to understand where the problem comes from and how it should be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In the diffeomorphism invariant action (9), the constant energy −E has physical consequences because it is coupled to g via the term proportional to √gE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This is analogous to the cosmological constant coupled to gravity via the term proportional to � | det gµν|Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' On the other hand, the action (1) with (2) is not diffeomorphism invariant and hence does not contain √g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' As a consequence, adding a constant E to the Lagrangian (2) does not have any physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In the corresponding quantum theory described by the Schr¨odinger equation (5), the Hamiltonian is shifted by a constant value −E, which changes the phase of the quantum state by an additional phase factor eiEt/¯h, which does not have any physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The quantum ground state energy further shifts this value from E to Eeff as given by (38), but the new phase factor eiEefft/¯h still does not have any physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Hence the conclusion is very similar to that in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The toy version of the cosmological constant problem originates from taking the diffeomorphism invariance too seriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' When one takes into account that this invariance is only emergent at the classical level, while fundamental quantum theory does not have this invariance, the toy cosmological constant problem disappears in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 9 5 Black hole and firewall 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='1 The model Consider a subsystem described by only two degrees of freedom q(t) = {x(t), y(t)}, and suppose that the subsystem is invariant under rotations in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Suppose also that E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Under these conditions, the action (9) reduces to ˜A = � dt√g �m( ˙x2 + ˙y2) 2g − V (x, y) � , (44) where V (x, y) = V (x2 +y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Due to the rotational symmetry, it is convenient to work in polar coordinates z = � x2 + y2, ϕ = arctgy x, (45) with ranges z ∈ [0, ∞), ϕ ∈ [0, 2π), (46) where the values ϕ = 0 and ϕ = 2π are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Note that z is the usual radial coordinate, but we denote it with z, rather than with r, for the reasons that will become clear later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus the action (44) can be written as ˜A = � dt√g �m( ˙z2 + z2 ˙ϕ2) 2g − V (z2) � , (47) and the corresponding constraint (10) reduces to m( ˙z2 + z2 ˙ϕ2) 2g + V (z2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (48) To get an interesting solution of the constraint, let us suppose that the potential V (z2) for small z has a form of an inverted harmonic oscillator V (z2) = −kz2 2 , (49) with k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus, assuming in addition that ϕ(t) = 0 and choosing the gauge g(t) = 1, (50) the constraint (48) finally reduces to m ˙z2 2 − kz2 2 = 0, (51) which is a differential equation for z(t) �dz(t) dt �2 = γ2z2(t), (52) where γ = � k/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' We will see that (52) describes a motion analogous to the radial motion of a particle around a black hole with a horizon at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='2 Analogy with a black hole The solution of the differential equation (52) is z(t) = z(0)e±γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (53) The solution z(t) = z(0)e−γt can be visualized as radial infalling towards z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The infalling exponentially slows down as z = 0 is approached, and it takes an infinite time t to reach z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Likewise, the solution z(t) = z(0)eγt is a time inversion of the infalling, it describes an escaping from small z towards z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, if it starts from z(0) = 0, then it can never escape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' it remains trapped at z(t) = 0 forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This behavior is very much analogous to infalling towards the black hole, or escaping from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particular, it takes an infinite time to reach the black hole horizon, from the point of view of observer staying at a fixed non-zero distance from the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Also, an object initially at the horizon can never escape from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' We see that the point z = 0 is analogous to the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Moreover, the analogy with black holes does not stop here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The solution (53) is obtained in the gauge (50), but the theory is diffeomorphism invariant under time reparametrizations (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus we can introduce a new time variable t′ defined im- plicitly by e−γt = 1 − γt′, (54) so the infalling solution z(t) = z(0)e−γt can be written as z(t(t′)) = z(0)[1 − γt′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (55) Now the point z = 0 is reached after a finite time t′ = 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Furthermore, the solution (55) can be extended to negative values of z (this is the reason why we denote it with z, rather than with r), reached at times t′ > 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This is analogous to the Kruskal extension (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' [1, 2, 3]) of the Schwarzschild solution in general relativity, where in appropriate spacetime coordinates a freely falling object reaches the horizon after a finite time and the Schwarzschild solution is extended beyond the horizon, thus describing not only the black hole exterior, but also its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Hence, the region of negative z in the toy model is analogous to the black hole interior behind the Schwarzschild horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='3 Effective spacetime The analogy above can also be made more explicit by introducing an effective space- time metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The constraint (52) can be written as γ2z2dt2 − dz2 = 0, which can be interpreted as motion of a relativistic massless particle in a spacetime with the effective metric ds2 eff = Ω(t, z)[γ2z2dt2 − dz2], (56) where Ω(t, z) > 0 is an arbitrary conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This effective metric has a horizon at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particular, the metric in the square bracket has the same form as the Rindler metric [37, 1] ds2 Rindler = a2z2dt2 − dz2, (57) 11 associated with an observer at z = 1/a accelerating with proper acceleration a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The Rindler horizon at z = 0 is known to have many similarities with the black hole horizon [37, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Since (56) has a coordinate singularity at z = 0, we want to see what happens with this singularity after the coordinate transformation (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' By applying (54) to (56), we get ds2 eff = Ω � γ2z2dt′2 (1 − γt′)2 − dz2 � , (58) which is still singular at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, the singular quantity g′ 00 = Ωγ2z2 (1 − γt′)2 (59) is in fact regular along the infalling trajectory (55), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' g′ 00 traj = Ωγ2z2(0) (60) is regular provided that the initial position obeys z(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' A standard way to completely remove the coordinate singularity at the horizon z = 0 is to introduce the new spacetime coordinates T = z shγt, Z = z chγt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (61) Indeed, an elementary calculus shows that dT 2 − dZ2 = γ2z2dt2 − dz2, so (56) can be written as ds2 eff = Ω[dT 2 − dZ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (62) In these coordinates the relativistic massless particle obeys dT 2 − dZ2 = 0, so the infalling solution is Z(T) = Z(0) − T, (63) which corresponds to (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Now we want to express the position of the horizon z = 0 in the T, Z coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Inserting z = 0 into (61) gives (T, Z) = (0, 0), if t is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But what about the limit t → ±∞?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In this limit (61) gives Z/T = ±1 for any z, including the limit z → 0, so the two lines Z = ±T are also consistent with z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus the horizon is the union of the point (T, Z) = (0, 0) (corresponding to finite t) and the lines Z = ±T (corresponding to t → ±∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But this union is simply the two lines Z = ±T, so we conclude that the horizon is the two lines Z = ±T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The line Z = T is the future horizon, which is characteristic for a black hole, while the line Z = −T is the past horizon, which is characteristic for a white hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus we see that the infalling solution (63) crosses the future horizon Z = T and extends beyond the future horizon, which corresponds to the extension beyond the analogue horizon z = 0 in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Finally note that the effective spacetime metric can be introduced not only for the potential (49), but also for any potential V (x, y) in (44), provided that it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The constraint resulting from (44) is m( ˙x2 + ˙y2) 2g + V (x, y) = 0, (64) 12 which in the gauge g = 1 can be written as − 2V (x, y) m dt2 − dx2 − dy2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (65) This can be interpreted as motion of a relativistic massless particle in a spacetime with the effective metric ds2 eff = Ω(t, x, y) � −2V (x, y) m dt2 − dx2 − dy2 � , (66) where Ω(t, x, y) > 0 is an arbitrary conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This metric has the relativis- tic signature (+ − −), provided that V (x, y) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Taking Ω = 1 for convenience and defining the effective “Newtonian” gravitational potential φgrav(x, y) through the standard relation [3] g00(x, y) = 1 + 2φgrav(x, y), (67) we see that the potentials V and φgrav are related as φgrav(x, y) = −V (x, y) m − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (68) The important message of (68) is that φgrav corresponds to −V , rather than to V as one might naively expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particular, we see that a repulsive potential V such as (49) corresponds to an attractive gravitational potential φgrav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='4 The firewall We have seen that the solution (55) can be extended to negative values of z, and that this extension is analogous to the extension of black hole behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, in the toy model, the extension is conceptually problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' How can the extension to negative values of z be compatible with the fact that the z-coordinate was restricted to non-negative values by definition, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (46)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The answer is that it cannot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Only non-negative values of z are physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The region of space with negative z does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The motivation for extension to negative values of z has arisen from (55), which, in turn, has arisen from a new time coordinate introduced in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But the original model (1) with (2) is not diffeomorphism invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' it does not allow arbitrary redefinitions of the time coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' From this point of view, the gauge (50) is not merely an arbitrary choice, but the correct physical value of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The negative values of z have arisen from taking the diffeomorphism invariance too seriously, while this invariance is just an emergent feature resulting from a formalism that encoded the classical value of energy E into the action, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The conclusion above that there is no region behind z = 0 is completely classi- cal, it does not involve any quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Nevertheless, a semiclassical version resembling Hawking radiation can also be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Suppose that two entangled particles are created at z > 0, one infalling and the other escaping, thus mimicking 13 the Hawking pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Suppose also that the potential V (z2), given by (49) for small z, is defined for all z ≥ 0 as V (z2) = � −kz2/2 for z ≤ z0 −V0 for z ≥ z0, (69) where z0 = � 2V0 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (70) This potential can be visualized as a flat valley at the constant potential −V0 for z > z0, with a hill of height V0, radius z0, and the top at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' It mimics a stationary black hole approximated with flat geometry for r ≥ r0, which is justified if r0 is much larger than the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' To mimic a non-stationary evaporating black hole, we modify (69) and (70) to V (z2, t) = � −k(t)z2/2 for z < z0(t) −V0 for z ≥ z0(t), (71) z0(t) = � 2V0 k(t), (72) where k(t) is an increasing function that, after a large but finite time t∗, becomes infinite k(t∗) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus the radius z0(t) shrinks and becomes zero at time t∗, which mimics the shrinking of the evaporating black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The information paradox can now be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The peak of the infalling wave packet follows approximately the classical trajectory (55), thus entering the region behind z = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' behind the top of the hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But at late times t > t∗ the potential is V (z2) = −V0, so there is no hill and hence no region behind the top of the hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' It looks as if the infalling particle disappears at late times, so the remaining escaping particle in the mixed state seems to contradict unitarity of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This is the toy version of the black hole information paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The solution of the paradox is that the region behind z = 0 never existed in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' As we said, the motivation for extension to negative values of z originated from (55), which, in turn, originated from introducing a new time coordinate in (54), which, however, is not allowed in the fundamental theory without diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Remarkably, the non-existence of the region behind z = 0 in the toy model has an analogy in black hole physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' With a motivation to resolve the black hole information paradox [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] in semiclassical gravity, it has been proposed that the black hole interior does not exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' the black hole horizon represents a physical boundary called firewall [35, 36, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The problem with the firewall is to reconcile it with standard classical general relativity, which predicts that the black hole interior exists, and that the horizon is not a physical boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But such a standard view of classical general relativity is a consequence of the 4-dimensional diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Alternatively, if the 4-dimensional diffeomorphism invariance in general relativity is emergent in a way similar to the emergence of the 1-dimensional diffeomorphism invariance in our toy model, then the 14 4-dimensional diffeomorphism invariance should not be taken too seriously even in the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' If so, then the existence of the black hole interior resulting from the Kruskal extension should not be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Such an alternative view of classical gravity, if correct, makes the firewall perfectly compatible with classical physics, which resolves the firewall problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Hence the conclusion is similar to that in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The toy version of the firewall problem originates from taking the diffeomorphism invariance too seriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' When one takes into account that this invariance is only emergent, while the funda- mental theory does not have this invariance, the toy firewall problem disappears in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 6 Towards emergent 4-dimensional diffeomorphism invariance The motivation for studying the toy models with 1-dimensional diffeomorphism invari- ance is to teach us something about the real 4-dimensional diffeomorphism invariance, namely, about real classical, semiclassical and quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' So the question is, how the ideas of the toy models can be generalized to 4-dimensional diffeomorphism invariance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Unfortunately, we do not have a full answer to that question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' A full answer would be tantamount to having a full theory of quantum gravity, which, of course, we do not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Nevertheless, inspired by the toy models, we sketch an idea how such a generalization might look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' What we present here can be thought of as a gist of a research program based on a series of educated guesses1, which at the current level is very far from a fully developed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Our starting point of view is that the spacetime curvature emerges from a massless spin-2 field [38, 39, 40, 41, 42], and not the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Roughly, this means that in the formula gµν(x) = ηµν + φspin-2 µν (x), (73) relating the curved spacetime metric gµν(x) to the flat Minkowski metric ηµν and the massless spin-2 field φspin-2 µν (x), the quantities on the right-hand side are more fundamental than that on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Philosophically, such a view complies much better with string theory than with loop quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In the fundamental theory, the formula (73) is expected to be valid only in some approximative sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' We assume that there is some fundamental action A[φ] without diffeomorphism invariance, where φ = φ(x) is a collective symbol for all fundamental dynamical fields φ = {φmatt, φspin-2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (74) Here φmatt are the usual “matter” fields of spins 0, 1 2 and 1, the field φspin-2 is the massless spin-2 field, and the ellipses are possible other fields beyond the Standard 1“Educated guess” is (supposed to be) a well balanced term, between the over-pretentious “con- jecture” and over-cynical “wishful thinking”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 15 Model of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The x denotes a spacetime position in 4 or more dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' From the action A[φ] one can derive the symmetrized energy-momentum tensor Tµν[φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x], which is conserved when the equations of motion δA/δφ(x) = 0 (75) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In classical physics the fields φ(x) attain some definite values Φ(x), where Φ(x) is a definite solution of (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Thus we can define Eµν(x) ≡ Tµν[Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x], (76) which is a generalization of the definite energy E appearing in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' For example, in a classical vacuum in Minkowski spacetime, the Eµν(x) may take the form Eµν(x) = −Ληµν, (77) where Λ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' But whatever the Eµν(x) is, in classical physics we can always write Tµν[φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x] − Eµν(x) = 0, (78) which is a generalization of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In some limit one expects that Tµν[φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x] can be decomposed as Tµν[φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x] = T matt µν [φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x] + T spin-2 µν [φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' x] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' (79) With this decomposition, (78) looks very much like the Einstein equation (41) written in the non-geometric spin-2 language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Now the idea is to think of (78) as a constraint derived from a new action ˜A[φ, g], where g(x) = {gµν(x)} is a symmetric tensor field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' By analogy with (9), one expects that the new action ˜A[φ, g] is diffeomorphism invariant, so that the diffeomorphism- covariant equation δ ˜A/δgµν(x) = 0 (80) reduces to (78) when the gauge for gµν is chosen appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' One also expects that, in a certain limit, the action ˜A[φ, g] reduces to the usual gravitational action with the matter term, the Einstein-Hilbert term, and the cosmological term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This is, roughly, how the 4-dimensional diffeomorphism is expected to emerge at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' However, the fundamental action that needs to be quantized in this scheme is A[φ], not ˜A[φ, g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' With this approach, it it easy to see that there is no problem of time in quantum gravity, simply because the fundamental action A[φ] does not have a Hamiltonian constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The Hamiltonian H derived from A[φ] does not need to vanish on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Likewise, there is no cosmological constant problem, in the sense that energy (asso- ciated with H) of the quantum ground state does not have physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Finally, the quantum time evolution defined by e−iHt/¯h is unitary, so all quantum pro- cesses, including Hawking radiation, are compatible with unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Nevertheless, at this level, it is not clear how exactly the information paradox associated with Hawk- ing radiation resolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Since the quantum theory lacks diffeomorphism invariance, the firewall scenario discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='4 scenario seems plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In the same spirit, 16 since quantum gravity is not fundamentally geometrical in this picture, inherently geometrical proposals involving wormholes, such as ER=EPR [43] and black hole is- lands [44], seem less plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Nevertheless, at the current level of understanding of the ideas sketched above, it is impossible to make definite precise claims about the quantum nature of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 7 Discussion and conclusion In this paper we have constructed toy versions of the problem of time in quantum gravity, of the cosmological constant problem, and of the black hole firewall problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Within the models, the problems originate from taking the 1-dimensional diffeomor- phism invariance too seriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This 1-dimensional diffeomorphism invariance, real- ized as time-reparametrization invariance, is emergent, rather than fundamental, and when one takes it into account the problems disappear in a rather natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The problem of time disappears because quantum energy is uncertain in the absence of fundamental time-reparametrization invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The cosmological constant problem disappears because a shift of energy by a constant does not have physical conse- quences in the absence of fundamental time-reparametrization invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The black hole firewall problem disappears because a firewall at the horizon may be completely compatible with classical physics when the diffeomorphism invariance is interpreted as emergent, rather than fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Note also that the physical irrelevance of vacuum energy in the context of the cos- mological constant problem is compatible with the Casimir effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' The description of Casimir effect in terms of vacuum energy is just an effective macroscopic description, while the fundamental microscopic origin of Casimir effect lies in van der Waals forces [45, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particular, it can be understood in terms of a toy model [47] similar to that of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In our toy models, the solutions of the problems of time and of the cosmological constant are rather generic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' the solutions do not depend on details of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In particular, even though the cosmological constant problem is discussed for quantum harmonic oscillators, the solution of the problem works in essentially the same way for any other interaction V (q) that leads to a non-zero quantum ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' By contrast, our solution of the toy black hole firewall problem is not so generic, it depends on details of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Perhaps different models could suggest totally different solutions of the black hole information paradox, without any hints for the existence of firewalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Or perhaps some models would describe classical states resem- bling black holes, but without any hints how to solve the information paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' More research is needed to better understand how the lack of fundamental diffeomorphism invariance may, or may not, help to solve the information paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' More importantly, it is not at all clear whether such toy 1-dimensional ideas can, and should, be generalized to the real 4-dimensional diffeomorphism invariance of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' 6 we have sketched how such a generalization might look like, but it is far from a fully developed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Nevertheless, the conceptual simplicity of solutions of the toy problems seems suggestive, so we believe that this conceptual 17 simplicity could at least serve as a source of inspiration for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' In any case, we believe that our analysis of the toy models with emergent diffeo- morphism invariance may influence how physicists think about general relativity at an intuitive level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' A change of intuition may also induce new technical results and, hopefully, contribute to better understanding of semiclassical and quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Acknowledgements The author is grateful to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Juri´c for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' This work was supported by the Ministry of Science of the Republic of Croatia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E3T4oBgHgl3EQfUAr7/content/2301.04448v1.pdf'} +page_content=' Misner, K.' metadata={'source': 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b/BdE2T4oBgHgl3EQfngiP/content/tmp_files/2301.04009v1.pdf.txt @@ -0,0 +1,1075 @@ +On the Complexity of the Two-Stage Majority Rule* +Yongjie Yang +Chair of Economic Theory, Saarland University, Saarb¨ucken, Germany +yyongjiecs@gmial.com +Abstract +Sequential voting rules have been extensively used in parliamentary and legislative decision making. After observing +that the prevalent successive and the amendment rules fail several fundamental axioms, Horan and Sprumont [2021] +proposed very recently a two-stage sequential rule which satisfies a variety of desirable properties. This paper examines +this rule by investigating the complexity of AGENDA CONTROL, COALITION MANIPULATION, POSSIBLE WINNER, +NECESSARY WINNER, and eight standard election control problems. Our study offers a comprehensive understanding +of the complexity landscape of these problems. +keywords: parameterized complexity, successive rule, amendment rule, two-stage majority rule, NP-hard, W[2]-hard +1 +Introduction +Exploring the complexity of strategic voting problems has been being a vibrant topic in computational social choice (see, +e.g., [7, 17, 22, 25, 33]). The motivation is that malicious strategic voting may undermine election results, and it is widely +believed that complexity could serve as a barrier against strategic actions [3, 4]. In particular, to what extent a voting rule +resists strategic voting has been commonly recognized as an important factor to valuate the applicability of the rule. Over +the past three decades, the complexity of many different strategic voting problems under numerous voting rules has been +established [5, 20]. Needless to say, as long as a new meritorious voting rule in terms of axiomatic properties has emerged, +comparing it with existent rules with respect to their resistance degree to strategic voting becomes of great importance. +This paper aims to complete the complexity landscape of several strategic voting problems under a sequential voting +rule proposed recently by Horan and Sprumont [26]. Taking into as input preferences of voters over candidates and an +agenda over candidates (a linear order specifying the priorities of candidates being considered during the decision-making +process), a sequential rule outputs one candidate as the winner. Sequential rules are exceedingly useful in parliamentary +and legislative decision making. So far, the successive and the amendment rules are among the two most popular sequential +rules used in many countries [34]. However, these two rules fail several fundamental axioms from a theoretical point of +view. This motivates Horan and Sprumont [26] to study a new rule called two-stage majority rule (TSMR), which has been +shown to satisfy a variety of desirable axiomatic properties many of which are failed by the successive and the amendment +rules. +The work of Horan and Sprumont [26] naturally raises the question of whether the newly proposed rule is comparable +to the successive and the amendment rules in terms of their resistance to strategic voting. This paper aims to answer this +question. In addition, we also study two winner determination problems in the setting where only partial information on +voters’ preferences are available. Our main contributions are as follows. +(1) We study the AGENDA CONTROL problem, which models the scenario where an external agent empowered to set +the agenda attempts to make a distinguished candidate the winner. +(2) We study the COALITION MANIPULATION problem in which a set of voters, called manipulators, aim to make a +distinguished candidate the winner by coordinating their votes. +(3) We study eight standard election control problems, namely, CCAV, CCDV, CCAC, CCDC, DCAV, DCDV, +DCAC, and DCDC, where “CC”/“DC” stands for “constructive control”/“destructive control”, the third letter +“A”/“D” stands for “adding”/“deleting”, and the last letter “V”/“C” stands for “voters”/“candidates”. These prob- +lems model the scenario where a powerful external agent aims to make a distinguished candidate the winner (con- +structive) or not the winner (destructive) by adding or deleting a limited number of voters or candidates. +(4) We study the POSSIBLE WINNER and the NECESSARY WINNER problems under TSMR. These two problems are +relevant to the setting where only partial information on the preferences of voters and agenda are known. POSSIBLE +WINNER consists in determining which candidates have positive chances to win at least one completion of the +partial input, and NECESSARY WINNER consists in determining which candidates necessarily win regardless of the +missing information. +*A preliminary version will appear in the proceedings of AAMAS 2023. +1 +arXiv:2301.04009v1 [cs.GT] 10 Jan 2023 + +(5) For the above problems, we offer a comprehensive (parameterized) complexity landscape. Particularly, for the eight +election control problems, we study both the special case where the given distinguished candidate p is the first one, +and the case where p is the last one in the agenda. We refer to Table 1 for a summary of our concrete results as well +as previous results for the successive rule and the amendment rule. +Table 1: A summary of the complexity of many voting problems under several sequential rules. Our main results are in +bold face. In the table, “first”, “last”, and “last” mean that the distinguished candidate is respectively the first one, the +last one, and not the last one in the agenda. P-results spanning two rows hold for the general case, i.e., that they hold +regardless of the position of the distinguished candidate in the agenda. In addition, m is the number of candidates, n is the +number of votes, nrg is the number of registered votes, and k is the solution size. +CCAV +CCDV +CCAC +CCDC +TSMR +first W[2]-h (k +nrg, Thm. 3) W[2]-h (k,n−k, Thms. 5, 6) W[2]-h (k, Thm. 9) +P (Thm. 10) +last W[2]-h (k +nrg, Thm. 4) W[2]-h (k,n−k, Thms. 7, 8) +immune (Cor. 1) +successive +[29, 45] +first +P +P +immune +W[1]-h (k, m−k) +last +W[1]-h (k +nrg) +W[2]-h (k) +W[2]-h (k) +P +amendment +first +W[1]-h (k +nrg) +W[1]-h (k) +immune +P +P +last +W[2]-h (k +nrg) +W[2]-h (k) +DCAV +DCDV +DCAC +DCDC +TSMR +last W[2]-h (k +nrg, Thm. 11) W[2]-h (k,n−k, Thms. 12, 13) P (Thm. 14) +P (Cor. 3) +last +P [4] +P [4] +successive +[29, 45] +first +P +P +W[2]-h (k) +immune +last +P +W[1]-h (k,m−k) +amendment +first +P +P +P +immune +last +W[1]-h (k) +W[2]-h (k) +P +AGENDA CONTROL COALITION MANIPULATION +POSSIBLE WINNER +NECESSARY WINNER +TSMR +P (Thm. 1) +P (Thm. 2) +NP-h (Thms. 16, 17) +P (Thm. 15) +successive +[8] +P +P +NP-h +P +amendment +P +P +NP-h +coNP-h +1.1 +Related Works +AGENDA CONTROL is arguably one of the most popular problems in the setting of sequential rules and has a long +history of study (see, e.g., [6, 32]). However, the complexity of AGENDA CONTROL was only first studied recently [8]. +It should be pointed out that the complexity of some analogous problems in the setting of knockout tournaments has +been studied earlier [1, 3, 4, 11, 30, 39, 40]. COALITION MANIPULATION is a natural generalization of the well-known +MANIPULATION problem [3], and was first studied by Conitzer, Sandholm, and Lang [12]. We refer to [5, 13, 36, 37, 38] +for detailed results on the complexity of this problem for many traditional rules (i.e., voting rules like Borda, Maximin, +etc., which do not need an agenda to determine the winner). The constructive control problems were first studied by +Bartholdi, Tovey, and Trick [4], and their destructive counterparts were initiated by Hemaspaandra et al. [24]. Heretofore +the complexity of these problems for many rules has been extensively investigated. We refer to the book chapters [5, 20] +for important progress by 2016, and refer to [19, 33, 42, 43, 44] for some recent new results. The complexity of POSSIBLE +WINNER and NECESSARY WINNER for the successive and the amendment rules has been studied by Bredereck et al. [8]. +These two problems for traditional voting rules were first studied by Konczak and Lang [28], and the complexity of the +problems for many rules has been subsequently established [9, 10, 41]. +1.2 +Organization +The remainder of the paper is organized as follows. In Section 2, we give the formal definitions of important notions used +in the paper. Then, in Section 3, we unfold our concrete results for the strategic problems including AGENDA CONTROL, +COALITION MANIPULATION, and the eight standard election control problems. Then, we study the POSSIBLE WINNER +and the NECESSARY WINNER problems in Section 4. Finally, Section 5 summarizes our results and layouts some topics +for future research. +2 +Preliminaries +We assume the reader is familiar with basic notions in graph theory, complexity theory, and parameterized complexity +theory [2, 14, 15, 35]. +2 + +Let [i] be the set of positive integers equal to or smaller than i. For a binary relation R, we often use xRy to denote +(x,y) ∈ R. +2.1 +Graphs +An undirected graph is a tuple G = (N,A) where N is a set of vertices and A is a set of edges. An edge between two +vertices v and v′ is denoted by {v,v′}. We use ΓG(v) to denote the set of neighbors of v in G, i.e., ΓG(v) = {v′ ∈ N : +{v,v′} ∈ A}. +A digraph is a tuple G = (N,A) where N is a set of vertices and A is a set of arcs. Each arc from a vertex a to a vertex b +is denoted by (a,b). The set of inneighbors of a vertex a in G is Γ− +G(a) = {b ∈ N : (b,a) ∈ A}, and the set of outneighbors +of a in G is Γ+ +G(a) = {b ∈ N : (a,b) ∈ A}. When it is clear which graph G is discussed, we drop the index G from the +notions. An oriented graph is a digraph so that between every two vertices there is at most one arc. +For a graph G (be it directed or undirected) and a subset S of vertices, the subgraph of G induced by S is denoted +by G[S]. +2.2 +Elections and Voting Rules +An election is a tuple (C,V) of a set of candidates C and a multiset of votes V where every ≻∈ V is defined as a linear +order over C. For two candidates c,c′ ∈ C, we say that c is ranked before c′ in a vote ≻ if c ≻ c′. In addition, we say that c +is ranked immediately before c′ if c ≻ c′ and there are no other candidates ranked between them. A vote ≻ specifies the +preference of a voter casting ≻ where a is preferred to b if a is ranked before b. For notational brevity, we sometimes +write a preference in the format of a sequence of candidates from the most preferred one to the least preferred one. For +instance, by saying a vote with the preference a b c, we mean that a is ranked before b, and b ranked before c in the vote. +An agenda ▷ is a linear order over C. For c ∈ C, we call candidates before c in ▷ the predecessors of c, and call +those after c the successors of c. A sequential rule τ maps each election (C,V) and an agenda ▷ to a single candidate +τ(C,V,▷) ∈ C, the winner. +For c,c′ ∈ C, we use nV(c,c′) to denote the number of votes in V ranking c before c′. We say c beats (resp. ties) c′ +with respect to V if nV(c,c′) > nV(c′,c) (resp. nV(c,c′) = nV(c′,c)). A candidate is a weak Condorcet winner if it is not +beaten by anyone else. In addition, a candidate is a Condorcet winner if it beats all the other candidates. The majority +graph of an election E = (C,V), denoted GE, is an oriented graph with the vertex set C, and there is an arc from c ∈ C to +c′ ∈ C if and only if nV(c,c′) > nV(c′,c). +• Two-stage majority rule (TSMR) This procedure takes two steps. Let G denote the majority graph of (C,V). +Moreover, let G1 be the subdigraph of G with only forward arcs with respect to ▷, i.e., G1 takes C as the vertex set +and there is an arc from c to c′ in G1 if and only if c▷c′ and there is an arc from c to c′ in G. Let C′ ⊆ C be the set +of candidates without inneighbors in G1. Then, the procedure returns the right-most candidate in C′ as the winner, +i.e., the c ∈ C′ such that c′ ▷c for all c′ ∈ C′ \{c}. +We also give the formal definitions of the successive and amendment rules as they are closely related to our discussions. +• Successive For a candidate c ∈ C and a subset C′ ⊆ C \ {c}, we say c beats C′ if there is a strict majority of votes +each of which ranks c before all candidates in C′. The successive winner is the first one who beats the set of all her +successors. +• Amendment This procedure takes |C| rounds, where each round determines a temporary winner. Precisely, the +winner of the first round is the first candidate in the agenda. The winner of round i where i ≥ 2 is determined as +follows. Let c be the winner of round i−1, and let c′ be the i-th candidate in the agenda. The winner of round i is c +if c beats c′, and is c′ otherwise. The amendment winner is the winner of the last round. +We note that the successive rule and the amendment rule have been also studied under several other names (cf. [6, 21]). +Example 1. Let C = {a,b,c,d}, and let V be a set of three votes respectively with the preferences b d c a, c a b d, and +a d b c. The majority graph of (C,V), three different agendas, and the winners under different rules and agendas are +shown below. For TSMR, arcs NOT in G1 (backward arcs with respect to ▷i) are drawn as dashed lines. +a +b +c +d +agenda ▷1 +a +b +c +d +agenda ▷2 +a +b +c +d +agenda ▷3 +▷1 +▷2 +▷3 +TSMR +a +b +a +successive +d +a +d +amendment +d +a +c +winners +The first and the last candidates in the agenda are somehow related to (weak) Condorcet winner, as summarized below. +Observation 1. For an election (C,V) and an agenda ▷ over C, the following hold. +3 + +(1) The first or the second candidate in ▷ is the amendment winner of (C,V) if and only if it is the Condorcet winner +of (C,V). +(2) The last one in ▷ is the TSMR winner of (C,V) if and only if it is a weak Condorcet winner of (C,V). +(3) If the successive winner of (C,V) is the first one in ▷, then the successive winner is also the Condorcet winner +of (C,V). +(4) If the first candidate in ▷ is the Condorcet winner of (C,V), then it is also the TSMR winner of (C,V). +(5) If the last one in ▷ is a weak Condorcet winner of (C,V), it is also the successive and the amendment winner +of (C,V). +(6) The converses of (3)–(5) do not necessarily hold. +2.3 +Other Useful Notions +Throughout the paper, unless stated otherwise, for a set S we use −→S to denote an arbitrary but fixed linear order over S. +Once such an −→S is used, ←−S denotes then the reverse of −→S . For S′ ⊆ S, we use −→S [S′] to denote −→S restricted to S′, and +use −→S \S′ to denote −→S [S\S′]. +2.4 +Problem Formulations +For a sequential voting rule τ, we study the following problems defined in [8]. +AGENDA CONTROL +Given: +An election (C,V) and a distinguished candidate p ∈ C. +Question: Is there an agenda ▷ over C so that p is the winner of (C,V,▷) with respect to τ, i.e., p = τ(C,V,▷)? +COALITION MANIPULATION +Given: +An election (C,V), a distinguished candidate p ∈ C, an agenda ▷ over C, and a positive integer k. +Question: Is there a multiset V ′ of k votes over C so that p = τ(C,V ∪V ′,▷)? +For a partial order R over a set X, a linear extension of R is a linear order over X containing R, i.e., a linear order R′ so +that (x,y) ∈ R implies (x,y) ∈ R′ for all x,y ∈ X. +A partial election is a tuple (C,V) where V is a multiset of partial orders over C. An election (C,V ′) is a completion of +a partial election (C,V) if V ′ and V are one-to-one correspondence so that every v′ ∈ V ′ is a linear extension of its image. +A partial agenda over C is a partial order over C. +POSSIBLE WINNER +Given: +A partial election (C,V), a distinguished candidate p ∈ C, and a partial agenda ▷ over C. +Question: Is there a completion (C,V ′) of (C,V) and a linear extension ▷′ of ▷ so that p = τ(C,V ′,▷)? +NECESSARY WINNER +Given: +A partial election (C,V), a distinguished candidate p ∈ C, and a partial agenda ▷ over C +Question: Is p the τ winner of every completion of (C,V,▷), i.e., p = τ(C,V ′,▷′) for all (C,V ′) being a completion +of (C,V) and ▷′ being a linear extension of ▷? +We also study eight standard control problems which are special cases of the following problems. +CONSTRUCTIVE MULTIMODE CONTROL +Given: +An election (C∪D,V ∪W) with a set C of (registered) candidates,1 a set D of unregistered candidates, a mul- +tiset V of registered votes, a multiset W of unregistered votes, a distinguished candidate p ∈ C, an agenda ▷ +over C∪D, and four integers kAV, kDV, kAC, and kDC such that kAV ≤ |W|, kDV ≤ |V|, kAC ≤ |D|, and kDC ≤ |C|. +Question: Are there V ′ ⊆V, W ′ ⊆W, C′ ⊆C\{p}, and D′ ⊆ D such that |V ′| ≤ kDV, |W ′| ≤ kAV, |C′| ≤ kDC, |D′| ≤ kAC, +and p wins ((C \C′)∪D′,(V \V ′)∪W ′,▷′) with respect to τ where ▷′ is ▷ restricted to (C \C′)∪D′? +In DESTRUCTIVE MULTIMODE CONTROL, we have the same input as CONSTRUCTIVE MULTIMODE CONTROL, +and are asked whether there are V ′, W ′, C′, and D′ as in the above definition such that p is not the τ winner of ((C \C′)∪ +D′,(V \V ′)∪W ′,▷′). +The eight standard control problems studied in the paper are special cases of CONSTRUCTIVE MULTIMODE CONTROL +and DESTRUCTIVE MULTIMODE CONTROL. The specifications of the eight standard control problems are summarized +in Table 2. +4 + +Table 2: Special cases of CONSTRUCTIVE/DESTRUCTIVE MULTIMODE CONTROL. Here, X is either CC standing for +constructive control or DC standing for destructive control. +problems +restrictions +XAV +kAC = kDC = kDV = 0, D = /0 +XAC +kDC = kAV = kDV = 0, W = /0 +XDV +kAC = kDC = kAV = 0, D = W = /0 +XDC +kAC = kAV = kDV = 0, D = W = /0 +For simplicity, when we study a problem in Table 2, we use k to denote the integer in the input not required to be 0, and +omit components in the input requested to be 0 or /0. For example, an instance of CCAV is written as ((C,V ∪W), p,▷,k), +where k represents kAV. +Our hardness results are based on reductions from the following problem. +RED-BLUE DOMINATING SET (RBDS) +Given: +A bipartite graph G with bipartition (R,B) where vertices in R and B are referred to as red vertices and blue +vertices respectively, and a positive integer κ ≤ |B|. +Question: Is there a subset B′ ⊆ B of cardinality κ that dominates R, i.e., |B′| = κ and every vertex in R has at least one +neighbor from B′ in the graph G? +RBDS is NP-hard [23], and from a parameterized complexity point of view it is W[2]-complete with respect to κ [16]. +2.5 +Remarks +Most previous studies make the assumption that there are no ties in elections (see, e.g., [26, 29]). Our results are presented +without this assumption, but all of them still hold when the no-tie assumption is made. This is clear for polynomial-time +solvability results. Regarding hardness results for voter control problems, some of our reductions can be slightly adapted +to show the same hardness if the no-tie assumption is adopted, and others directly apply to the case with the no-tie +assumption. We note that in these problems the no-tie assumption means that after the addition or the deletion of votes +there are no ties. All our other reductions directly apply to the case with the no-tie assumption, because in these reductions +the elections constructed do not admit ties and the feasible solutions do not remove the assumption. +All our reductions take polynomial time. Therefore, a problem shown to be W[2]-hard in the paper is also NP-hard. +We won’t explicitly state the NP-hardness in the corresponding theorems. +3 +Strategic Problems +In this section, we study the complexity of many strategic voting problems for TSMR. +3.1 +Agenda Control and Manipulation +We first present a P-algorithm for AGENDA CONTROL. +Theorem 1. Agenda Control for TSMR is in P. +Proof. Let I = ((C,V), p) be an instance of AGENDA CONTROL. Let G be the majority graph of (C,V). We construct an +agenda ▷ as follows. Let A = C \(Γ− +G(p)∪{p}) be the set of candidates which beat or tie with p with respect to V. We +fill all candidates from A in any arbitrary order before p in the agenda ▷. Then, we fill candidates from Γ− +G(p) into the +agenda iteratively as follows. First, let S = A. In each iteration we compute the set S′ = Γ+ +G(S), and fill candidates from S′ +in the subsequent |S′| positions in the agenda ▷ after those from S. Then, we update S := S∪S′. The iterations terminate +until S′ defined above turned out to be empty. +After the iterations terminate, if all candidates C are in the agenda ▷, p is the TSMR winner of (C,V) with respect +to ▷. Thus, in this case, we conclude that I is a Yes-instance. If, however, there are still some candidates not filled in the +agenda, we conclude that I is a No-instance. The reason is as follows. By the above iterations, in this case it holds that (1) +none of C \S is beaten by anyone from S, and (2) everyone in C \S beats p. Condition (2) entails everyone in C \S being +after p in the agenda. However, as long as this is the case, Condition (1) warrants the winning of someone from C\S. +For COALITION MANIPULATION, we have again a P-algorithm. +Theorem 2. Coalition Manipulation for TSMR is in P. +5 + +Proof. Let I = (C,V), p,▷,k) be an instance of COALITION MANIPULATION. Let B be the set of predecessors of p, and +let B′ be the set of successors of p in the agenda ▷. Let V ′ be the multiset of k votes with the same preference p −→ +B −→ +B′, +where −→ +B and −→ +B′ are respectively the linear orders over B and B′ consistent with ▷, i.e., −→ +B = ▷[B] and −→ +B′ = ▷[B′]. If p is +the TSMR winner of (C,V ∪V ′,▷), we conclude that I is a Yes-instance; otherwise, we conclude that I is a No-instance. +The algorithm clearly runs in polynomial time. It remains to prove its correctness. To this end, we assume that I is a +Yes-instance, and to complete the proof it suffices to show that I has a feasible solution V ′ so that every vote in V ′ has the +same preference p −→ +B −→ +B′. Observe first that I has a feasible solution where p is ranked in the first place in all votes. Let U +be a feasible solution of I where p is in the top in all votes in U. If U equals V ′ defined above, we are done. Otherwise, we +show below how to transform U into V ′ without destroying the feasibility of the solution. If there exists at least one vote +≻∈ U and two candidates b ∈ B and b′ ∈ B′ so that b′ is ranked immediately before b in ≻, we do the following. Let ≻′ +be the vote obtained from ≻ by swapping b and b′, and let U′ = U \{≻}∪{≻′}. It is easy to verify that every candidate +who is beaten by at least one of her predecessors with respect to V ∪U is also beaten by at least one of her predecessors +with respect to V ∪U′, and everyone who is beaten by p with respect to V ∪U is still beaten by p with respect to V ∪U′. +Therefore, p still wins after the swap of b and b′. After the swapping operations are exhaustively applied, we obtain a +feasible solution W of I where p is ranked in the top, and all candidates in B are ranked before all candidates in B′ in +very vote of W. If W = V ′, we are done. Otherwise, there exists at least one vote ≻∈ W such that one of the following +conditions holds: +• ∃a,b ∈ B s.t. a is ranked immediately before b in ≻ and b▷a; +• ∃a′,b′ ∈ B′ s.t. a′ is ranked immediately before b′ in ≻ and b′ ▷a′. +Then, analogous to the above discussion, we can swap a and b (resp. a′ and b′) in ≻ without changing the winning status +of p. After the swapping operations are exhaustively used, we eventually obtain V ′. +3.2 +Constructive Controls +In this section, we study constructive control problems for TSMR. We first present results for control by adding/deleting +votes. We show that these problems are W[2]-hard with respect to several meaningful parameters, for both the special +case where the distinguished candidate is the first one in the agenda and the case where the distinguished candidate is the +last one in the agenda. +Theorem 3. CCAV for TSMR is W[2]-hard with respect to the number of added votes plus the number of registered votes. +Moreover, this holds even when the distinguished candidate is the first one in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G = (R∪B,A),κ) be an instance of RBDS. We construct +an instance of CCAV for TSMR as follows. We create for each vertex in G a candidate denoted by the same symbol +for simplicity. In addition, we create a candidate p. Let C = B ∪ R ∪ {p}. The agenda is ▷ = (p,−→ +B ,−→ +R ). We create the +following registered votes: +• κ votes with the preference ←− +B ←− +R p; and +• one vote with the preference ←− +R p ←− +B . +Let V be the multiset of the above κ + 1 registered votes. We create |B| unregistered votes corresponding to B. In +particular, for each b ∈ B, we create one vote ≻b with the preference +p +�←− +R \ΓG(b) +� +b +�←− +R [ΓG(b)] +� �←− +B \{b} +� +. +Let W be the set of the above |B| unregistered votes. Finally, we set k = κ. The instance of CCAV for TSMR is +((C,V ∪W), p,▷,k). In the following we show the correctness of the reduction. +(⇒) Suppose that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R in G. Let W ′ = {≻b: b ∈ B′} be the +set of the κ unregistered votes corresponding to B′. We show below that p becomes the TSMR winner of the election +E = (C,V ∪W ′). Obviously, |V ∪W ′| = 2κ +1. As one of the registered votes ranks p before B, and all the κ votes in W ′ +rank p before B too, there are κ +1 votes in V ∪W ′ ranking p before B. So, none of B is the winner of E . Let us consider +a candidate r ∈ R. Note that there are κ registered votes which rank B before R. As B′ dominates R, there is at least one +b ∈ B′ so that r ∈ ΓG(b). By the definition of ≻b, b is ranked before r in ≻b. Therefore, there are in total κ +1 votes in +V ∪W ′ which rank b before r, precluding the winning of r. As this holds for all r ∈ R, and all candidates from B are before +all candidates from R in the agenda ▷, none of R is the winner either. This leaves only the possibility that p is the winner. +(⇐) Suppose that there exists a subset W ′ ⊆ W of at most κ votes so that p is the TSMR winner of (C,V ∪W ′). +Observe that W ′ must contain exactly κ votes since otherwise someone in B precludes p from winning. Observe that all +candidates in R beat p with respect to V ∪W ′ no matter which votes are contained in W ′. Furthermore, everyone in R +beats all her predecessors in R with respect to V ∪W ′. So, if p wins (C,V ∪W ′) it must be that every r ∈ R is beaten by +6 + +someone in B. This implies that for every r ∈ R, there is at least one vote in W ′ which ranks some b ∈ B before r. By the +construction of the unregistered votes, this vote must be ≻b such that b dominates r. it follows that B′ = {b ∈ B :≻b∈ W ′} +dominates R. This implies that the RBDS instance is a Yes-instance. +Now we consider the case where the distinguished candidate is the last one in the agenda. Recall that the last one +in the agenda is the TSMR winner if and only if it is a weak Condorcet winner (Observation 1). The W[1]-hardness of +CCAV for Condorcet winner established by Liu et al. [29] can be adapted to show the same hardness for weak Condorcet +winner2. We strengthen the result by establishing a W[2]-hard reduction, excluding the possibility of being complete +to W[1]. +Theorem 4. CCAV for TSMR is W[2]-hard with respect to the number of added votes plus the number of registered votes +even when the distinguished candidate is the last one in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G,κ) be an instance of RBDS, where G = (R∪B,A) is a +bipartite graph. We create an instance of CCAV as follows. The candidate set is C = R∪{p,q}. Let ▷ = (−→ +R ,q, p). We +create a multiset V of κ registered votes as follows: +• κ −1 votes with the preference q p −→ +R ; +• one vote with the preference q −→ +R p. +For each b ∈ B we create one unregistered vote ≻b with the preference +�−→ +R \ΓG(b) +� +p +�−→ +R [ΓG(b)] +� +q. For a given +B′ ⊆ B, let W(B′) = {≻b: b ∈ B} be the multiset of unregistered votes corresponding to B′. Let k = κ. The instance of +CCAV is ((C,V ∪W(B)), p,▷,k). It remains to show the correctness of the reduction. +(⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R. Let E = (C,V ∪W(B′)). We show that +the CCAV instance is a Yes-instance by showing that p is the TSMR winner of E . First, observe that p ties q in E . As B′ +dominates R, for every r ∈ R there is at least one b ∈ B′ which dominates r. This implies that in the vote ≻b∈ W(B′), p is +ranked before r, and hence p is not beaten by r in E . As p is the last one in the agenda, it follows that p wins E . +(⇐) Assume that there exists B′ ⊆ B such that |B′| ≤ k = κ and p is the TSMR winner of E = (C,V ∪W(B′)). +This means that p is not beaten by anyone else in E . Therefore, |B′| = k, since otherwise q beats p. It follows that +|V ∪W(B′)| = 2κ. Let r ∈ R. As we have exactly κ −1 registered votes ranking p before r in V, there is at least one b ∈ B′ +so that p is ranked before r in the vote ≻b. By the definition of ≻b, this implies that b dominates r. It follows that B′ +dominates R. Thus, the RBDS instance is a Yes-instance. +Let us move on to constructive control by deleting votes. In this case we have two natural parameters: the solution +size k and its dual parameter n−k where n is the number of votes. We show that the problem is W[2]-hard with respect +to both parameters, even when the distinguished candidate is the first or the last one in the agenda. The following four +theorems summarize these results. +Theorem 5. CCDV for TSMR is W[2]-hard with respect to the number of deleted votes even when the distinguished +candidate is the first one in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G,κ) be an instance of RBDS where G = (B ∪ R,A) is +a bipartite graph. We assume that G does not contain any isolated vertices, κ ≥ 4, and every red vertex is of degree ℓ +where ℓ ≥ 1. These assumptions do not change the W[2]-hardness of the problem. 3 We construct an instance of CCDV +as follows. The candidate set is C = R∪{p,q,q′}, and the agenda is ▷ = (p,q′,−→ +R ,q). We create the following six groups +of votes: +• a multiset V1 of ℓ+1 votes with the preference +q′ p q ←− +R ; +• a multiset V2 of κ +ℓ−2 votes with the preference +q p ←− +R q′; +• a multiset V3 of |B|−κ +1 votes with the preference +←− +R p q q′; +2For this, we mean the problem of determining if we can add a limited number of votes to make a particular candidate a weak Condorcet winner. +3The assumption that G does not contain any isolated vertices and κ ≥ 4 are clear. If an instance does not satisfy the second assumption, we can +obtain an equivalent instance by the following operation: letting ℓ be the maximum degree of vertices in R, for each red vertex r ∈ R of degree strictly +smaller than ℓ, we create new degree-1 vertices adjacent only to r until r has degree exactly ℓ. An important observation for the equivalency to the two +instances is that there is an optimal solution (a subset B′ ⊆ B dominating R with the minimum cardinality) of the new instance which does not contain +any of the newly introduced degree-1 vertices. +7 + +• a singleton V4 of one vote with the preference +←− +R q p q′; +• a multiset V5 of κ −2 votes with the preference +←− +R q′ p q; +• for every blue vertex b ∈ B, we create one vote ≻b with the preference +q q′ �←− +R [ΓG(b)] +� +p +�←− +R \ΓG(b) +� +. +Let V denote the multiset of the above 2|B| + κ + 2ℓ − 1 votes. For a given B′ ⊆ B, let V(B′) = {≻b: b ∈ B′} be the +multiset of votes created for vertices in B′. We complete the construction by setting k = κ. The instance of CCDV is +((C,V), p,▷,k) which can be constructed in polynomial time. It remains to show the correctness of the reduction. +(⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R. Let E = (C,V \V(B′)). We show below +that p is the TSMR winner of E with respect to the agenda ▷. To this end, it suffices to show that p beats everyone else +in E . Let r ∈ R. As B′ dominates R, there exists b ∈ B′ such that b dominates r, and thus ≻b ranks r before p. As there are in +total |B|−ℓ votes in V(B) ranking p before r, we know that there are at least |B|−ℓ−κ +1 votes in V(B)\V(B′) ranking p +before r. As all votes in V1 ∪V2 rank p before all candidates in R, there are at least |B|−ℓ−κ +1+ℓ+κ +ℓ−1 = |B|+ℓ +votes ranking p before r in E . As |V \V(B′)| = 2|B|+2ℓ−2, we know that p beats r in E . It is easy to verify that there +are |B|+ℓ votes ranking p before q and q′ in V \V(B′), meaning that p beats both q and q′ in E too. In summary, p beats +everyone else in the election E and hence is the winner of E . +(⇐) Assume that there exists V ′ ⊆ V such that |V ′| ≤ k = κ and p is the TSMR winner of E = (C,V \V ′). Observe +that by the construction of the votes and the assumption that κ ≥ 4, no matter which at most k votes are contained in V ′, +every candidate in C \ {p} beats all her predecessors in C \ {p}. Then, as p is the first candidate in the agenda and p +wins E , we know that p beats all the other candidates. It follows that V ′ and V1 ∪V3 ∪V5 are disjoint and |V ′| = κ, since +otherwise p cannot beat q in E . Similarly, it holds that V ′ and V2 ∪V4 are disjoint, since otherwise p cannot beat q′. As +a consequence, it holds that V ′ ⊆ V(B). Without loss of generality, let B′ ⊆ B be such that V(B′) = V ′. We claim that B′ +dominates R. Assume, for the sake of contradiction, that this is not the case. Let r ∈ R be a red vertex not dominated by +any vertex in B′. Then, by the construction of the votes, all votes in V(B′) rank p before r. This implies that there are in +total at most |B| − ℓ − κ + |V1 ∪V2| = |B| + ℓ − 1 votes ranking p before r in E . In other words, p is beaten by r in E . +However, in this case p cannot be the TSMR winner of E , a contradiction. +Theorem 6. CCDV for TSMR is W[2]-hard with respect to the number of votes not deleted even when the distinguished +candidate is the first one in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G,κ) be an instance of RBDS where G = (B ∪ R,A) +is a bipartite graph. As in the proof of Theorem 5, we assume that every red vertex has degree exactly ℓ for some +positive integer ℓ. We construct an instance of CCDV as follows. The candidate set is C = R∪{p,q}, and the agenda is +▷ = (p,−→ +R ,q). We create the following three groups of votes: +• a multiset V1 of κ votes with the preference p q ←− +R ; +• a singleton V2 of one vote with the preference ←− +R p q; +• for every blue vertex b ∈ B, one vote ≻b with the preference +q +�←− +R \ΓG(b) +� +p +�←− +R [ΓG(b)] +� +. +Let V denote the multiset of the above |B|+κ +1 votes. For a given B′ ⊆ B, we use V(B′) = {≻b: b ∈ B′} to denote the +multiset of votes corresponding to B′. We complete the construction by setting k = |B| − κ. The instance of CCDV is +((C,V), p,▷,k), which can be constructed in polynomial time. It remains to show the correctness of the reduction. +(⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R. Let E = (C,V1 ∪V(B′)). We show below +that p is the TSMR winner of E with respect to the agenda ▷. To this end, it suffices to show that p beats everyone else +in E . Let r ∈ R. As B′ dominates R, there is at least one b ∈ B′ such that b dominates r, and hence ≻b ranks p before r. +Therefore, in total there are κ + 1 votes in E ranking p before r. Clearly, there are κ + 1 votes in E ranking p before q. +As |V1 ∪V(B′)| = 2κ +1, we know that p beats all the other candidates in E , and hence p is the winner of E . +(⇐) Assume that there exists V ′ ⊆ V such that |V ′| ≤ k = |B| − κ and p is the TSMR winner of the election E = +(C,V \V ′). Observe first that V ′ ⊆ V(B) and |V ′| = k, since otherwise q is not beaten by any of her predecessors, leading +to q winning E , a contradiction. So, without loss of generality, let B′ ⊆ B be such that |B′| = k = |B|−κ and V(B′) = V ′. +Let B = B \ B′. Obviously, |B| = κ and |V \V ′| = 2κ + 1. By the construction of the votes, no matter which k votes +are contained in V(B′), everyone from C \ {p} beats all her predecessors in C \ {p}. As p is the first candidate in the +8 + +agenda, the winning of p in E implies that p beats all the other candidates. We claim that B dominates R. Assume, for the +sake of contradiction, that this is not the case. Let r ∈ R be a red vertex not dominated by any vertex in B. Then, by the +construction of the votes, all votes in V(B) rank r before p. As the only vote in V2 also ranks r before p, there are in total +|B|+1 = κ +1 votes ranking r before p in E , contradicting that p beats r in E . +Theorem 7. CCDV for TSMR is W[2]-hard with respect to the number of deleted votes. This holds even if the distin- +guished candidate is the last one in the agenda. +Proof. We prove the theorem by a reduction from RBDS. Let (G,κ) be an instance of RBDS, where G = (R∪B,A) is a +bipartite graph. We assume that G does not contain any isolated vertices, κ ≥ 4, and every red vertex is of degree ℓ where +ℓ ≥ 1. These assumptions do not change the W[2]-hardness of the problem.4 Let C = R∪{p,q}, and let ▷ be an agenda +over C where p is the last one (the relative orders of other candidates do not matter). We create the following 2|B|+2ℓ+κ +votes in V: +• |B|+1 votes with the preference ←− +R p q; +• ℓ+κ votes with the preference q p ←− +R ; +• ℓ−1 votes with the preference p q ←− +R ; and +• for each blue vertex b ∈ B, one vote ≻b with the preference +q +�←− +R [ΓG(b)] +� +p +�←− +R \ΓG(b) +� +. +For a given B′ ⊆ B, let V(B′) = {≻b: b ∈ B′} be the multiset of votes corresponding to B′. Finally, we set k = κ. The +instance of CCDV is ((C,V), p,▷,k). In the following, we prove the correctness of the reduction. +(⇒) Assume that there exists B′ ⊆ B of cardinality κ such that B′ dominates R. Let E = (C,V \V(B′)). Clearly, +|V \V(B′)| = 2|B| + 2ℓ. We show below that p is not beaten by anyone else in E and hence is the TSMR winner of E . +As all votes in V(B′) rank q before p, it holds that nV\V(B′)(p,q) = (|B| + 1) + (ℓ − 1) = |B| + ℓ, meaning that p ties q +in E . Moreover, as B′ dominates R, for every r ∈ R, there exists b ∈ B′ dominating r. By the construction of the votes, r is +ranked before p in the vote ≻b∈ V(B′). It follows that at most κ −1 votes in V(B′) rank p before r. By the construction +of the votes, we know that there are at least (ℓ + κ) + (ℓ − 1) + (|B| − ℓ) − (κ − 1) = |B| + ℓ votes ranking p before r +in V \V(B′), implying that p ties r in E . +(⇐) Assume there exists V ′ ⊆ V such that |V ′| ≤ k and p is the TSMR winner of E = (C,V \V ′) with respect to ▷. +As p is the last one in the agenda, it holds that p beats or ties everyone else in E . As a consequence, all votes in V ′ must +rank q before p and, moreover, it must be that |V ′| = k = κ, since otherwise p is beaten by q in E . There are two groups of +votes ranking q before p: those corresponding to the blue vertices, and those with the preference q p ←− +R . We may assume +that all votes in V ′ are from V(B). Indeed, if V ′ contained some vote with the preference q p ←− +R , we can obtain another +feasible solution V ′′ from V ′ by replacing this vote with any vote in V(B)\V ′. Let r ∈ R. As nV(r, p) = (|B|+1)+ℓ and +|V \V ′| = 2|B| + 2ℓ, we know that there is at least one vote ≻b∈ V ′ which ranks r before p. By the reduction, we know +that the vertex b corresponding to ≻b dominates r. It is clear now that B′ = {b ∈ B :≻b∈ V ′} dominates R, implying that +the RBDS instance is a Yes-instance. +Theorem 8. CCDV for TSMR is W[2]-hard with respect to the number of votes not deleted. This holds even when the +distinguished candidate is the last one in the agenda. +Proof. We prove the theorem by a reduction from RBDS. Let (G,κ) be an instance of RBDS, where G is a bipartite +graph with the vertex bipartition (R,B). We create an instance of CCDV as follows. Let C = R ∪ {q, p}. Let ▷ be an +agenda over C where p is in the last position. We create the following votes: +• a multiset V1 of κ −1 votes with the preference p q −→ +R ; +• a singleton V2 of one vote with the preference −→ +R p q; and +• for each blue vertex b ∈ B, one vote ≻b with the preference +q +�−→ +R \ΓG(b) +� +p +�−→ +R [ΓG(b)] +� +. +4The assumption that G does not contain any isolated vertices and κ ≥ 4 are clear. If an instance does not satisfy the second assumption, we can +obtain an equivalent instance by the following operation: letting ℓ be the maximum degree of vertices in R, for each red vertex r ∈ R of degree strictly +smaller than ℓ, we create new degree-1 vertices adjacent only to r until r has degree exactly ℓ. An important observation for the equivalency to the two +instances is that there is an optimal solution (a subset B′ ⊆ B dominating R with the minimum cardinality) of the new instance which does not contain +any of the newly introduced degree-1 vertices. +9 + +For a given B′ ⊆ B, we use V(B′) = {≻b: b ∈ B′} to denote the set of votes created for the blue vertices in B′. Let +V = V1 ∪V2 ∪V(B). Clearly, |V| = |B|+κ. Finally, let k = |B|−κ. The instance of CCDV is ((C,V), p,▷,k). We prove +the correctness as follows. +(⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R. Let V ′ = V1 ∪V2 ∪V(B′), and let +E = (C,V ′). We claim that p is the TSMR winner of E . As p is the last candidate in the agenda, it suffices to show that p +is not beaten by any other candidates in E . It is clear that p ties q in E . Let r ∈ R be a red vertex. As B′ dominates R, +there exists b ∈ B′ dominating r. From the construction of the votes, p is ranked before r in the vote ≻b. Therefore, there +are at least |V1|+1 = κ votes ranking p before r in V ′, implying that p is not beaten by r. As this holds for all r ∈ R, the +correctness for this direction follows. +(⇐) Assume that there exists V ′ ⊆ V so that |V ′| ≥ 2κ and p is the TSMR winner of (C,V ′). As |V1| + |V2| = κ +and all votes in V(B) rank q in the first place, it must be that (V1 ∪V2) ⊆ V ′ and V ′ contains exactly κ votes from V(B), +since otherwise q will be the winner of (C,V ′), contradicting the winning of p. Let V(B′) = V ′ ∩V(B), where B′ ⊆ B. +As just discussed, |V(B′)| = κ. We claim that B′ dominates R. Suppose for contradiction that this is not the case. Then, +there exists r ∈ R not dominated by any vertex in B′. From the construction of the votes, r is ranked before p in all votes +of V(B′). Together with the vote in V2, there are κ +1 votes in V ′ ranking r before p, meaning that r beats p. However, in +this case, p cannot be the winner of (C,V ′), a contradiction. As |B′| = κ, the RBDS instance is a Yes-instance. +Let us now explore the complexity landscape of constructive control by adding or deleting candidates. Unlike voter +controls, we have only one hardness result as stated in the following theorem. +Theorem 9. CCAC for TSMR is W[2]-hard with respect to the number of added candidates. This holds even when the +distinguished candidate is the first one in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G = (R∪B,A),κ) be an instance of RBDS. We construct +an instance of CCAC for TSMR as follows. For each vertex in G we create one candidate denoted by the same symbol +for notational simplicity. In addition, we create a distinguished candidate p. Let C = R∪{p} and let D = B. Besides, let +k = κ and let ▷ = (p,−→ +B ,−→ +R ). We create a multiset V of votes in some way so that +• everyone in R beats all her predecessors in R∪{p}; +• p beats everyone in B; and +• for each r ∈ R and each b ∈ B, if b dominates r in G, then b beats r; otherwise, r beats b. +By the famous McGarvey’s theorem [31] such votes can be constructed in polynomial time. The instance of CCAC +for TSMR is ((C ∪D,V), p,▷,k). +The correctness of the reduction is easy to see. In particular, if there exists B′ ⊆ B of κ vertices dominating R, then +after adding the candidates corresponding to B′, every r ∈ R has at least one predecessor from B′ who beats her, excluding +the winning of r. Candidates in B′ cannot win as they are beaten by p. Therefore, after adding these candidates, p becomes +the winner. If, however, the RBDS instance is a No-instance, no matter which at most k candidates from B are added, +there is at least one candidate in R who beats all her predecessors in the resulting election. In this case we cannot add at +most k candidates to make p the winner. +When the distinguished candidate is the last one in the agenda, we have the following corollary as a consequence of +Observation 1 and the immunity of weak Condorcet to CCAC [4]. +Corollary 1. If the distinguished candidate is the last in the agenda, TSMR is immune to CCAC. +For CCDC, a greedy P-algorithm can be easily obtained. +Theorem 10. CCDC for TSMR is in P. +Proof. Let I = ((C,V), p,▷,k) be an instance of CCDC. To solve I, we first remove all predecessors of p in ▷ who beat p +with respect to V. Then, we iteratively remove each successor c of p so that c is not beaten by any of her predecessors. +After the removals, p becomes the TSMR winner. We conclude that I is a Yes-instance if and only if at most k candidates +are removed in total. +3.3 +Destructive Controls +Now we start the exploration on destructive control problems. One may expect more tractability results, because destruc- +tive controls are generally easy to solve compared with their constructive counterparts. Nevertheless, let us start with a +hardness result. +Theorem 11. DCAV for TSMR is W[2]-hard with respect to the number of added votes plus the number of registered +votes. Moreover, this holds even when the distinguished candidate is the first one in the agenda. +10 + +Proof. We prove the theorem via a reduction from RBDS. Let (G = (R∪B,A),κ) be an instance of RBDS. We construct +an instance of DCAV for TSMR as follows. Let C = R∪{p,q} and let ▷ = (p,−→ +R ,q). We create the following registered +votes: +• κ −1 votes with the preference p q −→ +R . +• two votes with the preference p −→ +R q. +• one vote with the preference q p −→ +R . +Let V be the multiset of the above κ +2 registered votes. The unregistered votes are created according to B. In particular, +for each b ∈ B, we create one vote ≻b with the preference +�−→ +R \ΓG(b) +� +q p +�−→ +R [ΓG(b)] +� +. +For a given B′ ⊆ B, let W(B′) = {≻b: b ∈ B′} be the multiset of unregistered votes corresponding to B′. For simplicity, +let W = W(B) be the set of the above |B| unregistered votes. Let k = κ. The instance of DCAV is ((C,V ∪W), p,▷,k). +We prove the correctness of the reduction as follows. +(⇒) Suppose that there is a B′ ⊆ B of κ vertices which dominate R in G. Then, one can check that q beats or ties every +other candidate with respect to V ∪W(B′), implying that q is the winner of (C,V ∪W(B′)). Thus, in this case the instance +of DCAV is a Yes-instance. +(⇐) Suppose that there exists a subset W ′ ⊆W of at most k votes so that p is not the TSMR winner of E = (C,V ∪W ′). +Observe that no matter which at most k votes are contained in W ′, p beats all candidates in R, implying that the only +candidate which is able to preclude p from winning is q. As q is the last candidate in the agenda ▷, q is the winner if +and only if q beats or ties everyone else. This implies that W ′ contains exactly κ votes since otherwise p beats q in E . +Moreover, for each r ∈ R, at least one vote in W ′ ranks q before r. By the construction of the unregistered votes, an +unregistered vote ≻b ranks q before r if and only if b dominates r in G. This implies that the set of vertices corresponding +to W ′ dominates R, and hence the instance of RBDS is a Yes-instance. +It is known that DCAV and DCDV for weak Condorcet winner is polynomial-time solvable [24]. By Observation 1, +we have the following corollary. +Corollary 2 ([24]). DCAV and DCDV for TSMR are in P if the distinguished candidate is in the last position of the +agenda. +However, the complexity of DCDV increases if the distinguished candidate is not the last one in the agenda. +Theorem 12. DCDV for TSMR is W[2]-hard with respect to the number of deleted votes. This holds as long as the +distinguished candidate is not the last one in the agenda. +Proof. The reduction is the same as the one in the proof of Theorem 7 with only the difference that q is the distinguished +candidate. The correctness hinges upon the fact that no matter which at most k votes are deleted, q beats all candidates +in R, which leaves p the unique candidate preventing q from winning and, moreover, this holds as long as q is not the last +one in the agenda. +Parameterizing by the dual parameter of the solution size yields the same result. +Theorem 13. DCDV is W[2]-hard with respect to the number of votes not deleted. This holds as long as the distinguished +candidate is not the last one in the agenda. +Proof. The reduction is the same as the one in the proof of Theorem 8 with only the difference that q is the distinguished +candidate. The correctness arguments are the same as in the proof of Theorem 12. +For destructive control by modifying candidates, we have polynomial-time solvability results, regardless of the posi- +tion of the distinguished candidate in the agenda. +Theorem 14. DCAC for TSMR is in P. +Proof. Let I = ((C ∪D,V), p,▷,k) be an instance of DCAC. We assume that k ≥ 1 and p is the winner of (C,V), since +otherwise I can be solved trivially. Our algorithm goes as follows. +As p wins (C,V), p is not beaten by any of her predecessors, and each successor c ∈ C \ {p} of p is beaten by at +least one of c’s predecessors. If there exists c ∈ D which is before p in the agenda and beats p, we conclude that I is a +Yes-instance because p does not win (C ∪{c},V). Additionally, if there exists c ∈ D so that p▷c, and c is not beaten by +any of her predecessors in C, we also determine I to be a Yes-instance, since p does not win (C ∪{c},V). If neither of the +two cases occurs, then no matter which unregistered candidates are added, p remains the winner. Therefore, in this case, +we conclude that I is a No-instance. +The following result is a consequence of Theorem 10. +Corollary 3. DCDC for TSMR is in P. +11 + +4 +Possible and Necessary Winner +In this section, we study NECESSARY WINNER and POSSIBLE WINNER for TSMR. Bredereck et al. [8] showed that +except NECESSARY WINNER for the successive rule which is polynomial-time solvable, other cases of the two problems +for the successive and the amendment rules are computationally hard (NP-hardness for POSSIBLE WINNER and coNP- +hardness for NECESSARY WINNER). We show below that TSMR behaves the same as the successive rule in terms +their complexity of determining possible and necessary winners, though the proofs for these results for the two rules are +different. +Theorem 15. Necessary Winner for TSMR is in P. +Proof. Let I = ((C,V), p,▷) be an instance of NECESSARY WINNER. We determine if there is a completion of (C,V) +and a completion of the agenda ▷ so that p is not the TSMR winner of the completion. Note that p is not the winner if +and only if +(1) either some of her predecessor beats her, +(2) or some of her successor c is not beaten by any of the predecessors of c. +We consider first if there is a completion leading to the occurrence of Case 1. For this purpose, let B = {c ∈ C \{p} : +(p,c) ̸∈ ▷} be the set of all candidates that can be predecessors of p in some completion of ▷. We consider candidates +in B one by one, and for each considered c ∈ B, we greedily complete the preference profile to determine if there exists at +least one completion so that c beats p. More precisely, for every partial vote ≻∈ V such that (p,c) ̸∈≻, we complete it so +that c is ranked before p. If in the completion of (C,V) obtained this way c beats p, we conclude that I is a No-instance. +If we cannot draw the conclusion that I is a No-instance above, we consider whether it is possible to male the second +case happen. To this end, we enumerate all candidates which can be successors of p in some completion of the partial +agenda. More precisely, these candidates are those in B′ = {c ∈ C \ {p} : (c, p) ̸∈ ▷}. For each enumerated c ∈ B′, we +compute the minimum set Ac of candidates that can be successors of c under the restriction that p is before c in the agenda, +and then we greedily complete the preference profile to check if they can be completed so that c is not beaten by anyone +in Ac. More precisely, for each enumerated c ∈ B′, we compute Ac = {c′ ∈C : (c′,c) ∈ ▷}, and for each partial vote ≻∈V, +we complete ≻ so that c is ranked as higher as possible, i.e., we complete ≻ so that c is ranked below all candidates in +{c′ ∈ C : (c′,c) ∈≻} and is above all the other candidates. If in the completion c is not beaten by anyone from Ac, we +conclude that I is a No-instance. +If none of the above enumerations provides us a conclusion that I is a No-instance, we conclude that I is a Yes- +instance. +Unlike the above problems, we show that POSSIBLE WINNER becomes NP-hard. +Theorem 16. Possible Winner for TSMR is NP-hard, even if the given agenda is complete and the distinguished candidate +is the first one in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G,κ) be an instance of RBDS where G is a bipartite +graph with the partition (B,R). We assume that G does not contain any isolated vertices, and all vertices in R have the +same degree ℓ where ℓ ≥ 1. We create an instance of POSSIBLE WINNER for TSMR as follows. Let C = R∪{p,q} and +let ▷ = (p,q,−→ +R ). We create five groups of votes as follows, where only the first group of votes are incomplete: +• for each b ∈ B, one partial vote ≻b with the following partial preference +�←− +R [ΓG(b)] +� +p +�←− +R \ΓG(b) +� +and +q +�←− +R \ΓG(b) +� +; +• a multiset V1 of |B| votes with the preference ←− +R q p; +• a multiset V2 of 2ℓ+κ votes with the preference q ←− +R p; +• a multiset V3 of ℓ+2κ +1 votes with the preference ←− +R p q; +• a multiset V4 of ℓ+κ votes with the preference p q ←− +R . +Let V(B) = {≻b: b ∈ B} be the set of the |B| partial votes in the first group. Let V be the multiset of the above 2|B| + +4ℓ + 4κ + 1 votes, and let V(B) = V \V(B). The instance of POSSIBLE WINNER is ((C,V), p,▷). Clearly, the above +construction can be done in polynomial time. We show below that the RBDS instance is a Yes-instance if and only if the +constructed POSSIBLE WINNER instance is a Yes-instance. +(⇒) Suppose that there is a subset B′ ⊆ B such that |B′| = κ and B′ dominates R. We complete each ≻b where b ∈ B +as follows: +• if b ∈ B′, we complete it as q +�←− +R [ΓG(b)] +� +p +�←− +R \ΓG(b) +� +, +12 + +• otherwise, we complete it as +�←− +R [ΓG(b)] +� +p q +�←− +R \ΓG(b) +� +. +It is fairly easy to verify that with respect to the completion p beats q, and q beats all candidates in R. Then, by the +definition of the agenda, p is the TSMR winner with respect to the above completion of (C,V). +(⇐) Suppose that there is a completion V ′ of V(B) so that p wins the completion E = (C,V(B) ∪V ′) of (C,V). +Observe that in all completions of (C,V), everyone in R beats all her predecessors in R ∪ {p}. Then, by the definition +of the agenda, and the fact that p wins E , it holds that (1) q beats all candidates in R, and (2) q is beaten by p in E . +As V(B) contains exactly 2ℓ+3κ +1 votes (those in V3 ∪V4) ranking p before q, Condition (2) implies that there are at +least |B| − κ votes in V ′ ranking p before q. Let B′ be the subset of B corresponding to votes in V ′ ranking p before q, +and let B′′ = B \ B′. Clearly, |B′′| ≤ κ. We show below that Condition (1) implies that B′′ dominates R. For the sake +of contradiction, assume that there exists r ∈ R not dominated by any vertex in B′′. In other words, all the ℓ neighbors +of r in G are contained in B′. This implies that there are ℓ votes in V ′ (the ℓ completions of votes corresponding to the ℓ +neighbors of r) ranking r before q. Together with the |B|+ℓ+2κ +1 votes (V1 ∪V3) in V(B) ranking r before q, we have +|B| + 2ℓ + 2κ + 1 votes ranking r before q, implying that r beats q in E . However, this is impossible since otherwise r +beats all her predecessors in E which contradicts that p wins E . This completes the proof that B′′ dominates R. Then, +from |B′′| ≤ κ, we know that the RBDS instance is a Yes-instance. +Our reduction in the proof of Theorem 16 is completely different from those used in [8] for showing the NP-hardness +of POSSIBLE WINNER for the successive and the amendment rules. In fact, their reductions are from the INDEPENDENT +SET and VERTEX COVER problems, while our reduction is from RBDS. Moreover, in their reductions for POSSIBLE +WINNER under the successive and the amendment rules the distinguished candidate is respectively the penultimate and +the third candidates in the agenda. Our reduction can be adapted to show the NP-hardness of POSSIBLE WINNER for +TSMR when the distinguished candidate is the i-th candidate in the agenda for every constant i, by adding i−1 dummy +candidates before p in the agenda, and ranking all of them below all the other candidates in all votes. +Notice that POSSIBLE WINNER for TSMR becomes polynomial-time solvable if the given agenda is complete and p +is the last one in the agenda. This follows from Observation 1 and the polynomial-time solvability of determining if a +partial election can be completed so that a candidate becomes a (weak) Condorcet winner [28].5 By Observation 1, the +result in [28] also implies that POSSIBLE WINNER for the amendment rule becomes polynomial-time solvable if the given +agenda is complete and p is in the top-2 positions , and their algorithm also applies to the determination for the winning +of a particular candidate as a weak Condorcet winner. So, there is a radical complexity shift for the amendment rule as +the distinguished candidate moves from the second place to the third place in the agenda. Our next result also reveals a +seamless complexity shift for TSMR as p moves from the last position just one position up. +Theorem 17. Possible Winner for TSMR is NP-hard even when the given agenda is complete with the distinguished +candidate being the penultimate candidate in the agenda. +Proof. We prove the theorem via a reduction from RBDS. Let (G,κ) be an instance of RBDS where G = (B ∪ R,A) is +a bipartite graph and 1 ≤ κ ≤ |B|. Similar to the previous proofs, we assume that every red vertex has degree exactly ℓ +where ℓ > 0 in the graph G. We construct an instance of POSSIBLE WINNER as follows. Let C = R ∪ {p,q,q′} and let +▷ = (q′,−→ +R , p,q). We create five groups of votes where only the first group of contains partial votes. +• For every b ∈ B, we create one partial vote ≻b with the following partial preference +�−→ +R \ΓG(b) +� +q′ +and +q p +�−→ +R [ΓG(b)] +� +. +Let V(B) be the set of the |B| partial votes corresponding to B. +• We create a multiset V1 of |B|+1 votes with the preference +q′ q −→ +R p. +• We create a multiset V2 of 2κ votes with the preference +q p −→ +R q′. +• We create a multiset V3 of κ votes with the preference +q p q′ −→ +R . +5The result in [28] is for Condorcet winner but the algorithm also accommodates weak Condorcet winner. +13 + +• Finally, we create a multiset V4 of κ votes with the preference +−→ +R p q′ q. +Let V be the multiset of the above 2|B| + 4κ + 1 votes, and let V(B) = V \V(B). The instance of POSSIBLE WINNER +is ((C,V), p,▷) which can be constructed in polynomial time. In the following, we prove that the RBDS instance is a +Yes-instance if and only if the constructed instance of POSSIBLE WINNER is a Yes-instance. +(⇒) Suppose that there is a subset B′ ⊆ B such that |B′| = κ, and B′ dominates R. We complete each vote ≻b∈ V(B) +as follows. +• if b ∈ B′, we complete it as +�−→ +R \ΓG(b) +� +q′ q p +�−→ +R [ΓG(b)] +� +, +• otherwise, we complete it as +q p +�−→ +R \ΓG(b) +� +q′ �−→ +R [ΓG(b)] +� +. +It is easy to verify that after completing votes as above, p beats all her predecessors in ▷, and q is beaten by her predeces- +sor q′, which implies that p is the TSMR winner of the completion. +(⇐) Assume that there is a completion V ′ of V(B) so that p wins the election E = (C,V(B) ∪V ′). Observe that no +matter how we complete the votes, q beats all her predecessors except q′. As p wins E , it must be that q′ beats q in E . +This implies that there are at least κ partial votes in V(B) which are completed so that q′ is ranked before q. There is only +one such completion for each partial vote ≻b∈ V(B), i.e., the completion with the preference +�−→ +R \ΓG(b) +� +q′ q p +�−→ +R [ΓG(b)] +� +. +Let B′ ⊆ B be such that the partial votes corresponding to B′ are completed this way. As just discussed, |B′| ≥ κ. Without +loss of generality, let us assume that |B′| = κ +t for some nonnegative integer t. Observe further that as p wins E and |V| +is odd, p beats all candidates in R. For every r ∈ R, there are in total 3κ votes in V(B) (precisely, votes in V2 ∪V3) which +rank p before r. This implies there are at least |B| − κ + 1 completions of partial votes in V(B) which rank p before r. +Then, from |B \ B′| = |B| − κ −t, it follows that there are at least t + 1 completions of partial votes corresponding to B′ +where p is ranked before r. By the definitions of these completions, p is ranked before r in a completion corresponding +to some b ∈ B′ if and only if r is a neighbor of b in G. Therefore, every r ∈ R has at least t + 1 neighbors in B′ in the +graph G. Then, by removing any arbitrary t vertices from B′, we obtain a κ-subset of B that dominate R, and hence the +RBDS instance is a Yes-instance. +It would be interesting to see if similar complexity shift also applies to the successive rule. This mounts to determining +the complexity of POSSIBLE WINNER for the successive rule when the agenda is compete with the distinguished candidate +being the last one. We leave it as an open question. +5 +Conclusion +We conducted the (parameterized) complexity of many well-motivated voting problems under the recently proposed voting +rule TSMR, with respect to the solution size and the dual parameters. We obtained fruitful results including polynomial- +time solvability results, NP-hardness results, W[1]-hardness results, and W[2]-hardness results. Particularly, many of our +hardness results hold even when the distinguished candidate is the first or the last one in the agenda. Our exploration +offers a complete picture of the complexity of these problems under TSMR, enabling us to compare TSMR with the +successive and the amendment rules. See Table 1. Our results indicate that TSMR resists most of the control problems, +but is vulnerable to agenda control and coalition manipulation. In addition, we showed that NECESSARY WINNER is +polynomial-time solvable while POSSIBLE WINNER turned out to be NP-hard. Compared with previous works, our study +suggests that TSMR behaves at least well as the other two important sequential rules regarding their resistance to strategic +voting problems, and their complexity of calculating possible and necessary winners. We point out that our exploration +is a pure theoretic analysis, and whether many problems are hard to solve in specific practical settings demands further +investigation. For more details, we refer to Table 1. +An important topic for future research is to investigate if restricting the preference domains (e.g., single-peaked/crossing +preferences, top-monotonicity preferences, etc.) radically changes the complexity. We refer to [18, 27] for a comprehen- +sive survey on many restricted preference domains. +14 + +References +[1] Aziz, H., Gaspers, S., Mackenzie, S., Mattei, N., Stursberg, P., Walsh, T.: Fixing balanced knockout and double +elimination tournaments. Artif. Intell. 262, 1–14 (2018) +[2] Bang-Jensen, J., Gutin, G.Z. (eds.): Classes of Directed Graphs. Springer Monographs in Mathematics. Springer +(2018) +[3] Bartholdi III, J.J., Tovey, C.A., Trick, M.A.: The computational difficulty of manipulating an election. Soc. Choice +Welfare 6(3), 227–241 (1989) +[4] Bartholdi III, J.J., Tovey, C.A., Trick, M.A.: How hard is it to control an election? Math. Comput. 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In: AAMAS, pp. 1768–1770 (2022) +16 + diff --git a/BdE2T4oBgHgl3EQfngiP/content/tmp_files/load_file.txt b/BdE2T4oBgHgl3EQfngiP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4b12214f8db813ac40c629e2b3224cba8ad8f7d --- /dev/null +++ b/BdE2T4oBgHgl3EQfngiP/content/tmp_files/load_file.txt @@ -0,0 +1,1036 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf,len=1035 +page_content='On the Complexity of the Two-Stage Majority Rule* Yongjie Yang Chair of Economic Theory, Saarland University, Saarb¨ucken, Germany yyongjiecs@gmial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='com Abstract Sequential voting rules have been extensively used in parliamentary and legislative decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' After observing that the prevalent successive and the amendment rules fail several fundamental axioms, Horan and Sprumont [2021] proposed very recently a two-stage sequential rule which satisfies a variety of desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This paper examines this rule by investigating the complexity of AGENDA CONTROL, COALITION MANIPULATION, POSSIBLE WINNER, NECESSARY WINNER, and eight standard election control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our study offers a comprehensive understanding of the complexity landscape of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' keywords: parameterized complexity, successive rule, amendment rule, two-stage majority rule, NP-hard, W[2]-hard 1 Introduction Exploring the complexity of strategic voting problems has been being a vibrant topic in computational social choice (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', [7, 17, 22, 25, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The motivation is that malicious strategic voting may undermine election results, and it is widely believed that complexity could serve as a barrier against strategic actions [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In particular, to what extent a voting rule resists strategic voting has been commonly recognized as an important factor to valuate the applicability of the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Over the past three decades, the complexity of many different strategic voting problems under numerous voting rules has been established [5, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Needless to say, as long as a new meritorious voting rule in terms of axiomatic properties has emerged, comparing it with existent rules with respect to their resistance degree to strategic voting becomes of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This paper aims to complete the complexity landscape of several strategic voting problems under a sequential voting rule proposed recently by Horan and Sprumont [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Taking into as input preferences of voters over candidates and an agenda over candidates (a linear order specifying the priorities of candidates being considered during the decision-making process), a sequential rule outputs one candidate as the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Sequential rules are exceedingly useful in parliamentary and legislative decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' So far, the successive and the amendment rules are among the two most popular sequential rules used in many countries [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, these two rules fail several fundamental axioms from a theoretical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This motivates Horan and Sprumont [26] to study a new rule called two-stage majority rule (TSMR), which has been shown to satisfy a variety of desirable axiomatic properties many of which are failed by the successive and the amendment rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The work of Horan and Sprumont [26] naturally raises the question of whether the newly proposed rule is comparable to the successive and the amendment rules in terms of their resistance to strategic voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This paper aims to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, we also study two winner determination problems in the setting where only partial information on voters’ preferences are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our main contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (1) We study the AGENDA CONTROL problem, which models the scenario where an external agent empowered to set the agenda attempts to make a distinguished candidate the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (2) We study the COALITION MANIPULATION problem in which a set of voters, called manipulators, aim to make a distinguished candidate the winner by coordinating their votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (3) We study eight standard election control problems, namely, CCAV, CCDV, CCAC, CCDC, DCAV, DCDV, DCAC, and DCDC, where “CC”/“DC” stands for “constructive control”/“destructive control”, the third letter “A”/“D” stands for “adding”/“deleting”, and the last letter “V”/“C” stands for “voters”/“candidates”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' These prob- lems model the scenario where a powerful external agent aims to make a distinguished candidate the winner (con- structive) or not the winner (destructive) by adding or deleting a limited number of voters or candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (4) We study the POSSIBLE WINNER and the NECESSARY WINNER problems under TSMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' These two problems are relevant to the setting where only partial information on the preferences of voters and agenda are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' POSSIBLE WINNER consists in determining which candidates have positive chances to win at least one completion of the partial input, and NECESSARY WINNER consists in determining which candidates necessarily win regardless of the missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A preliminary version will appear in the proceedings of AAMAS 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='04009v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='GT] 10 Jan 2023 (5) For the above problems, we offer a comprehensive (parameterized) complexity landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Particularly, for the eight election control problems, we study both the special case where the given distinguished candidate p is the first one, and the case where p is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We refer to Table 1 for a summary of our concrete results as well as previous results for the successive rule and the amendment rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Table 1: A summary of the complexity of many voting problems under several sequential rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our main results are in bold face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In the table, “first”, “last”, and “last” mean that the distinguished candidate is respectively the first one, the last one, and not the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' P-results spanning two rows hold for the general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', that they hold regardless of the position of the distinguished candidate in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, m is the number of candidates, n is the number of votes, nrg is the number of registered votes, and k is the solution size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCAV CCDV CCAC CCDC TSMR first W[2]-h (k +nrg, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3) W[2]-h (k,n−k, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 5, 6) W[2]-h (k, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 9) P (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 10) last W[2]-h (k +nrg, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 4) W[2]-h (k,n−k, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 7, 8) immune (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 1) successive [29, 45] first P P immune W[1]-h (k, m−k) last W[1]-h (k +nrg) W[2]-h (k) W[2]-h (k) P amendment first W[1]-h (k +nrg) W[1]-h (k) immune P P last W[2]-h (k +nrg) W[2]-h (k) DCAV DCDV DCAC DCDC TSMR last W[2]-h (k +nrg, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 11) W[2]-h (k,n−k, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 12, 13) P (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 14) P (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3) last P [4] P [4] successive [29, 45] first P P W[2]-h (k) immune last P W[1]-h (k,m−k) amendment first P P P immune last W[1]-h (k) W[2]-h (k) P AGENDA CONTROL COALITION MANIPULATION POSSIBLE WINNER NECESSARY WINNER TSMR P (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 1) P (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2) NP-h (Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 16, 17) P (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 15) successive [8] P P NP-h P amendment P P NP-h coNP-h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='1 Related Works AGENDA CONTROL is arguably one of the most popular problems in the setting of sequential rules and has a long history of study (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', [6, 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, the complexity of AGENDA CONTROL was only first studied recently [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It should be pointed out that the complexity of some analogous problems in the setting of knockout tournaments has been studied earlier [1, 3, 4, 11, 30, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' COALITION MANIPULATION is a natural generalization of the well-known MANIPULATION problem [3], and was first studied by Conitzer, Sandholm, and Lang [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We refer to [5, 13, 36, 37, 38] for detailed results on the complexity of this problem for many traditional rules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', voting rules like Borda, Maximin, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', which do not need an agenda to determine the winner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The constructive control problems were first studied by Bartholdi, Tovey, and Trick [4], and their destructive counterparts were initiated by Hemaspaandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Heretofore the complexity of these problems for many rules has been extensively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We refer to the book chapters [5, 20] for important progress by 2016, and refer to [19, 33, 42, 43, 44] for some recent new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The complexity of POSSIBLE WINNER and NECESSARY WINNER for the successive and the amendment rules has been studied by Bredereck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' These two problems for traditional voting rules were first studied by Konczak and Lang [28], and the complexity of the problems for many rules has been subsequently established [9, 10, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='2 Organization The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In Section 2, we give the formal definitions of important notions used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, in Section 3, we unfold our concrete results for the strategic problems including AGENDA CONTROL, COALITION MANIPULATION, and the eight standard election control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, we study the POSSIBLE WINNER and the NECESSARY WINNER problems in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Finally, Section 5 summarizes our results and layouts some topics for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2 Preliminaries We assume the reader is familiar with basic notions in graph theory, complexity theory, and parameterized complexity theory [2, 14, 15, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2 Let [i] be the set of positive integers equal to or smaller than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a binary relation R, we often use xRy to denote (x,y) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='1 Graphs An undirected graph is a tuple G = (N,A) where N is a set of vertices and A is a set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An edge between two vertices v and v′ is denoted by {v,v′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We use ΓG(v) to denote the set of neighbors of v in G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', ΓG(v) = {v′ ∈ N : {v,v′} ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A digraph is a tuple G = (N,A) where N is a set of vertices and A is a set of arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Each arc from a vertex a to a vertex b is denoted by (a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The set of inneighbors of a vertex a in G is Γ− G(a) = {b ∈ N : (b,a) ∈ A}, and the set of outneighbors of a in G is Γ+ G(a) = {b ∈ N : (a,b) ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' When it is clear which graph G is discussed, we drop the index G from the notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An oriented graph is a digraph so that between every two vertices there is at most one arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a graph G (be it directed or undirected) and a subset S of vertices, the subgraph of G induced by S is denoted by G[S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='2 Elections and Voting Rules An election is a tuple (C,V) of a set of candidates C and a multiset of votes V where every ≻∈ V is defined as a linear order over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For two candidates c,c′ ∈ C, we say that c is ranked before c′ in a vote ≻ if c ≻ c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, we say that c is ranked immediately before c′ if c ≻ c′ and there are no other candidates ranked between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A vote ≻ specifies the preference of a voter casting ≻ where a is preferred to b if a is ranked before b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For notational brevity, we sometimes write a preference in the format of a sequence of candidates from the most preferred one to the least preferred one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For instance, by saying a vote with the preference a b c, we mean that a is ranked before b, and b ranked before c in the vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An agenda ▷ is a linear order over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For c ∈ C, we call candidates before c in ▷ the predecessors of c, and call those after c the successors of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A sequential rule τ maps each election (C,V) and an agenda ▷ to a single candidate τ(C,V,▷) ∈ C, the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For c,c′ ∈ C, we use nV(c,c′) to denote the number of votes in V ranking c before c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We say c beats (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' ties) c′ with respect to V if nV(c,c′) > nV(c′,c) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' nV(c,c′) = nV(c′,c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A candidate is a weak Condorcet winner if it is not beaten by anyone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, a candidate is a Condorcet winner if it beats all the other candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The majority graph of an election E = (C,V), denoted GE, is an oriented graph with the vertex set C, and there is an arc from c ∈ C to c′ ∈ C if and only if nV(c,c′) > nV(c′,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Two-stage majority rule (TSMR) This procedure takes two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let G denote the majority graph of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Moreover, let G1 be the subdigraph of G with only forward arcs with respect to ▷, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', G1 takes C as the vertex set and there is an arc from c to c′ in G1 if and only if c▷c′ and there is an arc from c to c′ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C′ ⊆ C be the set of candidates without inneighbors in G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, the procedure returns the right-most candidate in C′ as the winner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', the c ∈ C′ such that c′ ▷c for all c′ ∈ C′ \\{c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We also give the formal definitions of the successive and amendment rules as they are closely related to our discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Successive For a candidate c ∈ C and a subset C′ ⊆ C \\ {c}, we say c beats C′ if there is a strict majority of votes each of which ranks c before all candidates in C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The successive winner is the first one who beats the set of all her successors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Amendment This procedure takes |C| rounds, where each round determines a temporary winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Precisely, the winner of the first round is the first candidate in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The winner of round i where i ≥ 2 is determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let c be the winner of round i−1, and let c′ be the i-th candidate in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The winner of round i is c if c beats c′, and is c′ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The amendment winner is the winner of the last round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We note that the successive rule and the amendment rule have been also studied under several other names (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' [6, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = {a,b,c,d}, and let V be a set of three votes respectively with the preferences b d c a, c a b d, and a d b c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The majority graph of (C,V), three different agendas, and the winners under different rules and agendas are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For TSMR, arcs NOT in G1 (backward arcs with respect to ▷i) are drawn as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a b c d agenda ▷1 a b c d agenda ▷2 a b c d agenda ▷3 ▷1 ▷2 ▷3 TSMR a b a successive d a d amendment d a c winners The first and the last candidates in the agenda are somehow related to (weak) Condorcet winner, as summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For an election (C,V) and an agenda ▷ over C, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3 (1) The first or the second candidate in ▷ is the amendment winner of (C,V) if and only if it is the Condorcet winner of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (2) The last one in ▷ is the TSMR winner of (C,V) if and only if it is a weak Condorcet winner of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (3) If the successive winner of (C,V) is the first one in ▷, then the successive winner is also the Condorcet winner of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (4) If the first candidate in ▷ is the Condorcet winner of (C,V), then it is also the TSMR winner of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (5) If the last one in ▷ is a weak Condorcet winner of (C,V), it is also the successive and the amendment winner of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (6) The converses of (3)–(5) do not necessarily hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='3 Other Useful Notions Throughout the paper, unless stated otherwise, for a set S we use −→S to denote an arbitrary but fixed linear order over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Once such an −→S is used, ←−S denotes then the reverse of −→S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For S′ ⊆ S, we use −→S [S′] to denote −→S restricted to S′, and use −→S \\S′ to denote −→S [S\\S′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='4 Problem Formulations For a sequential voting rule τ, we study the following problems defined in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' AGENDA CONTROL Given: An election (C,V) and a distinguished candidate p ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Question: Is there an agenda ▷ over C so that p is the winner of (C,V,▷) with respect to τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', p = τ(C,V,▷)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' COALITION MANIPULATION Given: An election (C,V), a distinguished candidate p ∈ C, an agenda ▷ over C, and a positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Question: Is there a multiset V ′ of k votes over C so that p = τ(C,V ∪V ′,▷)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a partial order R over a set X, a linear extension of R is a linear order over X containing R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', a linear order R′ so that (x,y) ∈ R implies (x,y) ∈ R′ for all x,y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A partial election is a tuple (C,V) where V is a multiset of partial orders over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An election (C,V ′) is a completion of a partial election (C,V) if V ′ and V are one-to-one correspondence so that every v′ ∈ V ′ is a linear extension of its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' A partial agenda over C is a partial order over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' POSSIBLE WINNER Given: A partial election (C,V), a distinguished candidate p ∈ C, and a partial agenda ▷ over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Question: Is there a completion (C,V ′) of (C,V) and a linear extension ▷′ of ▷ so that p = τ(C,V ′,▷)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' NECESSARY WINNER Given: A partial election (C,V), a distinguished candidate p ∈ C, and a partial agenda ▷ over C Question: Is p the τ winner of every completion of (C,V,▷), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', p = τ(C,V ′,▷′) for all (C,V ′) being a completion of (C,V) and ▷′ being a linear extension of ▷?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We also study eight standard control problems which are special cases of the following problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CONSTRUCTIVE MULTIMODE CONTROL Given: An election (C∪D,V ∪W) with a set C of (registered) candidates,1 a set D of unregistered candidates, a mul- tiset V of registered votes, a multiset W of unregistered votes, a distinguished candidate p ∈ C, an agenda ▷ over C∪D, and four integers kAV, kDV, kAC, and kDC such that kAV ≤ |W|, kDV ≤ |V|, kAC ≤ |D|, and kDC ≤ |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Question: Are there V ′ ⊆V, W ′ ⊆W, C′ ⊆C\\{p}, and D′ ⊆ D such that |V ′| ≤ kDV, |W ′| ≤ kAV, |C′| ≤ kDC, |D′| ≤ kAC, and p wins ((C \\C′)∪D′,(V \\V ′)∪W ′,▷′) with respect to τ where ▷′ is ▷ restricted to (C \\C′)∪D′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In DESTRUCTIVE MULTIMODE CONTROL, we have the same input as CONSTRUCTIVE MULTIMODE CONTROL, and are asked whether there are V ′, W ′, C′, and D′ as in the above definition such that p is not the τ winner of ((C \\C′)∪ D′,(V \\V ′)∪W ′,▷′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The eight standard control problems studied in the paper are special cases of CONSTRUCTIVE MULTIMODE CONTROL and DESTRUCTIVE MULTIMODE CONTROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The specifications of the eight standard control problems are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 4 Table 2: Special cases of CONSTRUCTIVE/DESTRUCTIVE MULTIMODE CONTROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Here, X is either CC standing for constructive control or DC standing for destructive control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' problems restrictions XAV kAC = kDC = kDV = 0, D = /0 XAC kDC = kAV = kDV = 0, W = /0 XDV kAC = kDC = kAV = 0, D = W = /0 XDC kAC = kAV = kDV = 0, D = W = /0 For simplicity, when we study a problem in Table 2, we use k to denote the integer in the input not required to be 0, and omit components in the input requested to be 0 or /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For example, an instance of CCAV is written as ((C,V ∪W), p,▷,k), where k represents kAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our hardness results are based on reductions from the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' RED-BLUE DOMINATING SET (RBDS) Given: A bipartite graph G with bipartition (R,B) where vertices in R and B are referred to as red vertices and blue vertices respectively, and a positive integer κ ≤ |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Question: Is there a subset B′ ⊆ B of cardinality κ that dominates R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', |B′| = κ and every vertex in R has at least one neighbor from B′ in the graph G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' RBDS is NP-hard [23], and from a parameterized complexity point of view it is W[2]-complete with respect to κ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='5 Remarks Most previous studies make the assumption that there are no ties in elections (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', [26, 29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our results are presented without this assumption, but all of them still hold when the no-tie assumption is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This is clear for polynomial-time solvability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Regarding hardness results for voter control problems, some of our reductions can be slightly adapted to show the same hardness if the no-tie assumption is adopted, and others directly apply to the case with the no-tie assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We note that in these problems the no-tie assumption means that after the addition or the deletion of votes there are no ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' All our other reductions directly apply to the case with the no-tie assumption, because in these reductions the elections constructed do not admit ties and the feasible solutions do not remove the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' All our reductions take polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, a problem shown to be W[2]-hard in the paper is also NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We won’t explicitly state the NP-hardness in the corresponding theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3 Strategic Problems In this section, we study the complexity of many strategic voting problems for TSMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='1 Agenda Control and Manipulation We first present a P-algorithm for AGENDA CONTROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Agenda Control for TSMR is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let I = ((C,V), p) be an instance of AGENDA CONTROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let G be the majority graph of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We construct an agenda ▷ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let A = C \\(Γ− G(p)∪{p}) be the set of candidates which beat or tie with p with respect to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We fill all candidates from A in any arbitrary order before p in the agenda ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, we fill candidates from Γ− G(p) into the agenda iteratively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' First, let S = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In each iteration we compute the set S′ = Γ+ G(S), and fill candidates from S′ in the subsequent |S′| positions in the agenda ▷ after those from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, we update S := S∪S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The iterations terminate until S′ defined above turned out to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' After the iterations terminate, if all candidates C are in the agenda ▷, p is the TSMR winner of (C,V) with respect to ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Thus, in this case, we conclude that I is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If, however, there are still some candidates not filled in the agenda, we conclude that I is a No-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The reason is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the above iterations, in this case it holds that (1) none of C \\S is beaten by anyone from S, and (2) everyone in C \\S beats p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Condition (2) entails everyone in C \\S being after p in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, as long as this is the case, Condition (1) warrants the winning of someone from C\\S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For COALITION MANIPULATION, we have again a P-algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Coalition Manipulation for TSMR is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let I = (C,V), p,▷,k) be an instance of COALITION MANIPULATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let B be the set of predecessors of p, and let B′ be the set of successors of p in the agenda ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V ′ be the multiset of k votes with the same preference p −→ B −→ B′, where −→ B and −→ B′ are respectively the linear orders over B and B′ consistent with ▷, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', −→ B = ▷[B] and −→ B′ = ▷[B′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If p is the TSMR winner of (C,V ∪V ′,▷), we conclude that I is a Yes-instance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' otherwise, we conclude that I is a No-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The algorithm clearly runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It remains to prove its correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' To this end, we assume that I is a Yes-instance, and to complete the proof it suffices to show that I has a feasible solution V ′ so that every vote in V ′ has the same preference p −→ B −→ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe first that I has a feasible solution where p is ranked in the first place in all votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let U be a feasible solution of I where p is in the top in all votes in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If U equals V ′ defined above, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Otherwise, we show below how to transform U into V ′ without destroying the feasibility of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If there exists at least one vote ≻∈ U and two candidates b ∈ B and b′ ∈ B′ so that b′ is ranked immediately before b in ≻, we do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let ≻′ be the vote obtained from ≻ by swapping b and b′, and let U′ = U \\{≻}∪{≻′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is easy to verify that every candidate who is beaten by at least one of her predecessors with respect to V ∪U is also beaten by at least one of her predecessors with respect to V ∪U′, and everyone who is beaten by p with respect to V ∪U is still beaten by p with respect to V ∪U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, p still wins after the swap of b and b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' After the swapping operations are exhaustively applied, we obtain a feasible solution W of I where p is ranked in the top, and all candidates in B are ranked before all candidates in B′ in very vote of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If W = V ′, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Otherwise, there exists at least one vote ≻∈ W such that one of the following conditions holds: ∃a,b ∈ B s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a is ranked immediately before b in ≻ and b▷a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' ∃a′,b′ ∈ B′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a′ is ranked immediately before b′ in ≻ and b′ ▷a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, analogous to the above discussion, we can swap a and b (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a′ and b′) in ≻ without changing the winning status of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' After the swapping operations are exhaustively used, we eventually obtain V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='2 Constructive Controls In this section, we study constructive control problems for TSMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We first present results for control by adding/deleting votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show that these problems are W[2]-hard with respect to several meaningful parameters, for both the special case where the distinguished candidate is the first one in the agenda and the case where the distinguished candidate is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCAV for TSMR is W[2]-hard with respect to the number of added votes plus the number of registered votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Moreover, this holds even when the distinguished candidate is the first one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G = (R∪B,A),κ) be an instance of RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We construct an instance of CCAV for TSMR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create for each vertex in G a candidate denoted by the same symbol for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, we create a candidate p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = B ∪ R ∪ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The agenda is ▷ = (p,−→ B ,−→ R ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create the following registered votes: κ votes with the preference ←− B ←− R p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' and one vote with the preference ←− R p ←− B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V be the multiset of the above κ + 1 registered votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create |B| unregistered votes corresponding to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In particular, for each b ∈ B, we create one vote ≻b with the preference p �←− R \\ΓG(b) � b �←− R [ΓG(b)] � �←− B \\{b} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let W be the set of the above |B| unregistered votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Finally, we set k = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCAV for TSMR is ((C,V ∪W), p,▷,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In the following we show the correctness of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Suppose that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let W ′ = {≻b: b ∈ B′} be the set of the κ unregistered votes corresponding to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that p becomes the TSMR winner of the election E = (C,V ∪W ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Obviously, |V ∪W ′| = 2κ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As one of the registered votes ranks p before B, and all the κ votes in W ′ rank p before B too, there are κ +1 votes in V ∪W ′ ranking p before B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' So, none of B is the winner of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let us consider a candidate r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Note that there are κ registered votes which rank B before R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As B′ dominates R, there is at least one b ∈ B′ so that r ∈ ΓG(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the definition of ≻b, b is ranked before r in ≻b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, there are in total κ +1 votes in V ∪W ′ which rank b before r, precluding the winning of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As this holds for all r ∈ R, and all candidates from B are before all candidates from R in the agenda ▷, none of R is the winner either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This leaves only the possibility that p is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Suppose that there exists a subset W ′ ⊆ W of at most κ votes so that p is the TSMR winner of (C,V ∪W ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe that W ′ must contain exactly κ votes since otherwise someone in B precludes p from winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe that all candidates in R beat p with respect to V ∪W ′ no matter which votes are contained in W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Furthermore, everyone in R beats all her predecessors in R with respect to V ∪W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' So, if p wins (C,V ∪W ′) it must be that every r ∈ R is beaten by 6 someone in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that for every r ∈ R, there is at least one vote in W ′ which ranks some b ∈ B before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the construction of the unregistered votes, this vote must be ≻b such that b dominates r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' it follows that B′ = {b ∈ B :≻b∈ W ′} dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that the RBDS instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Now we consider the case where the distinguished candidate is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Recall that the last one in the agenda is the TSMR winner if and only if it is a weak Condorcet winner (Observation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The W[1]-hardness of CCAV for Condorcet winner established by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' [29] can be adapted to show the same hardness for weak Condorcet winner2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We strengthen the result by establishing a W[2]-hard reduction, excluding the possibility of being complete to W[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCAV for TSMR is W[2]-hard with respect to the number of added votes plus the number of registered votes even when the distinguished candidate is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS, where G = (R∪B,A) is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create an instance of CCAV as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The candidate set is C = R∪{p,q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let ▷ = (−→ R ,q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create a multiset V of κ registered votes as follows: κ −1 votes with the preference q p −→ R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' one vote with the preference q −→ R p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For each b ∈ B we create one unregistered vote ≻b with the preference �−→ R \\ΓG(b) � p �−→ R [ΓG(b)] � q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a given B′ ⊆ B, let W(B′) = {≻b: b ∈ B} be the multiset of unregistered votes corresponding to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let k = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCAV is ((C,V ∪W(B)), p,▷,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It remains to show the correctness of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let E = (C,V ∪W(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show that the CCAV instance is a Yes-instance by showing that p is the TSMR winner of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' First, observe that p ties q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As B′ dominates R, for every r ∈ R there is at least one b ∈ B′ which dominates r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that in the vote ≻b∈ W(B′), p is ranked before r, and hence p is not beaten by r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As p is the last one in the agenda, it follows that p wins E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Assume that there exists B′ ⊆ B such that |B′| ≤ k = κ and p is the TSMR winner of E = (C,V ∪W(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This means that p is not beaten by anyone else in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, |B′| = k, since otherwise q beats p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It follows that |V ∪W(B′)| = 2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As we have exactly κ −1 registered votes ranking p before r in V, there is at least one b ∈ B′ so that p is ranked before r in the vote ≻b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the definition of ≻b, this implies that b dominates r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It follows that B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Thus, the RBDS instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let us move on to constructive control by deleting votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In this case we have two natural parameters: the solution size k and its dual parameter n−k where n is the number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show that the problem is W[2]-hard with respect to both parameters, even when the distinguished candidate is the first or the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The following four theorems summarize these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCDV for TSMR is W[2]-hard with respect to the number of deleted votes even when the distinguished candidate is the first one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS where G = (B ∪ R,A) is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We assume that G does not contain any isolated vertices, κ ≥ 4, and every red vertex is of degree ℓ where ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' These assumptions do not change the W[2]-hardness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3 We construct an instance of CCDV as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The candidate set is C = R∪{p,q,q′}, and the agenda is ▷ = (p,q′,−→ R ,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create the following six groups of votes: a multiset V1 of ℓ+1 votes with the preference q′ p q ←− R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V2 of κ +ℓ−2 votes with the preference q p ←− R q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V3 of |B|−κ +1 votes with the preference ←− R p q q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 2For this, we mean the problem of determining if we can add a limited number of votes to make a particular candidate a weak Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3The assumption that G does not contain any isolated vertices and κ ≥ 4 are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If an instance does not satisfy the second assumption, we can obtain an equivalent instance by the following operation: letting ℓ be the maximum degree of vertices in R, for each red vertex r ∈ R of degree strictly smaller than ℓ, we create new degree-1 vertices adjacent only to r until r has degree exactly ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An important observation for the equivalency to the two instances is that there is an optimal solution (a subset B′ ⊆ B dominating R with the minimum cardinality) of the new instance which does not contain any of the newly introduced degree-1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 7 a singleton V4 of one vote with the preference ←− R q p q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V5 of κ −2 votes with the preference ←− R q′ p q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' for every blue vertex b ∈ B, we create one vote ≻b with the preference q q′ �←− R [ΓG(b)] � p �←− R \\ΓG(b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V denote the multiset of the above 2|B| + κ + 2ℓ − 1 votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a given B′ ⊆ B, let V(B′) = {≻b: b ∈ B′} be the multiset of votes created for vertices in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We complete the construction by setting k = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCDV is ((C,V), p,▷,k) which can be constructed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It remains to show the correctness of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let E = (C,V \\V(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that p is the TSMR winner of E with respect to the agenda ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' To this end, it suffices to show that p beats everyone else in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As B′ dominates R, there exists b ∈ B′ such that b dominates r, and thus ≻b ranks r before p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As there are in total |B|−ℓ votes in V(B) ranking p before r, we know that there are at least |B|−ℓ−κ +1 votes in V(B)\\V(B′) ranking p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As all votes in V1 ∪V2 rank p before all candidates in R, there are at least |B|−ℓ−κ +1+ℓ+κ +ℓ−1 = |B|+ℓ votes ranking p before r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As |V \\V(B′)| = 2|B|+2ℓ−2, we know that p beats r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is easy to verify that there are |B|+ℓ votes ranking p before q and q′ in V \\V(B′), meaning that p beats both q and q′ in E too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In summary, p beats everyone else in the election E and hence is the winner of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Assume that there exists V ′ ⊆ V such that |V ′| ≤ k = κ and p is the TSMR winner of E = (C,V \\V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe that by the construction of the votes and the assumption that κ ≥ 4, no matter which at most k votes are contained in V ′, every candidate in C \\ {p} beats all her predecessors in C \\ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, as p is the first candidate in the agenda and p wins E , we know that p beats all the other candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It follows that V ′ and V1 ∪V3 ∪V5 are disjoint and |V ′| = κ, since otherwise p cannot beat q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Similarly, it holds that V ′ and V2 ∪V4 are disjoint, since otherwise p cannot beat q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As a consequence, it holds that V ′ ⊆ V(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Without loss of generality, let B′ ⊆ B be such that V(B′) = V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We claim that B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Assume, for the sake of contradiction, that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R be a red vertex not dominated by any vertex in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, by the construction of the votes, all votes in V(B′) rank p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that there are in total at most |B| − ℓ − κ + |V1 ∪V2| = |B| + ℓ − 1 votes ranking p before r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In other words, p is beaten by r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, in this case p cannot be the TSMR winner of E , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCDV for TSMR is W[2]-hard with respect to the number of votes not deleted even when the distinguished candidate is the first one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS where G = (B ∪ R,A) is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As in the proof of Theorem 5, we assume that every red vertex has degree exactly ℓ for some positive integer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We construct an instance of CCDV as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The candidate set is C = R∪{p,q}, and the agenda is ▷ = (p,−→ R ,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create the following three groups of votes: a multiset V1 of κ votes with the preference p q ←− R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a singleton V2 of one vote with the preference ←− R p q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' for every blue vertex b ∈ B, one vote ≻b with the preference q �←− R \\ΓG(b) � p �←− R [ΓG(b)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V denote the multiset of the above |B|+κ +1 votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a given B′ ⊆ B, we use V(B′) = {≻b: b ∈ B′} to denote the multiset of votes corresponding to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We complete the construction by setting k = |B| − κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCDV is ((C,V), p,▷,k), which can be constructed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It remains to show the correctness of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let E = (C,V1 ∪V(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that p is the TSMR winner of E with respect to the agenda ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' To this end, it suffices to show that p beats everyone else in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As B′ dominates R, there is at least one b ∈ B′ such that b dominates r, and hence ≻b ranks p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, in total there are κ + 1 votes in E ranking p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Clearly, there are κ + 1 votes in E ranking p before q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As |V1 ∪V(B′)| = 2κ +1, we know that p beats all the other candidates in E , and hence p is the winner of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Assume that there exists V ′ ⊆ V such that |V ′| ≤ k = |B| − κ and p is the TSMR winner of the election E = (C,V \\V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe first that V ′ ⊆ V(B) and |V ′| = k, since otherwise q is not beaten by any of her predecessors, leading to q winning E , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' So, without loss of generality, let B′ ⊆ B be such that |B′| = k = |B|−κ and V(B′) = V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let B = B \\ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Obviously, |B| = κ and |V \\V ′| = 2κ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the construction of the votes, no matter which k votes are contained in V(B′), everyone from C \\ {p} beats all her predecessors in C \\ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As p is the first candidate in the 8 agenda, the winning of p in E implies that p beats all the other candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We claim that B dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Assume, for the sake of contradiction, that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R be a red vertex not dominated by any vertex in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, by the construction of the votes, all votes in V(B) rank r before p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As the only vote in V2 also ranks r before p, there are in total |B|+1 = κ +1 votes ranking r before p in E , contradicting that p beats r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCDV for TSMR is W[2]-hard with respect to the number of deleted votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This holds even if the distin- guished candidate is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem by a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS, where G = (R∪B,A) is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We assume that G does not contain any isolated vertices, κ ≥ 4, and every red vertex is of degree ℓ where ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' These assumptions do not change the W[2]-hardness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='4 Let C = R∪{p,q}, and let ▷ be an agenda over C where p is the last one (the relative orders of other candidates do not matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create the following 2|B|+2ℓ+κ votes in V: |B|+1 votes with the preference ←− R p q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' ℓ+κ votes with the preference q p ←− R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' ℓ−1 votes with the preference p q ←− R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' and for each blue vertex b ∈ B, one vote ≻b with the preference q �←− R [ΓG(b)] � p �←− R \\ΓG(b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a given B′ ⊆ B, let V(B′) = {≻b: b ∈ B′} be the multiset of votes corresponding to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Finally, we set k = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCDV is ((C,V), p,▷,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In the following, we prove the correctness of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Assume that there exists B′ ⊆ B of cardinality κ such that B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let E = (C,V \\V(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Clearly, |V \\V(B′)| = 2|B| + 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that p is not beaten by anyone else in E and hence is the TSMR winner of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As all votes in V(B′) rank q before p, it holds that nV\\V(B′)(p,q) = (|B| + 1) + (ℓ − 1) = |B| + ℓ, meaning that p ties q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Moreover, as B′ dominates R, for every r ∈ R, there exists b ∈ B′ dominating r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the construction of the votes, r is ranked before p in the vote ≻b∈ V(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It follows that at most κ −1 votes in V(B′) rank p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the construction of the votes, we know that there are at least (ℓ + κ) + (ℓ − 1) + (|B| − ℓ) − (κ − 1) = |B| + ℓ votes ranking p before r in V \\V(B′), implying that p ties r in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Assume there exists V ′ ⊆ V such that |V ′| ≤ k and p is the TSMR winner of E = (C,V \\V ′) with respect to ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As p is the last one in the agenda, it holds that p beats or ties everyone else in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As a consequence, all votes in V ′ must rank q before p and, moreover, it must be that |V ′| = k = κ, since otherwise p is beaten by q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' There are two groups of votes ranking q before p: those corresponding to the blue vertices, and those with the preference q p ←− R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We may assume that all votes in V ′ are from V(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Indeed, if V ′ contained some vote with the preference q p ←− R , we can obtain another feasible solution V ′′ from V ′ by replacing this vote with any vote in V(B)\\V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As nV(r, p) = (|B|+1)+ℓ and |V \\V ′| = 2|B| + 2ℓ, we know that there is at least one vote ≻b∈ V ′ which ranks r before p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the reduction, we know that the vertex b corresponding to ≻b dominates r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is clear now that B′ = {b ∈ B :≻b∈ V ′} dominates R, implying that the RBDS instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCDV for TSMR is W[2]-hard with respect to the number of votes not deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This holds even when the distinguished candidate is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem by a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS, where G is a bipartite graph with the vertex bipartition (R,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create an instance of CCDV as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = R ∪ {q, p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let ▷ be an agenda over C where p is in the last position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create the following votes: a multiset V1 of κ −1 votes with the preference p q −→ R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a singleton V2 of one vote with the preference −→ R p q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' and for each blue vertex b ∈ B, one vote ≻b with the preference q �−→ R \\ΓG(b) � p �−→ R [ΓG(b)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 4The assumption that G does not contain any isolated vertices and κ ≥ 4 are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If an instance does not satisfy the second assumption, we can obtain an equivalent instance by the following operation: letting ℓ be the maximum degree of vertices in R, for each red vertex r ∈ R of degree strictly smaller than ℓ, we create new degree-1 vertices adjacent only to r until r has degree exactly ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An important observation for the equivalency to the two instances is that there is an optimal solution (a subset B′ ⊆ B dominating R with the minimum cardinality) of the new instance which does not contain any of the newly introduced degree-1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 9 For a given B′ ⊆ B, we use V(B′) = {≻b: b ∈ B′} to denote the set of votes created for the blue vertices in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V = V1 ∪V2 ∪V(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Clearly, |V| = |B|+κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Finally, let k = |B|−κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCDV is ((C,V), p,▷,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the correctness as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Assume that there exists B′ ⊆ B such that |B′| = κ and B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V ′ = V1 ∪V2 ∪V(B′), and let E = (C,V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We claim that p is the TSMR winner of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As p is the last candidate in the agenda, it suffices to show that p is not beaten by any other candidates in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is clear that p ties q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let r ∈ R be a red vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As B′ dominates R, there exists b ∈ B′ dominating r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' From the construction of the votes, p is ranked before r in the vote ≻b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, there are at least |V1|+1 = κ votes ranking p before r in V ′, implying that p is not beaten by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As this holds for all r ∈ R, the correctness for this direction follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Assume that there exists V ′ ⊆ V so that |V ′| ≥ 2κ and p is the TSMR winner of (C,V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As |V1| + |V2| = κ and all votes in V(B) rank q in the first place, it must be that (V1 ∪V2) ⊆ V ′ and V ′ contains exactly κ votes from V(B), since otherwise q will be the winner of (C,V ′), contradicting the winning of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V(B′) = V ′ ∩V(B), where B′ ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As just discussed, |V(B′)| = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We claim that B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Suppose for contradiction that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, there exists r ∈ R not dominated by any vertex in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' From the construction of the votes, r is ranked before p in all votes of V(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Together with the vote in V2, there are κ +1 votes in V ′ ranking r before p, meaning that r beats p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, in this case, p cannot be the winner of (C,V ′), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As |B′| = κ, the RBDS instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let us now explore the complexity landscape of constructive control by adding or deleting candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Unlike voter controls, we have only one hardness result as stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCAC for TSMR is W[2]-hard with respect to the number of added candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This holds even when the distinguished candidate is the first one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G = (R∪B,A),κ) be an instance of RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We construct an instance of CCAC for TSMR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For each vertex in G we create one candidate denoted by the same symbol for notational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, we create a distinguished candidate p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = R∪{p} and let D = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Besides, let k = κ and let ▷ = (p,−→ B ,−→ R ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create a multiset V of votes in some way so that everyone in R beats all her predecessors in R∪{p};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' p beats everyone in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' and for each r ∈ R and each b ∈ B, if b dominates r in G, then b beats r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' otherwise, r beats b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the famous McGarvey’s theorem [31] such votes can be constructed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of CCAC for TSMR is ((C ∪D,V), p,▷,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The correctness of the reduction is easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In particular, if there exists B′ ⊆ B of κ vertices dominating R, then after adding the candidates corresponding to B′, every r ∈ R has at least one predecessor from B′ who beats her, excluding the winning of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Candidates in B′ cannot win as they are beaten by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, after adding these candidates, p becomes the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If, however, the RBDS instance is a No-instance, no matter which at most k candidates from B are added, there is at least one candidate in R who beats all her predecessors in the resulting election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In this case we cannot add at most k candidates to make p the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' When the distinguished candidate is the last one in the agenda, we have the following corollary as a consequence of Observation 1 and the immunity of weak Condorcet to CCAC [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If the distinguished candidate is the last in the agenda, TSMR is immune to CCAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For CCDC, a greedy P-algorithm can be easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' CCDC for TSMR is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let I = ((C,V), p,▷,k) be an instance of CCDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' To solve I, we first remove all predecessors of p in ▷ who beat p with respect to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, we iteratively remove each successor c of p so that c is not beaten by any of her predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' After the removals, p becomes the TSMR winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We conclude that I is a Yes-instance if and only if at most k candidates are removed in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='3 Destructive Controls Now we start the exploration on destructive control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' One may expect more tractability results, because destruc- tive controls are generally easy to solve compared with their constructive counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Nevertheless, let us start with a hardness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' DCAV for TSMR is W[2]-hard with respect to the number of added votes plus the number of registered votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Moreover, this holds even when the distinguished candidate is the first one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G = (R∪B,A),κ) be an instance of RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We construct an instance of DCAV for TSMR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = R∪{p,q} and let ▷ = (p,−→ R ,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create the following registered votes: κ −1 votes with the preference p q −→ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' two votes with the preference p −→ R q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' one vote with the preference q p −→ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V be the multiset of the above κ +2 registered votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The unregistered votes are created according to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In particular, for each b ∈ B, we create one vote ≻b with the preference �−→ R \\ΓG(b) � q p �−→ R [ΓG(b)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For a given B′ ⊆ B, let W(B′) = {≻b: b ∈ B′} be the multiset of unregistered votes corresponding to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For simplicity, let W = W(B) be the set of the above |B| unregistered votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let k = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of DCAV is ((C,V ∪W), p,▷,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the correctness of the reduction as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Suppose that there is a B′ ⊆ B of κ vertices which dominate R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, one can check that q beats or ties every other candidate with respect to V ∪W(B′), implying that q is the winner of (C,V ∪W(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Thus, in this case the instance of DCAV is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Suppose that there exists a subset W ′ ⊆W of at most k votes so that p is not the TSMR winner of E = (C,V ∪W ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe that no matter which at most k votes are contained in W ′, p beats all candidates in R, implying that the only candidate which is able to preclude p from winning is q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As q is the last candidate in the agenda ▷, q is the winner if and only if q beats or ties everyone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that W ′ contains exactly κ votes since otherwise p beats q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Moreover, for each r ∈ R, at least one vote in W ′ ranks q before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the construction of the unregistered votes, an unregistered vote ≻b ranks q before r if and only if b dominates r in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that the set of vertices corresponding to W ′ dominates R, and hence the instance of RBDS is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is known that DCAV and DCDV for weak Condorcet winner is polynomial-time solvable [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By Observation 1, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Corollary 2 ([24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' DCAV and DCDV for TSMR are in P if the distinguished candidate is in the last position of the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, the complexity of DCDV increases if the distinguished candidate is not the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' DCDV for TSMR is W[2]-hard with respect to the number of deleted votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This holds as long as the distinguished candidate is not the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The reduction is the same as the one in the proof of Theorem 7 with only the difference that q is the distinguished candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The correctness hinges upon the fact that no matter which at most k votes are deleted, q beats all candidates in R, which leaves p the unique candidate preventing q from winning and, moreover, this holds as long as q is not the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Parameterizing by the dual parameter of the solution size yields the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' DCDV is W[2]-hard with respect to the number of votes not deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This holds as long as the distinguished candidate is not the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The reduction is the same as the one in the proof of Theorem 8 with only the difference that q is the distinguished candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The correctness arguments are the same as in the proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For destructive control by modifying candidates, we have polynomial-time solvability results, regardless of the posi- tion of the distinguished candidate in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' DCAC for TSMR is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let I = ((C ∪D,V), p,▷,k) be an instance of DCAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We assume that k ≥ 1 and p is the winner of (C,V), since otherwise I can be solved trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our algorithm goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As p wins (C,V), p is not beaten by any of her predecessors, and each successor c ∈ C \\ {p} of p is beaten by at least one of c’s predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If there exists c ∈ D which is before p in the agenda and beats p, we conclude that I is a Yes-instance because p does not win (C ∪{c},V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Additionally, if there exists c ∈ D so that p▷c, and c is not beaten by any of her predecessors in C, we also determine I to be a Yes-instance, since p does not win (C ∪{c},V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If neither of the two cases occurs, then no matter which unregistered candidates are added, p remains the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, in this case, we conclude that I is a No-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The following result is a consequence of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' DCDC for TSMR is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 11 4 Possible and Necessary Winner In this section, we study NECESSARY WINNER and POSSIBLE WINNER for TSMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Bredereck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' [8] showed that except NECESSARY WINNER for the successive rule which is polynomial-time solvable, other cases of the two problems for the successive and the amendment rules are computationally hard (NP-hardness for POSSIBLE WINNER and coNP- hardness for NECESSARY WINNER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that TSMR behaves the same as the successive rule in terms their complexity of determining possible and necessary winners, though the proofs for these results for the two rules are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Necessary Winner for TSMR is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let I = ((C,V), p,▷) be an instance of NECESSARY WINNER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We determine if there is a completion of (C,V) and a completion of the agenda ▷ so that p is not the TSMR winner of the completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Note that p is not the winner if and only if (1) either some of her predecessor beats her, (2) or some of her successor c is not beaten by any of the predecessors of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We consider first if there is a completion leading to the occurrence of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For this purpose, let B = {c ∈ C \\{p} : (p,c) ̸∈ ▷} be the set of all candidates that can be predecessors of p in some completion of ▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We consider candidates in B one by one, and for each considered c ∈ B, we greedily complete the preference profile to determine if there exists at least one completion so that c beats p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' More precisely, for every partial vote ≻∈ V such that (p,c) ̸∈≻, we complete it so that c is ranked before p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If in the completion of (C,V) obtained this way c beats p, we conclude that I is a No-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If we cannot draw the conclusion that I is a No-instance above, we consider whether it is possible to male the second case happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' To this end, we enumerate all candidates which can be successors of p in some completion of the partial agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' More precisely, these candidates are those in B′ = {c ∈ C \\ {p} : (c, p) ̸∈ ▷}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For each enumerated c ∈ B′, we compute the minimum set Ac of candidates that can be successors of c under the restriction that p is before c in the agenda, and then we greedily complete the preference profile to check if they can be completed so that c is not beaten by anyone in Ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' More precisely, for each enumerated c ∈ B′, we compute Ac = {c′ ∈C : (c′,c) ∈ ▷}, and for each partial vote ≻∈V, we complete ≻ so that c is ranked as higher as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', we complete ≻ so that c is ranked below all candidates in {c′ ∈ C : (c′,c) ∈≻} and is above all the other candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If in the completion c is not beaten by anyone from Ac, we conclude that I is a No-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' If none of the above enumerations provides us a conclusion that I is a No-instance, we conclude that I is a Yes- instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Unlike the above problems, we show that POSSIBLE WINNER becomes NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Possible Winner for TSMR is NP-hard, even if the given agenda is complete and the distinguished candidate is the first one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS where G is a bipartite graph with the partition (B,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We assume that G does not contain any isolated vertices, and all vertices in R have the same degree ℓ where ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create an instance of POSSIBLE WINNER for TSMR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = R∪{p,q} and let ▷ = (p,q,−→ R ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create five groups of votes as follows, where only the first group of votes are incomplete: for each b ∈ B, one partial vote ≻b with the following partial preference �←− R [ΓG(b)] � p �←− R \\ΓG(b) � and q �←− R \\ΓG(b) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V1 of |B| votes with the preference ←− R q p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V2 of 2ℓ+κ votes with the preference q ←− R p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V3 of ℓ+2κ +1 votes with the preference ←− R p q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' a multiset V4 of ℓ+κ votes with the preference p q ←− R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V(B) = {≻b: b ∈ B} be the set of the |B| partial votes in the first group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V be the multiset of the above 2|B| + 4ℓ + 4κ + 1 votes, and let V(B) = V \\V(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of POSSIBLE WINNER is ((C,V), p,▷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Clearly, the above construction can be done in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that the RBDS instance is a Yes-instance if and only if the constructed POSSIBLE WINNER instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Suppose that there is a subset B′ ⊆ B such that |B′| = κ and B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We complete each ≻b where b ∈ B as follows: if b ∈ B′, we complete it as q �←− R [ΓG(b)] � p �←− R \\ΓG(b) � , 12 otherwise, we complete it as �←− R [ΓG(b)] � p q �←− R \\ΓG(b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is fairly easy to verify that with respect to the completion p beats q, and q beats all candidates in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, by the definition of the agenda, p is the TSMR winner with respect to the above completion of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Suppose that there is a completion V ′ of V(B) so that p wins the completion E = (C,V(B) ∪V ′) of (C,V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe that in all completions of (C,V), everyone in R beats all her predecessors in R ∪ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, by the definition of the agenda, and the fact that p wins E , it holds that (1) q beats all candidates in R, and (2) q is beaten by p in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As V(B) contains exactly 2ℓ+3κ +1 votes (those in V3 ∪V4) ranking p before q, Condition (2) implies that there are at least |B| − κ votes in V ′ ranking p before q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let B′ be the subset of B corresponding to votes in V ′ ranking p before q, and let B′′ = B \\ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Clearly, |B′′| ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We show below that Condition (1) implies that B′′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For the sake of contradiction, assume that there exists r ∈ R not dominated by any vertex in B′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In other words, all the ℓ neighbors of r in G are contained in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that there are ℓ votes in V ′ (the ℓ completions of votes corresponding to the ℓ neighbors of r) ranking r before q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Together with the |B|+ℓ+2κ +1 votes (V1 ∪V3) in V(B) ranking r before q, we have |B| + 2ℓ + 2κ + 1 votes ranking r before q, implying that r beats q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' However, this is impossible since otherwise r beats all her predecessors in E which contradicts that p wins E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This completes the proof that B′′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, from |B′′| ≤ κ, we know that the RBDS instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our reduction in the proof of Theorem 16 is completely different from those used in [8] for showing the NP-hardness of POSSIBLE WINNER for the successive and the amendment rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In fact, their reductions are from the INDEPENDENT SET and VERTEX COVER problems, while our reduction is from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Moreover, in their reductions for POSSIBLE WINNER under the successive and the amendment rules the distinguished candidate is respectively the penultimate and the third candidates in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our reduction can be adapted to show the NP-hardness of POSSIBLE WINNER for TSMR when the distinguished candidate is the i-th candidate in the agenda for every constant i, by adding i−1 dummy candidates before p in the agenda, and ranking all of them below all the other candidates in all votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Notice that POSSIBLE WINNER for TSMR becomes polynomial-time solvable if the given agenda is complete and p is the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This follows from Observation 1 and the polynomial-time solvability of determining if a partial election can be completed so that a candidate becomes a (weak) Condorcet winner [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='5 By Observation 1, the result in [28] also implies that POSSIBLE WINNER for the amendment rule becomes polynomial-time solvable if the given agenda is complete and p is in the top-2 positions , and their algorithm also applies to the determination for the winning of a particular candidate as a weak Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' So, there is a radical complexity shift for the amendment rule as the distinguished candidate moves from the second place to the third place in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our next result also reveals a seamless complexity shift for TSMR as p moves from the last position just one position up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Possible Winner for TSMR is NP-hard even when the given agenda is complete with the distinguished candidate being the penultimate candidate in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We prove the theorem via a reduction from RBDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let (G,κ) be an instance of RBDS where G = (B ∪ R,A) is a bipartite graph and 1 ≤ κ ≤ |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Similar to the previous proofs, we assume that every red vertex has degree exactly ℓ where ℓ > 0 in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We construct an instance of POSSIBLE WINNER as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let C = R ∪ {p,q,q′} and let ▷ = (q′,−→ R , p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create five groups of votes where only the first group of contains partial votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For every b ∈ B, we create one partial vote ≻b with the following partial preference �−→ R \\ΓG(b) � q′ and q p �−→ R [ΓG(b)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V(B) be the set of the |B| partial votes corresponding to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create a multiset V1 of |B|+1 votes with the preference q′ q −→ R p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create a multiset V2 of 2κ votes with the preference q p −→ R q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We create a multiset V3 of κ votes with the preference q p q′ −→ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 5The result in [28] is for Condorcet winner but the algorithm also accommodates weak Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 13 Finally, we create a multiset V4 of κ votes with the preference −→ R p q′ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let V be the multiset of the above 2|B| + 4κ + 1 votes, and let V(B) = V \\V(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' The instance of POSSIBLE WINNER is ((C,V), p,▷) which can be constructed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In the following, we prove that the RBDS instance is a Yes-instance if and only if the constructed instance of POSSIBLE WINNER is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇒) Suppose that there is a subset B′ ⊆ B such that |B′| = κ, and B′ dominates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We complete each vote ≻b∈ V(B) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' if b ∈ B′, we complete it as �−→ R \\ΓG(b) � q′ q p �−→ R [ΓG(b)] � , otherwise, we complete it as q p �−→ R \\ΓG(b) � q′ �−→ R [ΓG(b)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It is easy to verify that after completing votes as above, p beats all her predecessors in ▷, and q is beaten by her predeces- sor q′, which implies that p is the TSMR winner of the completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' (⇐) Assume that there is a completion V ′ of V(B) so that p wins the election E = (C,V(B) ∪V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe that no matter how we complete the votes, q beats all her predecessors except q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As p wins E , it must be that q′ beats q in E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies that there are at least κ partial votes in V(B) which are completed so that q′ is ranked before q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' There is only one such completion for each partial vote ≻b∈ V(B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', the completion with the preference �−→ R \\ΓG(b) � q′ q p �−→ R [ΓG(b)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Let B′ ⊆ B be such that the partial votes corresponding to B′ are completed this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' As just discussed, |B′| ≥ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Without loss of generality, let us assume that |B′| = κ +t for some nonnegative integer t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Observe further that as p wins E and |V| is odd, p beats all candidates in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For every r ∈ R, there are in total 3κ votes in V(B) (precisely, votes in V2 ∪V3) which rank p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This implies there are at least |B| − κ + 1 completions of partial votes in V(B) which rank p before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, from |B \\ B′| = |B| − κ −t, it follows that there are at least t + 1 completions of partial votes corresponding to B′ where p is ranked before r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' By the definitions of these completions, p is ranked before r in a completion corresponding to some b ∈ B′ if and only if r is a neighbor of b in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Therefore, every r ∈ R has at least t + 1 neighbors in B′ in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Then, by removing any arbitrary t vertices from B′, we obtain a κ-subset of B that dominate R, and hence the RBDS instance is a Yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' It would be interesting to see if similar complexity shift also applies to the successive rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' This mounts to determining the complexity of POSSIBLE WINNER for the successive rule when the agenda is compete with the distinguished candidate being the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We leave it as an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 5 Conclusion We conducted the (parameterized) complexity of many well-motivated voting problems under the recently proposed voting rule TSMR, with respect to the solution size and the dual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We obtained fruitful results including polynomial- time solvability results, NP-hardness results, W[1]-hardness results, and W[2]-hardness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Particularly, many of our hardness results hold even when the distinguished candidate is the first or the last one in the agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our exploration offers a complete picture of the complexity of these problems under TSMR, enabling us to compare TSMR with the successive and the amendment rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' See Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Our results indicate that TSMR resists most of the control problems, but is vulnerable to agenda control and coalition manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In addition, we showed that NECESSARY WINNER is polynomial-time solvable while POSSIBLE WINNER turned out to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Compared with previous works, our study suggests that TSMR behaves at least well as the other two important sequential rules regarding their resistance to strategic voting problems, and their complexity of calculating possible and necessary winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We point out that our exploration is a pure theoretic analysis, and whether many problems are hard to solve in specific practical settings demands further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' For more details, we refer to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' An important topic for future research is to investigate if restricting the preference domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', single-peaked/crossing preferences, top-monotonicity preferences, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=') radically changes the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' We refer to [18, 27] for a comprehen- sive survey on many restricted preference domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 14 References [1] Aziz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', Gaspers, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=', Suksompong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=': Fixing knockout tournaments with seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Raedt (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=') IJCAI, pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 62(1-2), 7–26 (2011) [38] Walsh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=': Where are the hard manipulation problems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Artif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' Res.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In: AAMAS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 1178–1186 (2017) [44] Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=': On the complexity of constructive control under nearly single-peaked preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In: ECAI, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 243–250 (2020) [45] Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=': On the complexity of controlling amendment and successive winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' In: AAMAS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} +page_content=' 1768–1770 (2022) 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQfngiP/content/2301.04009v1.pdf'} diff --git a/E9A0T4oBgHgl3EQfBP8f/content/tmp_files/2301.01972v1.pdf.txt b/E9A0T4oBgHgl3EQfBP8f/content/tmp_files/2301.01972v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9774726770f99d70f853056978251abd0f9ea61 --- /dev/null +++ b/E9A0T4oBgHgl3EQfBP8f/content/tmp_files/2301.01972v1.pdf.txt @@ -0,0 +1,2939 @@ +arXiv:2301.01972v1 [math.GT] 5 Jan 2023 +A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY +SO YAMAGATA +Abstract. Khovanov [11] introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones +polynomial. The theory was subsequently applied to the chromatic polynomial of graph [9], resulting in a categorification +known as the “chromatic homology”. Much as in the Khovanov homology, in the chromatic homology the chromatic poly- +nomial can be obtained by taking the Euler characteristic of the chromatic homology. In the present paper, we introduce a +combinatorial description of enhanced states that can be applied to analysis of the homology in an explicit way by hand. Using +the new combinatorial description, we show a splitting property of the chromatic homology. Finally, as an application of the +description, we compute the chromatic homology of the complete graph. +1. Introduction +Khovanov [11] introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones +polynomial. The theory was subsequently applied to the chromatic polynomial of graph [9], resulting in a categori- +fication known as the “chromatic homology”. Much as in the Khovanov homology, in the chromatic homology the +chromatic polynomial can be obtained by taking the Euler characteristic of the chromatic homology. Several results +on the chromatic homology have been obtained. In 2006, Helme-Guizon et al. [7] studied torsions in the chromatic +homology and presented a vanishing theorem of the homology based on their results. Specifically, they determined +which graphs have the homology that contains torsion. They also proved a thickness-type theorem for the homology +groups, and gave computations of the homology of polygon graphs with coefficients in the general algebra. A study by +Lawrance and Sazdanovic [14] showed that the torsion of the chromatic homology is of order two. The first group of +the homology was studied by Pabiniak et al. [15], and they also gave many interesting conjecture about the homology +with algebras other than A2 = Z/(x2). Helme-Guizon and colleagues [4] showed that the chromatic homology with +a rational coefficient can be determined by the chromatic polynomial, proving that the homologies of the “knight” +pair are isomorphic. In 2018, Sazdanovic and Scofield [17] studied the span of the homology and considered how the +homology changes when a cycle graph is added to the given graph. The chromatic homology with arbitral algebra +was observed in a study by Helme-Guizon and Rong [8]. Providing another perspective, homology theories for the +chromatic polynomial have also been observed [3], [19]. +The chromatic homology is interesting not only in itself but also in relation to other areas of study. The relation +to Hochschild homology was investigated by Przytycki [16], who showed that the Hochschild homology of the unital +algebra is isomorphic to the chromatic homology over the algebra of a cycle graph. With respect to the topology of +configuration spaces, Baranovsky and Sazdanovic [1] showed that the E1-term of the Bendersky-Gitler-type spectral +sequence converging to the homology of the graph configuration space is given by the chromatic complex. B¨okstedt and +Minuz [2] subsequently studied the relation between the work of Baranovsky and Sazdanovic [1] and Kriz’s rational +model for the configuration space [12]. +There are also variants of the chromatic homology. In an analysis by Jasso-Hernandez and Rong [10], the Tutte +homology was provided as a categorification of the Tutte polynomial. The categorification of the chromatic polynomial +of embedded graphs was studied iby Loebl and Moffatt [13]. The categorification of the Stanley’s chromatic symmetric +function was introduced by Sazdanovic and Yip [18]. As an analogy of the chromatic homology, Dancsco and Licata [5] +provided several homology theories for hyperplane arrangement as a categorification of several polynomials associated +with the combinatorics of hyperplane arrangement. In particular, it is easily seen that the characteristic homology, a +categorification of the characteristic polynomial, of the braid arrangement is isomorphic to the chromatic homology of +the complete graph. +2020 Mathematics Subject Classification. 57M15, 57M27, 05C15. +Key words and phrases. chromatic homology, chromatic polynomial, categorification, complete graph. +1 + +2 +SO YAMAGATA +In the present paper, we introduce a combinatorial description of enhanced states which would be useful to ana- +lyze the homology in an explicit way by hand. Using the description we show a splitting property of the chromatic +homology. More precisely, we show the following theorem. +Theorem 1.1 (Theorem 3.3). Let G be a graph and e be its edge that is not a bridge. Then, we have the following split +exact sequence +(1) +0 → Hi, j(G/e) → Hi+1, j(G) → Hi+1, j(G − e) → 0 +for i, j. +If we sum over j, we have the split exact sequence +(2) +0 → Hi(G/e) → Hi+1(G) → Hi+1(G − e) → 0 +for i. +This result would allow us to compute the chromatic homology in an inductive way. Actually, as an application +of the theorem, we can describe the chromatic homology of the complete graph recursively. The description of the +homology was firstly conjectured by Hasegawa and the author in [9]. +Theorem 1.2 (Conjecture 6.8 [9]). For n ≥ 4 the chromatic homology groups of a complete graph Kn are given as +(3) +Hi(Kn) = + +Z{n} +i = 0 +Hi−1(Kn−1)⊕(n−2) ⊕ Hi(Kn−1){1} +1 ≤ i ≤ n − 2 +0 +i ≥ n − 1. +Remark that Theorem 1.2 also gives the characteristic homology, introduced in [5], of the braid arrangement, which +would be the first result for the explicit calculation of the homology. +This paper is organized as follows. In Section 2, we recall basic notions from graph theory, and the construction +of the chromatic homology. In Section 3, we introduce the combinatorial description of enhanced states, and show +a splitting property of the chromatic homology. In Section 4, we compute the chromatic homology of the complete +graph. +2. Preliminaries +2.1. Graph and its chromatic polynomial. In this subsection let us recall the basic notions of graph theory. +Let G = (V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). If there is an order on the set E(G), the graph +is called ordered. Throughout this paper we assume the following. +• The graph G is connected; +• The vertices of G are indexed by {i ∈ N | 1 ≤ i ≤ #V(G)}; +• The graph G is ordered lexicographically with respect to pairs of numbers representing edges, i.e., for (i1i2), +(j1 j2) ∈ E(G), (i1i2) < (j1 j2) if (i1i2) < (j1 j2) as a lexicographic order. +Let us take an edge e ∈ E(G) of a graph G. We define the deletion of G denoted by G − e as a graph obtained by just +deleting e from G, and the contraction of G denoted by G/e as a graph obtained by collapsing two end vertices of e +into a single vertex along e. For a subset s ⊂ E(G), a spanning graph denoted by [G : s] is a graph (V(G), s). An edge +e ∈ E(G) is called a bridge if the number of connected components of G − e is one more than that of G. +For a positive integer λ define a coloring by a map c : [λ] → V(G) with a condition that c(i) � c(j), i, j ∈ [λ] if +(i, j) ∈ E(G). Let PG(λ) be the number of different colorings of a graph G using at most λ colors. For any graph G +the PG(λ) is a well-defined polynomial of λ known as the chromatic polynomial. It is well-known that the chromatic +polynomial satisfies the deletion-contraction relation, i.e., for any edge e ∈ E(G) the relation +(4) +PG(λ) = PG−e(λ) − PG/e(λ) +holds. + +A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY +3 +2.2. Chromatic homology. Let us review the construction of the chromatic homology. Most of the exposition here +is based on [9]. Let M = ⊕ j≥0M j be a graded Z-module, where {M j} denotes the set of homogeneous elements with +degree j. We call the power series +q dimM = +∞ +� +j=0 +q j · rank (M j) +the graded dimension of M, where rank (M j) = dimQ M j ⊗Z Q. For a graded Z-module M we define M{l} j = M j−l; +that is, all of the degrees are increased by l, and the module satisfies q dimM{l} = ql · q dimM. +Helme-Guizon and Rong [9] give two equivalent constructions of the chromatic homology. One is the cubic complex +construction, and the other is the enhanced state construction. For our purposes in this subsection, it is sufficient to +review only the latter construction. +Let G = (V(G), E(G)) be an ordered graph and s ⊂ E(G). Let E1, . . ., Ed be connected components of a spanning graph +[G : s]. Consider a map c : ∪d +h=1Eh → Z[x]/(x2) called the coloring which gives a color 1 or x on each component Eh, +h = 1, . . ., d of the graph G. We call the colored graph an enhanced state and denote it by S = (s, c). For an enhanced +state S = (s, c) define +i(S ) = #s, and j(S ) = #{h ∈ [d] | c(Eh) = x}. +Let Ci, j(G) be a Z-module generated by enhanced states S of G with i(S ) = i and j(S ) = j. We define the differential +di, j : Ci, j(G) → Ci+1, j(G) by +(5) +di, j(G) = +� +e∈E(G)\s +(−1)n(e)S e, +where n(e) is the number of edges in s that are ordered before e and S e = (se, ce) is an enhanced state defined as +follows. Let se = s ∪ {e} and E1, . . . , Ed be the components of [G : s]. If e is a bridge of Ea and Eb, a � b, then define +a map ce(Ea ∪ Eb ∪ {e}) = c(Ea)c(Eb). If e is not a bridge and an edge in some connected component Ea, then define +se = s ∪ {e} and ce(Ea ∪ {e}) = c(Ea). +Let Ci(G) = ⊕ j≥0Ci, j(G) and di = ⊕ j≥0di, j. Notice that the differential satisfies the property di+1di = 0, and thus +C(G) = (Ci(G), di) is a chain complex. With the above notations the group +(6) +Hi(G) = +Ker +� +di : Ci(G) → Ci+1(G) +� +Im �di−1 : Ci−1(G) → Ci(G)� +is called the graph homology or chromatic (graph) homology. In the present paper we call it simply chromatic +homology. For an enhanced state S = (s, c) of G/e, let ˜s = s∪{e} and ˜c be coloring of components of [G : ˜s]. Then, by +defining a map αi−1, j(S ) = (˜s, ˜c) and extending it linearly, we obtain a homomorphism αi−1, j : Ci−1, j(G/e) → Ci, j(G). +For an enhanced state S = (s, c) of G define a map βi, j : Ci, j(G) → Ci, j(G − e) in such a way that if e � s, then +βi, j(S ) = S , and if e ∈ s, then βi, j(S ) = 0. Again, by extending the map βi, j linearly we obtain the homomorphism +βi, j : Ci, j(G) → Ci, j(G − e). By summing over j we have homomorphisms αi : Ci−1(G/e) → Ci(G) and βi : Ci(G) → +Ci(G − e), respectively. We abbreviate the maps by α and β. The following lemma holds. +Lemma 2.1 (Lemma 3.1 [9]). α and β are chain maps such that 0 → Ci−1, j(G/e) +α−→ Ci, j(G) +β−→ Ci, j(G − e) → 0 is a +short exact sequence. +By the Zig-Zag lemma the following theorem holds. +Theorem 2.2 (Theorem 3.2 [9]). Given a graph G and an edge e of G, for each j there is a long exact sequence +0 → H0, j(G) +β∗ +−→ H0, j(G − e) +γ∗ +−→ H0, j(G/e) +α∗ +−−→ H1, j(G) +β∗ +−→ H1, j(G − e) +γ∗ +−→ +H1, j(G/e) → · · · → . . . Hi, j(G) +β∗ +−→ Hi, j(G − e) +γ∗ +−→ Hi, j(G/e) +α∗ +−−→ Hi+1, j(G) → . . . +If we sum over j, we have a degree-preserving long exact sequence: +0 → H0(G) +β∗ +−→ H0(G − e) +γ∗ +−→ H0(G/e) +α∗ +−−→ H1(G) +β∗ +−→ H1(G − e) +γ∗ +−→ +H1(G/e) → · · · → Hi(G) +β∗ +−→ Hi(G − e) +γ∗ +−→ Hi(G/e) +α∗ +−−→ Hi+1(G) → . . . + +4 +SO YAMAGATA +2.3. A combinatorial description of enhanced states. In this subsection, let us introduce a combinatorial description +of enhanced states. Let S = (s, c) be an enhanced state and E1, . . . , Ed1, P1, . . ., Pd2 be connected components of [G : s], +where each Eh, h = 1, . . ., d1 is a connected subgraph of [G : s] with at least one edge, and each Pk, k = 1, . . ., d2 +is a vertex. As an abuse of symbol let us denote the edge set E(Eh) by Eh. Using this notation, we can describe +enhanced states as follows. Order the components Eh, h = 1, . . ., d1 are followed by Pk, k = 1, . . ., d2 and separate +each component by the symbol “|” of the form E1 | . . . | Ed1 | P1 | . . . | Pd2. Remark that we do not make particular +assumptions about the ordering of the components. Put x above the component Eh or Pk if its corresponding component +is colored x. +Let G be a graph and S = (s, c) ∈ Ci, j(G) be an enhanced state. For the components E, E′, P, P′ of S and an edge +e ∈ E(G), we denote a new component obtained by adding the edge e to the component(s) as follows. +Ee : if e connects E itself; +�EE′�e : if e is a bridge connecting E and E′; +(EP)e : if e is a bridge connecting E and P; +�PP′�e : if e is a bridge connecting P and P′. +For fixed 1 ≤ i1 < · · · < it < · · · < ip ≤ d1, 1 ≤ k1 < · · · < kt′ < · · · < kq ≤ d2, p + q = j let S = (s, c) = +E1 | . . . | +x +Eit | . . . | Ed1 | P1 | . . . | +x +Pkt′ | . . . | Pd2 be an enhanced state of Ci, j(G), where �d1 +h=1 #Eh = i. For an edge +e ∈ E(G) \ �d1 +h=1 Eh we denote an enhanced state in which the edge e is added to S by S ∪ e. More precisely, S ∪ e is +one of the following: +(x) +Ee +a ≔ +� +E1 | . . . | +x +Eit | . . . | +(x) +Ee +a | . . . | Ed1 | P1 | . . . | +x +Pkt′ | . . . | Pd2 +� +if e connects Ea itself; +(x) +(EaEb)e ≔ +� +E1 | . . . | +x +Eit | . . . | +(x) +(EaEb)e | . . . | Ed1 | P1 | . . . | +x +Pkt′ | . . . | Pd2 +� +if e is a bridge connecting Ea and Eb; +(x) +(EaPα)e ≔ +� +E1 | . . . | +x +Eit | . . . | +(x) +(EaPα)e | . . . | Ed1 | P1 | . . . | +x +Pkt′ | . . . | Pd2 +� +if e is a bridge connecting Ea and Pα; +(x) +� +PαPβ +�e ≔ +E1 | . . . | +x +Eit | . . . | Ed1 | +(x) +� +PαPβ +�e | P1 | . . . | +x +Pkt′ | . . . | Pd2 + +if e = (PαPβ) ∈ E(G). +Remark 2.3. If any two components K, K′ are connected by a bridge e, then K and K′ are replaced by (KK′)e. If e +connects two components that are both colored x, then we regard the enhanced state S ∪ e as 0. +The following figures express the corresponding enhanced states S ∪ e. In the figures, each circle represents a +connected component of the enhanced state S and each point represents a vertex, both possibly with color x. +E1 +x +Eit +(x) +Ee +a +Ed1 +P1 +x +Pkt′ +Pd2 +· · · +· · · +· · · +· · · +· · · +e +Figure 1. +(x) +Ee +a +For a component E and two edges, e, f, of G we denote a component obtained by adding the two edges e, f to the +same component E by Ee, f. We denote the enhanced state obtained by adding two edges e, f in this order to S by +S ∪ e · f. For n ≥ 3, ((S ∪ e1) ∪ e2) · · · ∪ en = S ∪ e1 · e2 · . . . · en is determined inductively. +For an enhanced state S and distinct edges e, f we give an anti-commutative structure as follows: +(7) +S ∪ e · f = −S ∪ f · e. +This is compatible with the fact that the changing the order in which the edges are added results in a change in the +number of edges ordered before e or f. + +A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY +5 +E1 +x +Eit +(x) +Ea +(x) +Eb +Ed1 +P1 +x +Pkt′ +Pd2 +· · · +· · · +· · · +· · · +· · · +· · · +e +Figure 2. +(x) +(EaEb)e +E1 +x +Eit +(x) +Ea +Ed1 +P1 +x +Pkt′ +(x) +Pα +Pd2 +· · · +· · · +· · · +· · · +· · · +· · · +e +Figure 3. +(x) +(EaPα)e +E1 +x +Eit +Ed1 +P1 +x +Pkt′ +(x) +Pα +(x) +Pβ +Pd2 +· · · +· · · +· · · +· · · +· · · +· · · +e +Figure 4. +(x) +(PαPβ)e +For a component E we denote its vertex set by v(E), and a full subgraph of G with vertex set v(E) by Fv(E); that is, +Fv(E) is a subgraph (E(Fv(E)), V(Fv(E))) of G defined by E(Fv(E)) = {(ab) | a, b ∈ v(E), (ab) ∈ E(G)}, V(Fv(E)) = v(E). +We denote the graph Fv(E) and its edge set E(Fv(E)) by the same symbol FE for simplicity. For components E1, E2 +define a new graph E1 ∧ E2 as a graph (V(E1 ∧ E2), E(E1 ∧ E2)), where V(E1 ∧ E2) = v(E1) ∪ v(E2), E(E1 ∧ E2) = +E1 ∪ E2 ∪ {(ab) ∈ E(G) | a ∈ v(E1), b ∈ v(E2)}. +With the above notations we introduce a combinatorial description of a differential ∂i, j : Ci, j(G) → Ci+1, j(G) as +follows. +∂i, j � +E1 | . . . | +x +Eit | . . . | Ed1 | P1 | . . . | +x +Pkt′ | . . . | Pd2 +� +(8) += +� +1≤a≤d1 +� +e∈FEa \Ea +(−1)n(e) (x) +Ee +a ∪ e + +� +1≤a 5𝜎) of AME for 42 sources, +which reduces to safe detection of AME for 27 sources once the +potential contribution of thick free-free emission from ultra compact +H ii regions has been integrated to the analysis. In this work, we +complete and revisit the sample of sources observable from the +Northern hemisphere. For this we use the QUIJOTE-MFI wide- +survey maps (Rubiño-Martín et al. 2023), which are crucial to pin +down the AME spectrum at low frequencies, thence allowing a more +reliable separation between the AME and free-free amplitudes (e.g., +Poidevin et al. 2019) than previous works, which systematically have +overestimated the free-free emission and underestimated the AME +amplitude. Some of the sections in the present article closely follow +those in PIRXV. In such cases we tried to use similar section names +so that the reader can easily refer to the information provided by +PIRXV and, as much as possible, we tried to avoid redundancy with +their explanations. All the calculations made for our analysis are +independent of those done by PIRXV. +The structure of the article is as follows: the data used for the +analysis are presented in Section 2. The sample selection and fitting +procedure used for the Spectral Energy Distribution (SED) anal- +ysis are detailed in Section 3. Consistency checks obtained from +the comparison of our method with that used by PIRXV are also +presented in that Section. The significance of the AME detection +obtained from our analysis, potential contamination by UCH ii re- +gions and robustness and validation of our method are discussed in +Section 4. Statistics on the parameters characterizing the sample of +regions that passed the validation tests are investigated in Section 5. +A discussion is given in Section 6. Our results and conclusions are +summarized in Section 7. Additional plots showing low Spearman +rank correlation coefficients (SRCCs) between some of the param- +eters obtained from the modelling of the SEDs, and mentioned in +some of the above sections, are presented in Appendix A. All the +parameters estimates obtained from the modelling of the SEDs, and +additional information, obtained on the full sample, are tabulated +in Appendix B. All the plots of the SEDs and the multicomponents +models are shown in Appendix C. Finally, a summary of the SR- +CCs obtained between all the pairs of parameters used to model the +SEDs are given in Appendix D. +2 +DATA +The maps used in this analysis are listed in Table 1. Details about +the maps are given in the following subsections. +2.1 +QUIJOTE Data +The data used at frequencies 11, 13, 17 and 19 GHz come from +the first release of the QUIJOTE wide survey maps (Rubiño-Martín +et al. 2023). These maps were obtained from 9 200 h of data col- +lected over 6 years of observations from 2012 to 2018 with the +Multi-Frequency Instrument (MFI) on the first QUIJOTE telescope, +from the Teide Observatory in Tenerife, Canary Islands, Spain at +an altitude of 2 400 meters above sea level, at 28.3◦ N and 16.5◦ +W. These observations were performed at constant elevations and +with the telescope continuously spinning around the azimuth axis +(the so-called “nominal mode”) to obtain daily maps of the full +northern sky. After combination of all these data we obtained maps +covering ∼ 70% of the sky and with sensitivities in total intensity +between 60 and 200 𝜇K/deg, depending on the horn and frequency +and sensitivites, down to ∼ 35𝜇K/deg, in polarisation. Full details +on these maps, and multiple characterisation and validation tests, +are given in Rubiño-Martín et al. (2023), while the general MFI +data processing pipeline will be described in Génova-Santos et al. +(2023). +The MFI consists of 4 horns, two of them (horns 1 and 3) +covering a 10-14 GHz band with two outputs channels centred at 11 +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +3 +Frequency +Wavelength +Telescope/ +Angular Resolution +Original +Calibration +References +[GHz] +[mm] +survey +[′] +Units +Uncertaintiy [%] +0.408 +735.42 +JB/Eff/Parkes +≈ 60 +[KRJ] +10 +Haslam et al. (1982) +Remazeilles et al. (2015) +0.820 +365.91 +Dwingeloo +72 +[KRJ] +10 +Berkhuijsen (1972) +1.420 +211.30 +Stockert/Villa-Elisa +36 +[KRJ] +10 +Reich (1982) +Reich & Reich (1986) +Reich et al. (2001) +11.1 +28.19 +QUIJOTE +55.4 +[mKCMB] +5 +Rubiño-Martín et al. (2023) +12.9 +23.85 +QUIJOTE +55.8 +[mKCMB] +5 +Rubiño-Martín et al. (2023) +16.8 +18.24 +QUIJOTE +38.9 +[mKCMB] +5 +Rubiño-Martín et al. (2023) +18.8 +16.32 +QUIJOTE +40.3 +[mKCMB] +5 +Rubiño-Martín et al. (2023) +22.8 +13.16 +WMAP 9-yr +≈ 49 +[mKCMB] +3 +Bennett et al. (2013) +28.4 +10.53 +𝑃𝑙𝑎𝑛𝑐𝑘 LFI +32.29 +[KCMB] +3 +Planck Collaboration et al. (2016a) +33.0 +9.09 +WMAP 9-yr +≈ 40 +[mKCMB] +3 +Bennett et al. (2013) +40.6 +7.37 +WMAP 9-yr +≈ 31 +[mKCMB] +3 +Bennett et al. (2013) +44.1 +6.80 +𝑃𝑙𝑎𝑛𝑐𝑘 LFI +27 +[KCMB] +3 +Planck Collaboration et al. (2016a) +60.8 +4.94 +WMAP 9-yr +≈ 21 +[mKCMB] +3 +Bennett et al. (2013) +70.4 +4.27 +𝑃𝑙𝑎𝑛𝑐𝑘 LFI +13.21 +[KCMB] +3 +Planck Collaboration et al. (2016a) +93.5 +3.21 +WMAP 9-yr +≈ 13 +[mKCMB] +3 +Bennett et al. (2013) +100 +3.00 +𝑃𝑙𝑎𝑛𝑐𝑘 HFI +9.68 +[KCMB] +3 +Planck Collaboration et al. (2016a) +143 +2.10 +𝑃𝑙𝑎𝑛𝑐𝑘 HFI +7.30 +[KCMB] +3 +Planck Collaboration et al. (2016a) +217 +1.38 +𝑃𝑙𝑎𝑛𝑐𝑘 HFI +5.02 +[KCMB] +3 +Planck Collaboration et al. (2016a) +353 +0.85 +𝑃𝑙𝑎𝑛𝑐𝑘 HFI +4.94 +[KCMB] +3 +Planck Collaboration et al. (2016a) +545 +0.55 +𝑃𝑙𝑎𝑛𝑐𝑘 HFI +4.83 +[MJy/sr] +6.1 +Planck Collaboration et al. (2016a) +857 +0.35 +𝑃𝑙𝑎𝑛𝑐𝑘 HFI +4.64 +[MJy/sr] +6.4 +Planck Collaboration et al. (2016a) +1249 +0.24 +COBE-DIRBE +≈ 40 +[MJy/sr] +11.9 +Hauser et al. (1998) +2141 +0.14 +COBE-DIRBE +≈ 40 +[MJy/sr] +11.9 +Hauser et al. (1998) +2998 +0.10 +COBE-DIRBE +≈ 40 +[MJy/sr] +11.9 +Hauser et al. (1998) +Table 1. List of surveys and maps used in our analysis. +and 13 GHz, and two other ones (horns 2 and 4) covering the 16- +20 GHz band with two output channels at 17 and 19 GHz (Génova- +Santos et al. 2023). Due to a malfunctioning of horn 1 in polarization +during some periods, all the scientific QUIJOTE papers associated +with this release make use of horn 3 only at 11 and 13 GHz. Although +this paper uses intensity data only, we follow the same criterion and +use only horn 3, which is much better characterised1. At 17 and 19 +GHz we combine data from horns 2 and 4 through a weighted mean, +using predefined constant weights2 (Rubiño-Martín et al. 2023). +Finally, it must be noted that, due to the use of the same low-noise +amplifiers, the noises from the lower and upper frequency bands of +each horn are significantly correlated (see Section 4.3.3 in Rubiño- +Martín et al. 2023). In principle this correlation should be accounted +for in any scientific analysis that uses spectral information. However, +we have checked that neglecting them introduces a small effect on +the results presented in this paper. AME parameters are the most +affected, and we have checked that accounting for this correlation +introduces differences in these parameters that are typically below +the 3% level. Therefore, for the sake of simplicity we decided to use +the four frequency points (nominal frequencies 11.1, 12.9, 16.8 and +18.8 GHz) in the analysis as independent data points. We assume a +5% overall calibration uncertainty of the QUIJOTE MFI data, which +is added in quadrature to the statistical error bar. There is compelling +1 Note that the analysis in intensity presented in this paper benefits from a +sufficiently large signal-to-noise ratio and therefore a good characterisation +of systematics is more relevant. +2 Instead of doing a pixel-by-pixel combination at the map level, we extract +flux densities independently and combine the derived flux densities. +evidence that this 5 % value, which is driven by uncertainties in the +calibration models, is sufficiently conservative (Génova-Santos et al. +2023; Rubiño-Martín et al. 2023). +2.2 +Ancillary Data +2.2.1 +Low frequency ancillary data +At low frequencies we use a destriped version (Platania et al. 2003) +of the all-sky 408 MHz map of Haslam et al. (1982), the Dwingeloo +survey map at 0.820 GHz of Berkhuijsen (1972), and the 1.420 GHz +map of Reich (1982). Since our study is focused on compact candi- +date AME sources we prefer to use the all-sky 408 MHz destriped +map of Haslam et al. (1982). The Platania et al. (2003) version of +this map is used for consistency with previous QUIJOTE papers, +but we have checked that the results are consistent with those ob- +tained using the map provided by Remazeilles et al. (2015). The +Jonas et al. (1998) map at 2.326 GHz, which was used in PIRXV, +measures 𝐼 + 𝑄. Therefore it would lead to residuals in polarised +regions, and we prefer not to use it. +Some of the considered sources are not well sampled or not +included in the footprint of some of the ancillary maps. Therefore, +for a given source a map is used only if all pixels within a circular +region of 3◦ radius are covered. We noted that, for a subset of +compact sources, the map at 1.420 GHz shows a misscentering of +the emission by more than half a degree with respect to other low- +frequency maps. For that reason we prefer not to use that map in the +analysis of G059.42−00.21, G061.47+00.11 and G099.60+03.70. +The 1.420 GHz map is calibrated to the full beam, and therefore we +apply the full-beam to main-beam recalibration factor of 1.55 for +MNRAS 000, 1–36 (2023) + +4 +F. Poidevin et al. +compact sources derived by Reich & Reich (1988). Overall, we +assume a 10 % uncertainty in the radio data at low frequency, which +encompasses intrinsic calibration uncertainties as well as issues +related with beam uncertainties and recalibration factors. +2.2.2 +WMAP Maps +At frequencies of 23, 33, 41, 61, and 94 GHz, we use the intensity +maps from the 9-year data release of the WMAP satellite (Ben- +nett et al. 2013). All the maps were retrieved from the LAMBDA +database.3 For all the maps we assume a 3% overall calibration +uncertainty. The uncertainty in WMAP’s amplitude calibration is +much better, however here we use 3% to account for other system- +atic effects like uncertainties in the beams or bandpasses (which in +turn lead to uncertainties in the colour corrections) that will have a +direct effect on our derived flux densities. +2.2.3 +Planck Maps +Below 100 GHz intensity maps are available at frequencies 28, 44, +and 70 GHz. They were obtained with the Low-Frequency Instru- +ment (LFI) on board of the Planck satellite (Planck Collaboration +et al. 2016a). We use the second public release version of the in- +tensity maps as provided by the Planck Legacy Archive (PLA4). +Above 100 GHz we use the second data release version of the in- +tensity maps obtained with the High-Frequency Instrument (HFI) +on board the Planck satellite (Planck Collaboration et al. 2016a) at +frequencies centred at 100, 143, 217, 353, 545, and 857 GHz. We +have checked that using the third data release (PR3) leads to differ- +ences in the derived flux densities typically below 0.3 % for most +of the frequencies and therefore have no impact in the final results +presented in this paper. The Type 1 CO maps (Planck Collabora- +tion et al. 2014b) were used to correct the 100, 217, and 353 GHz +intensity maps for contamination introduced by the CO rotational +transition lines (1-0), (2-1) and (3-2), respectively. We assume an +overall calibration uncertainty of 3 % for the LFI data, and also +for the HFI data at frequencies lower than or equal to 353 GHz, a +value of 6.1% at 545 GHz, and a value of 6.4% at 857 GHz (Planck +Collaboration et al. 2016b). +2.2.4 +High frequency ancillary data +In the FIR range we use the Zodi-Subtracted Mission Average +(ZSMA) COBE-DIRBE maps (Hauser et al. 1998) at 240 𝜇m +(1249 GHz), 140 𝜇m (2141 GHz), and 100 𝜇m (2997 GHz). We as- +sume an 11.9% overall calibration uncertainty in the data at these +frequencies5. +3 +SAMPLE SELECTION AND SED FITTING +In the following section we describe the process followed to build +the sample of the candidate compact Galactic AME sources. De- +tails about aperture photometry used to build the SEDs are given in +3 Legacy +Archive +for +Microwave +Background +Data +Analysis, +http://lambda.gsfc.nasa.gov/. +4 Planck Legacy Archive (PLA) http://pla.esac.esa.int/pla/. +5 11.9% is the calibration uncertainty for the 240 𝜇m according to Hauser +et al. (1998), and we consider the same value for all bands. +Section 3.2. The modelling used to analyse the SED of each candi- +date AME source is detailed in Section 3.3. Finally, a consistency +test is investigated and a comparison of our analysis, including the +QUIJOTE maps, with the analysis obtained by Planck Collabora- +tion et al. (2014a) on the sample of sources common to both studies +is given in Section 3.4. +3.1 +AME sources sample +To build the sample of candidate AME sources, we use the list +of sources selected and discussed in PIRXV as a reference. In +their work, this list was obtained by using 3 different methods. +One method was to identify sources already known from the lit- +erature and add them to a sample. Another method was to pro- +duce a 1◦-smoothed map of residuals at 28.4 GHz, by subtracting +off synchrotron, free-free, thermal dust, and CMB components. A +5◦-smoothed version of this map was also created and subtracted +from the 1◦-map in order to minimise diffuse emission. Bright and +relatively compact sources were then identified in that map. In a +third method, an initial sample was built by using the SExtrac- +tor (Bertin & Arnouts 1996) software to detect bright sources in +the 70 GHz Planck CMB-subtracted map. This sample was cross- +correlated with 28.4 GHz and 100 GHz catalogs obtained using the +same technique. The output catalog was filtered to remove sources +associated with radio galaxies, including a small number of known +bright supernova remnants and planetary nebulae. Visual inspection +was conducted on preliminary SEDs obtained from the 1◦-smoothed +maps in order to filter out the regions that were not showing a peak +at 30 GHz on scales ≲ 2◦ and to define the final sample of 98 +candidate AME sources analysed and discussed in PIRXV. +Of these 98 sources, 42 are well observed at all QUIJOTE +frequencies of the MFI wide survey and are therefore included in +our sample. Additional sources that are not included in the sample +analysed by PIRXV have been identified from catalogs and lists of +molecular clouds regions available in the literature. This was done +with the SCUPOL catalog that compiles thermal dust polarimetry +information on small scales (≈ 14′′) provided by Matthews et al. +(2009), with the list of molecular clouds toward which Zeeman +measurements provide magnetic field line-of-sight (LOS) estimates +obtained by Crutcher (1999), and with the molecular cloud cata- +log of Lee et al. (2016). In this way 10 additional candidate AME +sources have been identified. The maps of these sources that are +not already included in PIRXV’s catalog were inspected by eye at +all available frequencies between 0.4 GHz and 3000 GHz and pre- +liminary SEDs were built in order to look for the presence of a +bump in the frequency range 10 – 60 GHz. The location of the fi- +nal sample of candidate AME regions selected for our analysis is +shown superimposed on the QUIJOTE 11 GHz Galactic full sky +map in Figure 1. Their names, coordinates and additional informa- +tion are displayed in Table 2. The final sample contains a total of 52 +sources. QUIJOTE-MFI intensity maps at 11, 13, 17 and 19 GHz +and WMAP 22.7 GHz intensity maps are displayed in Figure 2 for +a sample of sources. Each source clearly shows similar intensity +distribution patterns across the different frequency survey. +3.2 +Aperture photometry +In this work we conduct a component separation analysis of the +various components in intensity contributing to the total emission +of each source based on a SED analysis. In intensity this method +consists in calculating the total emission of a given source at each +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +5 +Source Name +Glon +Glat +Region Type +Other Name(𝑎) +References +𝜎AME +𝜎AME +[◦] +[◦] +PIRXV +This Work +G010.19−00.32 +10.19 +−0.32 +SNR +Kes62. Synch. SNR9.9−0.8 +1 +3.4𝑆 +2.6𝑆𝑆 +G010.84−02.59 +10.84 +−2.59 +MC +GGD 27 +2 +... +4.8𝑆𝑆 +G011.11−00.12 +10.60 +−0.12 +MC +G011.11−0.12 +2 +... +2.5𝑆𝑆 +G012.80−00.19 +12.80 +−0.19 +SNR +W33 +1 +2.7 +1.2𝐿𝐷 +G015.06−00.69 +15.06 +−0.69 +MC +M17 +1 +1.9 +8.0𝑆→𝑆𝑆 +G017.00+00.85 +17.00 +0.85 +MC +M16 +1,2 +5.3 +6.0𝑆 +G037.79−00.11 +37.79 +−0.11 +SNR +W47 +1 +3.4 +7.6𝑆→𝑆𝑆 +G040.52+02.53 +40.52 +2.53 +MC/HII +W45 +1 +0.2 +12.9𝑆→𝑆𝑆 +G041.03−00.07 +41.03 +−0.07 +MC +SDC G41.003−0.097 +4 +... +7.9𝑆 +G043.20−00.10 +43.20 +−0.10 +MC +W49 +1,3 +5.3 +8.3𝑆→𝑆𝑆 +G045.47+00.06 +45.47 +0.06 +SNR +NRAO601 +1 +5.9 +15.6𝑆→𝑆𝑆 +G049.14−00.60 +49.14 +−0.60 +MC/HII +W51 +2 +... +22.9𝑆→𝑆𝑆 +G059.42−00.21 +59.42 +−0.21 +MC/HII +W55 +1 +7.0 +8.7𝑆→𝑆𝑆 +G061.47+00.11 +61.47 +0.11 +MC/HII +HII LBN061.50+00.29. SH2−88 +1 +1.9 +4.1𝑆𝑆 +G062.98+00.05 +62.98 +0.05 +MC +S89 +1 +7.5 +6.1𝑆→𝑆𝑆 +G070.14+01.61 +70.14 +1.61 +Cluster +NGC 6857 +4 +... +3.1𝐵𝐷 +G071.59+02.85 +71.59 +2.85 +MC/HII +s101 +1 +1.8 +4.8𝑆𝑆 +G075.81+00.39 +75.81 +0.39 +MC/HII +HII GAL075.84+00.40. SH2−105. Cyg 2N +1 +2.5 +5.9𝑆→𝑆𝑆 +G076.38−00.62 +76.38 +−0.62 +MC/HII +S106 +1,3 +... +3.9𝐵𝐷 +G078.57+01.00 +78.57 +1.00 +MC/HII +LDN 889 +2,3 +... +...𝐵𝐷 +G081.59+00.01 +81.59 +0.01 +MC/HII +DR23/DR21 +1,2 +1.3 +17.9𝑆 +G084.68−00.58 +84.68 +−0.58 +MC +DOBASHI 2732 +4 +... +18.8𝑆 +G085.00+04.20 +84.90 +3.80 +MC/HII +LBN 084.97+04.21 +4 +... +21.1𝑆 +G093.02+02.76 +93.02 +2.76 +MC/HII +HII GAL093.06+2.81 +1 +1.6 +21.0𝑆→𝑆𝑆 +G094.47−01.53 +94.47 +−1.53 +MC/HII +LDN 1059 +1 +0.6 +4.1𝑆𝑆 +G098.00+01.47 +98.00 +1.47 +MC/HII +RNe GM1-12, DNe TGU H582 +1 +6.1 +17.2𝑆→𝑆𝑆 +G099.60+03.70 +99.60 +3.70 +MC +LDN1111 +1 +0.6 +3.0𝑆𝑆 +G102.88−00.69 +102.88 +−0.69 +MC/HII +LDN1161/1163 +1 +2.5 +10.9𝑆 +G107.20+05.20 +107.20 +5.20 +MC +S140 +1,2 +9.9 +27.8𝑆→𝑆𝑆 +G110.25+02.58 +110.25 +2.58 +MC/HII +HII G110.2+02.5. LBN110.11+02.44 +1 +3.4 +2.7𝑆𝑆 +G111.54+00.81 +111.54 +0.81 +Open Cluster +NGC 7538 +2 +... +10.8𝑆→𝑆𝑆 +G118.09+04.96 +118.09 +4.96 +SNR +NGC 7822 +1 +... +14.2𝑆 +G123.13−06.27 +123.13 +−6.27 +MC/HII +S184 +2 +... +25.2𝑆→𝑆𝑆 +G133.27+09.05 +133.27 +9.05 +MC +LDN 1358/1355/1357 +1 +8.5𝑆 +11.1𝐵𝐷 +G133.74+01.22 +133.74 +1.22 +MC +W3 +1 +1.5 +24.8𝑆→𝑆𝑆 +G142.35+01.35 +142.35 +1.35 +MC +DNe TGU H942, DOBASHI 3984 +1 +9.5𝑆 +8.4𝑆 +G151.62−00.28 +151.62 +−0.28 +MC/HII +HII SH2−209 +1 +1.5 +11.4𝑆→𝑆𝑆 +G160.26-18.62 +160.26 +−18.62 +MC +Perseus +1,2 +17.4𝑆 +19.2𝑆 +G160.60−12.05 +160.60 +−12.05 +MC +NGC 1499 (California nebula) +1 +5.1𝑆 +12.6𝑆 +G173.56−01.76 +173.56 +−1.76 +Open Cluster +NGC 1893 +1 +0.8 +4.4𝑆𝑆 +G173.62+02.79 +173.62 +2.79 +Cluster +S235 +1 +5.6 +15.5𝑆→𝑆𝑆 +G190.00+00.46 +190.00 +0.46 +MC/HII +NGC 2174/2175 +1 +7.4 +29.3𝑆→𝑆𝑆 +G192.34−11.37 +192.34 +−11.37 +MC +LDN 1582/1584 +1 +12.3𝑆 +12.5𝐵𝐷 +G192.60−00.06 +192.60 +−0.06 +Cluster +S255 +1 +4.3 +7.9𝑆→𝑆𝑆 +G201.62+01.63 +201.62 +1.63 +MC +LDN 1608/1609 +1 +7.4𝑆 +27.3𝑆 +G203.24+02.08 +203.24 +2.08 +MC/HII +LDN 1613 +1,2 +8.3𝑆 +15.8𝑆 +G208.80−02.65 +208.80 +−2.65 +MC/HII +S280–LBN 970 +1 +2.0 +1.9𝐿𝐷 +G239.40−04.70 +239.40 +−4.70 +MC +LDN 1667, HII LBN1059, V VY Cma +1 +9.9𝑆 +16.5𝑆 +G351.31+17.28 +351.31 +17.28 +MC/HII +HII LBN1105/1104 +1 +5.3𝑆 +32.9𝑆 +G353.05+16.90 +353.05 +16.90 +MC +Rho Ophiuchi, AME-G353.05+16.901 +1,3 +29.8𝑆 +27.3𝑆→𝑆𝑆 +G353.97+15.79 +353.97 +15.79 +MC +In Ophiuchus +1 +10.9𝑆 +10.6𝑆 +G355.63+20.52 +355.63 +20.52 +MC +In Rho Ophiuchus +1 +13.3𝑆 +17.0𝐵𝐷 +Table 2. List of sources. References: 1: Planck Collaboration et al. (2014a) (PIRXV), 2: Matthews et al. (2009), 3: Crutcher (1999), 4: Lee et al. (2016). Note: +(𝑎) information retrieved from the Simbad database (http://simbad.u-strasbg.fr/simbad//). Sources such that 𝜎AME from PIRXV are greater than +𝜎AME from this work are shown in bold. Superscript symbols in last two columns are 𝑆 for “significant” AME detection, 𝑆𝑆 for “semi-significant” AME +detection, 𝑆→𝑆𝑆 for “significant” AME detection reclassified as “semi-significant” AME detection (see text for details), 𝐿𝐷 for low detection of AME and, +𝐵𝐷 for bad detection because of a bad fit of the AME, of the free-free or of the thermal dust component. See section 4.3 for details. +MNRAS 000, 1–36 (2023) + +6 +F. Poidevin et al. +Figure 1. AME sources location in the Galaxy displayed on top of the QUIJOTE-MFI 11 GHz wide survey map at 1 degree resolution. Coordinates are listed +in Table 2. The map is centred at position (𝑙, 𝑏) = (120◦, 0◦). +frequency. Once a SED has been calculated one can use modelling +to assess the fraction of the total intensity emission associated with +the different components (synchrotron, free–free, thermal dust, and +AME) at all frequencies. SED modelling analysis has been widely +used in the literature (e.g., Watson et al. 2005; Planck Collaboration +et al. 2011; López-Caraballo et al. 2011; Planck Collaboration et al. +2014a; Génova-Santos et al. 2015, 2017; Poidevin et al. 2019). +The maps of pixel size 𝑁side = 512 in the HEALPix6 pixeliza- +tion scheme (see Górski et al. 2005) are first smoothed to 1◦. To +calculate the total emission at each frequency, the maps in CMB +thermodynamic units (KCMB) are first converted to Rayleigh-Jeans +(RJ) units (KRJ) at the central frequency, then all the maps are con- +verted to units of Jy pixel−1 using 𝑆 = 2𝑘𝑏𝑇RJΩ𝜈2/𝑐2, where 𝑘𝑏 is +the Boltzmann constant, 𝑇RJ, is the Rayleigh-Jeans temperature, Ω +is the solid angle of the pixel, 𝜈 is the frequency and 𝑐 is the speed +of light. The pixels are then summed in the aperture covering the +region of interest to obtain an integrated flux density. An estimate +of the background is subtracted using a median estimator of pixels +lying in the region defined as the background region. +In Section 3.4 we provide some comparisons with the results +obtained by PIRXV. To do so, we use the same apertures and annu- +6 https://sourceforge.net/projects/healpix/ +lus used in that paper, i.e. 𝑟APERTURE = 60 +′, 𝑟ANNULUS(IN) = 80 +′ +and 𝑟ANNULUS(OUT) = 100 +′. This method, also used in previous +works, relies on the pixel-to-pixel scatter in the background annulus +to obtain an estimate of the uncertainty in the flux density estimate. +This technique is straightforward in the case of uncorrelated noise. +However, in our case there is pixel-to-pixel correlated noise, due +to instrumental 1/f noise and to beam-averaged sky background +fluctuations, whose correlation function is not easy to be reliably +characterised. We instead apply aperture photometry at the central +position of each source in the standard manner, and then the calcu- +lations are repeated eight times such that we perform flux-density +integrations on eight independent disks of radius 𝑟APERTURE = 30 +′ +with central coordinates distributed along a circle with radius 2◦ +around the source (as shown in Figure 2). The final uncertainty is +obtained from the scatter of these eight flux-density estimates. This +procedure is used for all sources except for the California region +for which the background structure is complex and was producing +bad fits such that 𝜈AME= 60.0 ±0.0 GHz, i.e. the prior upper limit. +For that region we therefore use the same aperture and background +annulus as in PIRXV and we expect our uncertainties on the fluxes +of this region to be slightly underestimated. +MNRAS 000, 1–36 (2023) + +AME sources location map +0) +mk +-5 +20QUIJOTE-MFI wide-survey Galactic AME sources +7 +Figure 2. Subsample of 1◦ smoothed intensity maps in Galactic coordinates. Fom left to right: QUIJOTE-MFI intensity maps at 11 GHz (horn 3), 13 GHz +(horn 3), 17 GHz (horn 2 and 4), 19 GHz (horn 2 and 4), and WMAP intensity maps at 23 GHz. From top to bottom the sources shown are the well-known +Galactic supernova remnant NRAO601 (G045.47+00.06), star forming region W49 (G043.20-00.10), Perseus molecular cloud (G160.26-18.62) and the cluster +S235 (G173.62+02.79). In each plot the central circle shows the aperture used to obtain the density flux estimates. The eight dashed circles show the positions +of the apertures used to calculate the uncertainties on these fluxes as explained in section 3.2. +3.3 +Model fitting +For each source the flux density 𝑆 from the aperture photometry is +fitted by a simple model consisting of the free-free, synchrotron (if +appropriate), thermal dust, AME and CMB components: +𝑆total = 𝑆ff + 𝑆sync + 𝑆td + 𝑆AME + 𝑆CMB. +(1) +The free–free spectrum shape is fixed and the free-free flux +density, 𝑆ff, is calculated from the brightness temperature, 𝑇ff, using +the expression: +𝑆ff = 2𝑘𝑏𝑇ffΩ𝜈2 +𝑐2 +, +(2) +where Ω is the solid angle of our 60 +′ aperture. The brightness +temperature is calculated with the expression: +𝑇ff = 𝑇e(1 − 𝑒−𝜏ff), +(3) +MNRAS 000, 1–36 (2023) + +8 +F. Poidevin et al. +where, following Draine (2011) the optical depth, 𝜏ff, is given by +𝜏ff = 5.468 × 10−2𝑇−1.5 +e +𝜈−2 +9 EM𝑔ff, +(4) +where 𝑇e is the electron temperature in Kelvin, 𝜈9 is the frequency +in GHz units, EM is the Emission Measure in units of pc cm−6, and +𝑔ff is the Gaunt factor, which is approximated as: +𝑔ff = ln(exp[5.960 − +√ +3 +𝜋 ln(Zi𝜈9T−3/2 +e,4 )] + e), +(5) +where the charge is assumed to be 𝑍𝑖 = 1 (i.e., hydrogen plasma) +and 𝑇e,4 is in units of 104 K. Our best estimate for the electron +temperature is the median value of the Commander template within +the aperture used on each source (Planck Collaboration et al. 2016c). +These values lie in the range 5 458–7 194 K. The only remaining free +parameter associated with the free–free component is the free–free +amplitude, which can be parameterized by the effective EM. +Equation 4 tells that the turnover frequency that marks the +transition between the optically-thick and optically-thin regimes +(𝜏ff ≈ 1) depends on the emission measure (as EM1/2) and on the +electron temperature. In order to properly trace the degeneration +between the free-free amplitude and the turnover frequency, instead +of working with integrated quantities we would have to reconstruct +EM along individual lines of sight inside each region and then +integrate. Given the non-linear dependency of the flux density on +EM, the two procedures are not equivalent, and this typically results +in our fitted spectra having smaller turnover frequencies. For this +reason, in cases where the data clearly shows the turnover frequency +to be above 0.408 GHz (see e.g. G015.06 − 00.69 in Figure C1), +in order to avoid the free-free (AME) amplitude to be biased low +(high) we do not use in the fit the points with frequencies below +1.42 GHz (depicted in these cases by a blue asterisk in Figure C1). +The synchrotron component is fitted by a single power law +given by: +𝑆sync = 𝑆synch,1GHz · +� +𝜈 +GHz +� 𝛼synch,int , +(6) +where the two parameters that are fitted for are the spectral in- +dex, 𝛼synch,int, and the amplitude at 1 GHz, 𝑆synch,1GHz. This +synchrotron component is included in the fits only for a few +sources (G010.19 − 00.32, G012.80 − 00.19, G037.79 − 00.11, +G040.52 + 02.53, G041.03 − 00.07 and G045.47 + 00.06) as in- +dicated in Table B1. This choice was based on the slope of the +low-frequency flux densities. The first three and the last of these +sources are SNRs, as listed in Table 2. There is yet another source +classified as SNR in our sample, G118.09 + 04.96. However the +low-frequency data do not show any hint of synchrotron emission +in this source, and actually the addition of this component to the fit +has no impact on the fitted AME spectrum. +The CMB is modelled using the differential of a blackbody at +𝑇CMB = 2.7255 K (Fixsen 2009): +𝑆CMB = 𝜂 2𝑘𝑏Ω𝜈2 +𝑐2 +Δ𝑇CMB, +(7) +where 𝜂 = 𝑥2·exp(𝑥)/(exp(𝑥) − 1)2 and 𝑥 = ℎ𝜈/(𝑘𝑏𝑇CMB) is the +conversion between thermodynamic and RJ brightness temperature, +and Δ𝑇CMB is the CMB fluctuation temperature in thermodynamic +units. +Spinning dust models have many free parameters, which are +extremely difficult to constrain jointly. As a result, using a phenome- +logical model, which traces well the data and the typical spinning +dust models, is common practice in the field. In this work the AME +component is fitted by the phenomenological model consisting of +an empirical log-normal approximation, first proposed by Steven- +son (2014). The log-normal model is described by the following +equation: +𝑆AME = 𝐴AME · exp +� +−1 +2 · +� ln(𝜈/𝜈AME) +WAME +�2 +, +� +(8) +where the three free parameters are the width of the parabola𝑊AME, +the peak frequency 𝜈AME, and the amplitude of the parabola at the +peak frequency 𝐴AME. Some previous works (e.g., Génova-Santos +et al. 2017) have used a different phenomelogical model proposed by +Bonaldi et al. (2007). However we note that in this model the AME +peak frequency and the AME width are not independent parameters. +Hence, we prefer to use the Stevenson (2014) model, which does +not have this coupling. +The thermal dust emission is modelled by a single-component +modified blackbody relation of the form, +𝑆td = 𝜏250(𝜈/1200GHz)𝛽dustB𝜈(Tdust), +(9) +where 𝜏250 is the averaged dust optical depth at 250 𝜇m, 𝛽dust is the +averaged thermal dust emissivity, and 𝐵𝜈 is the Planck’s law of the +black-body radiation at the temperature, 𝑇dust, which is the averaged +dust temperature. +The fit procedure includes priors on some of the parameters and +consists of a minimization process using non-linear least-squares fit- +ting in Interactive Data Language (IDL) with MPFIT (Markwardt +2009). The errors on the fitted parameters in this method are com- +puted from the input data covariance, and neither the goodness of +the fit nor parameter degeneracies are taken into account. It must +then be noted that parameter errors are sometimes underestimated. +This is the case for instance when it is hard to separate the free-free +and the spinning dust components. In those cases the errors on EM +and 𝐴AME will tend to be underestimated. A more reliable error +estimate would require full sampling of the probability distribution +and will be considered in future similar studies. Such a method +should help to refine our results but would not change our main +conclusions. +Flat priors are used on the following list of parameters: 𝑇dust, +𝛽dust, Δ𝑇CMB, 𝐴AME, 𝜈AME and 𝑊AME. Dust temperatures, 𝑇dust, +are allowed in the temperature range 10–35 K while dust index +emissivities, 𝛽dust, are allowed in the range 1.2–2.5. Both priors +are representative of average dust physical conditions in the diffuse +interstellar medium (ISM) and molecular clouds. The CMB fluctu- +ation temperatures, Δ𝑇CMB, are allowed to vary in the temperature +range ±125 K. This range of values is representative of the CMB +fluctuation temperatures one can expect when operating aperture +photometry including a background subtraction. The AME ampli- +tude, 𝑊AME, is allowed to vary in the range 0–300 Jy. The AME +frequency, 𝜈AME, is allowed to vary in the frequency range 10– +60 GHz, and for the width parameter 𝑊AME, we use a prior 0.2–1.0. +While spinning dust models computed for representative ISM en- +vironments (Draine & Lazarian 1998; Ali-Haïmoud et al. 2009) +typically have maximum widths corresponding to 𝑊AME ≈ 0.7 we +prefer not to be so strongly model constrained and allow for slightly +wider AME spectra. More details on the effect of the priors used +to model the AME are discussed in Section 4.3 and Table 3, in +Section 5.1.5, and in Section 5.1.6. +Colour corrections for QUIJOTE, WMAP, Planck and DIRBE, +which depend on the fitted spectral models, have been applied using +an iterative procedure that involves calls to a specifically developed +software package. This code, which will be described in more detail +in Génova-Santos et al. (2023), uses as input the fitted spectral +model in each iteration, which is convolved with the experiment +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +9 +bandpass. Colour corrections are typically ≲ 2% for QUIJOTE, +WMAP and Planck-LFI, and ≲ 10% for Planck-HFI and DIRBE, +which have considerably larger bandwidths. Colour corrections for +low-frequency surveys, which have much narrower bandpasses, are +not necessary. +3.4 +Comparison with AME sources previously characterized +in Planck Intermediate Results XV +Before making an analysis of the full sample of 52 candidate AME +sources displayed in Table 2 we first compare the results obtained +with a multicomponent analysis of the SEDs calculated on the sam- +ple of 42 sources already studied by PIRXV. The AME model used +by PIRXV assumes a spinning dust model corresponding to the +warm ionized medium (WIM) with a peak at 28.1 GHz to give the +generic shape for which only the amplitude of the peak and the peak +frequency were fitted for. This horizontal shift in frequency is arti- +ficial, as the WIM model, with the parameters that have been used +do produce that model, predicts a precise value for 𝜈peak. On the +contrary, as explained before, the AME model used in our analysis +is a phenomenological model with three parameters including one +parameter to fit for the width of the bump of the AME. +To build the SEDs in the same way as PIRXV, as mentioned +before, we use an aperture of radius 60′ and an annulus of inter- +nal and external radii of sizes 80′ and 100′, respectively. For this +comparison, we then use the parameters obtained by PIRXV on the +CMB and thermal dust components as fixed input parameters and +then we fit our model of AME, free-free and synchrotron (in the +cases where the synchrotron was considered in the fits by PIRXV, +i.e. on sources G010.19−00.32, G012.80−00.19, G037.79−00.11, +G045.47 + 00.06 and G118.09 + 04.96). From these fits we calcu- +late the AME significance (𝜎AME) as the ratio of the flux density +of AME at the frequency peak position divided by the uncertainty +on this estimate. The results are displayed in Figure 3 (a). Three +points show a higher AME significance in PIRXV than in our anal- +ysis (data shown with red colour in the plots). Overall, however, +our analysis shows that for most of the sources the AME amplitude, +and its significance are higher once the QUIJOTE data are included +(data shown with black colour in the plot). This trend can be ex- +plained by the level of free-free detection to be generally higher in +the PIRXV analysis than in our component separation analysis as +shown in Figure 3 (b). This point is also confirmed by the higher +level of AME obtained with our analysis compared to the level of +AME detected by PIRXV at a frequency of 28.4 GHz as displayed in +Figure 3 (c). In this plot AME 𝑆28.4 +resid is the AME flux obtained from +the modelling at 28.4 GHz. This general trend is consistent with the +results obtained by Génova-Santos et al. (2017), by Poidevin et al. +(2019) and by Fernandez-Torreiro et al. (2023), and confirms that +the QUIJOTE-MFI data are crucial to help breaking the inevitable +degeneracy between the AME and the free-free that occurs when +only data above 23 GHz are used in regions with AME peak flux +densities close to this frequency. From Fig. 3 (d) it is also clear that +the inclusion of QUIJOTE data favours lower AME peak frequen- +cies, which are found to be on average around 4 GHz smaller than +in PIRXV. It is also worth stressing that the addition of QUIJOTE +data clearly leads to a more precise characterisation of the emission +models in the 10 − 60 GHz frequency range. We find on average +errors smaller by ≈ 30 % on 𝐸𝑀 and 𝐴AME, by ≈ 70 % on 𝑊AME, +by ≈ 60 % on 𝜈AME and even by 10 % on 𝛽dust and 𝑇dust. +To test that our interpretation of the results is not model- +dependent we repeated the analysis described above with the model +proposed by Bonaldi et al. (2007). The final plots are very similar +to those displayed in Figure 3 meaning that the higher level of de- +tection of AME comes from the addition of the QUIJOTE maps +at 10–20 GHz. In addition to this, our model should provide more +reliable estimates of the AME peak frequency thanks to it being +fully independent on the AME width. +4 +REGIONS OF AME +In the following sections we describe the level of detection of AME +derived from the modelling analysis of the SEDs (Section 4.1) +and their possible contamination by UCH ii regions (Section 4.2). +From this analysis we define the final sample of candidate AME +sources that will be used for further statistical studies. Additional +calculations used to test the robustness and validate this sample are +given in Section 4.3. +4.1 +Significance of AME detections in our sample +In order to make a study of the detection of AME in the 52 sources +from our sample we first produced a series of intensity maps at +all available frequencies. The maps were inspected and removed if +some pixels were showing no data in the aperture or annulus areas; +this process affecting more specifically low frequency maps. +The component separation was operated by including fits for +the free-free, the AME, the thermal dust and the CMB components. +The synchrotron component was also included in the six sources +indicated in Section 3.3. Each SED was then inspected by eye and it +was found that most of the sources were showing the detection of a +bump in the frequency range 10–60 GHz. Some examples of SEDs +in intensity are shown in Figure 4. +The histogram displayed in Figure 5 shows the distribution of +the significance of the AME detection, 𝜎AME. Following PIRXV we +define the sources with 𝜎AME > 5 as “significant AME sources”, +the sources with 2 < 𝜎AME < 5 as “semi significant AME sources”, +and the sources with 𝜎AME < 2 as “non AME detections”. Some of +the “significant AME sources” are re-classified as “semi-significant +AME sources” as will be discussed in the next section. The con- +cerns regarding modelling problems and systematic errors for a few +sources are discussed in Section 4.3. +4.2 +Ultra-Compact Hii regions +Ultra-Compact H ii regions (UCH ii) could bias AME detections +and change the free-free typical behaviour. It is therefore impor- +tant to assess their possible impact on our analysis. UCH ii with +EM ≳ 107 cm−6pc are expected to produce optically thick free- +free emission up to 10 GHz or higher (Kurtz 2002, 2005). To +take into account possible contamination of our sample by emis- +sion from arcsec resolution point sources (Wood & Churchwell +1989a) that are not AME in nature we follow the method used +in PIRXV as illustrated by their Figure 5. To this aim we catalog +all the IRAS points sources retrieved from the IRAS Point Source +Catalog (PSC)7 that lie inside the 2◦ diameter circular apertures +of our sample. These sources are classified as a function of their +colour-colour index defined by the logarithm of flux ratios obtained +in several bands. The PSC UCH ii potential candidates tend to have +7 See +the +link +to +the +IRAS +Faint +Source +Catalog, +Ver- +sion +2.0 +in +the +HEASARC +Catalog +Resources +Index, +https://heasarc.gsfc.nasa.gov/W3Browse/iras/iraspsc.html +MNRAS 000, 1–36 (2023) + +10 +F. Poidevin et al. +Figure 3. Comparison between the results obtained with our analysis and in PIRXV for the AME significance 𝜎AME defined as the ratio of the flux density of +AME at the frequency peak position divided by the uncertainty on this estimate (a), the emission measure EM (b), the residual AME flux density at 28.4 GHz +(c) and the AME peak frequency (d). Our analysis includes the QUIJOTE-MFI data. The data shown in red correspond to sources for which the significance +of the AME detection is higher in PIRXV than in our analysis. +ratios log10(𝑆60/𝑆12) ≥ 1.30 and log10(𝑆25/𝑆12) ≥ 0.57 (Wood +& Churchwell 1989b). They are identified accordingly. Kurtz et al. +(1994) measured the ratio of 100 𝜇m to 2 cm (15 GHz) flux den- +sities and found it lies in the range 1000–400000, with no UCH ii +regions having 𝑆100𝜇𝑚/𝑆2cm < 1000. Following PIRXV we use +this relation to put limits on the 15 GHz maximum flux densities +that could be emitted by candidate UCH ii regions encountered in +the apertures used for measuring the flux densities of our sample of +sources. The fluxes at 100 𝜇m of the PSC sources are summed up +toward each aperture and then divided by 1000 to get an estimate of +the the maximum UCH ii flux density at 15 GHz, 𝑆UCHII +max +, towards +each candidate AME source. From the multicomponent fits, the flux +densities at 15 GHz (or 2 cm) are calculated and compared to these +maximum UCH ii flux densities. The distribution is shown in Fig- +ure 6 where the maximum UCH ii flux densities are plotted against +the 15 GHz flux densities obtained with our analysis. If a candidate +AME source detected with more than 5𝜎 has a residual AME flux +density at 15 GHz lower than 25% of the maximum UCH ii flux +density then it is re-classified as “semi-significant”, as indicated in +Table 2. We believe that this is a very conservative approach, in a +way that many of these re-classified sources are actually “signifi- +cant” AME detections. UCH ii contributions to the 30 GHz excess +have been recently investigated by Rennie et al. (2021) on a small +sample of galactic H ii regions using data from the 5 GHz COR- +NISH catalog. The study rejects such regions as the cause of the +AME excess. +4.3 +Robustness and validation +The significance of AME detection, defined by the parameter 𝜎AME, +discussed in section 4.1, is an important indicator reflecting the abil- +ity of our analysis to detect and fit any excess of emission observed in +the frequency range 10–60 GHz; whether such a bump is potentially +dominated by UCH ii regions or not (Section 4.2). The significance +of AME detection obtained on each source, though, is also depen- +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +11 +Figure 4. SED of the sample of regions shown in Figure 2. The QUIJOTE intensity flux densities are shown with red points, and the WMAP, Planck, and +DIRBE intensity flux densities are shown with green, blue, and yellow< points, respectively. The low frequency points are shown in pale blue. The result of +the multicomponent fit is illustrated by the continuous black curve. The fit to the AME component is shown with the dashed red line. The fit to the free–free +component is shown with the dashed blue line. The fit to the thermal dust component is shown with the dashed yellow line. The fit to the CMB component is +shown with the dashed green line. A zoom on the AME bump is shown in the subpanel. Residuals to the fits are shown in the bottom plots. +DR23/DR21 +𝐴AME +𝜎AME +𝜈AME +𝑊AME +Δ𝑇CMB +𝐴AME priors +𝜈AME priors +𝑊AME priors +𝜒2 +red +[G081.59+00.01] +[Jy] +[GHz] +[Jy] +[Jy] +[GHz] +See plot on Figure 7, left +99.4 ± 5.7 +17.4 +36.8 ±40.5 +1.8 ± 1.2 +-1.2 ± 69.6 +[0 , 300] +[10 , 60] +[0.2 , 2.5] +0.18 +See plot on Figure 7, right +94.0 ± 5.2 +18.1 +26.3 ± 3.6 +1.0 +125.0 +[0 , 300] +[10 , 60] +[0.2 , 1.0] +0.20 +Table 3. Fit parameters of the AME and CMB components obtained with different priors on the AME width parameter, 𝑊AME. Note that in the case of stronger +priors the best-fit values for 𝑊AME and Δ𝑇CMB are found in the border of the prior. The corresponding plots are shown in Figure 7. +dent on the overall accuracy of the multicomponent fit obtained over +the full frequency spectrum considered in the analysis. +In order to explore the stability of the fitting procedure we +made a number of tests to check that our main results are not af- +fected by our fitting method and assumptions. This includes relaxing +the assumed calibration uncertainty and changing the sizes of the +aperture and annulus radius. Overall we were able to fit all the 4 +or 5 components on 46 sources from the 52 sources included in +the initial sample, or in other words the multicomponent fit was +converging on all the components considered to fit each of the 46 +sources. +The SPDust2 models (see Ali-Haïmoud et al. 2009; Ali- +Haimoud 2010) for cold neutral medium, dark cloud, molecular +cloud, warm ionized medium and warm neutral medium have widths +lying in the range [0.4 − 0.7] while in practice slightly wider distri- +butions could be expected (see discussion in Section 3.3). To take +MNRAS 000, 1–36 (2023) + +12 +F. Poidevin et al. +Figure 5. Histogram of the AME significance values (𝜎AME) for the sample +of 52 sources. The 5𝜎 limit is shown with the vertical dashed line. Sources +that are significant and have a maximum contribution from UCH ii regions, +𝑓 UCH II +max +< 0.25, are shown as the filled histogram. +Figure 6. Estimated maximum contribution from UCH ii regions against +15 GHz AME residual flux density. The most significant AME sources +(𝜎AME > 5 and 𝑆residual +15 +> 0.25 × 𝑆UCHII +max +) are shown as red diamond +symbols, while non-AME regions (𝜎AME < 2) are shown as dark cross +symbols. “Semi-significant” AME sources (𝜎AME =2–5) are shown as blue +triangle symbols. “Significant” AME regions that have a potentially large +contribution from UCH ii +(𝑆residual +15 +< 0.25 × 𝑆UCHII +max +) are re-classed as +“semi-significant” and are highlighted by blue diamonds. The data shown +with red diamond symbols are the “Significant” AME regions such that +𝑆residual +15 +> 0.25 × 𝑆UCHII +max +, if this information is available. Regions with +no matched UCH II regions are set to 0.01 for visualization and lie on the +bottom of the plot. The dashed lines correspond to different maximum frac- +tions of UCH ii flux density: 1, 10, 25 (solid line), and 100% of the 15 GHz +residual flux density. +this into account the uniform priors used on the AME parameters +are: 10 < 𝜈AME < 60 GHz, and 0.2 < 𝑊AME < 1.0. Such assump- +tions on the values allowed to be taken by 𝑊AME are important +to keep realistic AME detections. An example of the effect of the +priors is shown in Figure 7 where multicomponent fits obtained on +the DR23/DR21 maps are displayed. The plot on the left shows +the fit on the AME component with priors on 𝑊AME such that +0.2 < 𝑊AME < 2.5, while the plot on the right displays the AME +fit component with priors on 𝑊AME such that 0.2 < 𝑊AME < 1.0. +The AME fit parameters obtained in both cases are given in Ta- +ble 3. In the case of loose priors on 𝑊AME the AME component +shows an excessively wide looking bump, even if the improvement +in the goodness of the fit is marginal (see the values of the 𝜒2 +red in +Table 3). Such a broad spectrum cannot be reproduced by spinning +dust models for environments with reasonable physical parame- +ters, so models like this might be deemed as physically unrealistic. +This demonstrates the need for setting realistic priors on the fits +to overcome the problem with fit degeneracies. Finally, as it was +commented in Section 3.3, our methodology for error estimation do +not properly grasp those parameter degeneracies, leading in some +cases to an underestimation of the error (see the too small error of +𝜈AME in the case of strong prior in Table 3). +As a final test we repeated the analyses with more stringent +priors such that 0.4 < 𝑊AME < 0.7 and 16 GHz < 𝜈AME < 60 GHz, +and found that this does not have a strong impact on the derived +results. In particular, we found differences typically smaller than +5 % in 𝜈AME and typically smaller than 20 % in 𝐴AME. +Our final sample follows the superscript symbols given in the +last column in Table 2. A total of 6 sources (labelled as “BD”) +considered as bad detections of AME because of a bad fit of the +AME, of the free-free or of the thermal dust component, are not +considered on a statistical basis. On the other hand, statistics are +given for the sample which we refer to as the selected sample (46 +sources). This data set includes sources with low or poor AME de- +tection (2 sources, labelled as “LD”), with “semi-significant” AME +detection (29 sources labelled as “SS”, including 20 “significant” +AME sources reclassified as "semi-significant AME sources") and +with “significant" AME detections (15 sources labelled as “S”). +Statistics are also given on the sample of “semi-significant” AME +detections and on the sample of “significant” AME detections. The +selected sample includes a total of 7 sources with fits reaching the +prior upper limit on 𝑊AME and such that, the uncertainty on this +parameter is, 𝜎𝑊AME = 0. These sources are included in the sam- +ple of AME well-detected 44 sources (i.e., the sample including +“semi-significant” and “significant” AME detections). +5 +STATISTICAL STUDY OF AME SOURCES +Along this section we study the statistical properties of the physical +parameters of the sample discussed in the previous section, with +the aim of better understanding the physical and environmental +conditions of the AME sources, as well as to obtain insights about +the nature of the carriers that cause the AME. The parameter values +used to model the components estimated from the analysis of the +SEDs in intensity are given in Tables B1 and B2. The method used +to calculate the flux densities does not take into account the effect +of the signal integration through the thickness of the clouds as well +as across the area sustended by each telescope. This limitation will +be taken into account, as much as possible, in the interpretation of +the results. +5.1 +Nature of the sources +In this section we focus our analysis on the parameters used to model +the AME and some of the thermal dust component parameters. This +includes the relative strength of the ISRF, which is estimated from +the fitted thermal dust parameters. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +13 +Figure 7. Two multicomponent fits of the DR23/DR21 region. Colours and symbols definitions are the same as in Figure 4. Left: fits obtained with priors on +the AME parameters such that 10 < 𝜈AME < 60 GHz, and 0.2 < 𝑊AME < 2.5. Right: Same as left but with 0.2 < 𝑊AME < 1. A discussion about the choice +of priors is given in section 4.3. The AME and CMB components fit parameters obtained in each case are displayed in Table 3. +5.1.1 +AME fraction at 28.4 GHz +As a first step we investigate the fraction of the total flux density +at 28.4 GHz that is produced by AME under the expectation that +free-free and AME are the dominant sources of radiation at this +frequency. For this we calculated the residual AME flux density +at 28.4 GHz, 𝑆28 +res, by subtracting to the measured flux density at +this same frequency all the other components and propagating their +uncertainties. The histogram of this quantity is plotted in Figure 8 +and shows that regardless of whether the sources are classified as +“significant” or “semi-significant”, the contribution of the AME +flux density goes from a few per cent to almost 100 per cent of +the total flux density. This result is different from that obtained by +PIRXV who found that in their sample the sources classified as +“significant” AME sources were mainly showing 𝑆28 +res/𝑆28 > 50 %, +while the remaining sources classified as “semi-significant” were +lying in the lower part of the histogram such that 𝑆28 +res/𝑆28 ≲ 50 %. +All in all, the majority of the sources in our selected sample show +𝑆28 +res/𝑆28 < 50 %. This result could come from the AME peak +frequency distribution which is found to be about 4 GHz lower than +by PIRXV. This result will be presented in Section 5.1.5. +5.1.2 +Dust properties +The distribution of the thermal dust temperature, 𝑇dust, against the +thermal dust emissivisity, 𝛽dust obtained from the SEDs multicom- +ponent fits are displayed in Figure 9. The expected anti-correlation +that is discussed and analysed in many works (e.g., Paradis et al. +2014) is also seen in the plot. +An apparent sequence in the IRAS colours given by the +12𝜇m/25𝜇m and 60𝜇m/100𝜇m ratios can also be expected from +previous studies of H ii regions (Chan & Fich 1995; Boulanger +et al. 1988), and external galaxies (Helou 1986) showing an anti- +correlation between the two ratios. The interpretation relates to the +spatial distribution of different grain populations as a function of +the Inter-Stellar Radiation Field (ISRF) intensity. This trend was +obtained for the sample of sources discussed by PIRXV. We find +a result similar to their analysis but our plots shown in Figure 10 +Figure 8. Histogram of the AME fraction 𝑆28 +res/𝑆28 at 28.4 GHz. The selected +sample is shown as the unfilled histogram. The “significant” AME detection +sample is shown with the hatched area. +presents a lower dynamic range of the colour ratio 60 𝜇m/100 𝜇m +than the one from their analysis. Our sample probes line-of-sights +(LOSs) with colour ratios 60 𝜇m/100 𝜇m lying in the range 0.2– +0.7, which is the range in which PIRXV found most of their sources +classified as “significant” AME detections and not expected to be +dominated by UCH ii region emission. +5.1.3 +Dust optical depth +The sources of our sample are distributed across regions of differ- +ent optical depths. In order to understand how this parameter could +help us to build up a picture of the distribution of the parameters +used to fit the AME components classified as “semi-significant” +or “significant”, in Figure 11 we show the variations of the peak +MNRAS 000, 1–36 (2023) + +14 +F. Poidevin et al. +Figure 9. Distribution of the thermal dust temperature, 𝑇dust, against the +thermal dust emissivisity, 𝛽dust obtained from the SEDs multicomponent +fits. The “significant” AME detection sample is shown with red diamonds. +The “semi-significant” AME detection sample is shown with blue triangles. +Low AME detections are shown in black. +Figure 10. Colour–colour plot of IRAS 12𝜇m/25𝜇m against 60𝜇m/100𝜇m +for the sample of sources. Symbols and colours definition are the same as in +Figure 9. +AME flux density, 𝐴AME, as a function of the thermal dust optical +depth at 250 𝜇m, 𝜏250, obtained from the fits of the thermal dust +components. One can see a clear trend showing an increase of the +maximum AME flux density with the quantity of thermal dust mat- +ter encountered along the LOSs. The Spearman Rank Correlation +Coefficient (SRCC) of that distribution is rs = 0.80 ± 0.04. This is +not a surprise, as a strong spatial correlation was already observed +between the AME and thermal dust, when AME was first detected +(see Kogut 1996; Leitch et al. 1997), and it is well established +that the interstellar medium is pervaded by a complex non-uniform +distribution of thermal dust material, a fraction of which spatially +correlates with the spiral arms structure of the Galaxy (e.g. Marshall +et al. 2006; Lallement et al. 2019) toward which many sources of +Figure 11. Distribution of the AME peak flux density AAME against 𝜏250. +All selected data are displayed. Symbols and colours definition are the same +as in Figure 9. +our sample are located (see Figure 1). In addition, no correlation is +observed between the AME peak frequencies and the thermal dust +optical depths at 250 𝜇m, (see Figure A1). Similarily, no correlation +is observed between the width of the parabola used to fit the AME +and the thermal dust optical depth (see Figure A2). One can clearly +see in that plot the cases for which the AME width reaches the upper +limit of the prior 𝑊AME = 1. These cases are not restrained to a +specific range of the thermal dust optical depth parameter, which +means that the AME detections with 𝑊AME = 1 are not expected to +depend on this parameter. +5.1.4 +The interstellar radiation field: G0 +Another important parameter that is useful to describe the physics +of the several environments towards AME regions is the relative +strength of the ISRF, 𝐺0 (see Mathis et al. 1983). AME carriers +are believed to be tiny particles lying in the bottom part of the +interstellar dust grain size spectrum (𝑎 ≲ 1 nm) (possibly includ- +ing Polycyclic Aromatic Hydrocarbons or PAHs). Their chemical +properties, physical coherence and total charge could vary over time +and from one environment to another, and therefore depend on the +relative strength of the ISRF. Therefore, having our estimation of +𝐺0 is very useful to explore possible relations with the parameters +used to model the AME component detected at the SED level. An +estimation of 𝐺0 can be obtained from the equilibrium dust temper- +ature of the big dust grains (𝑇BG) compared to the average value of +17.5 K (see Mathis et al. 1983), with the relation: +𝐺0 = +� 𝑇BG +17.5K +�4+𝛽BG +, +(10) +where 𝛽BG is the spectral index associated with the opacity of the big +grains. In the following, we assume 𝑇BG ≈ 𝑇dust, where 𝑇dust is the +averaged temperature of the thermal dust component obtained from +the fit on each region. As in PIRXV, we also assume a constant value +𝛽BG = 2. We note that using 𝛽BG ≈ 𝛽dust could also be considered, +but would not change the conclusions of our analysis. +The correlation between the AME fraction at 28.4 GHz (de- +fined as the residual AME flux density at 28.4 GHz divided by the +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +15 +Figure 12. AME fraction at 28.4 GHz as a function of the estimated 𝐺0. +Symbols and colours definition are the same as in Figure 9. +total flux density at 28.4 GHz) and G0 is shown in Figure 12. The +data show a decrease of the AME fraction as a function of G0. This +trend is similar to the one obtained by PIRXV in their analysis and +seems to be dependent of the considered subsets. In our analysis the +slope of the “significant” AME detection data sample is of order +𝛾 = −0.48, while the slope of “semi-significant” AME detection +data sample is of order 𝛾 = −0.61. We point out that the uncer- +tainties of the values of the slopes we estimated are large, ≈ 0.8 +for both “significant” and “semi-significant” AME detections data +points, which prevents a full and fair comparison with results from +previous studies. Our slopes, though, can be compared to the slope +of 𝛾 = −0.11 ± 0.04 obtained by PIRXV on their strongest AME +sources sample (see their Figure 15 and section 5.1.4), and to the +slope of 𝛾 = −0.59 ± 0.11 obtained on their semi-significant AME +sources. All in all, our results agree with those of PIRXV within the +uncertainties. Differences in the slopes estimates can be explained +by the different sample sizes (half-sky versus full sky coverage) and +by the introduction of the QUIJOTE data in our analysis. +5.1.5 +Peak frequency of AME +Among the three parameters used to fit the AME components in +our sample, one is the peak frequency, which is allowed to vary +in the frequency range 10–60 GHz. Such a degree of freedom is +important since it allows to get better final fits. It has also been +shown in previous works that one can expect the frequency of AME +to vary from one source to the other, or even within the same +region (Cepeda-Arroita et al. 2021). The histogram of the AME peak +frequency calculated for the selected sample is shown in Figure 13. +The Gaussian fit to the distribution provides a mean frequency and +dispersion given by 23.6 ± 3.6 GHz. The hashed histogram shows +the distribution of the “significant” AME sources sample peaking +around the weighted mean frequency. PIRXV found their sample +of AME sources to peak in the range 20–35 GHz, with a weighted +mean of 27.9 GHz, a bit higher than our mean value, the main reason +of this difference being that flux densities in the frequency range +10–20 GHz were not available in their analysis. In fact, the addition +of QUIJOTE-MFI data clearly helps reducing the uncertainty in the +determination of 𝜈AME, thanks to allowing to trace the down-turn of +Figure 13. Histogram of the AME peak frequency in bins of size 2 GHz. +The selected sample is shown as the unfilled histogram. The “significant” +AME detection sample is shown with the hashed area. A Gaussian fit to the +histogram is shown with the dashed-line. +the AME spectrum at low frequencies. Our average error on 𝜈AME +is 3.4 GHz, and when we repeat our analysis excluding QUIJOTE- +MFI data we get an average error of 7.5 GHz (see also discussion in +Section 6.3). On the other hand our analysis of G160.60-12.05 (the +California nebula/NGC 1499) recovers an AME peak frequency +at 49.1 ±38.5 GHz, which is consistent with values obtained in +previous analyses (Planck Collaboration et al. 2011, 2014a). The +uncertainty on our estimate is quite large because the free-free +dominates at 𝜈 <100 GHz making the width of the AME bump +poorly constrained and the fitted parameters strongly degenerated. +On top of that the circular aperture that we use may not be optimal +in this case where the emission is elongated and pretty extended. +5.1.6 +Width of the AME bump +In addition to the maximum flux density and peak frequency pa- +rameters, the third parameter used to fit the AME components is +the width of the parabola, 𝑊AME (see Equation 8). The allowed +range in the fit was 0.2–1 and the initial value was 𝑊AME = 0.5 +for all sources, this value being the expected average value from the +SPDust2 models. The histogram of our fitted values is displayed in +Figure 14. As discussed previously, the multicomponent fits lead- +ing to output fit parameters of values 𝑊AME = 1 and 𝜎𝑊AME = 0 +are cases reaching the prior upper limit value, and this artificially +leads to a higher number of sources lying in the last bin of the +histogram. The selected sample is shown as the whole histogram. +The single-dashed histogram shows the same distribution without +the prior dominated AME detections. This distribution has a mean +and dispersion given by, 𝑊AME = 0.58 ± 0.61. The distribution +looks rather flat, and far from Gaussian, which is reflected in the +large error bar of the Gaussian fit. This in fact illustrates that 𝑊AME +is maybe the worst constrained parameter in our fit, due to large +degeneracies with other parameters. +This result is obtained with a bin of size 0.1 and would need a +higher sample for one to drive strong conclusions on a statistical ba- +sis. Indeed using a bin size of 0.2 the whole histogram looks rather +like a normal distribution without any clear peak. Statistically, we +MNRAS 000, 1–36 (2023) + +16 +F. Poidevin et al. +Figure 14. Histogram of the width of the AME component parameterized +by 𝑊AME (see Equation 8) in bins of size 0.1. The selected sample is shown +as the whole histogram. The “constrained” AME detections are shown with +the unfilled histogram. The “significant” AME detection sample is shown +with the double harshed area. The gaussian fit to the histogram is shown +with the dashed-line. +find that 𝑊AME does not correlate with the free-free component EM +parameter. Neither do we find any correlation between 𝑊AME and +any of the thermal dust parameters. On the other hand we observe a +mild correlation of 𝑊AME with the AME emissivity (𝐴AME/𝜏250). +A detailed definition of the AME emissivity will be given in Sec- +tion 6.2 where these results will be discussed. +5.1.7 +Width of AME bump and peak frequency of AME +The three parameters describing the parabola used to fit the AME +flux density bump (see Equation 8) are independent from each +other. With this model any correlation found between the AME +peak frequency and the parabola width parameter could therefore +be indicative of the physics underlying the description of the AME +carriers. We checked that neither a negative nor a positive correlation +can be seen between the two parameters. As shown in Table D1, +all the samples (selected, “semi-significant” and “significant”) are +showing SRCCs consistent with a null correlation. These results +show that the width and the peak frequency of the AME component +are fully independent from each other, although this conclusion +could be affected by the fact that, in some cases, 𝑊AME seems to be +poorly constrained in our analysis. +5.2 +Dust correlations +In this section we focus on the thermal dust component with the +aim to better understand its relation with the AME component. We +also consider high frequency maps at 100 𝜇m, 60 𝜇m and 12 𝜇m, +since these data have the potential to provide information about +some of the candidate AME carriers (i.e., spinning dust, PAHs or +fullerenes). +5.2.1 +Dust flux densities at 100 𝜇m, 60 𝜇m, 25 𝜇m and 12 𝜇m +Following the spatial correlation observed between AME and the +thermal dust emission when AME was first discovered, many stud- +ies have explored and discussed the possibility that AME carriers +are spinning dust grains in nature (e.g., Draine & Lazarian 1998, +1999; Ali-Haïmoud et al. 2009), i.e., possibly a specific subclass +of the dust grain population spectrum. A look to various dust grain +emission templates should therefore be useful to explore if any spe- +cific correlation exists between the maximum AME flux densities +and the flux densities of thermal dust observed at 100 𝜇m, 60 𝜇m, +25 𝜇m and 12 𝜇m. Such plots are shown in Figure 15 (top row) +and the strength of the correlations described by their SRCCs are +given in Table 4. We find very strong correlations between the AME +flux densities and the thermal dust flux densities at 100 𝜇m, 60 𝜇m, +25 𝜇m and 12 𝜇m. This result is consistent with the one obtained by +PIRXV from their analysis. +If the AME carriers are spinning dust grains, the AME compo- +nent is expected to be quite insensitive to the ISRF relative strength, +𝐺0 (Ali-Haïmoud et al. 2009; Ysard & Verstraete 2010) while on +the contrary the thermal dust grains population is expected to be +sensitive to it, mainly because the UV radiation should control their +temperature. If that was true one would expect better correlations +between the maximum AME flux densities and the flux densities of +thermal dust observed at 100 𝜇m, 60 𝜇m, 25 𝜇m and 12 𝜇m, once +they are normalized by 𝐺0. This has been discussed in some pre- +vious analysis (e.g., Ysard & Verstraete 2010). The plots obtained +once the thermal dust fluxes are normalized by 𝐺0 are shown in Fig- +ure 15 (bottom row) and the strength of the correlations described by +their SRCCs are given between parenthesis in Table 4. Contrary to +what was found on their sample by PIRXV, normalizing the thermal +dust templates by 𝐺0 leads to less tight correlations. These results +suggest that the AME carriers could be coupled to the thermal dust +grain components rather than to a dust grain population relatively +insensitive to 𝐺0. On the other hand the dust grain size distribution +is very sensitive to the ISRF, as well as to other parameters such +as the dipole moments of PAHs (Ali-Haïmoud et al. 2009), mean- +ing that the interpretation of the results obtained with plots such as +those given in Figure 15 may be complicated. The role of 𝐺0 will +be discussed further in Section 5.4. +5.2.2 +Thermal Dust peak flux densities +The size of the aperture used to build the SEDs could introduce a +coupling between some of the thermal dust parameters 𝜏250, 𝑇dust +and 𝛽dust due to a possible range of degeneracy at the fit level +between these parameters. In order to circumvent this problem, +that could mislead the interpretation of some of the correlations +discussed above, we looked at the distribution between the flux +densities at the peak of the AME bumps and at the maximum of +the thermal dust components. This is shown in Figure 16 where it +can be seen a correlation between the two flux components at their +maximum. The slope of a power-law fit to the selected sample is +0.96 and almost consistent with 1 as shown with the dark solid line +on the plot. The SRRC between the two parameters is equal to 0.89 +± 0.05. +5.2.3 +Thermal Dust radiance +The radiance of a component is defined as the integral of the +flux density of that component over the full spectral range, ℜ = +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +17 +Figure 15. Top row: AME peak flux density as a function of the 100 𝜇m (panel a), 60 𝜇m (panel b) and 12 𝜇m (panel c) flux density. Bottom row: same as top +row but after the infrared tracers of dust have been divided by 𝐺0 (panel d, e and f, respectively). Symbols and colours definition are the same as in Figure 9. +Power-law fits to the full set are shown with back solid lines. SRCCs are given in Table 4. +Wavelentgh +SRCC +SRCC +SRCC +selected sample +AME significant +AME semi-significant +100 𝜇m +0.87± 0.04 ( 0.84± 0.05 ) +0.86± 0.02 ( 0.65± 0.08 ) +0.89± 0.03 ( 0.88± 0.05) +60 𝜇m +0.84± 0.04 ( 0.86± 0.05 ) +0.82± 0.03 ( 0.65± 0.08 ) +0.88± 0.03 ( 0.90± 0.05) +25 𝜇m +0.85± 0.04 ( 0.81± 0.05 ) +0.65± 0.03 ( 0.43± 0.07 ) +0.90± 0.03 ( 0.90± 0.05) +12 𝜇m +0.80± 0.04 ( 0.70± 0.05 ) +0.39± 0.04 ( 0.19± 0.07 ) +0.89± 0.03 ( 0.85± 0.06) +Table 4. Spearman rank correlation coefficients (SRCCs) between the AME maximum flux densities and the IR/submm flux densities. The values displayed +between parentheses are the SRCCs obtained once the IR/submm flux densities are divided by the IRSFs estimates G0. +∫ +∞ +−∞ 𝑆(𝜈)𝑑𝜈. In this work, all radiances were calculated by inte- +grating the fitted models between 0.4 and 3000 GHz, which is the +frequency range where all the maps used in this analysis are available +(see Table 1). Some studies have shown strong correlations between +the dust radiance and the AME amplitude at the peak frequency +(Hensley et al. 2016; Hensley & Draine 2017). The distribution of +both components for our sample is shown in Figure 17, (top). A +good correlation is observed between the two variables of the se- +lected sample, with a SRCC of 0.89 ± 0.05, and a power-law slope +consistent with 1. This tight correlation suggests a strong coupling +between the big dust grains expected to be the main contributors +to the dust grain radiance considered here (i.e., in the wavelength +range 𝜆 > 100𝜇m). Figure 17 (bottom) shows the distribution of the +AME radiance ℜAME as a function of the dust radiance ℜtd. In that +case a lower correlation is observed between the two parameters +with a SRCC of 0.70 ± 0.06. +We believe that the reason why the AME amplitude correlates +better than the AME radiance is because the latter is quite sensitive +to 𝑊AME, and this parameter has large error bars due to not being +very well constrained by our fit (see section 5.1.6). This said, these +two correlations can be interpreted using two different views. A +first one is that the AME model used to fit the data and designed +to approximate the spectrum of the spinning dust emission is not +fully appropriate to capture the contribution of the AME carriers, +or that in some regions it is difficult to properly disentangle the +AME contribution from the free-free and thermal dust contributions. +Another view could be that if the AME model used to fit the data +is good enough to capture the AME components accurately, then +the dust radiance of PAHs and/or Very Small Grains (VSGs) could +represent a relatively large contribution of the total dust radiance at +wavelengths greater than 100 𝜇m. +5.3 +AME emissivity +As discussed above, strong spatial correlations were found between +the AME emission and thermal dust emission when AME was first +detected (see Kogut 1996; Leitch et al. 1997). In order to build +a picture of the distribution of the AME emission along the third +spatial dimension (i.e., the line-of-sight, LOS), further works have +defined the AME emissivity as the ratio between the AME inten- +sity and the column density, for which the optical depth at a given +wavelength is often used as a proxy (see Dickinson et al. 2018, and +discussion and references therein). In order to make comparisons +with results discussed in the literature we first show in Figure 18 the +MNRAS 000, 1–36 (2023) + +18 +F. Poidevin et al. +Figure 16. Maximum AME flux density versus maximum thermal dust flux +density. Symbols and colours definition are the same as in Figure 9. The +solid line represents a fitted power-law model to the data. +distribution of the AME flux density obtained by subtracting to the +measured flux density at this same frequency all the other compo- +nents (defined as the residual flux density at 28.4 GHz) normalized +by the 100 𝜇m flux density (𝑆res +28.4GHz/𝑆100𝜇m), as a function of the +AME detection significance. In this case the 100 𝜇m flux density is +expected to be optically thin for a given dust temperature and com- +position and is used as a proxy to probe the column density of dust +along the LOSs. 𝑆res +28.4GHz/𝑆100𝜇m is in the range (0.05 − 9) × 10−4 +with a weighted mean of (4.2 ± 0.3) × 10−4 and an unweighted +average of (3.5 ± 1.6) × 10−4 (significant AME sample). These +values are consistent with each other. They are smaller than the +unweighted average value of (5.8 ± 0.7) × 10−4 of PIRXV and than +the 6.2 × 10−4 value of Davies et al. (2006) but are higher than +the weighted average of (2.5 ± 0.2) × 10−4obtained in PIRXV and +than the value of about 1.1 × 10−4 obtained by Todorović et al. +(2010) on a sample of H ii regions. The differences between our +estimates and those obtained by PIRXV could partially come from +the different samples used in each study. Our sample only covers +the North hemisphere sky while the analysis of PIRXV includes +also sources in the Southern hemisphere. Different error treatment +may also affect the weighted averages. Regardless of these issues, +we have applied a one-to-one comparison between our flux density +ratios and those reported in PIRXV in the subsample of 42 common +sources. When we represent the former against the latter and fit the +data to a straight line we find a slope of 0.76, meaning that we find +≈ 30% higher emissivities. This is a consequence of the increase of +the AME amplitude as a result of the inclusion of QUIJOTE data +(see Figure 3c and related discussion in section 6.3). A summary of +these results is given in Table 5. +The small range of values of the ratio of the AME residual flux +density at 28.4 GHz to the flux density at 100 𝜇m suggests that a +power-law index of order 1 could be expected between the two flux +density distributions. This is indeed what the best-fitting power-law +confirms as it yields a power-law index of 1.04 ± 0.21 in tension +with the power-law index of 0.67 ± 0.03 obtained by PIRXV on +their sample. Similarily, the best-fitting power-law index between +the AME residual flux density at 28.4 GHz and the dust optical +depth at a wavelength of 250 𝜇m, 𝜏250, yields a power-law index of +Figure 17. Top: AME flux density at peak frequency, 𝐴AME, as a function +of the thermal dust radiance, ℜtd. Bottom: AME radiance, ℜAME, as a +function of thermal dust radiance, ℜtd. Symbols and colours definition are +the same as in Figure 9. The solid lines represent fitted power-law models +to the data. +Sample +𝑆res +28.4GHz/𝑆100𝜇m [×10−4] +unweighted mean +weighted mean +This work - selected sample +2.5 ± 1.7 +3.7 ± 0.1 +This work - semi-significant +2.1 ± 1.5 +3.2 ± 0.1 +This work - significant +3.5 ± 1.6 +4.2 ± 0.3 +PIRXV - significant +5.8 ± 0.7 +2.5 ± 0.2 +Todorović et al. (2010) +1.1 ± - +... ± ... +Davies et al. (2006) +6.2 ± - +... ± ... +Table 5. Comparison of the AME flux densities normalized by the 100 𝜇m +flux densities obtained in this work and in previous studies. +1.13 ± 0.22 in agreement with the power-law index of 1.03 ± 0.03 +obtained by PIRXV. The results obtained by PIRXV were inferring +an AME mainly proportional to the column density estimate, i.e., to +the amount of material along the LOS. This is what we find whether +we consider the 100 𝜇m map or the 𝜏250 parameters as proxies of +the column density. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +19 +Figure 18. AME emissivity against AME significance. Symbols and their +colours definition are the same as in Figure 9. +Figure 19. Variations of the AME emissivity with the relative strength of +the ISRF, 𝐺0. Symbols and colours definition are the same as in Figure 9. +The power-law fit obtained on the selected sample is plotted with the black +line. +5.4 +Role of the ISRF +The ISRF is strongly coupled to the nature of the various phases +encountered in the ISM defined in terms of gas temperature and +matter density. The UV light produced by the population of stars +pervading the ISM is absorbed by the dust grain populations and +re-radiated in the IR. The ISRF therefore plays an important dy- +namic role since it will affect the chemical composition of the ISM +material, the dust grain distribution as well as the lifetime of the +small dust grain and complex molecule populations (see Jones et al. +2013). It is therefore interesting to investigate the existence of possi- +ble relationships between the relative strength of the ISRF, 𝐺0, and +the parameters describing the AME component derived from the +SEDs analysis. For this we looked at the distribution of the AME +emissivity, now defined as 𝐴AME/𝜏250, the AME peak frequency, +𝜈AME, and the AME bump width parameter, 𝑊AME, as a function +of 𝐺0. The plots are shown in Figures 19, Figure A4 and Figure A5, +respectively. We find poor correlations between 𝐺0 and the AME +parameters 𝜈AME and 𝑊AME. On the other hand, we find a SRCC +of rs = 0.68±0.08 between the AME emissivity and 𝐺0 parameters +for the selected sample (Figure 19). This distribution can be fitted +by a power-law of index of about 0.8 as shown with the black line in +Figure 19. Since we derived the relative strength of the ISRF, 𝐺0, +by using the thermal dust grain temperature, 𝑇dust, obtained from +the SED grey body fits, and by assuming a maximum and constant +thermal dust emissivity, 𝛽dust = 2 (see equation 10), the SRCCs +obtained between the 𝐴AME/𝜏250 and 𝐺0 parameter distributions +and between the 𝐴AME/𝜏250 and 𝑇dust parameter distributions are +by construction identical. Similarly, the introduction of the SEDs +fit estimates of 𝛽dust in the calculation of 𝐺0 only changes SRCCs +values by less than one percent. This means that the AME flux den- +sities obtained at the peak frequency are mainly correlated with the +combination of the dust optical depth, 𝜏250, and the thermal dust +temperature 𝑇dust parameters. This result is in agreement with the +strong correlation obtained between the AME peak flux densities +and 𝜏250, and with the 100𝜇m thermal dust fluxes discussed in the +previous section. +In the above we have considered that a good proxy of the rel- +ative strength of the ISRF is given by 𝐺0, which is a function of +the thermal dust temperature, 𝑇dust. The EM is another interesting +parameter associated with hot phases of the ISM, i.e., ionized re- +gions. In our sample one can expect electron temperatures lying +in the range 5 458–7 194 K as from the electron temperature map +provided by Planck Collaboration et al. (2016c). Inside molecular +clouds, the ionized regions produced by stellar radiation are ex- +pected to represent a fraction of the whole volume associated with +the clouds. Not all the sources displayed in Table 2 are only molecu- +lar cloud regions in nature but they all have thermal dust along their +LOSs, which is a component strongly correlated with the AME +component. In this context we show in Figure 20 the distribution +of the free-free EM parameter as a function of 𝐺0. The plot shows +only a poor correlation between the two parameters, this being also +illustrated by the low correlation coefficient, SRCC= 0.30 ± 0.06, +found between the two parameters. This lack of correlation would +indicate that the AME emissivity does not correlate significantly +with the EM free-free emission parameter at Galactic scales. +5.5 +Free-Free correlations +In our study the EM of the free-free does not correlate with the +AME emissivity estimated by 𝐴AME/𝜏250. On the other hand, a +mild correlation is observed between the amplitude of the AME +at the peak frequency, 𝐴AME, and the EM. This is shown on the +plot displayed at the top panel in Figure 21, with a SRCC between +the two parameters of 0.66 ± 0.0.5. Since a strong correlation is +observed between 𝐴AME and the emission of the thermal dust at the +peak frequency, 𝑆TD,PEAK, this also means that a correlation can +be expected between EM and, 𝑆TD,PEAK. This is shown in the plot +displayed in the bottom panel of Figure 21. In that case the SRCC +between the two parameters of the selected dataset is 0.64 ± 0.04. +In the interpretation of these results it must be taken into ac- +count that our EMs are estimated directly from integrated flux den- +sities, and given the non-linear dependency between the two, those +estimates could not be representative of the real averaged EMs of +each region, as it was already commented in section 3.3. This could +indeed contribute to smear out any underlying real correlation. In +addition, the fact that the correlation in the top panel of Figure 21 +is only seen for the sources with highest AME amplitudes could be +a hint that there could be a selection effect, in such a way that when +MNRAS 000, 1–36 (2023) + +20 +F. Poidevin et al. +Figure 20. Free-free Emission Measure (EM) parameter as a function of the +relative strength of the ISRF, 𝐺0. Symbols and colours definition are the +same as in Figure 9. +the free-free is high the AME can only be detected when it is also +very high. In order to better understand this, in Figure 22 we plot +𝐴AME as a function of the flux density of the free-free at 𝜈AME. +The one-to-one relation is displayed by the solid line while the one- +to-ten relation is shown with the dashed line. Given that calibration +uncertainties are of order 5 − 10% the lack of sources below the +one-to-ten line could in fact tell that the AME cannot be separated +when it is less than 10% of the free-free. On the contrary, the plot +also shows that there are a few regions (like the Perseus and 𝜌 oph +molecular clouds, respectively G160.26-18.62 and G353.05+16.90) +with more AME than free-free. +It must also be taken into account that our SED multicom- +ponent fit is subject to an anti-correlation between the AME and +free-free amplitudes which may contribute to worsening the cor- +relation observed in Figure 22. This parameter degeneracy, which +upcoming 5 GHz data from the C-BASS experiment (Jones et al. +2018) will help to break, is clearly seen in MCMC analyses like +those presented in Cepeda-Arroita et al. (2021) and in Fernandez- +Torreiro et al. (2023). +6 +DISCUSSION +In this section we summarize our results suggesting that the AME +carriers may be preferentially located in cold rather than in hot +phases of the ISM. Some limitations of our modelling of the AME +component are then discussed, followed by a comparisons of our +results with those from previous works. +6.1 +Does AME originate from the Cold ISM Phase ? +In the last sections we searched for correlations between some of +the parameters obtained from the multicomponent fits of the AME +component and ISM tracers including the flux densities obtained at +12, 25, 60 and 100 𝜇m. Interestingly, we find that the flux densities +obtained at the peak frequency of the AME bumps show strong +correlation with the flux densities at 100, 60 and 25 𝜇m, with a +small loss of correlation with the flux densities at 12 𝜇m. On the +Figure 21. Top: AME flux at the peak frequency versus free-free emission +measure. Bottom: Thermal dust flux at the peak frequency versus free- +free emission measure. Symbols and colours definition are the same as in +Figure 9. +other hand, once these four flux densities tracers are normalized by +the relative strength of the ISRF, 𝐺0, the correlations with 𝐴AME +are found to be about a few to ten percent lower in the high fre- +quency bands. These results could discard tiny dust particles (PAHs +or VSGs in nature) as AME carriers, if such particles are poorly +sensitive to the relative strength of the ISRF. For this reason we +explored in more detail possible relationships between the AME +component parameters with dust modelling parameters, with 𝐺0, +as well as with the free-free component parameters. Table 6 gives +a summary of some of the most relevant SRCCs obtained from +the previous analysis in this respect. They could help to shed light +on some existing physical relationships between the astrophysical +components. +From spectral energy distribution analysis of the sample of 46 +good candidate AME sources the strongest correlation is found be- +tween the maximum flux density of the thermal dust, 𝑆TD,peak, and +of the AME peak, 𝐴AME (Figure 16). A lower correlation is found +between the AME emissivity, 𝐴AME/𝜏250, and the interstellar radi- +ation field relative strength, 𝐺0 (Figure 19), and a mild correlation +is obtained between 𝐴AME and the free-free EM (Figure 21, top). +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +21 +Figure 22. AME flux at the peak frequency versus free-free flux at the AME +peak frequency. Symbols and colours definition are the same as in Figure 9. +The one-to-one relation is displayed by the solid line and the one-to-ten +relation is shown with the dashed line. +On the other hand no correlation is found between 𝐴AME/𝜏250 and +𝐸𝑀 (see end of Section 5.4), and neither between the AME peak +frequency, 𝜈AME, and 𝐺0 (Figure A4). As discussed in the previ- +ous section, averaging effects in our estimates of EM, as well as +a selection effect associated with only the brightest AME sources +being detected above very high free-free amplitudes, could have an +impact on the tentative correlation seen between EM and 𝐴AME. On +the other hand the correlation found between 𝐴AME and 𝑆TD,peak is +expected to be real since these two components are associated with +distinct wavelength ranges with poor overlap between each other. +Since there is a null correlation between 𝐴AME/𝜏250 and 𝐸𝑀, this +means that 𝐴AME/𝜏250, which also correlates with the dust grain +emissivity, 𝑆TD,peak/𝜏250, is rather driven by 𝐺0, which in turn is +a function of the thermal dust temperature approximated by 𝑇dust +obtained from the modelling. In other words the interstellar radi- +ation field still could be the main driver of the AME in terms of +spinning dust excitation mechanisms, but the spinning dust could +be more likely associated with cold phases of the ISM rather than +to hot phases associated with free-free radiation. +6.2 +AME components characterization +From the results obtained with the multicomponent fit analysis we +tested the level of independency between the parameters used to +fit the AME. This model is the analytical approximation of the +spectrum of spinning dust emission proposed by Stevenson (2014). +Indeed, we find null or very low correlations between parameters, +𝐴AME and 𝜈AME, 𝜈AME and 𝑊AME, and 𝑊AME and 𝐴AME. On +the other hand, we find a small correlation between 𝑊AME and +𝐴AME/𝜏250. The distribution of these two parameters is shown in +Figure 23. By definition, the AME emissivity depends on the total +amount of material along the LOS as estimated by 𝜏250, and this +correlation means that, on average, 𝐴AME/𝑊AME is not directly +proportional to 𝜏250. Testing this result using a physical AME mod- +elling is out of the scope of this work, but could be investigated in +future analyses. On the other hand, in a previous section we dis- +Figure 23. AME emissivity against the width of the AME parabola model, +𝑊AME. Symbols and colours definition are the same as in Figure A3. +cussed the strong correlation obtained between 𝐴AME and the dust +radiance, ℜDust. Put all together these results favor a strong cou- +pling between the peak AME flux densities and the total amount of +dust probed at 250 𝜇m, but only a fraction of the total amount of +material would be at the origin of the AME radiance. +6.3 +Comparison with previous works +The main differences found in this work with respect to the results +discussed in PIRXV have been discussed along the previous sec- +tions. Below we compare and discuss our results with those from +other works. +Using hierarchical Bayesian inference and full dust spectral +energy distribution (SED) modelling, Bell et al. (2019) argue that, +on angular scales of approximately 1◦, AME in 𝜆 Orionis correlates +more strongly with PAH mass than with total dust mass, giving +support for a spinning PAH hypothesis within this region. Here, on +similar angular scales, we find a better correlation with the 100 𝜇m +dust template than with the 12 𝜇m dust template giving hints that, +on Galactic scale, the dust grain components producing AME are +more likely associated with the cold ISM. This hypothesis is also +supported by the strong correlation we find between the maximum +flux density of the AME components with the dust radiance ob- +tained from the integration of the dust flux models at wavelengths +lower than 100 𝜇m. This result may suffer the lack of modelling, in +this work, at wavelengths shorter than 100 𝜇m though, but it sug- +gests that the AME carriers are spatially closely associated with the +thermal dust components. +Cepeda-Arroita et al. (2021) discuss AME spectral variations +in the 𝜆 Orionis region with mild correlation between the AME +peak frequency and the free-free emission measure, and strong cor- +relation between the thermal dust temperature and the free-free +emission measure. Their results obtained at 1◦-angular scale give +an overall picture consistent with spinning dust where the local +radiation field plays a key role. In our analysis we find mild and +null correlations between the AME peak frequency distribution and +the thermal dust temperature, or the free-free emission measure, +respectively. At face value, our result obtained at similar angular +scale tends to discard the free-free emission as the main driver of +MNRAS 000, 1–36 (2023) + +22 +F. Poidevin et al. +Variable 1 +Variable 2 +SRCC +SRCC +SRCC +Power-Law Slope(a) +Figure +selected sample +AME significant +AME semi-significant +selected sample +𝐴AME [Jy] +𝑆TD,peak [Jy] +0.88± 0.05 +0.82± 0.07 +0.91± 0.04 +0.96±1.56 +16 +𝐴AME [Jy] +ℜDust +0.88± 0.05 +0.85± 0.08 +0.90± 0.05 +0.95±2.37 +17 (top) +𝑊AME +𝐴AME/𝜏250 [Jy] +0.66± 0.12 +0.64± 0.18 +0.57± 0.15 +... +23 +𝐴AME/𝜏250 [Jy] +𝐺0 or Ttd +0.68± 0.08 +0.87± 0.07 +0.62± 0.11 +0.78±0.94 +19 +𝑆TD,peak [Jy] +EM [cm−6 pc] +0.64±0.03 +0.50± 0.08 +0.64± 0.04 +0.43±0.16 +21 (bottom) +𝐴AME [Jy] +EM [cm−6 pc] +0.59± 0.05 +0.65± 0.11 +0.55± 0.03 +1.42±0.89 +21 (top) +ℜAME +ℜtd𝑥10−4 +0.70± 0.14 +0.66± 0.23 +0.73± 0.17 +1.57±4.32 +17 (bottom) +𝜈AME [GHz] +𝐺0 or Ttd +0.40± 0.12 +0.21± 0.22 +0.60± 0.15 +-0.05±0.56 +A4 +𝐺0 or Ttd +EM [cm−6 pc] +0.30± 0.06 +0.49± 0.13 +0.09± 0.07 +... +20 +𝑊AME +𝐺0 or Ttd +0.23± 0.12 +0.57± 0.18 +0.07± 0.15 +... +A5 +𝜈AME [GHz] +EM [cm−6 pc] +0.06± 0.11 +0.27± 0.20 +-0.24± 0.14 +... +... +𝐴AME/𝜏250 [Jy] +EM [cm−6 pc] +0.01± 0.07 +0.46± 0.14 +-0.21± 0.07 +... +... +𝐴AME [Jy] +𝐺0 or Ttd +-0.16± 0.06 +0.26± 0.10 +-0.33± 0.07 +... +... +Table 6. Selection of Spearman rank correlation coefficients (SRCCs) between several model parameters in decreasing strength for the selected sample. (a): +Slopes obtained from linear fits in log–log space. +the excitation of the AME carriers. On the other hand, our anal- +ysis is obtained on a sample of sources distributed on a Galactic +scale. This makes direct comparisons with results obtained on in- +dividual regions quite difficult. One should also bear in mind that +some of the correlations obtained at low angular resolutions break +down on finer angular scales. E.g. Casassus et al. (2006) discuss +31 GHz Cosmic Background Imager (CBI) observations of LDN +1622; Casassus et al. (2008) discuss similar observations of the 𝜌 +Oph molecular cloud; Arce-Tord et al. (2020) discuss 𝜌 Oph 4.5 ar- +cmin resolution observations at 31 GHz with CBI 2; and Casassus +et al. (2021) discuss ATCA high resolution observations of the 𝜌 +Oph West photo-dissociation region suggesting spectral variations +that could be explained with two different cut offs on PAHs popu- +lations with the SPDust model. Actually, these studies demonstrate +that finer angular resolution observations are important to identify +the physical regions where spectral variations occur. +From another perspective, Bernstein et al. (2020) discuss +fullerenes based modelling of AME in 14 different regions. The +models are calibrated using the well studied LDN 1622 dark cloud +physical conditions. The rotational temperatures are of the order +of the dust grains temperatures for most of the regions, suggest- +ing that in this scenario the AME carriers are associated with cold +ISM phases. This result could support our discussion above (i.e. +that AME emissivity correlates slightly with the dust temperature +while not with EM). Our study is focused on high column density +regions pervaded by molecular clouds, i.e., including cold neutral +medium (CNM) phases, mainly located along the Galactic plane. +Using a completely different method, Hensley et al. (2021) investi- +gated the relationship between the CNM, the AME and the abun- +dance of PAHs over large areas associated with diffuse ISM regions +(𝑁HI < 4 × 1020 cm−2) at high Galactic latitudes (| 𝑏 |> 30◦). +Their study shows that the CNM fraction strongly correlates with +the fraction of dust in PAHs, and that PAHs preferentially reside in +cold and relatively dense phases of the gas. If PAHs are indeed at +the origin of the AME probed in our work, they could also pref- +erentially be associated with cold phases of the ISM, i.e., with the +CNM. +Finally, we point out that AME has been detected in other +galaxies. The first detection of AME in another galaxy, namely, +NGC 6946, was reported by Murphy et al. (2010). Detection of +AME has also been reported by Murphy et al. (2018) in NGC 4725B +using VLA data. In a following work Murphy et al. (2020) discussed +complementary ALMA observations on NGC 4725B that show dis- +crepancy with expected thermal dust component making the inter- +pretation of the results quite puzzling. In our study we sampled +the AME component over several AME candidate regions in our +Galaxy. The results show a distribution of peak frequencies close +to 25 GHz which is consistent with the average peak frequency ob- +served by Battistelli et al. (2019) on M31. Here, the relatively low +resolution used in our study allows to sample our galaxy at about +kiloparsec scales or lower. This is an asset allowing more straight- +forward comparisons with results obtained on close-by galaxies +sampled at kiloparsec scales (see for example Figure 1 in Murphy +et al. 2010, for comparison with our Figure 1). +7 +SUMMARY +In this work we revisited the approach proposed by PIRXV and their +analysis of the multicomponent parameters obtained on Galactic +candidates AME sources on the full sky at 1◦-angular scales. The +main difference with their work comes from the inclusion of flux +densities provided by the QUIJOTE-MFI wide survey maps at 11, +13, 17 and 19 GHz covering the northern hemisphere. These maps +allow generally improved detections, a better separation of the AME +and the free-free components and a better characterizations of the +AME spectra observed between 10 GHz and 60 GHz on a sample +of 46 sources. From our analysis we find the following: +• The distribution of the AME peak frequency has a weighted +mean frequency and dispersion of 23.6 ± 3.6 GHz, about 4 GHz +lower than the mean value obtained by PIRXV on their full-sky +sample. Our result demonstrates the importance of using low fre- +quency data in the range 10–20 GHz to properly characterize the +AME bump turnover. The value is in agreement with estimates +obtained on nearby spiral galaxies. +• The strongest correlations, of order 88%, are found between +the thermal dust peak flux density, and of the AME peak flux den- +sity, and between the AME peak flux density and the thermal dust +radiance. +• Mild correlation coefficients of order 66-68 per cent are found +between the AME emissivity (defined as 𝐴AME/𝜏250) and the width +of the AME component, as well as between the AME emissivity +and the interstellar radiation field relative strength. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +23 +• A mild correlation of order 59% is found between the AME +peak flux density and the free-free EM, but this could be affected +by averaging effects in the calculation of EM, as well as by the +fact that only very bright AME sources would be clearly detected +above strong free-free emission, whose determination is subject to +uncertainties associated with calibration errors of order 10%. +• No correlation is found between the AME emissivity, +𝐴AME/𝜏250, and the free-free radiation EM. +• No significant correlation is observed between the peak fre- +quencies of the AME and the thermal dust components as it has +been reported in the case of Lambda Orionis in a previous study by +Cepeda-Arroita et al. (2021). +From our analysis we conclude that the interstellar radiation +field still can be the main driver of the intensity of the AME toward +spinning dust excitation mechanisms, but it is not clear whether +spinning dust would be most likely associated with cold phases +of the interstellar medium rather than with hot phases dominated +by free-free radiation. Future data over large sky fractions coming +from projects currently under development like C-BASS (Jones +et al. 2018), TFGI (Rubiño-Martín et al. (2012b), and see also the +introduction in Rubiño-Martín et al. (2023)) and MFI2 (Hoyland +et al. 2022) should help to clarify these aspects and to further refine +similar statistical analyses. +ACKNOWLEDGEMENTS +We thank the staff of the Teide Observatory for invaluable +assistance in the commissioning and operation of QUIJOTE. +The QUIJOTE experiment is being developed by the Instituto +de Astrofisica de Canarias (IAC), the Instituto de Fisica de +Cantabria (IFCA), and the Universities of Cantabria, Manch- +ester and Cambridge. Partial financial support was provided +by the Spanish Ministry of Science and Innovation under +the projects AYA2007-68058-C03-01, AYA2007-68058-C03-02, +AYA2010-21766-C03-01, AYA2010-21766-C03-02, AYA2014- +60438-P, ESP2015-70646-C2-1-R, AYA2017-84185-P, ESP2017- +83921-C2-1-R, +AYA2017-90675-REDC +(co-funded +with +EU +FEDER funds), PGC2018-101814-B-I00, PID2019-110610RB- +C21, PID2020-120514GB-I00, IACA13-3E-2336, IACA15-BE- +3707, EQC2018-004918-P, the Severo Ochoa Programs SEV- +2015-0548 and CEX2019-000920-S, the Maria de Maeztu Pro- +gram MDM-2017-0765, and by the Consolider-Ingenio project +CSD2010-00064 (EPI: Exploring the Physics of Inflation). We +acknowledge support from the ACIISI, Consejeria de Economia, +Conocimiento y Empleo del Gobierno de Canarias and the European +Regional Development Fund (ERDF) under grant with reference +ProID2020010108. This project has received funding from the Eu- +ropean Union’s Horizon 2020 research and innovation program un- +der grant agreement number 687312 (RADIOFOREGROUNDS). +FP acknowledges the European Commission under the Marie +Sklodowska-Curie Actions within the European Union’s Horizon +2020 research and innovation programme under Grant Agree- +ment number 658499 (PolAME). FP acknowledges support from +the Spanish State Research Agency (AEI) under grant numbers +PID2019-105552RB-C43. FG acknowledges funding from the Eu- +ropean Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agree- +ment No 101001897). EdlH acknowledge partial financial support +from the Concepción Arenal Programme of the Universidad de +Cantabria. BR-G acknowledges ASI-INFN Agreement 2014-037- +R.0. DT acknowledges the support from the Chinese Academy of +Sciences President’s International Fellowship Initiative, Grant N. +2020PM0042. 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Symbols and colours definition are the same as +in Figure 9. +APPENDIX A: ADDITIONAL PLOTS +A few more Figures, all showing a lack of correlation between some +of the modelling parameters, are displayed in this appendix for the +interested reader. +• Figure A1 shows the variations of the AME peak frequency, +𝜈AME, as a function of the proxy of the thermal dust material, 𝜏250, +as discussed in section 5.1.3. The distribution of 𝜈AME seems to be +independent of the quantity of matter along the LOSs. +• Figure A2 shows the Variations of the AME characteristic +width,𝑊AME, as a function of the proxy of the thermal dust material, +𝜏250, as discussed in section 5.1.3. This AME parameter also seems +to be independent of the quantity of matter along the LOSs. +• Figure A3 shows the Variations of the AME peak frequency, +𝜈AME, as a function of the width of the parabola, 𝑊AME, as dis- +cussed in section 5.1.7. As expected from the formalism used to +model AME (see Stevenson 2014), this plot confirms that the two +parameters are independent from each other. +• Figure A4 and Figure A5 show the distribution of the AME +peak frequency and of the AME parabola width parameter with +the relative strength of the ISRF, 𝐺0, respectively, as discussed in +section 5.4. The two AME parameters show no dependence on 𝐺0. +APPENDIX B: SED MULTICOMPONENT FIT +PARAMETERS +The parameters obtained on each source from the multicomponent +analysis are displayed in Table B1 and Table B2. The name of the +sources are given in column one from each Table. The parameters +used to model the synchrotron component, and the free-free com- +ponent, are given in columns 2 and 3 from Table B1, respectively. +Columns 4 to 6 of the same table give the parameters used to model +the thermal dust grain component, while column 7 gives the relative +strength of the ISRF derived using 𝑇dust displayed in column 5. The +parameter used to model the CMB component is found in the last +column of Table B1. All the parameters related to the modelling of +AME are given in Table B2. The maximum fraction of emission that +Figure A2. Distribution of the width of the AME WAME against 𝜏250. All +selected data are displayed. Symbols and colours definition are the same as +in Figure 9. +Figure A3. Variations of the AME peak frequency 𝜈AME as a function of +the width of the parabola, 𝜎AME,𝜈. Symbols and colours definition are the +same as in Figure 9. +could be attributed to UCH ii regions is given in column 8 of that +table, while the reduced 𝜒2 of the multicomponent fits are given in +the last column. +APPENDIX C: SEDS OF THE FULL SAMPLE +The plots of the SEDs obtained on each of the 52 candidate AME +sources of the sample are displayed in Figures C1–C7. In each plot, +the QUIJOTE intensity flux densities are shown with red filled cir- +cles, and the WMAP, Planck, and DIRBE intensity flux densities are +shown with green, blue, and yellow opened diamonds, respectively. +The low frequency points used to fit the models are shown in pale +blue, and in blue if they were not included in the fitting procedure. +The result of the multicomponent fit is illustrated by the continu- +MNRAS 000, 1–36 (2023) + +26 +F. Poidevin et al. +Source Name +𝛼synch,int +EM +𝜏250 +𝑇dust +𝛽dust +G0 +Δ𝑇CMB +[cm−6 pc] +[×105] +[K] +[𝜇K] +G010.19-00.32 +-0.41 ± 0.27 +1969.0 ± 86.8 +307.0 ± 35.2 +20.6 ± 0.6 +1.90 ± 0.06 +2.68 ± 0.48 +125.0 (PUL) +G010.84-02.59 +... +244.2 ± 17.1 +29.8 ± 3.3 +20.7 ± 0.6 +1.68 ± 0.06 +2.73 ± 0.46 +7.3 ± 20.5 +G011.11-00.12 +... +2671.4 ± 111.7 +312.3 ± 38.6 +20.6 ± 0.7 +1.88 ± 0.07 +2.67 ± 0.51 +-28.2 ± 115.6 +G012.80-00.19 +-0.22 ± 0.05 +1328.8 ± 118.4 +413.0 ± 41.1 +19.9 ± 0.5 +1.92 ± 0.05 +2.13 ± 0.32 +125.0 (PUL) +G015.06-00.69 +... +7265.6 ± 63.8 +193.1 ± 16.6 +23.1 ± 0.5 +1.79 ± 0.04 +5.23 ± 0.74 +125.0 (PUL) +G017.00+00.85 +... +399.9 ± 72.9 +173.7 ± 34.3 +19.4 ± 0.9 +1.89 ± 0.12 +1.86 ± 0.53 +-74.7 ± 151.9 +G037.79-00.11 +-1.33 ± 0.51 +1812.5 ± 55.6 +237.8 ± 28.9 +20.2 ± 0.6 +1.81 ± 0.07 +2.38 ± 0.44 +69.5 ± 72.9 +G040.52+02.53 +-0.62 ± 0.03 +6.9 ± 3.8 +25.5 ± 2.1 +20.2 ± 0.4 +1.75 ± 0.05 +2.34 ± 0.28 +69.3 ± 10.3 +G041.03-00.07 +-0.80 ± 0.14 +534.3 ± 34.8 +163.6 ± 17.1 +19.1 ± 0.5 +1.83 ± 0.06 +1.71 ± 0.26 +125.0 (PUL) +G043.20-00.10 +... +1442.5 ± 40.2 +151.6 ± 19.0 +20.4 ± 0.7 +1.74 ± 0.07 +2.53 ± 0.48 +67.5 ± 62.3 +G045.47+00.06 +-1.37 ± 0.18 +383.9 ± 24.0 +113.3 ± 10.1 +20.3 ± 0.5 +1.71 ± 0.05 +2.41 ± 0.33 +125.0 (PUL) +G049.14-00.60 +... +3479.9 ± 35.5 +81.1 ± 8.5 +24.3 ± 0.8 +1.72 ± 0.06 +7.22 ± 1.34 +42.2 ± 38.1 +G059.42-00.21 +... +286.9 ± 18.4 +55.4 ± 11.0 +19.2 ± 0.9 +1.77 ± 0.12 +1.72 ± 0.48 +-60.4 ± 29.1 +G061.47+00.11 +... +311.4 ± 8.1 +23.4 ± 2.0 +21.9 ± 0.5 +1.57 ± 0.05 +3.81 ± 0.51 +8.0 ± 10.6 +G062.98+00.05 +... +243.1 ± 8.2 +27.2 ± 2.7 +20.3 ± 0.5 +1.66 ± 0.06 +2.42 ± 0.37 +5.2 ± 11.4 +G070.14+01.61 +... +272.4 ± 26.6 +... +10.0 (PLL) +2.11 ± 2.99 +0.03 (PLL) +65.8 ± 54.9 +G071.59+02.85 +... +215.4 ± 17.1 +15.0 ± 4.3 +21.5 ± 1.6 +1.57 ± 0.18 +3.42 ± 1.54 +-12.8 ± 32.9 +G075.81+00.39 +... +324.7 ± 12.5 +39.9 ± 3.4 +19.8 ± 0.4 +1.65 ± 0.05 +2.09 ± 0.27 +125.0 (PUL) +G076.38-00.62 +... +... +11.3 ± 2.7 +24.3 ± 1.7 +1.78 ± 0.17 +7.09 ± 2.93 +56.5 ± 38.5 +G078.57+01.00 +... +1835.0 ± 74.2 +32.4 ± 6.2 +23.9 ± 1.3 +1.46 ± 0.09 +6.53 ± 2.19 +-125.0 (PLL) +G081.59+00.01 +... +2541.5 ± 44.1 +60.9 ± 5.6 +25.3 ± 0.7 +1.56 ± 0.04 +9.17 ± 1.54 +125.0 (PUL) +G084.68-00.58 +... +1201.3 ± 15.4 +35.7 ± 2.6 +21.9 ± 0.4 +1.52 ± 0.04 +3.81 ± 0.45 +-1.3 ± 23.2 +G085.00+04.20 +... +435.2 ± 6.6 +9.3 ± 1.1 +23.1 ± 0.8 +1.74 ± 0.09 +5.30 ± 1.10 +86.2 ± 11.0 +G093.02+02.76 +... +497.2 ± 6.6 +24.0 ± 1.6 +21.5 ± 0.4 +1.47 ± 0.04 +3.46 ± 0.37 +-19.6 ± 9.7 +G094.47-01.53 +... +257.0 ± 3.7 +4.3 ± 0.8 +23.2 ± 1.2 +1.57 ± 0.13 +5.36 ± 1.64 +-6.1 ± 5.6 +G098.00+01.47 +... +104.1 ± 3.5 +6.2 ± 0.6 +21.5 ± 0.5 +1.46 ± 0.05 +3.43 ± 0.50 +-20.9 ± 6.4 +G099.60+03.70 +... +506.0 ± 7.6 +3.1 ± 0.8 +25.9 ± 2.0 +1.67 ± 0.17 +10.45 ± 4.80 +-34.5 ± 9.1 +G102.88-00.69 +... +332.9 ± 5.9 +8.9 ± 1.2 +23.4 ± 0.9 +1.41 ± 0.08 +5.66 ± 1.25 +-83.2 ± 11.7 +G107.20+05.20 +... +216.5 ± 5.6 +19.4 ± 2.3 +22.8 ± 0.8 +1.56 ± 0.06 +4.88 ± 0.97 +-1.7 ± 11.8 +G110.25+02.58 +... +271.0 ± 12.9 +10.8 ± 2.4 +25.4 ± 1.7 +1.52 ± 0.13 +9.26 ± 3.81 +-52.8 ± 30.4 +G111.54+00.81 +... +284.8 ± 13.8 +23.3 ± 3.0 +22.6 ± 0.8 +1.74 ± 0.08 +4.69 ± 1.02 +117.3 ± 16.2 +G118.09+04.96 +... +1096.0 ± 10.6 +8.9 ± 1.2 +27.7 ± 1.3 +1.53 ± 0.08 +15.63 ± 4.30 +18.2 ± 13.0 +G123.13-06.27 +... +140.4 ± 1.7 +3.7 ± 0.3 +24.6 ± 0.6 +1.34 ± 0.04 +7.76 ± 1.09 +5.5 ± 3.4 +G133.27+09.05 +... +... +35.5 ± 6.1 +15.4 ± 0.6 +1.69 ± 0.09 +0.46 ± 0.10 +0.2 ± 13.1 +G133.74+01.22 +... +1546.5 ± 25.3 +27.4 ± 3.1 +26.6 ± 1.0 +1.52 ± 0.06 +12.44 ± 2.77 +68.9 ± 25.9 +G142.35+01.35 +... +3.1 ± 2.5 +36.4 ± 5.1 +17.7 ± 0.6 +1.67 ± 0.08 +1.06 ± 0.20 +25.6 ± 13.1 +G151.62-00.28 +... +311.3 ± 4.2 +9.3 ± 0.6 +23.1 ± 0.4 +1.22 ± 0.03 +5.23 ± 0.59 +-12.5 ± 7.3 +G160.26-18.62 +... +46.4 ± 3.4 +27.7 ± 5.2 +19.8 ± 1.0 +1.60 ± 0.10 +2.12 ± 0.61 +19.8 ± 16.1 +G160.60-12.05 +... +343.8 ± 6.4 +2.8 ± 0.9 +25.9 ± 2.6 +1.58 ± 0.20 +10.57 ± 6.25 +-35.3 ± 8.5 +G173.56-01.76 +... +383.3 ± 5.9 +2.0 ± 0.3 +28.3 ± 1.6 +1.26 ± 0.10 +17.94 ± 6.07 +-26.6 ± 7.6 +G173.62+02.79 +... +172.9 ± 4.3 +28.7 ± 2.4 +20.3 ± 0.5 +1.48 ± 0.04 +2.44 ± 0.33 +-26.6 ± 8.6 +G190.00+00.46 +... +238.6 ± 4.7 +24.1 ± 2.5 +21.4 ± 0.6 +1.46 ± 0.05 +3.37 ± 0.58 +5.1 ± 9.5 +G192.34-11.37 +... +... +11.0 ± 2.0 +18.7 ± 0.8 +1.79 ± 0.11 +1.47 ± 0.39 +12.6 ± 8.7 +G192.60-00.06 +... +77.9 ± 3.1 +8.2 ± 0.9 +22.1 ± 0.6 +1.72 ± 0.07 +4.09 ± 0.70 +-13.4 ± 6.0 +G201.62+01.63 +... +145.7 ± 3.5 +8.5 ± 1.1 +20.6 ± 0.7 +1.70 ± 0.09 +2.70 ± 0.53 +60.9 ± 6.4 +G203.24+02.08 +... +113.4 ± 3.6 +24.9 ± 1.9 +18.8 ± 0.3 +1.53 ± 0.04 +1.56 ± 0.17 +-26.4 ± 8.1 +G208.80-02.65 +... +100.5 ± 6.7 +7.3 ± 2.3 +20.1 ± 1.6 +1.59 ± 0.20 +2.28 ± 1.07 +15.5 ± 19.2 +G239.40-04.70 +... +63.0 ± 3.7 +10.2 ± 1.7 +19.0 ± 0.7 +1.64 ± 0.10 +1.62 ± 0.38 +9.2 ± 7.7 +G351.31+17.28 +... +107.1 ± 3.1 +7.7 ± 1.3 +26.8 ± 1.4 +1.56 ± 0.10 +12.95 ± 4.05 +1.7 ± 8.2 +G353.05+16.90 +... +41.5 ± 4.9 +47.8 ± 4.5 +22.7 ± 0.6 +1.57 ± 0.04 +4.75 ± 0.79 +40.9 ± 12.7 +G353.97+15.79 +... +28.8 ± 4.7 +39.5 ± 7.0 +19.7 ± 0.9 +1.61 ± 0.09 +2.02 ± 0.57 +-25.4 ± 17.1 +G355.63+20.52 +... +... +15.7 ± 2.8 +17.6 ± 0.7 +1.76 ± 0.11 +1.05 ± 0.26 +52.5 ± 8.1 +Table B1. SEDs Fit parameters. (PUL): prior upper limit. (PLL): prior lower limit. +ous black curve. The fit to the AME component is shown with the +dashed red line. The fit to the free–free component is shown with +the dashed blue line. The fit to the thermal dust component is shown +with the dashed yellow line. The fit to the CMB component is shown +with the dashed green line. +APPENDIX D: ADDITIONAL TABLES +All the SRCCs obtained between the several parameters used to +model the free-free, AME, and thermal dust components are dis- +played in Tables D1–D5. In additional to several plots used to study +correlations between some of the parameters, these tables have been +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +27 +Source Name +𝐴AME +𝜎AME +𝜈AME +𝑊AME +𝑆28.4 +resid +𝑆28.4 +resid +𝑆100𝜇m +𝑓 UCHII +max +[15𝐺𝐻 𝑧] +𝜒2 +red +[Jy] +[GHz] +[log[GHz]] +[Jy] +G010.19-00.32 +66.7 ± 26.0 +2.6 +23.5 ± 7.9 +0.5 ± 0.7 +62.8 ± 14.2 +0.8 ± 0.2 +3.073 +0.4 +G010.84-02.59 +13.3 ± 2.7 +4.8 +20.6 ± 2.2 +0.3 ± 0.1 +7.8 ± 2.6 +1.8 ± 0.7 +0.482 +0.0 +G011.11-00.12 +40.6 ± 16.4 +2.5 +23.7 ± 3.5 +0.3 ± 0.2 +33.9 ± 6.8 +0.4 ± 0.1 +10.818 +0.4 +G012.80-00.19 +44.0 ± 36.2 +1.2 +25.8 ± 6.2 +0.3 ± 0.4 +42.0 ± 4.5 +0.4 ± 0.1 +5.904 +0.3 +G015.06-00.69 +57.7 ± 7.2 +8.0 +27.0 ± 6.0 +0.6 ± 0.3 +57.5 ± 17.0 +0.6 ± 0.2 +2.275 +0.2 +G017.00+00.85 +91.8 ± 15.4 +6.0 +30.3 ± 9.3 +1.0 ± 0.5 +91.6 ± 31.7 +3.6 ± 1.4 +0.044 +0.3 +G037.79-00.11 +56.0 ± 7.4 +7.6 +21.8 ± 2.9 +0.5 ± 0.3 +48.1 ± 14.0 +0.9 ± 0.3 +0.458 +0.4 +G040.52+02.53 +12.5 ± 1.0 +12.9 +24.9 ± 5.6 +0.8 ± 0.2 +12.3 ± 4.1 +2.7 ± 1.0 +0.831 +0.0 +G041.03-00.07 +51.8 ± 6.6 +7.9 +21.5 ± 2.5 +0.6 ± 0.3 +46.3 ± 14.0 +2.2 ± 0.8 +0.175 +0.2 +G043.20-00.10 +45.7 ± 5.5 +8.3 +21.7 ± 2.5 +0.5 ± 0.2 +40.4 ± 12.0 +1.5 ± 0.5 +1.437 +0.4 +G045.47+00.06 +62.4 ± 4.0 +15.6 +21.7 ± 2.1 +0.8 ± 0.2 +58.5 ± 19.3 +2.8 ± 1.1 +0.628 +0.2 +G049.14-00.60 +76.2 ± 3.3 +22.9 +17.2 ± 2.4 +1.0 (PUL) +67.1 ± 24.6 +1.3 ± 0.5 +1.047 +0.2 +G059.42-00.21 +20.9 ± 2.4 +8.7 +20.2 ± 2.3 +0.6 ± 0.2 +17.2 ± 5.4 +2.5 ± 0.9 +0.350 +0.7 +G061.47+00.11 +5.3 ± 1.3 +4.1 +25.3 ± 2.8 +0.3 ± 0.1 +5.0 ± 1.1 +0.8 ± 0.2 +10.855 +0.1 +G062.98+00.05 +7.4 ± 1.2 +6.1 +23.1 ± 1.8 +0.3 ± 0.1 +6.1 ± 1.6 +1.5 ± 0.5 +1.087 +0.2 +G070.14+01.61 +11.0 ± 3.6 +3.1 +23.0 ± 29.3 +1.0 (PUL) +10.8 ± 3.1 +3.8 ± 1.2 +2.820 +1.7 +G071.59+02.85 +12.7 ± 2.7 +4.8 +24.5 ± 7.7 +0.6 ± 0.4 +12.4 ± 3.4 +3.0 ± 1.0 +0.007 +0.3 +G075.81+00.39 +19.1 ± 3.2 +5.9 +19.9 ± 5.3 +0.7 ± 0.4 +16.8 ± 5.1 +2.6 ± 0.9 +0.575 +0.1 +G076.38-00.62 +15.5 ± 4.0 +3.9 +24.5 ± 14.3 +0.9 ± 0.9 +15.2 ± 4.5 +2.3 ± 0.8 +1.195 +0.1 +G078.57+01.00 +... +... +... +... +... +... +... +0.7 +G081.59+00.01 +94.0 ± 5.2 +17.9 +26.3 ± 3.6 +1.0 (PUL) +93.7 ± 34.3 +2.4 ± 1.0 +0.000 +0.2 +G084.68-00.58 +34.6 ± 1.8 +18.8 +22.5 ± 3.8 +1.0 (PUL) +33.7 ± 12.1 +3.4 ± 1.4 +0.000 +0.1 +G085.00+04.20 +16.7 ± 0.8 +21.1 +19.5 ± 2.9 +1.0 (PUL) +15.5 ± 5.6 +4.8 ± 1.9 +0.010 +0.1 +G093.02+02.76 +13.4 ± 0.6 +21.0 +22.6 ± 2.1 +0.7 ± 0.1 +12.7 ± 4.2 +2.3 ± 0.8 +0.724 +0.1 +G094.47-01.53 +2.2 ± 0.5 +4.1 +32.6 ± 3.3 +0.3 ± 0.1 +1.9 ± 0.5 +1.4 ± 0.4 +31.097 +0.1 +G098.00+01.47 +10.8 ± 0.6 +17.2 +27.1 ± 4.5 +1.0 (PUL) +10.8 ± 4.0 +8.2 ± 3.4 +0.314 +0.2 +G099.60+03.70 +3.0 ± 1.0 +3.0 +35.9 ± 4.7 +0.2 ± 0.2 +1.8 ± 0.6 +0.7 ± 0.2 +133.803 +0.3 +G102.88-00.69 +7.5 ± 0.7 +10.9 +26.7 ± 8.1 +0.8 ± 0.3 +7.5 ± 2.5 +3.1 ± 1.1 +0.018 +0.2 +G107.20+05.20 +18.5 ± 0.7 +27.8 +25.6 ± 2.7 +0.7 ± 0.1 +18.3 ± 6.1 +2.6 ± 1.0 +1.008 +0.4 +G110.25+02.58 +7.6 ± 2.8 +2.7 +38.7 ± 31.0 +0.6 ± 0.6 +6.7 ± 2.1 +0.8 ± 0.3 +11.067 +0.6 +G111.54+00.81 +21.4 ± 2.0 +10.8 +24.2 ± 4.3 +0.8 ± 0.3 +20.9 ± 6.7 +2.1 ± 0.8 +2.730 +0.3 +G118.09+04.96 +12.9 ± 0.9 +14.2 +30.2 ± 9.8 +1.0 (PUL) +12.9 ± 4.8 +1.5 ± 0.6 +0.003 +0.3 +G123.13-06.27 +4.6 ± 0.2 +25.2 +26.8 ± 2.6 +0.7 ± 0.1 +4.6 ± 1.5 +3.7 ± 1.4 +0.459 +0.1 +G133.27+09.05 +4.9 ± 0.4 +11.1 +21.9 ± 6.2 +0.7 ± 0.3 +4.6 ± 1.5 +8.4 ± 3.1 +0.055 +1.3 +G133.74+01.22 +54.4 ± 2.2 +24.8 +24.1 ± 3.3 +1.0 (PUL) +53.7 ± 19.6 +2.3 ± 0.9 +0.553 +0.3 +G142.35+01.35 +6.7 ± 0.8 +8.4 +20.6 ± 2.7 +0.6 ± 0.2 +5.7 ± 1.8 +2.4 ± 0.8 +0.200 +0.6 +G151.62-00.28 +5.6 ± 0.5 +11.4 +29.9 ± 3.9 +0.5 ± 0.1 +5.6 ± 1.7 +2.7 ± 0.9 +1.301 +0.1 +G160.26-18.62 +14.7 ± 0.8 +19.2 +25.6 ± 1.5 +0.5 ± 0.1 +14.4 ± 4.2 +3.2 ± 1.1 +0.064 +1.3 +G160.60-12.05 +9.3 ± 0.7 +12.6 +49.1 ± 38.5 +0.9 ± 0.5 +7.7 ± 3.7 +4.1 ± 2.1 +0.000 +0.8 +G173.56-01.76 +2.7 ± 0.6 +4.4 +26.4 ± 7.0 +0.6 ± 0.4 +2.7 ± 0.7 +2.1 ± 0.7 +0.055 +0.2 +G173.62+02.79 +9.6 ± 0.6 +15.5 +24.7 ± 1.1 +0.4 ± 0.1 +9.1 ± 2.5 +2.6 ± 0.8 +1.631 +0.3 +G190.00+00.46 +14.3 ± 0.5 +29.3 +22.2 ± 1.4 +0.7 ± 0.1 +13.4 ± 4.4 +3.1 ± 1.1 +0.848 +0.4 +G192.34-11.37 +9.1 ± 0.7 +12.5 +24.9 ± 4.0 +0.7 ± 0.2 +9.0 ± 2.9 +9.1 ± 3.3 +0.000 +0.8 +G192.60-00.06 +3.8 ± 0.5 +7.9 +20.8 ± 1.4 +0.3 ± 0.1 +2.4 ± 0.8 +1.0 ± 0.4 +0.719 +0.1 +G201.62+01.63 +10.0 ± 0.4 +27.3 +21.6 ± 2.8 +0.9 ± 0.2 +9.5 ± 3.3 +6.5 ± 2.5 +0.056 +0.1 +G203.24+02.08 +7.1 ± 0.4 +15.8 +26.9 ± 5.0 +0.7 ± 0.2 +7.1 ± 2.3 +3.6 ± 1.3 +0.022 +0.2 +G208.80-02.65 +3.4 ± 1.8 +1.9 +17.2 ± 23.4 +0.9 ± 1.7 +3.0 ± 0.7 +4.3 ± 1.2 +0.025 +0.2 +G239.40-04.70 +6.9 ± 0.4 +16.5 +21.2 ± 2.0 +0.7 ± 0.1 +6.3 ± 2.1 +6.9 ± 2.5 +0.000 +0.3 +G351.31+17.28 +12.0 ± 0.4 +32.9 +20.0 ± 1.3 +0.7 ± 0.1 +10.6 ± 3.6 +1.5 ± 0.6 +0.036 +0.1 +G353.05+16.90 +20.0 ± 0.7 +27.3 +28.8 ± 1.2 +0.5 ± 0.1 +20.0 ± 6.0 +1.3 ± 0.5 +0.997 +0.4 +G353.97+15.79 +13.1 ± 1.2 +10.6 +23.7 ± 1.2 +0.3 ± 0.1 +11.4 ± 3.1 +3.5 ± 1.1 +0.190 +0.3 +G355.63+20.52 +8.1 ± 0.5 +17.0 +23.5 ± 1.2 +0.5 ± 0.1 +7.5 ± 2.2 +7.7 ± 2.6 +0.000 +0.7 +Table B2. SEDs Fit parameters. (PUL): prior upper limit. +used to systematically identify the parameters showing the most +meaningful correlation factors, and guide our study. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–36 (2023) + +28 +F. Poidevin et al. +Variable 1 +Variable 2 +SRCC +SRCC +SRCC +selected sample +AME significant +AME semi-significant +𝐴AME [Jy] +𝜈AME [GHz] +−0.20± 0.11 +0.10± 0.22 +−0.52± 0.13 +𝐴AME [Jy] +𝑊AME +0.27± 0.12 +0.39± 0.20 +0.40± 0.14 +𝐴AME [Jy] +𝐴AME/𝜏250 [Jy] +−0.11± 0.08 +0.19± 0.09 +−0.27± 0.10 +𝐴AME [Jy] +ℜAME +0.88± 0.15 +0.77± 0.27 +0.91± 0.22 +𝜈AME [GHz] +𝑊AME +−0.06± 0.14 +0.27± 0.26 +−0.12± 0.17 +𝜈AME [GHz] +𝐴AME/𝜏250 [Jy] +0.18± 0.13 +0.02± 0.22 +0.31± 0.16 +𝜈AME [GHz] +ℜAME +−0.04± 0.15 +0.34± 0.25 +−0.33± 0.20 +𝑊AME +𝐴AME/𝜏250 [Jy] +0.66± 0.12 +0.64± 0.18 +0.57± 0.15 +𝑊AME +ℜAME +0.60± 0.15 +0.75± 0.27 +0.67± 0.20 +𝐴AME/𝜏250 [Jy] +ℜAME +0.19± 0.15 +0.43± 0.26 +−0.01± 0.19 +Table D1. Spearman rank correlation coefficients (SRCCs) between AME and AME parameters. +Variable 1 +Variable 2 +SRCC +SRCC +SRCC +selected sample +AME significant +AME semi-significant +𝐴AME [Jy] +Ttd or 𝐺0 +−0.16± 0.06 +0.26± 0.10 +−0.33± 0.07 +𝐴AME [Jy] +𝛽td +0.49± 0.08 +0.21± 0.14 +0.56± 0.09 +𝐴AME [Jy] +𝜏250 +0.81± 0.04 +0.53± 0.07 +0.85± 0.04 +𝐴AME [Jy] +𝑆TD,peak [Jy] +0.88± 0.05 +0.82± 0.07 +0.91± 0.04 +𝐴AME [Jy] +𝜈TD,peak [GHz] +0.04± 0.11 +0.34± 0.17 +−0.07± 0.15 +𝐴AME [Jy] +ℜtd𝑥10−4 +0.88± 0.05 +0.85± 0.08 +0.90± 0.05 +𝜈AME [GHz] +Ttd or 𝐺0 +0.40± 0.12 +0.21± 0.22 +0.60± 0.15 +𝜈AME [GHz] +𝛽td +−0.35± 0.13 +−0.35± 0.26 +−0.44± 0.16 +𝜈AME [GHz] +𝜏250 +−0.25± 0.12 +0.01± 0.23 +−0.57± 0.14 +𝜈AME [GHz] +𝑆TD,peak [Jy] +−0.14± 0.13 +0.11± 0.23 +−0.47± 0.14 +𝜈AME [GHz] +𝜈TD,peak [GHz] +0.34± 0.13 +0.19± 0.23 +0.42± 0.17 +𝜈AME [GHz] +ℜtd𝑥10−4 +−0.11± 0.13 +0.11± 0.21 +−0.42± 0.14 +𝑊AME +Ttd or 𝐺0 +0.23± 0.12 +0.57± 0.18 +0.07± 0.15 +𝑊AME +𝛽td +−0.24± 0.13 +−0.23± 0.25 +−0.21± 0.16 +𝑊AME +𝜏250 +−0.17± 0.12 +−0.18± 0.23 +−0.00± 0.16 +𝑊AME +𝑆TD,peak [Jy] +−0.05± 0.12 +0.22± 0.22 +0.08± 0.15 +𝑊AME +𝜈TD,peak [GHz] +0.18± 0.13 +0.68± 0.19 +0.00± 0.17 +𝑊AME +ℜtd𝑥10−4 +−0.01± 0.12 +0.28± 0.22 +0.12± 0.15 +𝐴AME/𝜏250 [Jy] +Ttd or 𝐺0 +0.68± 0.08 +0.87± 0.07 +0.62± 0.11 +𝐴AME/𝜏250 [Jy] +𝛽td +−0.52± 0.09 +−0.20± 0.20 +−0.64± 0.10 +𝐴AME/𝜏250 [Jy] +𝜏250 +−0.64± 0.08 +−0.64± 0.09 +−0.68± 0.10 +𝐴AME/𝜏250 [Jy] +𝑆TD,peak [Jy] +−0.41± 0.08 +−0.13± 0.13 +−0.54± 0.09 +𝐴AME/𝜏250 [Jy] +𝜈TD,peak [GHz] +0.54± 0.11 +0.88± 0.12 +0.40± 0.15 +𝐴AME/𝜏250 [Jy] +ℜtd𝑥10−4 +−0.34± 0.09 +−0.01± 0.15 +−0.48± 0.10 +ℜAME +Ttd or 𝐺0 +0.04± 0.16 +0.45± 0.27 +−0.20± 0.20 +ℜAME +𝛽td +0.31± 0.14 +0.05± 0.26 +0.36± 0.18 +ℜAME +𝜏250 +0.54± 0.14 +0.19± 0.23 +0.65± 0.17 +ℜAME +𝑆TD,peak [Jy] +0.68± 0.14 +0.60± 0.24 +0.72± 0.16 +ℜAME +𝜈TD,peak [GHz] +0.20± 0.16 +0.60± 0.28 +−0.01± 0.20 +ℜAME +ℜtd𝑥10−4 +0.70± 0.14 +0.66± 0.23 +0.73± 0.17 +Table D2. Spearman rank correlation coefficients (SRCCs) between AME and Thermal Dust parameters. +Variable 1 +Variable 2 +SRCC +SRCC +SRCC +selected sample +AME significant +free-free semi-significant +𝐴AME [Jy] +EM [cm−6/pc] +0.59± 0.04 +0.65± 0.11 +0.55± 0.03 +𝜈AME [GHz] +EM [cm−6/pc] +0.06± 0.11 +0.27± 0.20 +−0.24± 0.14 +𝑊AME +EM [cm−6/pc] +0.12± 0.10 +0.84± 0.13 +−0.02± 0.14 +𝐴AME/𝜏250 [Jy] +EM [cm−6/pc] +0.01± 0.07 +0.46± 0.14 +−0.21± 0.07 +ℜAME +EM [cm−6/pc] +0.59± 0.14 +0.90± 0.24 +0.42± 0.17 +Table D3. Spearman rank correlation coefficients (SRCCs) between AME and free-free parameters. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +29 +Variable 1 +Variable 2 +SRCC +SRCC +SRCC +selected sample +AME significant +free-free semi-significant +Ttd or 𝐺0 +EM [cm−6/pc] +0.30± 0.06 +0.49± 0.13 +0.09± 0.07 +𝛽td +EM [cm−6/pc] +0.28± 0.07 +−0.17± 0.16 +0.41± 0.07 +𝜏250 +EM [cm−6/pc] +0.43± 0.03 +0.10± 0.09 +0.51± 0.03 +𝑆TD,peak [Jy] +EM [cm−6/pc] +0.64± 0.03 +0.50± 0.08 +0.64± 0.04 +𝜈TD,peak [GHz] +EM [cm−6/pc] +0.45± 0.10 +0.62± 0.17 +0.29± 0.15 +ℜtd𝑥10−4 +EM [cm−6/pc] +0.66± 0.04 +0.56± 0.09 +0.65± 0.04 +Table D4. Spearman rank correlation coefficients (SRCCs) between thermal dust and free-free parameters. +Variable 1 +Variable 2 +SRCC +SRCC +SRCC +selected sample +AME significant +AME semi-significant +Ttd or 𝐺0 +𝛽td +−0.47± 0.09 +−0.44± 0.18 +−0.47± 0.10 +Ttd or 𝐺0 +𝜏250 +−0.52± 0.05 +−0.57± 0.10 +−0.58± 0.07 +Ttd or 𝐺0 +𝑆TD,peak [Jy] +−0.17± 0.07 +0.11± 0.14 +−0.32± 0.08 +Ttd or 𝐺0 +𝜈TD,peak [GHz] +0.92± 0.08 +0.97± 0.12 +0.85± 0.13 +Ttd or 𝐺0 +ℜtd𝑥10−4 +−0.07± 0.07 +0.22± 0.14 +−0.25± 0.09 +𝛽td +𝜏250 +0.60± 0.08 +0.34± 0.18 +0.69± 0.09 +𝛽td +𝑆TD,peak [Jy] +0.54± 0.08 +0.06± 0.17 +0.68± 0.09 +𝛽td +𝜈TD,peak [GHz] +−0.13± 0.12 +−0.31± 0.22 +0.01± 0.16 +𝛽td +ℜtd𝑥10−4 +0.50± 0.08 +0.03± 0.17 +0.65± 0.09 +𝜏250 +𝑆TD,peak [Jy] +0.90± 0.04 +0.69± 0.10 +0.92± 0.04 +𝜏250 +𝜈TD,peak [GHz] +−0.31± 0.11 +−0.51± 0.16 +−0.29± 0.14 +𝜏250 +ℜtd𝑥10−4 +0.85± 0.05 +0.61± 0.13 +0.88± 0.05 +𝑆TD,peak [Jy] +𝜈TD,peak [GHz] +0.06± 0.12 +0.18± 0.19 +−0.01± 0.15 +𝑆TD,peak [Jy] +ℜtd𝑥10−4 +0.99± 0.05 +0.99± 0.11 +0.99± 0.05 +𝜈TD,peak [GHz] +ℜtd𝑥10−4 +0.17± 0.12 +0.30± 0.19 +0.06± 0.15 +Table D5. Spearman rank correlation coefficients (SRCCs) between the thermal dust parameters. +Figure A4. Variations of the AME peak frequency with the relative strength +of the ISRF, 𝐺0. Symbols and colours definition are the same as in Figure 9. +Figure A5. Variations of the AME parabola width parameter with the rela- +tive strength of the ISRF, 𝐺0. Symbols and colours definition are the same +as in Figure 9. +MNRAS 000, 1–36 (2023) + +30 +F. Poidevin et al. +Figure C1. SED of the sample of regions discussed in this work. See caption of Figure 4 for symbols, lines and colours conventions. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +31 +Figure C2. Same as Figure C1. +MNRAS 000, 1–36 (2023) + +32 +F. Poidevin et al. +Figure C3. Same as Figure C1. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +33 +Figure C4. Same as Figure C1. +MNRAS 000, 1–36 (2023) + +34 +F. Poidevin et al. +Figure C5. Same as Figure C1. +MNRAS 000, 1–36 (2023) + +QUIJOTE-MFI wide-survey Galactic AME sources +35 +Figure C6. Same as Figure C1. +MNRAS 000, 1–36 (2023) + +36 +F. Poidevin et al. +Figure C7. Same as Figure C1. +MNRAS 000, 1–36 (2023) + diff --git a/G9E4T4oBgHgl3EQfgQ3v/content/tmp_files/load_file.txt b/G9E4T4oBgHgl3EQfgQ3v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1be633d8c5dd6fdd1f937e6bfe9aa0886f78f866 --- /dev/null +++ b/G9E4T4oBgHgl3EQfgQ3v/content/tmp_files/load_file.txt @@ -0,0 +1,4433 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf,len=4432 +page_content='MNRAS 000, 1–36 (2023) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 QUIJOTE Scientific Results – VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Galactic AME sources in the QUIJOTE-MFI Northern Hemisphere Wide-Survey F.' metadata={'source': 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López-Caraballo,1,2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Watson3, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Artal,4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ashdown,5,6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Barreiro,7 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Casas,7 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' de la Hoz,7,8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Fernández-Torreiro,1,2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Guidi,1,2,9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Herranz,7 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Hoyland,1,2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Lasenby,5,6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Martinez-Gonzalez,7 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Peel,1,2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Piccirillo,3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Rebolo,1,2,10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ruiz-Granados,1,2,11 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Tramonte,12,13,1,2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Vansyngel1,2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Vielva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' E-38205 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spain 2Departamento de Astrofísica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Universidad de La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' E-38206 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spain 3Jodrell Bank Centre for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Alan Turing Building,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Oxford Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Manchester M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' UK 4Departamento de Ingenieria de COMunicaciones (DICOM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Laboratorios de I+D de Telecomunicaciones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Plaza de la Ciencia s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' E-39005 Santander,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spain 5Astrophysics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Cavendish Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' J J Thomson Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Cambridge CB3 0HE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' UK 6Kavli Institute for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Cambridge CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' UK 7Instituto de Física de Cantabria (IFCA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' CSIC-Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' los Castros, s/n, E-39005 Santander, Spain 8Departamento de Física Moderna, Universidad de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' de los Castros s/n, 39005 Santander, Spain 9Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France 10Consejo Superior de Investigaciones Cientificas, E-28006 Madrid, Spain 11Departamento de Física.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Facultad de Ciencias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Universidad de Córdoba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Campus de Rabanales, Edif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Planta Baja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' E-14071 Córdoba, Spain 12Purple Mountain Observatory, CAS, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 Yuanhua Road, Qixia District, Nanjing 210034, China 13NAOC-UKZN Computational Astrophysics Center (NUCAC), University of Kwazulu-Natal, Durban 4000, South Africa Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The QUIJOTE-MFI Northern Hemisphere Wide-Survey has provided maps of the sky above declinations −30◦ at 11, 13, 17 and 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These data are combined with ancillary data to produce Spectral Energy Distributions in intensity in the frequency range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4–3 000 GHz on a sample of 52 candidate compact sources harbouring anomalous microwave emission (AME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We apply a component separation analysis at 1◦ scale on the full sample from which we identify 44 sources with high AME significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We explore correlations between different fitted parameters on this last sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' QUIJOTE-MFI data contribute to notably improve the characterisation of the AME spectrum, and its separation from the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In particular, ignoring the 10–20 GHz data produces on average an underestimation of the AME amplitude, and an overestimation of the free-free component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We find an average AME peak frequency of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 GHz, about 4 GHz lower than the value reported in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The strongest correlation is found between the peak flux density of the thermal dust and of the AME component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A mild correlation is found between the AME emissivity (𝐴AME/𝜏250) and the interstellar radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand no correlation is found between the AME emissivity and the free-free radiation Emission Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our statistical results suggest that the interstellar radiation field could still be the main driver of the intensity of the AME as regards spinning dust excitation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, it is not clear whether spinning dust would be most likely associated with cold phases of the interstellar medium rather than with hot phases dominated by free-free radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Key words: radiation mechanisms: general, thermal, non-thermal – ISM: clouds – photodis- sociation region (PDR) – radio continuum : ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ★ E-mail: fpoidevin@iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='es † E-mail: rgs@iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='es 1 INTRODUCTION A detailed knowledge of the sky emission properties in the fre- quency range ∼1–3 000 GHz, from low-frequency (LF) bands at © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05116v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='GA] 12 Jan 2023 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' which the Galactic synchrotron emission generally dominates, to high frequency (HF) bands at which the Galactic dust emission dominates, is crucial for a state-of-the art characterization of the Cosmic Microwave Background (CMB) radiation both in intensity and in polarization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Lite- BIRD Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Understanding the properties of the Galactic foregrounds is essential in order to measure a possi- bly intrinsic polarization signature in the CMB emission that could give insights about inflation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This task is considered to be a serious challenge by both the community of astronomers in quest of a B-mode detection (see Watts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' POLARBEAR Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The Simons Observatory Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Aiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The CMB-S4 Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' BICEP/Keck collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The LSPE collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' LiteBIRD Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2022) and the community of astronomers interested to understand the spatial and spectral variations of the Galactic emission (see Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Carretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In addition to synchrotron emission and thermal dust emis- sion, the Galactic sky also emits thermal bremmstrahlung or free- free radiation, a radiation produced by deceleration of electrons, and supposedly unpolarized (Rybicki & Lightman 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Trujillo-Bueno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Another type of radiation is the so-called Anomalous Microwave Emission (AME) that was discovered about twenty five years ago (see Leitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Kogut 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' de Oliveira-Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The AME is a diffuse component showing a spectral bump detected over almost the full sky in the frequency range 10– 60 GHz and peaking in flux density around a central frequency of ∼ 30 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this frequency range, the synchrotron and free-free emission can dominate over the AME emission while the thermal dust emission is expected to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The carriers and physical mechanisms producing AME are not conclusively known yet, how- ever theoretical emission mechanisms have been proposed based on phenomenological interpretations of correlations found between the AME radiation and other Galactic template components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A re- view of these aspects and of the proposed models in the literature is given by Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The main current paradigm is that electric dipole emission from very small fast rotating spinning dust grains out of thermal equilibrium could be the origin of this emis- sion (see Draine & Lazarian 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ali-Haïmoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ysard & Verstraete 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Recent advances on the de- velopment of another model, initially proposed by Jones (2009) and exploring the possibility that AME can be produced instead by thermal amorphous dust are discussed by Nashimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2020a) and Nashimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The majority of these models predict very low levels of polarisation for the AME, this being supported by observational data (López-Caraballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Given its twofold role as a CMB contaminant and as a source of information about the physics of the ISM, it is important to make progress on the study of the observational properties of AME, and confronting them with theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Galactic candidate AME sources were intensively discussed in Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2014a) (herefater PIRXV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In that work the analysis of a sample of 98 compact candidate AME sources distributed over the full sky provides significant detection (> 5𝜎) of AME for 42 sources, which reduces to safe detection of AME for 27 sources once the potential contribution of thick free-free emission from ultra compact H ii regions has been integrated to the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this work, we complete and revisit the sample of sources observable from the Northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For this we use the QUIJOTE-MFI wide- survey maps (Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023), which are crucial to pin down the AME spectrum at low frequencies, thence allowing a more reliable separation between the AME and free-free amplitudes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2019) than previous works, which systematically have overestimated the free-free emission and underestimated the AME amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some of the sections in the present article closely follow those in PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In such cases we tried to use similar section names so that the reader can easily refer to the information provided by PIRXV and, as much as possible, we tried to avoid redundancy with their explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All the calculations made for our analysis are independent of those done by PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The structure of the article is as follows: the data used for the analysis are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The sample selection and fitting procedure used for the Spectral Energy Distribution (SED) anal- ysis are detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Consistency checks obtained from the comparison of our method with that used by PIRXV are also presented in that Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The significance of the AME detection obtained from our analysis, potential contamination by UCH ii re- gions and robustness and validation of our method are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Statistics on the parameters characterizing the sample of regions that passed the validation tests are investigated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A discussion is given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our results and conclusions are summarized in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Additional plots showing low Spearman rank correlation coefficients (SRCCs) between some of the param- eters obtained from the modelling of the SEDs, and mentioned in some of the above sections, are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All the parameters estimates obtained from the modelling of the SEDs, and additional information, obtained on the full sample, are tabulated in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All the plots of the SEDs and the multicomponents models are shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Finally, a summary of the SR- CCs obtained between all the pairs of parameters used to model the SEDs are given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2 DATA The maps used in this analysis are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Details about the maps are given in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 QUIJOTE Data The data used at frequencies 11, 13, 17 and 19 GHz come from the first release of the QUIJOTE wide survey maps (Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These maps were obtained from 9 200 h of data col- lected over 6 years of observations from 2012 to 2018 with the Multi-Frequency Instrument (MFI) on the first QUIJOTE telescope, from the Teide Observatory in Tenerife, Canary Islands, Spain at an altitude of 2 400 meters above sea level, at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3◦ N and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5◦ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These observations were performed at constant elevations and with the telescope continuously spinning around the azimuth axis (the so-called “nominal mode”) to obtain daily maps of the full northern sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' After combination of all these data we obtained maps covering ∼ 70% of the sky and with sensitivities in total intensity between 60 and 200 𝜇K/deg, depending on the horn and frequency and sensitivites, down to ∼ 35𝜇K/deg, in polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Full details on these maps, and multiple characterisation and validation tests, are given in Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023), while the general MFI data processing pipeline will be described in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The MFI consists of 4 horns, two of them (horns 1 and 3) covering a 10-14 GHz band with two outputs channels centred at 11 MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 3 Frequency Wavelength Telescope/ Angular Resolution Original Calibration References [GHz] [mm] survey [′] Units Uncertaintiy [%] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='408 735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42 JB/Eff/Parkes ≈ 60 [KRJ] 10 Haslam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1982) Remazeilles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2015) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='820 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='91 Dwingeloo 72 [KRJ] 10 Berkhuijsen (1972) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='420 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30 Stockert/Villa-Elisa 36 [KRJ] 10 Reich (1982) Reich & Reich (1986) Reich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2001) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 QUIJOTE 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 [mKCMB] 5 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85 QUIJOTE 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 [mKCMB] 5 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24 QUIJOTE 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 [mKCMB] 5 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32 QUIJOTE 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 [mKCMB] 5 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 WMAP 9-yr ≈ 49 [mKCMB] 3 Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2013) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 𝑃𝑙𝑎𝑛𝑐𝑘 LFI 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='29 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 WMAP 9-yr ≈ 40 [mKCMB] 3 Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2013) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 WMAP 9-yr ≈ 31 [mKCMB] 3 Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2013) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80 𝑃𝑙𝑎𝑛𝑐𝑘 LFI 27 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='94 WMAP 9-yr ≈ 21 [mKCMB] 3 Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2013) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 𝑃𝑙𝑎𝑛𝑐𝑘 LFI 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 WMAP 9-yr ≈ 13 [mKCMB] 3 Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2013) 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 𝑃𝑙𝑎𝑛𝑐𝑘 HFI 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 143 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 𝑃𝑙𝑎𝑛𝑐𝑘 HFI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 217 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='38 𝑃𝑙𝑎𝑛𝑐𝑘 HFI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85 𝑃𝑙𝑎𝑛𝑐𝑘 HFI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='94 [KCMB] 3 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='55 𝑃𝑙𝑎𝑛𝑐𝑘 HFI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='83 [MJy/sr] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35 𝑃𝑙𝑎𝑛𝑐𝑘 HFI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64 [MJy/sr] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016a) 1249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24 COBE-DIRBE ≈ 40 [MJy/sr] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 Hauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1998) 2141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 COBE-DIRBE ≈ 40 [MJy/sr] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 Hauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1998) 2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 COBE-DIRBE ≈ 40 [MJy/sr] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 Hauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1998) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' List of surveys and maps used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' and 13 GHz, and two other ones (horns 2 and 4) covering the 16- 20 GHz band with two output channels at 17 and 19 GHz (Génova- Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Due to a malfunctioning of horn 1 in polarization during some periods, all the scientific QUIJOTE papers associated with this release make use of horn 3 only at 11 and 13 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Although this paper uses intensity data only, we follow the same criterion and use only horn 3, which is much better characterised1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' At 17 and 19 GHz we combine data from horns 2 and 4 through a weighted mean, using predefined constant weights2 (Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Finally, it must be noted that, due to the use of the same low-noise amplifiers, the noises from the lower and upper frequency bands of each horn are significantly correlated (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 in Rubiño- Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In principle this correlation should be accounted for in any scientific analysis that uses spectral information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' However, we have checked that neglecting them introduces a small effect on the results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME parameters are the most affected, and we have checked that accounting for this correlation introduces differences in these parameters that are typically below the 3% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Therefore, for the sake of simplicity we decided to use the four frequency points (nominal frequencies 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 GHz) in the analysis as independent data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We assume a 5% overall calibration uncertainty of the QUIJOTE MFI data, which is added in quadrature to the statistical error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' There is compelling 1 Note that the analysis in intensity presented in this paper benefits from a sufficiently large signal-to-noise ratio and therefore a good characterisation of systematics is more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2 Instead of doing a pixel-by-pixel combination at the map level, we extract flux densities independently and combine the derived flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' evidence that this 5 % value, which is driven by uncertainties in the calibration models, is sufficiently conservative (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Ancillary Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Low frequency ancillary data At low frequencies we use a destriped version (Platania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2003) of the all-sky 408 MHz map of Haslam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1982), the Dwingeloo survey map at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='820 GHz of Berkhuijsen (1972), and the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='420 GHz map of Reich (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Since our study is focused on compact candi- date AME sources we prefer to use the all-sky 408 MHz destriped map of Haslam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The Platania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2003) version of this map is used for consistency with previous QUIJOTE papers, but we have checked that the results are consistent with those ob- tained using the map provided by Remazeilles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The Jonas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1998) map at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='326 GHz, which was used in PIRXV, measures 𝐼 + 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Therefore it would lead to residuals in polarised regions, and we prefer not to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some of the considered sources are not well sampled or not included in the footprint of some of the ancillary maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Therefore, for a given source a map is used only if all pixels within a circular region of 3◦ radius are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We noted that, for a subset of compact sources, the map at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='420 GHz shows a misscentering of the emission by more than half a degree with respect to other low- frequency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For that reason we prefer not to use that map in the analysis of G059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21, G061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 and G099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='420 GHz map is calibrated to the full beam, and therefore we apply the full-beam to main-beam recalibration factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='55 for MNRAS 000, 1–36 (2023) 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' compact sources derived by Reich & Reich (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Overall, we assume a 10 % uncertainty in the radio data at low frequency, which encompasses intrinsic calibration uncertainties as well as issues related with beam uncertainties and recalibration factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 WMAP Maps At frequencies of 23, 33, 41, 61, and 94 GHz, we use the intensity maps from the 9-year data release of the WMAP satellite (Ben- nett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All the maps were retrieved from the LAMBDA database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 For all the maps we assume a 3% overall calibration uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The uncertainty in WMAP’s amplitude calibration is much better, however here we use 3% to account for other system- atic effects like uncertainties in the beams or bandpasses (which in turn lead to uncertainties in the colour corrections) that will have a direct effect on our derived flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 Planck Maps Below 100 GHz intensity maps are available at frequencies 28, 44, and 70 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' They were obtained with the Low-Frequency Instru- ment (LFI) on board of the Planck satellite (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We use the second public release version of the in- tensity maps as provided by the Planck Legacy Archive (PLA4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Above 100 GHz we use the second data release version of the in- tensity maps obtained with the High-Frequency Instrument (HFI) on board the Planck satellite (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2016a) at frequencies centred at 100, 143, 217, 353, 545, and 857 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We have checked that using the third data release (PR3) leads to differ- ences in the derived flux densities typically below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 % for most of the frequencies and therefore have no impact in the final results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The Type 1 CO maps (Planck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2014b) were used to correct the 100, 217, and 353 GHz intensity maps for contamination introduced by the CO rotational transition lines (1-0), (2-1) and (3-2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We assume an overall calibration uncertainty of 3 % for the LFI data, and also for the HFI data at frequencies lower than or equal to 353 GHz, a value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1% at 545 GHz, and a value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4% at 857 GHz (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 High frequency ancillary data In the FIR range we use the Zodi-Subtracted Mission Average (ZSMA) COBE-DIRBE maps (Hauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1998) at 240 𝜇m (1249 GHz), 140 𝜇m (2141 GHz), and 100 𝜇m (2997 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We as- sume an 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9% overall calibration uncertainty in the data at these frequencies5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3 SAMPLE SELECTION AND SED FITTING In the following section we describe the process followed to build the sample of the candidate compact Galactic AME sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' De- tails about aperture photometry used to build the SEDs are given in 3 Legacy Archive for Microwave Background Data Analysis, http://lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='gov/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 4 Planck Legacy Archive (PLA) http://pla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='int/pla/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9% is the calibration uncertainty for the 240 𝜇m according to Hauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1998), and we consider the same value for all bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The modelling used to analyse the SED of each candi- date AME source is detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Finally, a consistency test is investigated and a comparison of our analysis, including the QUIJOTE maps, with the analysis obtained by Planck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2014a) on the sample of sources common to both studies is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 AME sources sample To build the sample of candidate AME sources, we use the list of sources selected and discussed in PIRXV as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In their work, this list was obtained by using 3 different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' One method was to identify sources already known from the lit- erature and add them to a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Another method was to pro- duce a 1◦-smoothed map of residuals at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz, by subtracting off synchrotron, free-free, thermal dust, and CMB components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A 5◦-smoothed version of this map was also created and subtracted from the 1◦-map in order to minimise diffuse emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Bright and relatively compact sources were then identified in that map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In a third method, an initial sample was built by using the SExtrac- tor (Bertin & Arnouts 1996) software to detect bright sources in the 70 GHz Planck CMB-subtracted map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This sample was cross- correlated with 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz and 100 GHz catalogs obtained using the same technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The output catalog was filtered to remove sources associated with radio galaxies, including a small number of known bright supernova remnants and planetary nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Visual inspection was conducted on preliminary SEDs obtained from the 1◦-smoothed maps in order to filter out the regions that were not showing a peak at 30 GHz on scales ≲ 2◦ and to define the final sample of 98 candidate AME sources analysed and discussed in PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Of these 98 sources, 42 are well observed at all QUIJOTE frequencies of the MFI wide survey and are therefore included in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Additional sources that are not included in the sample analysed by PIRXV have been identified from catalogs and lists of molecular clouds regions available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This was done with the SCUPOL catalog that compiles thermal dust polarimetry information on small scales (≈ 14′′) provided by Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2009), with the list of molecular clouds toward which Zeeman measurements provide magnetic field line-of-sight (LOS) estimates obtained by Crutcher (1999), and with the molecular cloud cata- log of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this way 10 additional candidate AME sources have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The maps of these sources that are not already included in PIRXV’s catalog were inspected by eye at all available frequencies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz and 3000 GHz and pre- liminary SEDs were built in order to look for the presence of a bump in the frequency range 10 – 60 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The location of the fi- nal sample of candidate AME regions selected for our analysis is shown superimposed on the QUIJOTE 11 GHz Galactic full sky map in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Their names, coordinates and additional informa- tion are displayed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The final sample contains a total of 52 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' QUIJOTE-MFI intensity maps at 11, 13, 17 and 19 GHz and WMAP 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 GHz intensity maps are displayed in Figure 2 for a sample of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Each source clearly shows similar intensity distribution patterns across the different frequency survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Aperture photometry In this work we conduct a component separation analysis of the various components in intensity contributing to the total emission of each source based on a SED analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In intensity this method consists in calculating the total emission of a given source at each MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 5 Source Name Glon Glat Region Type Other Name(𝑎) References 𝜎AME 𝜎AME [◦] [◦] PIRXV This Work G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32 SNR Kes62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Synch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SNR9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4𝑆 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6𝑆𝑆 G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84−02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 MC GGD 27 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆𝑆 G011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 MC G011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5𝑆𝑆 G012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 SNR W33 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2𝐿𝐷 G015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 MC M17 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0𝑆→𝑆𝑆 G017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85 MC M16 1,2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0𝑆 G037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 SNR W47 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6𝑆→𝑆𝑆 G040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 MC/HII W45 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆→𝑆𝑆 G041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 MC SDC G41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='003−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='097 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆 G043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 MC W49 1,3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆→𝑆𝑆 G045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 SNR NRAO601 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6𝑆→𝑆𝑆 G049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 MC/HII W51 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆→𝑆𝑆 G059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 MC/HII W55 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7𝑆→𝑆𝑆 G061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 MC/HII HII LBN061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='50+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SH2−88 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝑆𝑆 G062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='98+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 MC S89 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝑆→𝑆𝑆 G070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61 Cluster NGC 6857 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝐵𝐷 G071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85 MC/HII s101 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆𝑆 G075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='39 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='39 MC/HII HII GAL075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SH2−105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Cyg 2N 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆→𝑆𝑆 G076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='38−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 MC/HII S106 1,3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝐵𝐷 G078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 MC/HII LDN 889 2,3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='𝐵𝐷 G081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01 MC/HII DR23/DR21 1,2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆 G084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 MC DOBASHI 2732 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆 G085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00+04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80 MC/HII LBN 084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97+04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝑆 G093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 MC/HII HII GAL093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0𝑆→𝑆𝑆 G094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47−01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 MC/HII LDN 1059 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝑆𝑆 G098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 MC/HII RNe GM1-12, DNe TGU H582 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2𝑆→𝑆𝑆 G099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60+03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70 MC LDN1111 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0𝑆𝑆 G102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 MC/HII LDN1161/1163 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆 G107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20+05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 MC S140 1,2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆→𝑆𝑆 G110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 MC/HII HII G110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' LBN110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='44 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7𝑆𝑆 G111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='54+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81 Open Cluster NGC 7538 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆→𝑆𝑆 G118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09+04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96 SNR NGC 7822 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2𝑆 G123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13−06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 MC/HII S184 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2𝑆→𝑆𝑆 G133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27+09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 MC LDN 1358/1355/1357 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5𝑆 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝐵𝐷 G133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='74+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 MC W3 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆→𝑆𝑆 G142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35 MC DNe TGU H942, DOBASHI 3984 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5𝑆 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4𝑆 G151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 MC/HII HII SH2−209 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4𝑆→𝑆𝑆 G160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 MC Perseus 1,2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4𝑆 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2𝑆 G160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 MC NGC 1499 (California nebula) 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1𝑆 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6𝑆 G173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56−01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 Open Cluster NGC 1893 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4𝑆𝑆 G173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 Cluster S235 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5𝑆→𝑆𝑆 G190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 MC/HII NGC 2174/2175 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆→𝑆𝑆 G192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 MC LDN 1582/1584 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5𝐵𝐷 G192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 Cluster S255 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆→𝑆𝑆 G201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='63 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='63 MC LDN 1608/1609 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4𝑆 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆 G203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 MC/HII LDN 1613 1,2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆 G208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80−02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65 MC/HII S280–LBN 970 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝐿𝐷 G239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40−04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70 MC LDN 1667, HII LBN1059, V VY Cma 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5𝑆 G351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='31+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='31 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 MC/HII HII LBN1105/1104 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆 G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90 353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90 MC Rho Ophiuchi, AME-G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='901 1,3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8𝑆 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆→𝑆𝑆 G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 MC In Ophiuchus 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9𝑆 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6𝑆 G355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='63+20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='63 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 MC In Rho Ophiuchus 1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3𝑆 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0𝐵𝐷 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' List of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' References: 1: Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2014a) (PIRXV), 2: Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2009), 3: Crutcher (1999), 4: Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Note: (𝑎) information retrieved from the Simbad database (http://simbad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='fr/simbad//).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Sources such that 𝜎AME from PIRXV are greater than 𝜎AME from this work are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Superscript symbols in last two columns are 𝑆 for “significant” AME detection, 𝑆𝑆 for “semi-significant” AME detection, 𝑆→𝑆𝑆 for “significant” AME detection reclassified as “semi-significant” AME detection (see text for details), 𝐿𝐷 for low detection of AME and, 𝐵𝐷 for bad detection because of a bad fit of the AME, of the free-free or of the thermal dust component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' See section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME sources location in the Galaxy displayed on top of the QUIJOTE-MFI 11 GHz wide survey map at 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Coordinates are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The map is centred at position (𝑙, 𝑏) = (120◦, 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Once a SED has been calculated one can use modelling to assess the fraction of the total intensity emission associated with the different components (synchrotron, free–free, thermal dust, and AME) at all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SED modelling analysis has been widely used in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' López-Caraballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2015, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The maps of pixel size 𝑁side = 512 in the HEALPix6 pixeliza- tion scheme (see Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2005) are first smoothed to 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To calculate the total emission at each frequency, the maps in CMB thermodynamic units (KCMB) are first converted to Rayleigh-Jeans (RJ) units (KRJ) at the central frequency, then all the maps are con- verted to units of Jy pixel−1 using 𝑆 = 2𝑘𝑏𝑇RJΩ𝜈2/𝑐2, where 𝑘𝑏 is the Boltzmann constant, 𝑇RJ, is the Rayleigh-Jeans temperature, Ω is the solid angle of the pixel, 𝜈 is the frequency and 𝑐 is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The pixels are then summed in the aperture covering the region of interest to obtain an integrated flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' An estimate of the background is subtracted using a median estimator of pixels lying in the region defined as the background region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 we provide some comparisons with the results obtained by PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To do so, we use the same apertures and annu- 6 https://sourceforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='net/projects/healpix/ lus used in that paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 𝑟APERTURE = 60 ′, 𝑟ANNULUS(IN) = 80 ′ and 𝑟ANNULUS(OUT) = 100 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This method, also used in previous works, relies on the pixel-to-pixel scatter in the background annulus to obtain an estimate of the uncertainty in the flux density estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This technique is straightforward in the case of uncorrelated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' However, in our case there is pixel-to-pixel correlated noise, due to instrumental 1/f noise and to beam-averaged sky background fluctuations, whose correlation function is not easy to be reliably characterised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We instead apply aperture photometry at the central position of each source in the standard manner, and then the calcu- lations are repeated eight times such that we perform flux-density integrations on eight independent disks of radius 𝑟APERTURE = 30 ′ with central coordinates distributed along a circle with radius 2◦ around the source (as shown in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The final uncertainty is obtained from the scatter of these eight flux-density estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This procedure is used for all sources except for the California region for which the background structure is complex and was producing bad fits such that 𝜈AME= 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 GHz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' the prior upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For that region we therefore use the same aperture and background annulus as in PIRXV and we expect our uncertainties on the fluxes of this region to be slightly underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) AME sources location map 0) mk 5 20QUIJOTE-MFI wide-survey Galactic AME sources 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Subsample of 1◦ smoothed intensity maps in Galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Fom left to right: QUIJOTE-MFI intensity maps at 11 GHz (horn 3), 13 GHz (horn 3), 17 GHz (horn 2 and 4), 19 GHz (horn 2 and 4), and WMAP intensity maps at 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From top to bottom the sources shown are the well-known Galactic supernova remnant NRAO601 (G045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06), star forming region W49 (G043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10), Perseus molecular cloud (G160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62) and the cluster S235 (G173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In each plot the central circle shows the aperture used to obtain the density flux estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The eight dashed circles show the positions of the apertures used to calculate the uncertainties on these fluxes as explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 Model fitting For each source the flux density 𝑆 from the aperture photometry is fitted by a simple model consisting of the free-free, synchrotron (if appropriate), thermal dust, AME and CMB components: 𝑆total = 𝑆ff + 𝑆sync + 𝑆td + 𝑆AME + 𝑆CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1) The free–free spectrum shape is fixed and the free-free flux density, 𝑆ff, is calculated from the brightness temperature, 𝑇ff, using the expression: 𝑆ff = 2𝑘𝑏𝑇ffΩ𝜈2 𝑐2 , (2) where Ω is the solid angle of our 60 ′ aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The brightness temperature is calculated with the expression: 𝑇ff = 𝑇e(1 − 𝑒−𝜏ff), (3) MNRAS 000, 1–36 (2023) 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' where, following Draine (2011) the optical depth, 𝜏ff, is given by 𝜏ff = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='468 × 10−2𝑇−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 e 𝜈−2 9 EM𝑔ff, (4) where 𝑇e is the electron temperature in Kelvin, 𝜈9 is the frequency in GHz units, EM is the Emission Measure in units of pc cm−6, and 𝑔ff is the Gaunt factor, which is approximated as: 𝑔ff = ln(exp[5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='960 − √ 3 𝜋 ln(Zi𝜈9T−3/2 e,4 )] + e), (5) where the charge is assumed to be 𝑍𝑖 = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', hydrogen plasma) and 𝑇e,4 is in units of 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our best estimate for the electron temperature is the median value of the Commander template within the aperture used on each source (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2016c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These values lie in the range 5 458–7 194 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The only remaining free parameter associated with the free–free component is the free–free amplitude, which can be parameterized by the effective EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Equation 4 tells that the turnover frequency that marks the transition between the optically-thick and optically-thin regimes (𝜏ff ≈ 1) depends on the emission measure (as EM1/2) and on the electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to properly trace the degeneration between the free-free amplitude and the turnover frequency, instead of working with integrated quantities we would have to reconstruct EM along individual lines of sight inside each region and then integrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Given the non-linear dependency of the flux density on EM, the two procedures are not equivalent, and this typically results in our fitted spectra having smaller turnover frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For this reason, in cases where the data clearly shows the turnover frequency to be above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='408 GHz (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' G015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 − 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 in Figure C1), in order to avoid the free-free (AME) amplitude to be biased low (high) we do not use in the fit the points with frequencies below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42 GHz (depicted in these cases by a blue asterisk in Figure C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The synchrotron component is fitted by a single power law given by: 𝑆sync = 𝑆synch,1GHz · � 𝜈 GHz � 𝛼synch,int , (6) where the two parameters that are fitted for are the spectral in- dex, 𝛼synch,int, and the amplitude at 1 GHz, 𝑆synch,1GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This synchrotron component is included in the fits only for a few sources (G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 − 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32, G012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80 − 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19, G037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 − 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11, G040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 + 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53, G041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 − 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 and G045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 + 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06) as in- dicated in Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This choice was based on the slope of the low-frequency flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The first three and the last of these sources are SNRs, as listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' There is yet another source classified as SNR in our sample, G118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 + 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' However the low-frequency data do not show any hint of synchrotron emission in this source, and actually the addition of this component to the fit has no impact on the fitted AME spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The CMB is modelled using the differential of a blackbody at 𝑇CMB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7255 K (Fixsen 2009): 𝑆CMB = 𝜂 2𝑘𝑏Ω𝜈2 𝑐2 Δ𝑇CMB, (7) where 𝜂 = 𝑥2·exp(𝑥)/(exp(𝑥) − 1)2 and 𝑥 = ℎ𝜈/(𝑘𝑏𝑇CMB) is the conversion between thermodynamic and RJ brightness temperature, and Δ𝑇CMB is the CMB fluctuation temperature in thermodynamic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spinning dust models have many free parameters, which are extremely difficult to constrain jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As a result, using a phenome- logical model, which traces well the data and the typical spinning dust models, is common practice in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this work the AME component is fitted by the phenomenological model consisting of an empirical log-normal approximation, first proposed by Steven- son (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The log-normal model is described by the following equation: 𝑆AME = 𝐴AME · exp � −1 2 · � ln(𝜈/𝜈AME) WAME �2 , � (8) where the three free parameters are the width of the parabola𝑊AME, the peak frequency 𝜈AME, and the amplitude of the parabola at the peak frequency 𝐴AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2017) have used a different phenomelogical model proposed by Bonaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' However we note that in this model the AME peak frequency and the AME width are not independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Hence, we prefer to use the Stevenson (2014) model, which does not have this coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The thermal dust emission is modelled by a single-component modified blackbody relation of the form, 𝑆td = 𝜏250(𝜈/1200GHz)𝛽dustB𝜈(Tdust), (9) where 𝜏250 is the averaged dust optical depth at 250 𝜇m, 𝛽dust is the averaged thermal dust emissivity, and 𝐵𝜈 is the Planck’s law of the black-body radiation at the temperature, 𝑇dust, which is the averaged dust temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit procedure includes priors on some of the parameters and consists of a minimization process using non-linear least-squares fit- ting in Interactive Data Language (IDL) with MPFIT (Markwardt 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The errors on the fitted parameters in this method are com- puted from the input data covariance, and neither the goodness of the fit nor parameter degeneracies are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' It must then be noted that parameter errors are sometimes underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is the case for instance when it is hard to separate the free-free and the spinning dust components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In those cases the errors on EM and 𝐴AME will tend to be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A more reliable error estimate would require full sampling of the probability distribution and will be considered in future similar studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Such a method should help to refine our results but would not change our main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Flat priors are used on the following list of parameters: 𝑇dust, 𝛽dust, Δ𝑇CMB, 𝐴AME, 𝜈AME and 𝑊AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Dust temperatures, 𝑇dust, are allowed in the temperature range 10–35 K while dust index emissivities, 𝛽dust, are allowed in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Both priors are representative of average dust physical conditions in the diffuse interstellar medium (ISM) and molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The CMB fluctu- ation temperatures, Δ𝑇CMB, are allowed to vary in the temperature range ±125 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This range of values is representative of the CMB fluctuation temperatures one can expect when operating aperture photometry including a background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The AME ampli- tude, 𝑊AME, is allowed to vary in the range 0–300 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The AME frequency, 𝜈AME, is allowed to vary in the frequency range 10– 60 GHz, and for the width parameter 𝑊AME, we use a prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' While spinning dust models computed for representative ISM en- vironments (Draine & Lazarian 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ali-Haïmoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2009) typically have maximum widths corresponding to 𝑊AME ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 we prefer not to be so strongly model constrained and allow for slightly wider AME spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' More details on the effect of the priors used to model the AME are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 and Table 3, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5, and in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Colour corrections for QUIJOTE, WMAP, Planck and DIRBE, which depend on the fitted spectral models, have been applied using an iterative procedure that involves calls to a specifically developed software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This code, which will be described in more detail in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023), uses as input the fitted spectral model in each iteration, which is convolved with the experiment MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 9 bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Colour corrections are typically ≲ 2% for QUIJOTE, WMAP and Planck-LFI, and ≲ 10% for Planck-HFI and DIRBE, which have considerably larger bandwidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Colour corrections for low-frequency surveys, which have much narrower bandpasses, are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 Comparison with AME sources previously characterized in Planck Intermediate Results XV Before making an analysis of the full sample of 52 candidate AME sources displayed in Table 2 we first compare the results obtained with a multicomponent analysis of the SEDs calculated on the sam- ple of 42 sources already studied by PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The AME model used by PIRXV assumes a spinning dust model corresponding to the warm ionized medium (WIM) with a peak at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 GHz to give the generic shape for which only the amplitude of the peak and the peak frequency were fitted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This horizontal shift in frequency is arti- ficial, as the WIM model, with the parameters that have been used do produce that model, predicts a precise value for 𝜈peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the contrary, as explained before, the AME model used in our analysis is a phenomenological model with three parameters including one parameter to fit for the width of the bump of the AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To build the SEDs in the same way as PIRXV, as mentioned before, we use an aperture of radius 60′ and an annulus of inter- nal and external radii of sizes 80′ and 100′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For this comparison, we then use the parameters obtained by PIRXV on the CMB and thermal dust components as fixed input parameters and then we fit our model of AME, free-free and synchrotron (in the cases where the synchrotron was considered in the fits by PIRXV, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' on sources G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32, G012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19, G037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11, G045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 + 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 and G118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 + 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From these fits we calcu- late the AME significance (𝜎AME) as the ratio of the flux density of AME at the frequency peak position divided by the uncertainty on this estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The results are displayed in Figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Three points show a higher AME significance in PIRXV than in our anal- ysis (data shown with red colour in the plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Overall, however, our analysis shows that for most of the sources the AME amplitude, and its significance are higher once the QUIJOTE data are included (data shown with black colour in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This trend can be ex- plained by the level of free-free detection to be generally higher in the PIRXV analysis than in our component separation analysis as shown in Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This point is also confirmed by the higher level of AME obtained with our analysis compared to the level of AME detected by PIRXV at a frequency of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz as displayed in Figure 3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this plot AME 𝑆28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 resid is the AME flux obtained from the modelling at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This general trend is consistent with the results obtained by Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2017), by Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2019) and by Fernandez-Torreiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023), and confirms that the QUIJOTE-MFI data are crucial to help breaking the inevitable degeneracy between the AME and the free-free that occurs when only data above 23 GHz are used in regions with AME peak flux densities close to this frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3 (d) it is also clear that the inclusion of QUIJOTE data favours lower AME peak frequen- cies, which are found to be on average around 4 GHz smaller than in PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' It is also worth stressing that the addition of QUIJOTE data clearly leads to a more precise characterisation of the emission models in the 10 − 60 GHz frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We find on average errors smaller by ≈ 30 % on 𝐸𝑀 and 𝐴AME, by ≈ 70 % on 𝑊AME, by ≈ 60 % on 𝜈AME and even by 10 % on 𝛽dust and 𝑇dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To test that our interpretation of the results is not model- dependent we repeated the analysis described above with the model proposed by Bonaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The final plots are very similar to those displayed in Figure 3 meaning that the higher level of de- tection of AME comes from the addition of the QUIJOTE maps at 10–20 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In addition to this, our model should provide more reliable estimates of the AME peak frequency thanks to it being fully independent on the AME width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 4 REGIONS OF AME In the following sections we describe the level of detection of AME derived from the modelling analysis of the SEDs (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1) and their possible contamination by UCH ii regions (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From this analysis we define the final sample of candidate AME sources that will be used for further statistical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Additional calculations used to test the robustness and validate this sample are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Significance of AME detections in our sample In order to make a study of the detection of AME in the 52 sources from our sample we first produced a series of intensity maps at all available frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The maps were inspected and removed if some pixels were showing no data in the aperture or annulus areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' this process affecting more specifically low frequency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The component separation was operated by including fits for the free-free, the AME, the thermal dust and the CMB components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The synchrotron component was also included in the six sources indicated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Each SED was then inspected by eye and it was found that most of the sources were showing the detection of a bump in the frequency range 10–60 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some examples of SEDs in intensity are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The histogram displayed in Figure 5 shows the distribution of the significance of the AME detection, 𝜎AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Following PIRXV we define the sources with 𝜎AME > 5 as “significant AME sources”, the sources with 2 < 𝜎AME < 5 as “semi significant AME sources”, and the sources with 𝜎AME < 2 as “non AME detections”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some of the “significant AME sources” are re-classified as “semi-significant AME sources” as will be discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The con- cerns regarding modelling problems and systematic errors for a few sources are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Ultra-Compact Hii regions Ultra-Compact H ii regions (UCH ii) could bias AME detections and change the free-free typical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' It is therefore impor- tant to assess their possible impact on our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' UCH ii with EM ≳ 107 cm−6pc are expected to produce optically thick free- free emission up to 10 GHz or higher (Kurtz 2002, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To take into account possible contamination of our sample by emis- sion from arcsec resolution point sources (Wood & Churchwell 1989a) that are not AME in nature we follow the method used in PIRXV as illustrated by their Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To this aim we catalog all the IRAS points sources retrieved from the IRAS Point Source Catalog (PSC)7 that lie inside the 2◦ diameter circular apertures of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These sources are classified as a function of their colour-colour index defined by the logarithm of flux ratios obtained in several bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The PSC UCH ii potential candidates tend to have 7 See the link to the IRAS Faint Source Catalog, Ver- sion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 in the HEASARC Catalog Resources Index, https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='gov/W3Browse/iras/iraspsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='html MNRAS 000, 1–36 (2023) 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Comparison between the results obtained with our analysis and in PIRXV for the AME significance 𝜎AME defined as the ratio of the flux density of AME at the frequency peak position divided by the uncertainty on this estimate (a), the emission measure EM (b), the residual AME flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz (c) and the AME peak frequency (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our analysis includes the QUIJOTE-MFI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The data shown in red correspond to sources for which the significance of the AME detection is higher in PIRXV than in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ratios log10(𝑆60/𝑆12) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30 and log10(𝑆25/𝑆12) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57 (Wood & Churchwell 1989b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' They are identified accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Kurtz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (1994) measured the ratio of 100 𝜇m to 2 cm (15 GHz) flux den- sities and found it lies in the range 1000–400000, with no UCH ii regions having 𝑆100𝜇𝑚/𝑆2cm < 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Following PIRXV we use this relation to put limits on the 15 GHz maximum flux densities that could be emitted by candidate UCH ii regions encountered in the apertures used for measuring the flux densities of our sample of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fluxes at 100 𝜇m of the PSC sources are summed up toward each aperture and then divided by 1000 to get an estimate of the the maximum UCH ii flux density at 15 GHz, 𝑆UCHII max , towards each candidate AME source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From the multicomponent fits, the flux densities at 15 GHz (or 2 cm) are calculated and compared to these maximum UCH ii flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The distribution is shown in Fig- ure 6 where the maximum UCH ii flux densities are plotted against the 15 GHz flux densities obtained with our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' If a candidate AME source detected with more than 5𝜎 has a residual AME flux density at 15 GHz lower than 25% of the maximum UCH ii flux density then it is re-classified as “semi-significant”, as indicated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We believe that this is a very conservative approach, in a way that many of these re-classified sources are actually “signifi- cant” AME detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' UCH ii contributions to the 30 GHz excess have been recently investigated by Rennie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2021) on a small sample of galactic H ii regions using data from the 5 GHz COR- NISH catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The study rejects such regions as the cause of the AME excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 Robustness and validation The significance of AME detection, defined by the parameter 𝜎AME, discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1, is an important indicator reflecting the abil- ity of our analysis to detect and fit any excess of emission observed in the frequency range 10–60 GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' whether such a bump is potentially dominated by UCH ii regions or not (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The significance of AME detection obtained on each source, though, is also depen- MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SED of the sample of regions shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The QUIJOTE intensity flux densities are shown with red points, and the WMAP, Planck, and DIRBE intensity flux densities are shown with green, blue, and yellow< points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The low frequency points are shown in pale blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The result of the multicomponent fit is illustrated by the continuous black curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the AME component is shown with the dashed red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the free–free component is shown with the dashed blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the thermal dust component is shown with the dashed yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the CMB component is shown with the dashed green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A zoom on the AME bump is shown in the subpanel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Residuals to the fits are shown in the bottom plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' DR23/DR21 𝐴AME 𝜎AME 𝜈AME 𝑊AME Δ𝑇CMB 𝐴AME priors 𝜈AME priors 𝑊AME priors 𝜒2 red [G081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01] [Jy] [GHz] [Jy] [Jy] [GHz] See plot on Figure 7, left 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ±40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 [0 , 300] [10 , 60] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 See plot on Figure 7, right 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 [0 , 300] [10 , 60] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Fit parameters of the AME and CMB components obtained with different priors on the AME width parameter, 𝑊AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Note that in the case of stronger priors the best-fit values for 𝑊AME and Δ𝑇CMB are found in the border of the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The corresponding plots are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' dent on the overall accuracy of the multicomponent fit obtained over the full frequency spectrum considered in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to explore the stability of the fitting procedure we made a number of tests to check that our main results are not af- fected by our fitting method and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This includes relaxing the assumed calibration uncertainty and changing the sizes of the aperture and annulus radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Overall we were able to fit all the 4 or 5 components on 46 sources from the 52 sources included in the initial sample, or in other words the multicomponent fit was converging on all the components considered to fit each of the 46 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The SPDust2 models (see Ali-Haïmoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ali- Haimoud 2010) for cold neutral medium, dark cloud, molecular cloud, warm ionized medium and warm neutral medium have widths lying in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7] while in practice slightly wider distri- butions could be expected (see discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' To take MNRAS 000, 1–36 (2023) 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Histogram of the AME significance values (𝜎AME) for the sample of 52 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The 5𝜎 limit is shown with the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Sources that are significant and have a maximum contribution from UCH ii regions, 𝑓 UCH II max < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25, are shown as the filled histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Estimated maximum contribution from UCH ii regions against 15 GHz AME residual flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The most significant AME sources (𝜎AME > 5 and 𝑆residual 15 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 × 𝑆UCHII max ) are shown as red diamond symbols, while non-AME regions (𝜎AME < 2) are shown as dark cross symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' “Semi-significant” AME sources (𝜎AME =2–5) are shown as blue triangle symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' “Significant” AME regions that have a potentially large contribution from UCH ii (𝑆residual 15 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 × 𝑆UCHII max ) are re-classed as “semi-significant” and are highlighted by blue diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The data shown with red diamond symbols are the “Significant” AME regions such that 𝑆residual 15 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 × 𝑆UCHII max , if this information is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Regions with no matched UCH II regions are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01 for visualization and lie on the bottom of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The dashed lines correspond to different maximum frac- tions of UCH ii flux density: 1, 10, 25 (solid line), and 100% of the 15 GHz residual flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' this into account the uniform priors used on the AME parameters are: 10 < 𝜈AME < 60 GHz, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 < 𝑊AME < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Such assump- tions on the values allowed to be taken by 𝑊AME are important to keep realistic AME detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' An example of the effect of the priors is shown in Figure 7 where multicomponent fits obtained on the DR23/DR21 maps are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The plot on the left shows the fit on the AME component with priors on 𝑊AME such that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 < 𝑊AME < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5, while the plot on the right displays the AME fit component with priors on 𝑊AME such that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 < 𝑊AME < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The AME fit parameters obtained in both cases are given in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In the case of loose priors on 𝑊AME the AME component shows an excessively wide looking bump, even if the improvement in the goodness of the fit is marginal (see the values of the 𝜒2 red in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Such a broad spectrum cannot be reproduced by spinning dust models for environments with reasonable physical parame- ters, so models like this might be deemed as physically unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This demonstrates the need for setting realistic priors on the fits to overcome the problem with fit degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Finally, as it was commented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3, our methodology for error estimation do not properly grasp those parameter degeneracies, leading in some cases to an underestimation of the error (see the too small error of 𝜈AME in the case of strong prior in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As a final test we repeated the analyses with more stringent priors such that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 < 𝑊AME < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 and 16 GHz < 𝜈AME < 60 GHz, and found that this does not have a strong impact on the derived results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In particular, we found differences typically smaller than 5 % in 𝜈AME and typically smaller than 20 % in 𝐴AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our final sample follows the superscript symbols given in the last column in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A total of 6 sources (labelled as “BD”) considered as bad detections of AME because of a bad fit of the AME, of the free-free or of the thermal dust component, are not considered on a statistical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, statistics are given for the sample which we refer to as the selected sample (46 sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This data set includes sources with low or poor AME de- tection (2 sources, labelled as “LD”), with “semi-significant” AME detection (29 sources labelled as “SS”, including 20 “significant” AME sources reclassified as "semi-significant AME sources") and with “significant" AME detections (15 sources labelled as “S”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Statistics are also given on the sample of “semi-significant” AME detections and on the sample of “significant” AME detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The selected sample includes a total of 7 sources with fits reaching the prior upper limit on 𝑊AME and such that, the uncertainty on this parameter is, 𝜎𝑊AME = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These sources are included in the sam- ple of AME well-detected 44 sources (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', the sample including “semi-significant” and “significant” AME detections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5 STATISTICAL STUDY OF AME SOURCES Along this section we study the statistical properties of the physical parameters of the sample discussed in the previous section, with the aim of better understanding the physical and environmental conditions of the AME sources, as well as to obtain insights about the nature of the carriers that cause the AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The parameter values used to model the components estimated from the analysis of the SEDs in intensity are given in Tables B1 and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The method used to calculate the flux densities does not take into account the effect of the signal integration through the thickness of the clouds as well as across the area sustended by each telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This limitation will be taken into account, as much as possible, in the interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Nature of the sources In this section we focus our analysis on the parameters used to model the AME and some of the thermal dust component parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This includes the relative strength of the ISRF, which is estimated from the fitted thermal dust parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 13 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Two multicomponent fits of the DR23/DR21 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Colours and symbols definitions are the same as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Left: fits obtained with priors on the AME parameters such that 10 < 𝜈AME < 60 GHz, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 < 𝑊AME < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Right: Same as left but with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 < 𝑊AME < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A discussion about the choice of priors is given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The AME and CMB components fit parameters obtained in each case are displayed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 AME fraction at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz As a first step we investigate the fraction of the total flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz that is produced by AME under the expectation that free-free and AME are the dominant sources of radiation at this frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For this we calculated the residual AME flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz, 𝑆28 res, by subtracting to the measured flux density at this same frequency all the other components and propagating their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The histogram of this quantity is plotted in Figure 8 and shows that regardless of whether the sources are classified as “significant” or “semi-significant”, the contribution of the AME flux density goes from a few per cent to almost 100 per cent of the total flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result is different from that obtained by PIRXV who found that in their sample the sources classified as “significant” AME sources were mainly showing 𝑆28 res/𝑆28 > 50 %, while the remaining sources classified as “semi-significant” were lying in the lower part of the histogram such that 𝑆28 res/𝑆28 ≲ 50 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All in all, the majority of the sources in our selected sample show 𝑆28 res/𝑆28 < 50 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result could come from the AME peak frequency distribution which is found to be about 4 GHz lower than by PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result will be presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Dust properties The distribution of the thermal dust temperature, 𝑇dust, against the thermal dust emissivisity, 𝛽dust obtained from the SEDs multicom- ponent fits are displayed in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The expected anti-correlation that is discussed and analysed in many works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Paradis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2014) is also seen in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' An apparent sequence in the IRAS colours given by the 12𝜇m/25𝜇m and 60𝜇m/100𝜇m ratios can also be expected from previous studies of H ii regions (Chan & Fich 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Boulanger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1988), and external galaxies (Helou 1986) showing an anti- correlation between the two ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The interpretation relates to the spatial distribution of different grain populations as a function of the Inter-Stellar Radiation Field (ISRF) intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This trend was obtained for the sample of sources discussed by PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We find a result similar to their analysis but our plots shown in Figure 10 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Histogram of the AME fraction 𝑆28 res/𝑆28 at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The selected sample is shown as the unfilled histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The “significant” AME detection sample is shown with the hatched area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' presents a lower dynamic range of the colour ratio 60 𝜇m/100 𝜇m than the one from their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our sample probes line-of-sights (LOSs) with colour ratios 60 𝜇m/100 𝜇m lying in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7, which is the range in which PIRXV found most of their sources classified as “significant” AME detections and not expected to be dominated by UCH ii region emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 Dust optical depth The sources of our sample are distributed across regions of differ- ent optical depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to understand how this parameter could help us to build up a picture of the distribution of the parameters used to fit the AME components classified as “semi-significant” or “significant”, in Figure 11 we show the variations of the peak MNRAS 000, 1–36 (2023) 14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Distribution of the thermal dust temperature, 𝑇dust, against the thermal dust emissivisity, 𝛽dust obtained from the SEDs multicomponent fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The “significant” AME detection sample is shown with red diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The “semi-significant” AME detection sample is shown with blue triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Low AME detections are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Colour–colour plot of IRAS 12𝜇m/25𝜇m against 60𝜇m/100𝜇m for the sample of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME flux density, 𝐴AME, as a function of the thermal dust optical depth at 250 𝜇m, 𝜏250, obtained from the fits of the thermal dust components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' One can see a clear trend showing an increase of the maximum AME flux density with the quantity of thermal dust mat- ter encountered along the LOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The Spearman Rank Correlation Coefficient (SRCC) of that distribution is rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is not a surprise, as a strong spatial correlation was already observed between the AME and thermal dust, when AME was first detected (see Kogut 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Leitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1997), and it is well established that the interstellar medium is pervaded by a complex non-uniform distribution of thermal dust material, a fraction of which spatially correlates with the spiral arms structure of the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Lallement et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2019) toward which many sources of Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Distribution of the AME peak flux density AAME against 𝜏250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All selected data are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' our sample are located (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In addition, no correlation is observed between the AME peak frequencies and the thermal dust optical depths at 250 𝜇m, (see Figure A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Similarily, no correlation is observed between the width of the parabola used to fit the AME and the thermal dust optical depth (see Figure A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' One can clearly see in that plot the cases for which the AME width reaches the upper limit of the prior 𝑊AME = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These cases are not restrained to a specific range of the thermal dust optical depth parameter, which means that the AME detections with 𝑊AME = 1 are not expected to depend on this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 The interstellar radiation field: G0 Another important parameter that is useful to describe the physics of the several environments towards AME regions is the relative strength of the ISRF, 𝐺0 (see Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME carriers are believed to be tiny particles lying in the bottom part of the interstellar dust grain size spectrum (𝑎 ≲ 1 nm) (possibly includ- ing Polycyclic Aromatic Hydrocarbons or PAHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Their chemical properties, physical coherence and total charge could vary over time and from one environment to another, and therefore depend on the relative strength of the ISRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Therefore, having our estimation of 𝐺0 is very useful to explore possible relations with the parameters used to model the AME component detected at the SED level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' An estimation of 𝐺0 can be obtained from the equilibrium dust temper- ature of the big dust grains (𝑇BG) compared to the average value of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 K (see Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1983), with the relation: 𝐺0 = � 𝑇BG 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5K �4+𝛽BG , (10) where 𝛽BG is the spectral index associated with the opacity of the big grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In the following, we assume 𝑇BG ≈ 𝑇dust, where 𝑇dust is the averaged temperature of the thermal dust component obtained from the fit on each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As in PIRXV, we also assume a constant value 𝛽BG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We note that using 𝛽BG ≈ 𝛽dust could also be considered, but would not change the conclusions of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The correlation between the AME fraction at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz (de- fined as the residual AME flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz divided by the MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 15 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME fraction at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz as a function of the estimated 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' total flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz) and G0 is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The data show a decrease of the AME fraction as a function of G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This trend is similar to the one obtained by PIRXV in their analysis and seems to be dependent of the considered subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In our analysis the slope of the “significant” AME detection data sample is of order 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='48, while the slope of “semi-significant” AME detection data sample is of order 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We point out that the uncer- tainties of the values of the slopes we estimated are large, ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 for both “significant” and “semi-significant” AME detections data points, which prevents a full and fair comparison with results from previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our slopes, though, can be compared to the slope of 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 obtained by PIRXV on their strongest AME sources sample (see their Figure 15 and section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4), and to the slope of 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 obtained on their semi-significant AME sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All in all, our results agree with those of PIRXV within the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Differences in the slopes estimates can be explained by the different sample sizes (half-sky versus full sky coverage) and by the introduction of the QUIJOTE data in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 Peak frequency of AME Among the three parameters used to fit the AME components in our sample, one is the peak frequency, which is allowed to vary in the frequency range 10–60 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Such a degree of freedom is important since it allows to get better final fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' It has also been shown in previous works that one can expect the frequency of AME to vary from one source to the other, or even within the same region (Cepeda-Arroita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The histogram of the AME peak frequency calculated for the selected sample is shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The Gaussian fit to the distribution provides a mean frequency and dispersion given by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The hashed histogram shows the distribution of the “significant” AME sources sample peaking around the weighted mean frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' PIRXV found their sample of AME sources to peak in the range 20–35 GHz, with a weighted mean of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 GHz, a bit higher than our mean value, the main reason of this difference being that flux densities in the frequency range 10–20 GHz were not available in their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In fact, the addition of QUIJOTE-MFI data clearly helps reducing the uncertainty in the determination of 𝜈AME, thanks to allowing to trace the down-turn of Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Histogram of the AME peak frequency in bins of size 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The selected sample is shown as the unfilled histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The “significant” AME detection sample is shown with the hashed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A Gaussian fit to the histogram is shown with the dashed-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' the AME spectrum at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our average error on 𝜈AME is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz, and when we repeat our analysis excluding QUIJOTE- MFI data we get an average error of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 GHz (see also discussion in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand our analysis of G160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 (the California nebula/NGC 1499) recovers an AME peak frequency at 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ±38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 GHz, which is consistent with values obtained in previous analyses (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2011, 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The uncertainty on our estimate is quite large because the free-free dominates at 𝜈 <100 GHz making the width of the AME bump poorly constrained and the fitted parameters strongly degenerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On top of that the circular aperture that we use may not be optimal in this case where the emission is elongated and pretty extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 Width of the AME bump In addition to the maximum flux density and peak frequency pa- rameters, the third parameter used to fit the AME components is the width of the parabola, 𝑊AME (see Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The allowed range in the fit was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2–1 and the initial value was 𝑊AME = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 for all sources, this value being the expected average value from the SPDust2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The histogram of our fitted values is displayed in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As discussed previously, the multicomponent fits lead- ing to output fit parameters of values 𝑊AME = 1 and 𝜎𝑊AME = 0 are cases reaching the prior upper limit value, and this artificially leads to a higher number of sources lying in the last bin of the histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The selected sample is shown as the whole histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The single-dashed histogram shows the same distribution without the prior dominated AME detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This distribution has a mean and dispersion given by, 𝑊AME = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The distribution looks rather flat, and far from Gaussian, which is reflected in the large error bar of the Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This in fact illustrates that 𝑊AME is maybe the worst constrained parameter in our fit, due to large degeneracies with other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result is obtained with a bin of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 and would need a higher sample for one to drive strong conclusions on a statistical ba- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Indeed using a bin size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 the whole histogram looks rather like a normal distribution without any clear peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Statistically, we MNRAS 000, 1–36 (2023) 16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Histogram of the width of the AME component parameterized by 𝑊AME (see Equation 8) in bins of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The selected sample is shown as the whole histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The “constrained” AME detections are shown with the unfilled histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The “significant” AME detection sample is shown with the double harshed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The gaussian fit to the histogram is shown with the dashed-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' find that 𝑊AME does not correlate with the free-free component EM parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Neither do we find any correlation between 𝑊AME and any of the thermal dust parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand we observe a mild correlation of 𝑊AME with the AME emissivity (𝐴AME/𝜏250).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A detailed definition of the AME emissivity will be given in Sec- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 where these results will be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 Width of AME bump and peak frequency of AME The three parameters describing the parabola used to fit the AME flux density bump (see Equation 8) are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' With this model any correlation found between the AME peak frequency and the parabola width parameter could therefore be indicative of the physics underlying the description of the AME carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We checked that neither a negative nor a positive correlation can be seen between the two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As shown in Table D1, all the samples (selected, “semi-significant” and “significant”) are showing SRCCs consistent with a null correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These results show that the width and the peak frequency of the AME component are fully independent from each other, although this conclusion could be affected by the fact that, in some cases, 𝑊AME seems to be poorly constrained in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Dust correlations In this section we focus on the thermal dust component with the aim to better understand its relation with the AME component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We also consider high frequency maps at 100 𝜇m, 60 𝜇m and 12 𝜇m, since these data have the potential to provide information about some of the candidate AME carriers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', spinning dust, PAHs or fullerenes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Dust flux densities at 100 𝜇m, 60 𝜇m, 25 𝜇m and 12 𝜇m Following the spatial correlation observed between AME and the thermal dust emission when AME was first discovered, many stud- ies have explored and discussed the possibility that AME carriers are spinning dust grains in nature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Draine & Lazarian 1998, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ali-Haïmoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2009), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', possibly a specific subclass of the dust grain population spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A look to various dust grain emission templates should therefore be useful to explore if any spe- cific correlation exists between the maximum AME flux densities and the flux densities of thermal dust observed at 100 𝜇m, 60 𝜇m, 25 𝜇m and 12 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Such plots are shown in Figure 15 (top row) and the strength of the correlations described by their SRCCs are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We find very strong correlations between the AME flux densities and the thermal dust flux densities at 100 𝜇m, 60 𝜇m, 25 𝜇m and 12 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result is consistent with the one obtained by PIRXV from their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' If the AME carriers are spinning dust grains, the AME compo- nent is expected to be quite insensitive to the ISRF relative strength, 𝐺0 (Ali-Haïmoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Ysard & Verstraete 2010) while on the contrary the thermal dust grains population is expected to be sensitive to it, mainly because the UV radiation should control their temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' If that was true one would expect better correlations between the maximum AME flux densities and the flux densities of thermal dust observed at 100 𝜇m, 60 𝜇m, 25 𝜇m and 12 𝜇m, once they are normalized by 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This has been discussed in some pre- vious analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Ysard & Verstraete 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The plots obtained once the thermal dust fluxes are normalized by 𝐺0 are shown in Fig- ure 15 (bottom row) and the strength of the correlations described by their SRCCs are given between parenthesis in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Contrary to what was found on their sample by PIRXV, normalizing the thermal dust templates by 𝐺0 leads to less tight correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These results suggest that the AME carriers could be coupled to the thermal dust grain components rather than to a dust grain population relatively insensitive to 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand the dust grain size distribution is very sensitive to the ISRF, as well as to other parameters such as the dipole moments of PAHs (Ali-Haïmoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2009), mean- ing that the interpretation of the results obtained with plots such as those given in Figure 15 may be complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The role of 𝐺0 will be discussed further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Thermal Dust peak flux densities The size of the aperture used to build the SEDs could introduce a coupling between some of the thermal dust parameters 𝜏250, 𝑇dust and 𝛽dust due to a possible range of degeneracy at the fit level between these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to circumvent this problem, that could mislead the interpretation of some of the correlations discussed above, we looked at the distribution between the flux densities at the peak of the AME bumps and at the maximum of the thermal dust components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is shown in Figure 16 where it can be seen a correlation between the two flux components at their maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The slope of a power-law fit to the selected sample is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96 and almost consistent with 1 as shown with the dark solid line on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The SRRC between the two parameters is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 Thermal Dust radiance The radiance of a component is defined as the integral of the flux density of that component over the full spectral range, ℜ = MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 17 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Top row: AME peak flux density as a function of the 100 𝜇m (panel a), 60 𝜇m (panel b) and 12 𝜇m (panel c) flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Bottom row: same as top row but after the infrared tracers of dust have been divided by 𝐺0 (panel d, e and f, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Power-law fits to the full set are shown with back solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SRCCs are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Wavelentgh SRCC SRCC SRCC selected sample AME significant AME semi-significant 100 𝜇m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='87± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='86± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='89± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05) 60 𝜇m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='86± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='82± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05) 25 𝜇m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='43± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05) 12 𝜇m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='39± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='89± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spearman rank correlation coefficients (SRCCs) between the AME maximum flux densities and the IR/submm flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The values displayed between parentheses are the SRCCs obtained once the IR/submm flux densities are divided by the IRSFs estimates G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ∫ +∞ −∞ 𝑆(𝜈)𝑑𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this work, all radiances were calculated by inte- grating the fitted models between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 and 3000 GHz, which is the frequency range where all the maps used in this analysis are available (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some studies have shown strong correlations between the dust radiance and the AME amplitude at the peak frequency (Hensley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Hensley & Draine 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The distribution of both components for our sample is shown in Figure 17, (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A good correlation is observed between the two variables of the se- lected sample, with a SRCC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05, and a power-law slope consistent with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This tight correlation suggests a strong coupling between the big dust grains expected to be the main contributors to the dust grain radiance considered here (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', in the wavelength range 𝜆 > 100𝜇m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 17 (bottom) shows the distribution of the AME radiance ℜAME as a function of the dust radiance ℜtd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In that case a lower correlation is observed between the two parameters with a SRCC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We believe that the reason why the AME amplitude correlates better than the AME radiance is because the latter is quite sensitive to 𝑊AME, and this parameter has large error bars due to not being very well constrained by our fit (see section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This said, these two correlations can be interpreted using two different views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A first one is that the AME model used to fit the data and designed to approximate the spectrum of the spinning dust emission is not fully appropriate to capture the contribution of the AME carriers, or that in some regions it is difficult to properly disentangle the AME contribution from the free-free and thermal dust contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Another view could be that if the AME model used to fit the data is good enough to capture the AME components accurately, then the dust radiance of PAHs and/or Very Small Grains (VSGs) could represent a relatively large contribution of the total dust radiance at wavelengths greater than 100 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 AME emissivity As discussed above, strong spatial correlations were found between the AME emission and thermal dust emission when AME was first detected (see Kogut 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Leitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to build a picture of the distribution of the AME emission along the third spatial dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', the line-of-sight, LOS), further works have defined the AME emissivity as the ratio between the AME inten- sity and the column density, for which the optical depth at a given wavelength is often used as a proxy (see Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2018, and discussion and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to make comparisons with results discussed in the literature we first show in Figure 18 the MNRAS 000, 1–36 (2023) 18 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Maximum AME flux density versus maximum thermal dust flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The solid line represents a fitted power-law model to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' distribution of the AME flux density obtained by subtracting to the measured flux density at this same frequency all the other compo- nents (defined as the residual flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz) normalized by the 100 𝜇m flux density (𝑆res 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4GHz/𝑆100𝜇m), as a function of the AME detection significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this case the 100 𝜇m flux density is expected to be optically thin for a given dust temperature and com- position and is used as a proxy to probe the column density of dust along the LOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 𝑆res 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4GHz/𝑆100𝜇m is in the range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 − 9) × 10−4 with a weighted mean of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3) × 10−4 and an unweighted average of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6) × 10−4 (significant AME sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These values are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' They are smaller than the unweighted average value of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7) × 10−4 of PIRXV and than the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 × 10−4 value of Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2006) but are higher than the weighted average of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2) × 10−4obtained in PIRXV and than the value of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 × 10−4 obtained by Todorović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2010) on a sample of H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The differences between our estimates and those obtained by PIRXV could partially come from the different samples used in each study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our sample only covers the North hemisphere sky while the analysis of PIRXV includes also sources in the Southern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Different error treatment may also affect the weighted averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Regardless of these issues, we have applied a one-to-one comparison between our flux density ratios and those reported in PIRXV in the subsample of 42 common sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' When we represent the former against the latter and fit the data to a straight line we find a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76, meaning that we find ≈ 30% higher emissivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is a consequence of the increase of the AME amplitude as a result of the inclusion of QUIJOTE data (see Figure 3c and related discussion in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A summary of these results is given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The small range of values of the ratio of the AME residual flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz to the flux density at 100 𝜇m suggests that a power-law index of order 1 could be expected between the two flux density distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is indeed what the best-fitting power-law confirms as it yields a power-law index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 in tension with the power-law index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 obtained by PIRXV on their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Similarily, the best-fitting power-law index between the AME residual flux density at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 GHz and the dust optical depth at a wavelength of 250 𝜇m, 𝜏250, yields a power-law index of Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Top: AME flux density at peak frequency, 𝐴AME, as a function of the thermal dust radiance, ℜtd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Bottom: AME radiance, ℜAME, as a function of thermal dust radiance, ℜtd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The solid lines represent fitted power-law models to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Sample 𝑆res 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4GHz/𝑆100𝜇m [×10−4] unweighted mean weighted mean This work - selected sample 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 This work - semi-significant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 This work - significant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 PIRXV - significant 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 Todorović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2010) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2006) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Comparison of the AME flux densities normalized by the 100 𝜇m flux densities obtained in this work and in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 in agreement with the power-law index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 obtained by PIRXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The results obtained by PIRXV were inferring an AME mainly proportional to the column density estimate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', to the amount of material along the LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is what we find whether we consider the 100 𝜇m map or the 𝜏250 parameters as proxies of the column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 19 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME emissivity against AME significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and their colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variations of the AME emissivity with the relative strength of the ISRF, 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The power-law fit obtained on the selected sample is plotted with the black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 Role of the ISRF The ISRF is strongly coupled to the nature of the various phases encountered in the ISM defined in terms of gas temperature and matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The UV light produced by the population of stars pervading the ISM is absorbed by the dust grain populations and re-radiated in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The ISRF therefore plays an important dy- namic role since it will affect the chemical composition of the ISM material, the dust grain distribution as well as the lifetime of the small dust grain and complex molecule populations (see Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' It is therefore interesting to investigate the existence of possi- ble relationships between the relative strength of the ISRF, 𝐺0, and the parameters describing the AME component derived from the SEDs analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For this we looked at the distribution of the AME emissivity, now defined as 𝐴AME/𝜏250, the AME peak frequency, 𝜈AME, and the AME bump width parameter, 𝑊AME, as a function of 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The plots are shown in Figures 19, Figure A4 and Figure A5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We find poor correlations between 𝐺0 and the AME parameters 𝜈AME and 𝑊AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, we find a SRCC of rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 between the AME emissivity and 𝐺0 parameters for the selected sample (Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This distribution can be fitted by a power-law of index of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 as shown with the black line in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Since we derived the relative strength of the ISRF, 𝐺0, by using the thermal dust grain temperature, 𝑇dust, obtained from the SED grey body fits, and by assuming a maximum and constant thermal dust emissivity, 𝛽dust = 2 (see equation 10), the SRCCs obtained between the 𝐴AME/𝜏250 and 𝐺0 parameter distributions and between the 𝐴AME/𝜏250 and 𝑇dust parameter distributions are by construction identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Similarly, the introduction of the SEDs fit estimates of 𝛽dust in the calculation of 𝐺0 only changes SRCCs values by less than one percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This means that the AME flux den- sities obtained at the peak frequency are mainly correlated with the combination of the dust optical depth, 𝜏250, and the thermal dust temperature 𝑇dust parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result is in agreement with the strong correlation obtained between the AME peak flux densities and 𝜏250, and with the 100𝜇m thermal dust fluxes discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In the above we have considered that a good proxy of the rel- ative strength of the ISRF is given by 𝐺0, which is a function of the thermal dust temperature, 𝑇dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The EM is another interesting parameter associated with hot phases of the ISM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', ionized re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In our sample one can expect electron temperatures lying in the range 5 458–7 194 K as from the electron temperature map provided by Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2016c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Inside molecular clouds, the ionized regions produced by stellar radiation are ex- pected to represent a fraction of the whole volume associated with the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Not all the sources displayed in Table 2 are only molecu- lar cloud regions in nature but they all have thermal dust along their LOSs, which is a component strongly correlated with the AME component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In this context we show in Figure 20 the distribution of the free-free EM parameter as a function of 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The plot shows only a poor correlation between the two parameters, this being also illustrated by the low correlation coefficient, SRCC= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06, found between the two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This lack of correlation would indicate that the AME emissivity does not correlate significantly with the EM free-free emission parameter at Galactic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 Free-Free correlations In our study the EM of the free-free does not correlate with the AME emissivity estimated by 𝐴AME/𝜏250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, a mild correlation is observed between the amplitude of the AME at the peak frequency, 𝐴AME, and the EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is shown on the plot displayed at the top panel in Figure 21, with a SRCC between the two parameters of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Since a strong correlation is observed between 𝐴AME and the emission of the thermal dust at the peak frequency, 𝑆TD,PEAK, this also means that a correlation can be expected between EM and, 𝑆TD,PEAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is shown in the plot displayed in the bottom panel of Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In that case the SRCC between the two parameters of the selected dataset is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In the interpretation of these results it must be taken into ac- count that our EMs are estimated directly from integrated flux den- sities, and given the non-linear dependency between the two, those estimates could not be representative of the real averaged EMs of each region, as it was already commented in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This could indeed contribute to smear out any underlying real correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In addition, the fact that the correlation in the top panel of Figure 21 is only seen for the sources with highest AME amplitudes could be a hint that there could be a selection effect, in such a way that when MNRAS 000, 1–36 (2023) 20 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Free-free Emission Measure (EM) parameter as a function of the relative strength of the ISRF, 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' the free-free is high the AME can only be detected when it is also very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In order to better understand this, in Figure 22 we plot 𝐴AME as a function of the flux density of the free-free at 𝜈AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The one-to-one relation is displayed by the solid line while the one- to-ten relation is shown with the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Given that calibration uncertainties are of order 5 − 10% the lack of sources below the one-to-ten line could in fact tell that the AME cannot be separated when it is less than 10% of the free-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the contrary, the plot also shows that there are a few regions (like the Perseus and 𝜌 oph molecular clouds, respectively G160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 and G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90) with more AME than free-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' It must also be taken into account that our SED multicom- ponent fit is subject to an anti-correlation between the AME and free-free amplitudes which may contribute to worsening the cor- relation observed in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This parameter degeneracy, which upcoming 5 GHz data from the C-BASS experiment (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2018) will help to break, is clearly seen in MCMC analyses like those presented in Cepeda-Arroita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2021) and in Fernandez- Torreiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 6 DISCUSSION In this section we summarize our results suggesting that the AME carriers may be preferentially located in cold rather than in hot phases of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some limitations of our modelling of the AME component are then discussed, followed by a comparisons of our results with those from previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Does AME originate from the Cold ISM Phase ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In the last sections we searched for correlations between some of the parameters obtained from the multicomponent fits of the AME component and ISM tracers including the flux densities obtained at 12, 25, 60 and 100 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Interestingly, we find that the flux densities obtained at the peak frequency of the AME bumps show strong correlation with the flux densities at 100, 60 and 25 𝜇m, with a small loss of correlation with the flux densities at 12 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Top: AME flux at the peak frequency versus free-free emission measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Bottom: Thermal dust flux at the peak frequency versus free- free emission measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' other hand, once these four flux densities tracers are normalized by the relative strength of the ISRF, 𝐺0, the correlations with 𝐴AME are found to be about a few to ten percent lower in the high fre- quency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These results could discard tiny dust particles (PAHs or VSGs in nature) as AME carriers, if such particles are poorly sensitive to the relative strength of the ISRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' For this reason we explored in more detail possible relationships between the AME component parameters with dust modelling parameters, with 𝐺0, as well as with the free-free component parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Table 6 gives a summary of some of the most relevant SRCCs obtained from the previous analysis in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' They could help to shed light on some existing physical relationships between the astrophysical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From spectral energy distribution analysis of the sample of 46 good candidate AME sources the strongest correlation is found be- tween the maximum flux density of the thermal dust, 𝑆TD,peak, and of the AME peak, 𝐴AME (Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A lower correlation is found between the AME emissivity, 𝐴AME/𝜏250, and the interstellar radi- ation field relative strength, 𝐺0 (Figure 19), and a mild correlation is obtained between 𝐴AME and the free-free EM (Figure 21, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 21 Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME flux at the peak frequency versus free-free flux at the AME peak frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The one-to-one relation is displayed by the solid line and the one-to-ten relation is shown with the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand no correlation is found between 𝐴AME/𝜏250 and 𝐸𝑀 (see end of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4), and neither between the AME peak frequency, 𝜈AME, and 𝐺0 (Figure A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As discussed in the previ- ous section, averaging effects in our estimates of EM, as well as a selection effect associated with only the brightest AME sources being detected above very high free-free amplitudes, could have an impact on the tentative correlation seen between EM and 𝐴AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand the correlation found between 𝐴AME and 𝑆TD,peak is expected to be real since these two components are associated with distinct wavelength ranges with poor overlap between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Since there is a null correlation between 𝐴AME/𝜏250 and 𝐸𝑀, this means that 𝐴AME/𝜏250, which also correlates with the dust grain emissivity, 𝑆TD,peak/𝜏250, is rather driven by 𝐺0, which in turn is a function of the thermal dust temperature approximated by 𝑇dust obtained from the modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In other words the interstellar radi- ation field still could be the main driver of the AME in terms of spinning dust excitation mechanisms, but the spinning dust could be more likely associated with cold phases of the ISM rather than to hot phases associated with free-free radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 AME components characterization From the results obtained with the multicomponent fit analysis we tested the level of independency between the parameters used to fit the AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This model is the analytical approximation of the spectrum of spinning dust emission proposed by Stevenson (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Indeed, we find null or very low correlations between parameters, 𝐴AME and 𝜈AME, 𝜈AME and 𝑊AME, and 𝑊AME and 𝐴AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, we find a small correlation between 𝑊AME and 𝐴AME/𝜏250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The distribution of these two parameters is shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' By definition, the AME emissivity depends on the total amount of material along the LOS as estimated by 𝜏250, and this correlation means that, on average, 𝐴AME/𝑊AME is not directly proportional to 𝜏250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Testing this result using a physical AME mod- elling is out of the scope of this work, but could be investigated in future analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, in a previous section we dis- Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AME emissivity against the width of the AME parabola model, 𝑊AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' cussed the strong correlation obtained between 𝐴AME and the dust radiance, ℜDust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Put all together these results favor a strong cou- pling between the peak AME flux densities and the total amount of dust probed at 250 𝜇m, but only a fraction of the total amount of material would be at the origin of the AME radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 Comparison with previous works The main differences found in this work with respect to the results discussed in PIRXV have been discussed along the previous sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Below we compare and discuss our results with those from other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Using hierarchical Bayesian inference and full dust spectral energy distribution (SED) modelling, Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2019) argue that, on angular scales of approximately 1◦, AME in 𝜆 Orionis correlates more strongly with PAH mass than with total dust mass, giving support for a spinning PAH hypothesis within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Here, on similar angular scales, we find a better correlation with the 100 𝜇m dust template than with the 12 𝜇m dust template giving hints that, on Galactic scale, the dust grain components producing AME are more likely associated with the cold ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This hypothesis is also supported by the strong correlation we find between the maximum flux density of the AME components with the dust radiance ob- tained from the integration of the dust flux models at wavelengths lower than 100 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result may suffer the lack of modelling, in this work, at wavelengths shorter than 100 𝜇m though, but it sug- gests that the AME carriers are spatially closely associated with the thermal dust components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Cepeda-Arroita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2021) discuss AME spectral variations in the 𝜆 Orionis region with mild correlation between the AME peak frequency and the free-free emission measure, and strong cor- relation between the thermal dust temperature and the free-free emission measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Their results obtained at 1◦-angular scale give an overall picture consistent with spinning dust where the local radiation field plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In our analysis we find mild and null correlations between the AME peak frequency distribution and the thermal dust temperature, or the free-free emission measure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' At face value, our result obtained at similar angular scale tends to discard the free-free emission as the main driver of MNRAS 000, 1–36 (2023) 22 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variable 1 Variable 2 SRCC SRCC SRCC Power-Law Slope(a) Figure selected sample AME significant AME semi-significant selected sample 𝐴AME [Jy] 𝑆TD,peak [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='82± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='91± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56 16 𝐴AME [Jy] ℜDust 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='95±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 17 (top) 𝑊AME 𝐴AME/𝜏250 [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 23 𝐴AME/𝜏250 [Jy] 𝐺0 or Ttd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='87± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='94 19 𝑆TD,peak [Jy] EM [cm−6 pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='50± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 21 (bottom) 𝐴AME [Jy] EM [cm−6 pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='55± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='89 21 (top) ℜAME ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='73± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32 17 (bottom) 𝜈AME [GHz] 𝐺0 or Ttd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='49± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 20 𝑊AME 𝐺0 or Ttd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' A5 𝜈AME [GHz] EM [cm−6 pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 𝐴AME/𝜏250 [Jy] EM [cm−6 pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 𝐴AME [Jy] 𝐺0 or Ttd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='33± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Selection of Spearman rank correlation coefficients (SRCCs) between several model parameters in decreasing strength for the selected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (a): Slopes obtained from linear fits in log–log space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' the excitation of the AME carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' On the other hand, our anal- ysis is obtained on a sample of sources distributed on a Galactic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This makes direct comparisons with results obtained on in- dividual regions quite difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' One should also bear in mind that some of the correlations obtained at low angular resolutions break down on finer angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Casassus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2006) discuss 31 GHz Cosmic Background Imager (CBI) observations of LDN 1622;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Casassus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2008) discuss similar observations of the 𝜌 Oph molecular cloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Arce-Tord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2020) discuss 𝜌 Oph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ar- cmin resolution observations at 31 GHz with CBI 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' and Casassus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2021) discuss ATCA high resolution observations of the 𝜌 Oph West photo-dissociation region suggesting spectral variations that could be explained with two different cut offs on PAHs popu- lations with the SPDust model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Actually, these studies demonstrate that finer angular resolution observations are important to identify the physical regions where spectral variations occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From another perspective, Bernstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2020) discuss fullerenes based modelling of AME in 14 different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The models are calibrated using the well studied LDN 1622 dark cloud physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The rotational temperatures are of the order of the dust grains temperatures for most of the regions, suggest- ing that in this scenario the AME carriers are associated with cold ISM phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This result could support our discussion above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' that AME emissivity correlates slightly with the dust temperature while not with EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our study is focused on high column density regions pervaded by molecular clouds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', including cold neutral medium (CNM) phases, mainly located along the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Using a completely different method, Hensley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2021) investi- gated the relationship between the CNM, the AME and the abun- dance of PAHs over large areas associated with diffuse ISM regions (𝑁HI < 4 × 1020 cm−2) at high Galactic latitudes (| 𝑏 |> 30◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Their study shows that the CNM fraction strongly correlates with the fraction of dust in PAHs, and that PAHs preferentially reside in cold and relatively dense phases of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' If PAHs are indeed at the origin of the AME probed in our work, they could also pref- erentially be associated with cold phases of the ISM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', with the CNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Finally, we point out that AME has been detected in other galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The first detection of AME in another galaxy, namely, NGC 6946, was reported by Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Detection of AME has also been reported by Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2018) in NGC 4725B using VLA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In a following work Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2020) discussed complementary ALMA observations on NGC 4725B that show dis- crepancy with expected thermal dust component making the inter- pretation of the results quite puzzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In our study we sampled the AME component over several AME candidate regions in our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The results show a distribution of peak frequencies close to 25 GHz which is consistent with the average peak frequency ob- served by Battistelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2019) on M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Here, the relatively low resolution used in our study allows to sample our galaxy at about kiloparsec scales or lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This is an asset allowing more straight- forward comparisons with results obtained on close-by galaxies sampled at kiloparsec scales (see for example Figure 1 in Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2010, for comparison with our Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 7 SUMMARY In this work we revisited the approach proposed by PIRXV and their analysis of the multicomponent parameters obtained on Galactic candidates AME sources on the full sky at 1◦-angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The main difference with their work comes from the inclusion of flux densities provided by the QUIJOTE-MFI wide survey maps at 11, 13, 17 and 19 GHz covering the northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' These maps allow generally improved detections, a better separation of the AME and the free-free components and a better characterizations of the AME spectra observed between 10 GHz and 60 GHz on a sample of 46 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From our analysis we find the following: The distribution of the AME peak frequency has a weighted mean frequency and dispersion of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 GHz, about 4 GHz lower than the mean value obtained by PIRXV on their full-sky sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Our result demonstrates the importance of using low fre- quency data in the range 10–20 GHz to properly characterize the AME bump turnover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The value is in agreement with estimates obtained on nearby spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The strongest correlations, of order 88%, are found between the thermal dust peak flux density, and of the AME peak flux den- sity, and between the AME peak flux density and the thermal dust radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Mild correlation coefficients of order 66-68 per cent are found between the AME emissivity (defined as 𝐴AME/𝜏250) and the width of the AME component, as well as between the AME emissivity and the interstellar radiation field relative strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 23 A mild correlation of order 59% is found between the AME peak flux density and the free-free EM, but this could be affected by averaging effects in the calculation of EM, as well as by the fact that only very bright AME sources would be clearly detected above strong free-free emission, whose determination is subject to uncertainties associated with calibration errors of order 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' No correlation is found between the AME emissivity, 𝐴AME/𝜏250, and the free-free radiation EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' No significant correlation is observed between the peak fre- quencies of the AME and the thermal dust components as it has been reported in the case of Lambda Orionis in a previous study by Cepeda-Arroita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' From our analysis we conclude that the interstellar radiation field still can be the main driver of the intensity of the AME toward spinning dust excitation mechanisms, but it is not clear whether spinning dust would be most likely associated with cold phases of the interstellar medium rather than with hot phases dominated by free-free radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Future data over large sky fractions coming from projects currently under development like C-BASS (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2018), TFGI (Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2012b), and see also the introduction in Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (2023)) and MFI2 (Hoyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2022) should help to clarify these aspects and to further refine similar statistical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the staff of the Teide Observatory for invaluable assistance in the commissioning and operation of QUIJOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The QUIJOTE experiment is being developed by the Instituto de Astrofisica de Canarias (IAC), the Instituto de Fisica de Cantabria (IFCA), and the Universities of Cantabria, Manch- ester and Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Partial financial support was provided by the Spanish Ministry of Science and Innovation under the projects AYA2007-68058-C03-01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AYA2007-68058-C03-02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AYA2010-21766-C03-01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AYA2010-21766-C03-02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AYA2014- 60438-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ESP2015-70646-C2-1-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AYA2017-84185-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ESP2017- 83921-C2-1-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' AYA2017-90675-REDC (co-funded with EU FEDER funds),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' PGC2018-101814-B-I00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' PID2019-110610RB- C21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' PID2020-120514GB-I00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' IACA13-3E-2336,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' IACA15-BE- 3707,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' EQC2018-004918-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' the Severo Ochoa Programs SEV- 2015-0548 and CEX2019-000920-S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' the Maria de Maeztu Pro- gram MDM-2017-0765,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' and by the Consolider-Ingenio project CSD2010-00064 (EPI: Exploring the Physics of Inflation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We acknowledge support from the ACIISI, Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference ProID2020010108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This project has received funding from the Eu- ropean Union’s Horizon 2020 research and innovation program un- der grant agreement number 687312 (RADIOFOREGROUNDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' FP acknowledges the European Commission under the Marie Sklodowska-Curie Actions within the European Union’s Horizon 2020 research and innovation programme under Grant Agree- ment number 658499 (PolAME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' FP acknowledges support from the Spanish State Research Agency (AEI) under grant numbers PID2019-105552RB-C43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' FG acknowledges funding from the Eu- ropean Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agree- ment No 101001897).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' EdlH acknowledge partial financial support from the Concepción Arenal Programme of the Universidad de Cantabria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' BR-G acknowledges ASI-INFN Agreement 2014-037- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' DT acknowledges the support from the Chinese Academy of Sciences President’s International Fellowship Initiative, Grant N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2020PM0042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' We acknowledge the use of data from the Planck /ESA mission, downloaded from the Planck Legacy Archive, and of the Legacy Archive for Microwave Background Data Analysis (LAMBDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Support for LAMBDA is provided by the NASA Of- fice of Space Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Some of the results in this paper have been derived using the HEALPix (Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2005) package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' DATA AVAILABILITY The QUIJOTE nominal mode maps, are expected to be made pub- licly available in the first QUIJOTE data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Other ancillary data employed in this work are publicly available and can be ac- cessed online as detailed in the paper text.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Churchwell E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', 1989b, ApJ, 340, 265 Ysard N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', Verstraete L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=', 2010, A&A, 509, A12 MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 25 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Distribution of the AME peak frequency 𝜈AME against 𝜏250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All selected data are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' APPENDIX A: ADDITIONAL PLOTS A few more Figures, all showing a lack of correlation between some of the modelling parameters, are displayed in this appendix for the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A1 shows the variations of the AME peak frequency, 𝜈AME, as a function of the proxy of the thermal dust material, 𝜏250, as discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The distribution of 𝜈AME seems to be independent of the quantity of matter along the LOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A2 shows the Variations of the AME characteristic width,𝑊AME, as a function of the proxy of the thermal dust material, 𝜏250, as discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This AME parameter also seems to be independent of the quantity of matter along the LOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A3 shows the Variations of the AME peak frequency, 𝜈AME, as a function of the width of the parabola, 𝑊AME, as dis- cussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' As expected from the formalism used to model AME (see Stevenson 2014), this plot confirms that the two parameters are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A4 and Figure A5 show the distribution of the AME peak frequency and of the AME parabola width parameter with the relative strength of the ISRF, 𝐺0, respectively, as discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The two AME parameters show no dependence on 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' APPENDIX B: SED MULTICOMPONENT FIT PARAMETERS The parameters obtained on each source from the multicomponent analysis are displayed in Table B1 and Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The name of the sources are given in column one from each Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The parameters used to model the synchrotron component, and the free-free com- ponent, are given in columns 2 and 3 from Table B1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Columns 4 to 6 of the same table give the parameters used to model the thermal dust grain component, while column 7 gives the relative strength of the ISRF derived using 𝑇dust displayed in column 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The parameter used to model the CMB component is found in the last column of Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All the parameters related to the modelling of AME are given in Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The maximum fraction of emission that Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Distribution of the width of the AME WAME against 𝜏250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' All selected data are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variations of the AME peak frequency 𝜈AME as a function of the width of the parabola, 𝜎AME,𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' could be attributed to UCH ii regions is given in column 8 of that table, while the reduced 𝜒2 of the multicomponent fits are given in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' APPENDIX C: SEDS OF THE FULL SAMPLE The plots of the SEDs obtained on each of the 52 candidate AME sources of the sample are displayed in Figures C1–C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In each plot, the QUIJOTE intensity flux densities are shown with red filled cir- cles, and the WMAP, Planck, and DIRBE intensity flux densities are shown with green, blue, and yellow opened diamonds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The low frequency points used to fit the models are shown in pale blue, and in blue if they were not included in the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The result of the multicomponent fit is illustrated by the continu- MNRAS 000, 1–36 (2023) 26 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Source Name 𝛼synch,int EM 𝜏250 𝑇dust 𝛽dust G0 Δ𝑇CMB [cm−6 pc] [×105] [K] [𝜇K] G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='48 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 (PUL) G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84-02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 G011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='51 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 G012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='80-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 G037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='51 1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 237.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='44 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 G040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 G041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='48 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 G045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 113.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='51 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 G062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='98+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 G070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 (PLL) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='99 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='93 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 G078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 (PLL) G081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 2541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 44.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 G107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20+05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 G110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 G111.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 G118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09+04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='96 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='63 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 G123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13-06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 G133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27+09.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 G133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='74+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='44 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='77 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 G142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35+01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 G151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 G160.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 G160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 G173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='94 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 G173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62+02.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='33 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 G190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 G192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='37 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='39 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 G192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 G239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40-04.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 G351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='31+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='95 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 G355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='63+20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SEDs Fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (PUL): prior upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (PLL): prior lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' ous black curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the AME component is shown with the dashed red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the free–free component is shown with the dashed blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the thermal dust component is shown with the dashed yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' The fit to the CMB component is shown with the dashed green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' APPENDIX D: ADDITIONAL TABLES All the SRCCs obtained between the several parameters used to model the free-free, AME, and thermal dust components are dis- played in Tables D1–D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' In additional to several plots used to study correlations between some of the parameters, these tables have been MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 27 Source Name 𝐴AME 𝜎AME 𝜈AME 𝑊AME 𝑆28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 resid 𝑆28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 resid 𝑆100𝜇m 𝑓 UCHII max [15𝐺𝐻 𝑧] 𝜒2 red [Jy] [GHz] [log[GHz]] [Jy] G010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 G010.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='0 G011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='9 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 G353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='79 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='7 Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SEDs Fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' (PUL): prior upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' used to systematically identify the parameters showing the most meaningful correlation factors, and guide our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) 28 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variable 1 Variable 2 SRCC SRCC SRCC selected sample AME significant AME semi-significant 𝐴AME [Jy] 𝜈AME [GHz] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 𝐴AME [Jy] 𝑊AME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='39± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 𝐴AME [Jy] 𝐴AME/𝜏250 [Jy] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 𝐴AME [Jy] ℜAME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='77± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='91± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 𝜈AME [GHz] 𝑊AME −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12± 0.' 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𝜈AME [GHz] ℜAME −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='33± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 𝑊AME 𝐴AME/𝜏250 [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝑊AME ℜAME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='75± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='67± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 𝐴AME/𝜏250 [Jy] ℜAME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='43± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spearman rank correlation coefficients (SRCCs) between AME and AME parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variable 1 Variable 2 SRCC SRCC SRCC selected sample AME significant AME semi-significant 𝐴AME [Jy] Ttd or 𝐺0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='33± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 𝐴AME [Jy] 𝛽td 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='49± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 𝐴AME [Jy] 𝜏250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='81± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='53± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 𝐴AME [Jy] 𝑆TD,peak [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='82± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='91± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 𝐴AME [Jy] 𝜈TD,peak [GHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝐴AME [Jy] ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 𝜈AME [GHz] Ttd or 𝐺0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝜈AME [GHz] 𝛽td −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='35± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='44± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 𝜈AME [GHz] 𝜏250 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 𝜈AME [GHz] ℜtd𝑥10−4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 𝑊AME Ttd or 𝐺0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝑊AME 𝛽td −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 𝑊AME 𝜏250 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='00± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 𝑊AME 𝑆TD,peak [Jy] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝑊AME 𝜈TD,peak [GHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} 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+page_content='87± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 𝐴AME/𝜏250 [Jy] 𝛽td −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 𝐴AME/𝜏250 [Jy] 𝜏250 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 𝐴AME/𝜏250 [Jy] 𝑆TD,peak [Jy] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='41± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='54± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 𝐴AME/𝜏250 [Jy] 𝜈TD,peak [GHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='54± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='40± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝐴AME/𝜏250 [Jy] ℜtd𝑥10−4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='48± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 ℜAME Ttd or 𝐺0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='45± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 ℜAME 𝛽td 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='31± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='36± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 ℜAME 𝜏250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='54± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 ℜAME 𝑆TD,peak [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='72± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 ℜAME 𝜈TD,peak [GHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 ℜAME ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='70± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='73± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 Table D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spearman rank correlation coefficients (SRCCs) between AME and Thermal Dust parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variable 1 Variable 2 SRCC SRCC SRCC selected sample AME significant free-free semi-significant 𝐴AME [Jy] EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='55± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 𝜈AME [GHz] EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='27± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 𝑊AME EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='84± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='02± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 𝐴AME/𝜏250 [Jy] EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='46± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 ℜAME EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='59± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='42± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 Table D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spearman rank correlation coefficients (SRCCs) between AME and free-free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 29 Variable 1 Variable 2 SRCC SRCC SRCC selected sample AME significant free-free semi-significant Ttd or 𝐺0 EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='49± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 𝛽td EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='28± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='41± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 𝜏250 EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='43± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='51± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 𝑆TD,peak [Jy] EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='50± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='64± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 𝜈TD,peak [GHz] EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='45± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='62± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='29± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 ℜtd𝑥10−4 EM [cm−6/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='66± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='56± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 Table D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spearman rank correlation coefficients (SRCCs) between thermal dust and free-free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variable 1 Variable 2 SRCC SRCC SRCC selected sample AME significant AME semi-significant Ttd or 𝐺0 𝛽td −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='44± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='47± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 Ttd or 𝐺0 𝜏250 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='52± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='57± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='58± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 Ttd or 𝐺0 𝑆TD,peak [Jy] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='32± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 Ttd or 𝐺0 𝜈TD,peak [GHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='92± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='97± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 Ttd or 𝐺0 ℜtd𝑥10−4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='25± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 𝛽td 𝜏250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='60± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='34± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 𝛽td 𝑆TD,peak [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='54± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='68± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 𝛽td 𝜈TD,peak [GHz] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='31± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 𝛽td ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='50± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='03± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='65± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='09 𝜏250 𝑆TD,peak [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='90± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='69± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='92± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='04 𝜏250 𝜈TD,peak [GHz] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='31± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='51± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='29± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='14 𝜏250 ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='85± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='61± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 𝑆TD,peak [Jy] 𝜈TD,peak [GHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='18± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 𝑆TD,peak [Jy] ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='99± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='99± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='99± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='05 𝜈TD,peak [GHz] ℜtd𝑥10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='17± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='30± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='06± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content='15 Table D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Spearman rank correlation coefficients (SRCCs) between the thermal dust parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variations of the AME peak frequency with the relative strength of the ISRF, 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Variations of the AME parabola width parameter with the rela- tive strength of the ISRF, 𝐺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Symbols and colours definition are the same as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) 30 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' SED of the sample of regions discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' See caption of Figure 4 for symbols, lines and colours conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 31 Figure C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Same as Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) 32 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Same as Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 33 Figure C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Same as Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) 34 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Same as Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) QUIJOTE-MFI wide-survey Galactic AME sources 35 Figure C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Same as Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023) 36 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Figure C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' Same as Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} +page_content=' MNRAS 000, 1–36 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E4T4oBgHgl3EQfgQ3v/content/2301.05116v1.pdf'} diff --git a/ItFKT4oBgHgl3EQfdy7h/content/tmp_files/2301.11822v1.pdf.txt b/ItFKT4oBgHgl3EQfdy7h/content/tmp_files/2301.11822v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c9d0ed4e6a0773130e6a7f8ada9177f06c3183e --- /dev/null +++ b/ItFKT4oBgHgl3EQfdy7h/content/tmp_files/2301.11822v1.pdf.txt @@ -0,0 +1,830 @@ +arXiv:2301.11822v1 [math.AP] 27 Jan 2023 +LAGRANGIAN STABILITY FOR A SYSTEM OF NON-LOCAL +CONTINUITY EQUATIONS UNDER OSGOOD CONDITION +MARCO INVERSI AND GIORGIO STEFANI +Abstract. We extend known existence and uniqueness results of weak measure solu- +tions for systems of non-local continuity equations beyond the usual Lipschitz regularity. +Existence of weak measure solutions holds for uniformly continuous vector fields and +convolution kernels, while uniqueness follows from a Lagrangian stability estimate under +an additional Osgood condition. +1. Introduction +1.1. Statement of the problem. For fixed T ∈ (0, +∞) and k, d ∈ N, we consider the +system of non-local continuity equations + + + +∂t̺i + div (̺i V i(t, x, ̺ ∗ ηi)) += +0, +t ∈ (0, T), x ∈ Rd, +̺i(0) += +¯̺i, +i = 1, . . . , k, +(1.1) +where the unknown ̺ = (̺1, . . . , ̺k) ∈ L∞([0, T]; M+(Rd)k) is a time-dependent k-vector +of non-negative Borel measures on Rd, the initial datum ¯̺ = (¯̺1, . . . , ¯̺k) ∈ M+(Rd)k is a +k-vector of non-negative Borel measures, +V = (V 1, . . . , V k) ∈ L∞([0, T]; Cb(Rd × Rk; Rd)k) +(1.2) +is a uniformly-in-time bounded continuous k-vector field and +ηi = (ηi,1, . . . , ηi,k) ∈ L∞([0, T]; Cb(Rd; Rk)) +(1.3) +is a uniformly-in-time bounded continuous k-vector of convolution kernels, with the con- +volution ̺ ∗ ηi = (̺1 ∗ ηi,1, . . . , ̺k ∗ ηi,k) occurring in the space variable only. +In the +entire paper, we frequently consider the 1-norm (i.e., the sum of the absolute values of +the entries) on both vectors and matrices. In particular, |̺| = |̺1| + · · · + |̺k| and thus +∥̺∥M = ∥̺1∥M+· · ·+∥̺k∥M for all ̺ ∈ M(Rd). When considering other norms, constants +depending on d and/or k may be dropped without notice. +Date: January 30, 2023. +2020 Mathematics Subject Classification. Primary 35L65. Secondary 34A30. +Key words and phrases. Non-local continuity equation, Lagrangian stability, Osgood condition. +Acknowledgements. The authors thank Gianluca Crippa for several useful comments on a preliminary +version of this work. The first-named author is partially funded by the SNF grant FLUTURA: Fluids, +Turbulence, Advection No. 212573. The second-named author is member of the Istituto Nazionale di +Alta Matematica (INdAM), Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Appli- +cazioni (GNAMPA), is partially supported by the INdAM–GNAMPA 2022 Project Analisi geometrica in +strutture subriemanniane, codice CUP_E55F22000270001, and has received funding from the European +Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program +(grant agreement No. 945655). +1 + +2 +M. INVERSI AND G. STEFANI +Solutions of the system (1.1) are understood in the usual distributional sense, which is +well-set thanks to (1.2) and (1.3). +Definition 1.1 (Weak solution). We say that ̺ ∈ L∞([0, T]; M+(Rd)k) is a weak solution +of the system (1.1) starting from the initial datum ¯̺ ∈ M+(Rd)k if +� T +0 +� +Rd +� +∂tϕ + V i(t, x, ̺ ∗ ηi) · ∇ϕ +� +d̺i(t, x) dt + +� +Rd ϕ(0, x) d¯̺i(x) = 0 +(1.4) +for each i = 1, . . ., k and any ϕ ∈ C∞ +c ([0, T) × Rd). +Any weak solution in the sense of Definition 1.1 admits a weakly continuous repre- +sentative in duality with the space C0(Rd) of continuous functions vanishing at infinity, +see [2, Lem. 8.1.2] and [1, 14]. So, from now on, we restrict our attention to weakly- +continuous weak solutions ̺ ∈ C([0, T]; M+(Rd)k−w∗) only. +The system (1.1) is used in several physical situations—for instance, pedestrian traffic, +sedimentation models and supply chains—to describe the time evolution of the density +of a vectorial quantity (e.g., pedestrians or particles), possibly concentrating in some +points or along hypersurfaces. Far from being complete, we refer the reader for example +to [4, 10–13, 16, 18, 21, 24, 25] for a panoramic on the related literature. Because of the +physical relevance of the system (1.1), here we deal with non-negative solutions only. +The system (1.1) can be also interpreted in the sense of the Control Theory. Indeed, +the convolution kernel η can be viewed as a non-local control for the (non-linear) PDE +in (1.1). Therefore, assuming V is fixed for simplicity, any stability result for the solutions +of the system (1.1) in terms of the convolution kernel η yields a continuous dependence +of the curve t �→ ̺t[η] solving (1.1) in terms of the control η. +The well-posedness of the system (1.1) was established in [14], provided that V and η +are bounded and Lipschitz continuous uniformly in time, namely, +V ∈ L∞([0, T]; Lipb(Rd × Rk; Rd)k) +and +η ∈ L∞([0, T]; Lipb(Rd; Rk)k). +(1.5) +The crucial ingredient of [14] is a stability estimate (in terms of the 1-Wasserstein distance +between two solutions, see [14, Prop. 4.2]) which, in turn, allows to obtain existence and +uniqueness of the solution of (1.1) via a fix point argument. The idea of exploiting the +Lipschitz regularity to gain stability of trajectories has been later applied to several other +related problems, see [5,7,9,17,23] and the references therein for instance. +1.2. Main results. The aim of the present note is to prove the well-posedness of the +system (1.1) under less restrictive assumptions than (1.5), that is, to extend the existence +and uniqueness result of [14] beyond the Lipschitz regularity. Our interest is motivated +by some recent works [1,3,6,15,19,20] dealing with non-Lipschitz velocity fields. +Our first main result deals with the existence of weak solutions of the system (1.1), in +the spirit of the celebrated Peano’s Theorem. To this aim, we consider the following struc- +tural hypotheses (where modulus of continuity means a non-decreasing concave function +vanishing continuously at zero): +(V ) The vector field V ∈ L∞([0, T]; Cb(Rd × Rk; Rd)k) satisfies +ess sup +t∈[0,T] +|V (t, x, u) − V (t, y, v)| ≤ ωV (|x − y| + |u − v|) +∀x, y ∈ Rd, u, v ∈ Rk, +(1.6) +where ωV : [0, +∞) → [0, +∞) is a modulus of continuity. + +LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION +3 +(η) For each i = 1, . . ., k, the convolution kernel ηi ∈ L∞([0, T]; C0(Rd; Rk)) satisfies +ess sup +t∈[0,T] +|ηi(t, x) − ηi(t, y)| ≤ ωη(|x − y|) +∀x, y ∈ Rd, +(1.7) +where ωη : [0, +∞) → [0, +∞) is a modulus of continuity. +Theorem 1.2 (Existence). If (V ) and (η) hold, then the system (1.1) admits a weak +solution starting from any given initial datum in M+(Rd)k. +To prove Theorem 1.2, we first consider the smoothed functions Vε and ηε and obtain +a weak solution ̺ε of the corresponding system (1.1) for all ε > 0 in virtue of the main +result of [14]. Then, we pass to the limit as ε → 0+ showing that ̺ε (weakly) converges +to a weak solution of the system (1.1). The needed a priori compactness is achieved via +an Aubin–Lion-type Lemma which is inspired by [15, Th. A.1]. +In order to achieve uniqueness of weak solutions of the system (1.1), we need to impose +a further Osgood condition on the composition of the two moduli of continuity of V and η: +(O) for each λ > 0, it holds +� +0+ +dr +ωV (r + λ ωη(r)) = +∞. +For example, condition (O) is satisfied as soon as ωV ◦ ωη is a log-linear function, such +as r| log r|, r log | log r| and similar, with r > 0 sufficiently small. +Our uniqueness result deals with Lagrangian weak solutions of the system (1.1). +Definition 1.3 (Lagrangian weak solution). A weak solution ̺ ∈ L∞([0, T]; M+(Rd)k) +of the system (1.1) starting from the initial datum ¯̺ ∈ M+(Rd)k is Lagrangian if +̺i(t, ·) = Xi(t, ·)#¯̺i, +i = 1, . . ., k, +where Xi : [0, T] × Rd → Rd is the (classical) solution of the ODE + + + + + +d +dt Xi(t, x) += +V i� +t, Xi(t, x), ̺ ∗ ηi(t, Xi(t, x)) +� +, +t ∈ (0, T), x ∈ Rd, +Xi(0, x) += +x, +x ∈ Rd. +(1.8) +Thanks to Proposition 1.4 below, the Osgood condition in (O) ensures the well-posed- +ness of the ODE in (1.8). +Proposition 1.4 (Associated vector field). Let assumptions (V ) and (η) be in force. If +̺ ∈ C([0, T]; M+(Rd)k−w∗) is a weak solution of the system (1.1) starting from the initial +datum ¯̺ ∈ M+(Rd)k, then the vector field +bi +V,η,̺(t, x) = V i� +t, x, ̺ ∗ ηi(t, x) +� +, +t ∈ [0, T], x ∈ Rd, i = 1, . . ., k, +(1.9) +appearing in (1.8) satisfies b ∈ L∞([0, T]; Cb(Rd; Rd)k) with +ess sup +t∈[0,T] +|bV,η,̺(t, x) − bV,η,̺(t, y)| ≲ ωV +� +|x − y| + ∥¯̺∥M ωη(|x − y|) +� +∀x, y ∈ Rd. +With the above notation, our main uniqueness result reads as follows. +Theorem 1.5 (Uniqueness). If (V ), (η) and (O) hold, then (1.1) admits a unique La- +grangian weak solution starting from any given initial datum in M+(Rd)k. + +4 +M. INVERSI AND G. STEFANI +The word “Lagrangian” in Theorem 1.5 can be dropped, since any weak solution of the +system (1.1) is in fact Lagrangian because of [1, Th. 1] (also see [8]) and of Proposition 1.4. +However, this regularity result is not at all elementary, so we prefer to state Theorem 1.5 +for Lagrangian solutions only in order to emphasize what is possible to achieve just relying +on our elementary approach. +The strategy of [14] exploits the linearity of ωη in an essential way. Indeed, the au- +thors need the Lipschitz continuity of η in order to recover the 1-Wasserstein distance +between two weak solutions of (1.1) in terms of its dual Kantorovich–Rubinstein formu- +lation (see [14, Lem. 4.1]). We do not know if the strategy of [14] can be adapted to deal +with a more general modulus of continuity ωη. +To overcome this issue, we adopt a different point of view, which is inspired by the +elementary uniqueness result achieved in the recent work [15]. Instead of controlling the +1-Wasserstein distance between two weak solutions of the system (1.1), we exploit their +Lagrangian property to quantitatively estimate the difference between the two associated +ODE flows, thus providing a Lagrangian stability of weak solutions from which Theo- +rem 1.5 immediately follows. +Theorem 1.6 (Lagrangian stability). Let V, U ∈ L∞([0, T]; Cb(Rd×Rk; Rd)k) satisfy (1.6) +with the same modulus of continuity ωV and let η, ν ∈ L∞([0, T]; C0(Rd; Rk)k) satisfy (1.7) +with the same same modulus of continuity ωη. Let ̺, σ ∈ C([0, T]; M+(Rd)k −w∗) be +two weak solutions of the system (1.1) starting from the initial data ¯̺, ¯σ ∈ M+(Rd)k, +with vector fields V, U and convolution kernels η, ν, respectively. Assume that ̺, σ are +Lagrangian, namely, ̺ = X(t, ·)#¯̺ and σ = Y (t, ·)#¯σ for t ∈ [0, T], where X, Y are the +flows solving the corresponding ODEs in (1.8). Then, there exists a modulus of continuity +Ω: [0, +∞) → [0, +∞), only depending on +T, ∥¯̺∥M, ∥¯σ∥M, ∥η∥L∞(C), ∥ν∥L∞(C), ωV , ωη, +such that +sup +t∈[0,T] +∥X(t, ·) − Y (t, ·)∥L∞ ≤ Ω +� +∥¯̺ − ¯σ∥M + ∥V − U∥L∞(C) + ∥ν − η∥L∞(C) +� +. +(1.10) +The modulus of continuity Ω in Theorem 1.6 can be explicitly computed as soon as one +can invert the integral function +GV,η,λ(r) = +� r +r0 +ds +ωV (s + λ ωη(s)), +r ≥ 0, r0 > 0, +(1.11) +naturally brought by the Osgood condition assumed in (O). In fact, the stability esti- +mate (1.10) follows by simply differentiating a localized integral distance between the flows +with respect to the time variable, and then applying the classical Bihari–LaSalle inequality +(see [22, Th. 2.3.1] for instance) with Osgood modulus of continuity r → ωV (r + λ ωη(r)), +for some specific parameter λ > 0 depending on ∥¯̺∥M and ∥¯σ∥M. +Theorem 1.6 clearly rephrases as a stability result of the flow of the ODE in (1.8). From +the point of view of Control Theory, the stability estimate in (1.10) yields a continuous +dependence of the (Lagrangian) solutions of the system (1.1), i.e., of the flows induced +by the corresponding ODE in (1.8), in terms of the (non-local) control given by the +convolution kernel, as well as of the velocity vector field and of the initial datum. + +LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION +5 +2. Proofs +2.1. Existence of weak solutions. To prove Theorem 1.2, we need some preliminary +results. We begin with an Aubin–Lions-type Lemma, which is inspired by [15, Th. A.1]. +Lemma 2.1 (Compactness). Let (̺n)n∈N ⊂ C([0, T]; M(Rd)−w∗) be such that +sup +n∈N ∥̺n∥L∞(M) < +∞. +(2.1) +Assume that, for each ϕ ∈ C∞ +c (Rd), the functions Fn[ϕ]: [0, T] → R, given by +Fn[ϕ](t) = +� +Rd ϕ d̺n(t, ·), +t ∈ [0, T], +are uniformly equicontinuous on [0, T], that is, +∀ε > 0 ∃δ > 0 : s, t ∈ [0, T], |s − t| < δ =⇒ sup +n∈N |Fn[ϕ](s) − Fn[ϕ](t)| < ε. +(2.2) +Then, there exist a subsequence (̺nk)k∈N and ̺ ∈ C([0, T], M(Rd)−w∗) such that +lim +k→+∞ sup +t∈[0,T] +���� +� +Rd ϕ d̺nk(t, ·) − +� +Rd ϕ d̺(t, ·) +���� = 0 +(2.3) +for all ϕ ∈ C0(Rd). +Proof. Let D ⊂ Cc(Rd) be a countable and dense set in C0(Rd). +In virtue of (2.1) +and (2.2), for each ϕ ∈ D the sequence (Fn[ϕ])n∈N is equibounded and equicontinuous +on [0, T]. By Ascoli–Arzelà Theorem and a standard diagonal argument, we can find a +subsequence (nk)k∈N such that, for each ϕ ∈ D, the sequence (Fnk[ϕ])k∈N is uniformly +convergent to some F[ϕ] ∈ C([0, T]), with +∥F[ϕ]∥L∞([0,T]) ≤ ∥ϕ∥L∞ sup +n∈N ∥̺n∥L∞(M). +(2.4) +By construction, the function ϕ �→ F[ϕ](t) is a continuous linear functional on D for +each t ∈ [0, T]. Thus, for each fixed t ∈ [0, T], we can extend the map ϕ �→ F[ϕ](t) to a +linear and continuous functional on C0(Rd) for which we keep the same notation. A plain +approximation argument readily proves that, for each ϕ ∈ C0(Rd), the map t �→ F[ϕ](t) +is continuous on [0, T] and satisfies (2.4). By Riesz’s Representation Theorem, for each +t ∈ [0, T] there exists a finite Borel measure ̺(t, ·) ∈ M(Rd) such that +F[ϕ](t) = +� +Rd ϕ d̺(t, ·) +for all ϕ ∈ C0(Rd), +so that ̺ ∈ C([0, T]; M(Rd)−w∗). Finally, in virtue of (2.1) and (2.4), for ϕ ∈ C0(Rd) +and ψ ∈ D, we can estimate +sup +t∈[0,T] +|Fnk[ϕ](t) − F[ϕ](t)| ≤ sup +t∈[0,T] +|Fnk[ψ](t) − F[ψ](t)| + 2 ∥ψ − ϕ∥L∞ sup +n∈N ∥̺n∥L∞(M) +and the desired (2.3) readily follows. +□ +In order to exploit Lemma 2.1, we need the following mass preservation property for +weak solutions of the system (1.1). + +6 +M. INVERSI AND G. STEFANI +Lemma 2.2 (Mass preservation). Let V and η be as in (1.2) and (1.3), respectively. If +̺ ∈ C([0, T]; M+(Rd)k−w∗) is a weak solution of the system (1.1) starting from the initial +datum ̺ ∈ M+(Rd)k, then +∥̺i(t, ·)∥M = ∥¯̺i∥M +(2.5) +for t ∈ [0, T] and i = 1, . . ., k. +Proof. Let i ∈ {1, . . ., k} be fixed. +By applying (1.4) to the test function ϕ(t, x) = +α(t) β(x), (t, x) ∈ [0, T] × Rd, where α ∈ C∞ +c ([0, T)) and β ∈ C∞ +c (Rd), we get +� T +0 +� +Rd +� +α′β + α V i(t, x, ̺ ∗ ηi) · ∇β +� +d̺i(t, ·) dt + α(0) +� +Rd β(x) d¯̺i = 0. +Since α ∈ C∞ +c ([0, T)) is arbitrary and ̺ ∈ C([0, T]; M+(Rd)k−w∗), we infer that +t �→ +� +Rd β d̺i(t, ·) ∈ AC1,1([0, T]; R) +(2.6) +with +� +Rd β d̺i(t, ·) = +� +Rd β d¯̺i + +� t +0 +� +Rd V i(s, ·, ̺ ∗ ηi) · ∇β d̺i(s, ·) ds +(2.7) +for all t ∈ [0, T]. Now let t ∈ [0, T] be fixed. We let (βR)R>0 ⊂ C∞ +c (Rd) be such that +βR ≥ 0, +supp βR ⊂ B2R, +βR = 1 on BR, +∥∇βR∥L∞ ≤ 2 +R +for all R > 0. By the Monotone Convergence Theorem, we infer that +lim +R→+∞ +� +Rd βR d̺i(t, ·) = ∥̺i(t, ·)∥M +as well as +lim +R→+∞ +� +Rd βR d¯̺i = ∥¯̺i∥M. +Since +���� +� t +0 +� +Rd V i(s, ·, ̺ ∗ ηi) · ∇βR d̺i(s, ·) ds +���� ≤ 2 +R ∥̺i∥L∞(M) ∥V i∥L∞(C) +for all R > 0, we get (2.5) by applying (2.7) to βR and passing to the limit as R → +∞. +□ +We are ready to prove our existence result. +Proof of Theorem 1.2. Let (ℓε)ε>0 ⊂ C∞ +c (Rd+k) and (ε)ε>0 ∈ C∞ +c (Rd) be two families of +standard non-negative mollifiers and set +V i,j +ε (t, ·) = V i,j(t, ·) ∗ ℓε, +ηi,j +ε = ηi,j(t, ·) ∗ ε, +where in both cases the (component-wise) convolution occur in the spatial variables only. +Since Vε and ηε clearly satisfy the Lipschitz property (1.5) for each ε > 0, by [14, Th. 1.1] +there exists a weak solution +̺ε ∈ C([0, T], M+(Rd)k−w∗) +of the system (1.1) starting from the initial datum ¯̺ ∈ M+(Rd)k, so that +� T +0 +� +Rd +� +∂tϕ + V i +ε (t, ·, ̺ε ∗ ηi +ε) · ∇ϕ +� +d̺i +ε(t, ·) dt + +� +Rd ϕ(0, ·) d¯̺i = 0 +(2.8) +for each i = 1, . . ., k and ε > 0 and ϕ ∈ C∞ +c ([0, T) × Rd). + +LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION +7 +Now let i ∈ {1, . . ., k} be fixed. We claim that (any sequence in) the family (̺i +ε)ε>0 +satisfies the assumptions (2.1) and (2.2) of Lemma 2.1. Indeed, from Lemma 2.2 we get +∥̺i +ε(t, ·)∥M = ∥¯̺i∥M +(2.9) +for all t ∈ [0, T] and ε > 0, from which (2.1) immediately follows. To prove (2.2), we +simply argue as in the proof of Lemma 2.2. Recalling (2.6) and (2.7), we easily recognize +that the time derivative of the function +Fε[β](t) = +� +Rd β(·) d̺i +ε(t, ·), +t ∈ [0, T], +(2.10) +is bounded by +���� +� +Rd V i +ε (t, x, ̺ε ∗ ηi +ε) · ∇β d̺i +ε(t, x) +���� ≤ ∥V i∥L∞(C) ∥∇β∥L∞ ∥¯̺i∥M +for a.e. t ∈ [0, T] and for each ε > 0. In particular, the family (Fε[β])ε>0 in (2.10) is +equi-Lipschitz and thus satisfies (2.2). +Therefore, by Lemma 2.1, we find a sequence +(̺εn)n∈N ⊂ C([0, T]; M+(Rd)k−w∗) and ̺ ∈ C([0, T]; M+(Rd)k−w∗) such that +lim +n→+∞ sup +t∈[0,T] +���� +� +Rd β d̺εn(t, ·) − +� +Rd β d̺(t, ·) +���� = 0 +(2.11) +for all β ∈ C0(Rd). +To conclude, we just need to prove that ̺ is a weak solution of (1.1) starting from the +initial datum ¯̺. We do so by passing to the limit in (2.8) along (εn)n∈N as n → +∞ for +each given ϕ ∈ C∞ +c ([0, +∞) × Rd). Indeed, on the one side, since +lim +n→+∞ +� +Rd ∂tϕ d̺i +εn(t, ·) = +� +Rd ∂tϕ d̺i(t, ·) +because of (2.11) and +���� +� +Rd ∂tϕ d̺i +εn(t, ·) +���� ≤ ∥∂tϕ∥L∞ ∥¯̺i∥M +because of (2.9), for all t ∈ [0, T], by the Dominated Convergence Theorem we infer that +lim +n→+∞ +� T +0 +� +Rd ∂tϕ d̺i +εn(t, ·) dt = +� T +0 +� +Rd ∂tϕ d̺i(t, ·) dt. +(2.12) +On the other side, since ηi(t, ·) ∈ C0(Rd) in virtue of the assumption (η), we have that +ηi +εn(t, ·) → ηi(t, ·) in C0(Rd) as n → +∞, so that +lim +n→+∞ +� +̺εn(t, ·) ∗ ηi +εn(t, ·) +� +(x) = lim +n→+∞ +� +Rd ηi +εn(t, x − y) d̺εn(t, y) += +� +Rd ηi(t, x − y) d̺(t, y) = +� +̺(t, ·) ∗ ηi(t, ·) +� +(x) +(2.13) +for each x ∈ Rd and all t ∈ [0, T] as a weak-strong convergent pair, due to (2.11). +Moreover, again in virtue of (2.9) and (η), we can estimate +∥̺εn(t, ·) ∗ ηi +εn(t, ·)∥ ≤ ∥̺i∥M ∥ηi∥L∞(C) +and +��� +� +̺εn(t, ·) ∗ ηi +εn(t, ·) +� +(x) − +� +̺εn(t, ·) ∗ ηi +εn(t, ·) +� +(y) +��� +≤ +� +Rd +���ηi +εn(t, x − ·) − ηi +εn(t, y − ·) +��� d̺εn(t, ·) ≤ ωη(|x − y|) ∥̺i∥M + +8 +M. INVERSI AND G. STEFANI +for all n ∈ N and t ∈ [0, T]. By Arzelà–Ascoli’s Theorem, we thus get that the pointwise +convergence in (2.13) must be uniform on compact sets in Rd, uniformly in t ∈ [0, T]. An +analogous argument relying on the assumption (V ) proves that also V i +εn(t, ·) → V i(t, ·) as +n → +∞ uniformly on compact sets in Rd, uniformly in t ∈ [0, T]. Again by (2.11), by +weak-strong convergence and by the Dominated Convergence Theorem, we hence get +lim +n→+∞ +� T +0 +� +Rd V i +εn(t, ·, ̺εn ∗ ηi +εn) · ∇ϕ d̺i +εn(t, ·) dt = +� T +0 +� +Rd V i(t, ·, ̺ ∗ ηi) · ∇ϕ d̺i(t, ·) dt. +(2.14) +Thus, the conclusion follows by combining (2.12) with (2.14). +□ +2.2. Lagrangian stability. We deal with the Lagrangian stability of weak solutions. We +begin with the proof of Proposition 1.4. +Proof of Proposition 1.4. Let t ∈ [0, T] be fixed. Given x, y ∈ Rd and i ∈ {1, . . ., k}, in +virtue of assumption (η) and of Lemma 2.2, we can estimate +|̺ ∗ ηi(t, x) − ̺ ∗ ηi(t, y)| ≤ +k +� +j=1 +� +Rd |ηi,j(t, x − z) − ηi,j(t, y − z)| d̺j(t, z) +≤ +k +� +j=1 +� +Rd ωη(|x − y|) d̺k(t, z) = ∥̺(t, ·)∥M ωη(|x − y|) += ∥¯̺∥M ωη(|x − y|). +Thus, thanks to assumption (V ), we get that +���V i� +t, x, ̺ ∗ ηi(t, x) +� +− V i� +t, y, ̺ ∗ ηi(t, y) +���� ≤ ωV +� +|x − y| + |̺ ∗ ηi(t, x) − ̺ ∗ ηi(t, y)| +� +≤ ωV +� +|x − y| + ∥¯̺∥M ωη(|x − y|) +� +and the conclusion immediately follows. +□ +We conclude our paper with the proof of Theorem 1.6. +Proof of Theorem 1.6. Let V, U, η, ν, ¯̺, ¯σ, X, Y and ̺, σ be as in the statement. +Fix +ζ ∈ C(Rd) with ζ ≥ 0 and +� +Rd ζ(x) dx = 1. Letting µ ∈ M+(Rd) be defined by µ = +|¯̺| + |¯σ| + ζ L d, we consider the quantity +Qζ(t) = +k +� +i=1 +− +� +Rd |Xi(t, ·) − Y i(t, ·)| dµ +for all t ∈ [0, T]. Note that t �→ Qζ(t) is well defined and Lipschitz, with Qζ(0) = 0 and +|Qζ(s) − Qζ(t)| ≤ k (∥U∥L∞(C) + ∥V ∥L∞(C)) |s − t| +for all s, t ∈ [0, T]. Therefore, for a.e. t ∈ [0, T], we can write +Q′ +ζ(t) ≤ +k +� +i=1 +− +� +Rd |V i(t, Xi, ̺ ∗ ηi(t, Xi)) − Ui(t, Y i, σ ∗ νi(t, Y i))| dµ +≤ +k +� +i=1 +(1)i + (2)i + (3)i + (4)i, + +LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION +9 +where (dropping the variables of X and Y for notational convenience) +(1)i = − +� +Rd |V i(t, Xi, ̺ ∗ ηi(t, Xi)) − V i(t, Y i, ̺ ∗ ηi(t, Y i))| dµ, +(2)i = − +� +Rd |V i(t, Y i, ̺ ∗ ηi(t, Y i)) − V i(t, Y i, σ ∗ ηi(t, Y i))| dµ, +(3)i = − +� +Rd |V i(t, Y i, σ ∗ ηi(t, Y i)) − V i(t, Y i, σ ∗ νi(t, Y i))| dµ, +(4)i = − +� +Rd |V i(t, Y i, σ ∗ νi(t, Y i)) − Ui(t, Y i, σ ∗ νi(t, Y i))| dµ. +We now estimate each term separately at a given t ∈ [0, T]. By Proposition 1.4 and +Jensen’s inequality, we can easily estimate the first term as +(1)i ≤ − +� +Rd ωV +� +|Xi − Y i| + ∥¯̺∥M ωη(|Xi − Y i|) +� +dµ +≤ ωV +� +− +� +Rd |Xi − Y i| dµ + ∥¯̺∥M ωη +� +− +� +Rd |Xi − Y i| dµ +�� +≤ ωV +� +Qζ(t) + ∥µ∥M ωη(Qζ(t)) +� +. +Concerning the second term, since +|(̺ − σ) ∗ ηi(t, x)| = +���� +� +Rd ηi(t, x − y) d(X#¯̺(y) − Y#¯σ(y)) +���� +≤ +k +� +j=1 +� +Rd |ηi,j(t, x − Xj) − ηi,j(t, x − Y j)| d¯̺j + +� +Rd |ηi,j(t, x − Y j)| d|¯̺j − ¯σj| +≤ +k +� +j=1 +� +Rd ωη(|Xj − Y j|) d¯̺j + ∥η∥L∞(C)∥¯̺j − ¯σj∥M +≤ +� +Rd ωη + + +k +� +j=1 +|Xj − Y j| + + d|¯̺| + ∥η∥L∞(C)∥¯̺ − ¯σ∥M +for all x ∈ Rd, again by Jensen’s inequality we get +(2)i ≤ − +� +Rd ωV +� +|(̺ − σ) ∗ ηi(t, Y i)| +� +dµ +≤ ωV +�� +Rd ωη +� k +� +i=1 +|Xi − Y i| +� +d|¯̺| + ∥η∥L∞(C)∥¯̺ − ¯σ∥M +� +≤ ωV +� +∥µ∥M ωη(Qζ(t)) + ∥η∥L∞(C)∥¯̺ − ¯σ∥M +� +≤ ωV +� +Qζ(t) + ∥µ∥M ωη(Qζ(t)) +� ++ ωV +� +∥η∥L∞(C)∥¯̺ − ¯σ∥M +� +. +The last two terms can be trivially estimated as +(3)i ≤ ωV +� +∥σ∥L∞(M) ∥η − ν∥L∞(C) +� += ωV +� +∥¯σ∥M ∥η − ν∥L∞(C) +� +≤ ωV +� +∥µ∥M ∥η − ν∥L∞(C) +� +thanks to Lemma 2.2, and +(4)i ≤ ∥V − U∥L∞(C). + +10 +M. INVERSI AND G. STEFANI +Putting everything altogether, we conclude that +Q′ +ζ(t) ≲ ωV +� +Qζ(t) + λ ωη(Qζ(t)) +� ++ M, +where λ = ∥̺∥M + ∥σ∥M + 1 and +M = ωV +� +∥η∥L∞(C)∥¯̺ − ¯σ∥M +� ++ ωV +� +λ ∥η − ν∥L∞(C)) +� ++ ∥V − U∥L∞(C). +At this point, we just need to recall the Osgood condition assumed in (O) and the integral +function in (1.11). Indeed, by the classical Bihari–LaSalle inequality (see [22, Th. 2.3.1] +for instance), we find a modulus of continuity Ω: [0, +∞) → [0, +∞), only depending on +T, ∥¯̺∥M, ∥¯σ∥M, ∥η∥L∞(C), ∥ν∥L∞(C), ωV , ωη, +such that +sup +t∈[0,T] +Qζ(t) ≤ Ω +� +∥¯̺ − ¯σ∥M + ∥V − U∥L∞(C) + ∥ν − η∥L∞(C) +� +. +(2.15) +We remark that Ω is independent of ζ, as long as we choose ζ ≥ 0 and ∥ζ∥L1 = 1. To +conclude, we choose a family (ζx0,ε)ε>0 of standard mollifiers around x0 ∈ Rd. Since the +flows X(t, ·), Y (t, ·) are continuous maps, we deduce that +lim +ε→0+ Qζx0,ε(t) = |X(t, x0) − Y (t, x0)|. +(2.16) +Thus, (1.10) follows from (2.15) and (2.16) and the proof is complete. +□ +References +[1] L. Ambrosio and P. Bernard, Uniqueness of signed measures solving the continuity equation for +Osgood vector fields, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19 (2008), no. 3, 237–245. +[2] L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability +measures, Second, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. +[3] L. Ambrosio, S. Nicolussi Golo, and F. Serra Cassano, Classical flows of vector fields with exponential +or sub-exponential summability (2022). Preprint, available at arXiv:2208.01381. +[4] D. Armbruster, D. Marthaler, C. Ringhofer, K. Kempf, and T. Jo, A continuum model for a re- +entrant factory, Oper. Res. 54 (2006), no. 5, 933–950. +[5] A. Bressan and W. 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Stefani, An elementary proof of existence and uniqueness for the Euler flow in +localized Yudovich spaces (2021). Preprint, available at arXiv:2110.15648v2. +[16] M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, +Nonlinearity 26 (2013), no. 10, 2777–2808. +[17] J. H. M. Evers, S. C. Hille, and A. Muntean, Measure-valued mass evolution problems with flux +boundary conditions and solution-dependent velocities, SIAM J. Math. Anal. 48 (2016), no. 3, 1929– +1953. +[18] A. Keimer and L. Pflug, Existence, uniqueness and regularity results on nonlocal balance laws, J. +Differential Equations 263 (2017), no. 7, 4023–4069. +[19] J. La, Regularity and drift by Osgood vector fields (2022). Preprint, available at arXiv:2206.14237v1. +[20] H. Li and D. Luo, A unified treatment for ODEs under Osgood and Sobolev type conditions, Bull. +Sci. Math. 139 (2015), no. 1, 114–133. +[21] A. Mackey, T. Kolokolnikov, and A. L. Bertozzi, Two-species particle aggregation and stability of +co-dimension one solutions, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 5, 1411–1436. +[22] B. G. Pachpatte, Inequalities for differential and integral equations, Mathematics in Science and +Engineering, vol. 197, Academic Press, Inc., San Diego, CA, 1998. +[23] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations +with source, Arch. Ration. Mech. Anal. 211 (2014), no. 1, 335–358. +[24] J. Rubinstein, Evolution equations for stratified dilute suspensions, Phys. Fluids A 2 (1990), no. 1, +3–6. +[25] K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Quart. Appl. Math. +57 (1999), no. 3, 573–600. +(M. Inversi) Department Mathematik und Informatik, Universität Basel, Spiegelgasse 1, +4051 Basel, Switzerland +Email address: marco.inversi@unibas.ch +(G. Stefani) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, +34136 Trieste (TS), Italy +Email address: gstefani@sissa.it or giorgio.stefani.math@gmail.com + diff --git a/ItFKT4oBgHgl3EQfdy7h/content/tmp_files/load_file.txt b/ItFKT4oBgHgl3EQfdy7h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f617bf368f382d8015ed143b14fd449bc3b8ea67 --- /dev/null +++ b/ItFKT4oBgHgl3EQfdy7h/content/tmp_files/load_file.txt @@ -0,0 +1,640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf,len=639 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11822v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='AP] 27 Jan 2023 LAGRANGIAN STABILITY FOR A SYSTEM OF NON-LOCAL CONTINUITY EQUATIONS UNDER OSGOOD CONDITION MARCO INVERSI AND GIORGIO STEFANI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We extend known existence and uniqueness results of weak measure solu- tions for systems of non-local continuity equations beyond the usual Lipschitz regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Existence of weak measure solutions holds for uniformly continuous vector fields and convolution kernels, while uniqueness follows from a Lagrangian stability estimate under an additional Osgood condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Statement of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' For fixed T ∈ (0, +∞) and k, d ∈ N, we consider the system of non-local continuity equations \uf8f1 \uf8f2 \uf8f3 ∂t̺i + div (̺i V i(t, x, ̺ ∗ ηi)) = 0, t ∈ (0, T), x ∈ Rd, ̺i(0) = ¯̺i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' , k, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) where the unknown ̺ = (̺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' , ̺k) ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k) is a time-dependent k-vector of non-negative Borel measures on Rd, the initial datum ¯̺ = (¯̺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' , ¯̺k) ∈ M+(Rd)k is a k-vector of non-negative Borel measures, V = (V 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' , V k) ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Cb(Rd × Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rd)k) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2) is a uniformly-in-time bounded continuous k-vector field and ηi = (ηi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' , ηi,k) ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Cb(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rk)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3) is a uniformly-in-time bounded continuous k-vector of convolution kernels, with the con- volution ̺ ∗ ηi = (̺1 ∗ ηi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' , ̺k ∗ ηi,k) occurring in the space variable only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' In the entire paper, we frequently consider the 1-norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', the sum of the absolute values of the entries) on both vectors and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' In particular, |̺| = |̺1| + · · · + |̺k| and thus ∥̺∥M = ∥̺1∥M+· · ·+∥̺k∥M for all ̺ ∈ M(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' When considering other norms, constants depending on d and/or k may be dropped without notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Primary 35L65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Secondary 34A30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Non-local continuity equation, Lagrangian stability, Osgood condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The authors thank Gianluca Crippa for several useful comments on a preliminary version of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The first-named author is partially funded by the SNF grant FLUTURA: Fluids, Turbulence, Advection No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 212573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The second-named author is member of the Istituto Nazionale di Alta Matematica (INdAM), Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Appli- cazioni (GNAMPA), is partially supported by the INdAM–GNAMPA 2022 Project Analisi geometrica in strutture subriemanniane, codice CUP_E55F22000270001, and has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 945655).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' INVERSI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' STEFANI Solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) are understood in the usual distributional sense, which is well-set thanks to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1 (Weak solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We say that ̺ ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k) is a weak solution of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial datum ¯̺ ∈ M+(Rd)k if � T 0 � Rd � ∂tϕ + V i(t, x, ̺ ∗ ηi) · ∇ϕ � d̺i(t, x) dt + � Rd ϕ(0, x) d¯̺i(x) = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4) for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k and any ϕ ∈ C∞ c ([0, T) × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Any weak solution in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1 admits a weakly continuous repre- sentative in duality with the space C0(Rd) of continuous functions vanishing at infinity, see [2, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2] and [1, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' So, from now on, we restrict our attention to weakly- continuous weak solutions ̺ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k−w∗) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) is used in several physical situations—for instance, pedestrian traffic, sedimentation models and supply chains—to describe the time evolution of the density of a vectorial quantity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', pedestrians or particles), possibly concentrating in some points or along hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Far from being complete, we refer the reader for example to [4, 10–13, 16, 18, 21, 24, 25] for a panoramic on the related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Because of the physical relevance of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1), here we deal with non-negative solutions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) can be also interpreted in the sense of the Control Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Indeed, the convolution kernel η can be viewed as a non-local control for the (non-linear) PDE in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Therefore, assuming V is fixed for simplicity, any stability result for the solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) in terms of the convolution kernel η yields a continuous dependence of the curve t �→ ̺t[η] solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) in terms of the control η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The well-posedness of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) was established in [14], provided that V and η are bounded and Lipschitz continuous uniformly in time, namely, V ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Lipb(Rd × Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rd)k) and η ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Lipb(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rk)k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5) The crucial ingredient of [14] is a stability estimate (in terms of the 1-Wasserstein distance between two solutions, see [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2]) which, in turn, allows to obtain existence and uniqueness of the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) via a fix point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The idea of exploiting the Lipschitz regularity to gain stability of trajectories has been later applied to several other related problems, see [5,7,9,17,23] and the references therein for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The aim of the present note is to prove the well-posedness of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) under less restrictive assumptions than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5), that is, to extend the existence and uniqueness result of [14] beyond the Lipschitz regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Our interest is motivated by some recent works [1,3,6,15,19,20] dealing with non-Lipschitz velocity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Our first main result deals with the existence of weak solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1), in the spirit of the celebrated Peano’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To this aim, we consider the following struc- tural hypotheses (where modulus of continuity means a non-decreasing concave function vanishing continuously at zero): (V ) The vector field V ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Cb(Rd × Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rd)k) satisfies ess sup t∈[0,T] |V (t, x, u) − V (t, y, v)| ≤ ωV (|x − y| + |u − v|) ∀x, y ∈ Rd, u, v ∈ Rk, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6) where ωV : [0, +∞) → [0, +∞) is a modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION 3 (η) For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k, the convolution kernel ηi ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' C0(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rk)) satisfies ess sup t∈[0,T] |ηi(t, x) − ηi(t, y)| ≤ ωη(|x − y|) ∀x, y ∈ Rd, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='7) where ωη : [0, +∞) → [0, +∞) is a modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2 (Existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' If (V ) and (η) hold, then the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) admits a weak solution starting from any given initial datum in M+(Rd)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2, we first consider the smoothed functions Vε and ηε and obtain a weak solution ̺ε of the corresponding system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) for all ε > 0 in virtue of the main result of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Then, we pass to the limit as ε → 0+ showing that ̺ε (weakly) converges to a weak solution of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The needed a priori compactness is achieved via an Aubin–Lion-type Lemma which is inspired by [15, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' In order to achieve uniqueness of weak solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1), we need to impose a further Osgood condition on the composition of the two moduli of continuity of V and η: (O) for each λ > 0, it holds � 0+ dr ωV (r + λ ωη(r)) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' For example, condition (O) is satisfied as soon as ωV ◦ ωη is a log-linear function, such as r| log r|, r log | log r| and similar, with r > 0 sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Our uniqueness result deals with Lagrangian weak solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3 (Lagrangian weak solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' A weak solution ̺ ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k) of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial datum ¯̺ ∈ M+(Rd)k is Lagrangian if ̺i(t, ·) = Xi(t, ·)#¯̺i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k, where Xi : [0, T] × Rd → Rd is the (classical) solution of the ODE \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 d dt Xi(t, x) = V i� t, Xi(t, x), ̺ ∗ ηi(t, Xi(t, x)) � , t ∈ (0, T), x ∈ Rd, Xi(0, x) = x, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8) Thanks to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4 below, the Osgood condition in (O) ensures the well-posed- ness of the ODE in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4 (Associated vector field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let assumptions (V ) and (η) be in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' If ̺ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k−w∗) is a weak solution of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial datum ¯̺ ∈ M+(Rd)k, then the vector field bi V,η,̺(t, x) = V i� t, x, ̺ ∗ ηi(t, x) � , t ∈ [0, T], x ∈ Rd, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='9) appearing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8) satisfies b ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Cb(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rd)k) with ess sup t∈[0,T] |bV,η,̺(t, x) − bV,η,̺(t, y)| ≲ ωV � |x − y| + ∥¯̺∥M ωη(|x − y|) � ∀x, y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' With the above notation, our main uniqueness result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5 (Uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' If (V ), (η) and (O) hold, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) admits a unique La- grangian weak solution starting from any given initial datum in M+(Rd)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' INVERSI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' STEFANI The word “Lagrangian” in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5 can be dropped, since any weak solution of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) is in fact Lagrangian because of [1, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 1] (also see [8]) and of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' However, this regularity result is not at all elementary, so we prefer to state Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5 for Lagrangian solutions only in order to emphasize what is possible to achieve just relying on our elementary approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The strategy of [14] exploits the linearity of ωη in an essential way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Indeed, the au- thors need the Lipschitz continuity of η in order to recover the 1-Wasserstein distance between two weak solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) in terms of its dual Kantorovich–Rubinstein formu- lation (see [14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We do not know if the strategy of [14] can be adapted to deal with a more general modulus of continuity ωη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To overcome this issue, we adopt a different point of view, which is inspired by the elementary uniqueness result achieved in the recent work [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Instead of controlling the 1-Wasserstein distance between two weak solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1), we exploit their Lagrangian property to quantitatively estimate the difference between the two associated ODE flows, thus providing a Lagrangian stability of weak solutions from which Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5 immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6 (Lagrangian stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let V, U ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Cb(Rd×Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rd)k) satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6) with the same modulus of continuity ωV and let η, ν ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' C0(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Rk)k) satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='7) with the same same modulus of continuity ωη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let ̺, σ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k −w∗) be two weak solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial data ¯̺, ¯σ ∈ M+(Rd)k, with vector fields V, U and convolution kernels η, ν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Assume that ̺, σ are Lagrangian, namely, ̺ = X(t, ·)#¯̺ and σ = Y (t, ·)#¯σ for t ∈ [0, T], where X, Y are the flows solving the corresponding ODEs in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Then, there exists a modulus of continuity Ω: [0, +∞) → [0, +∞), only depending on T, ∥¯̺∥M, ∥¯σ∥M, ∥η∥L∞(C), ∥ν∥L∞(C), ωV , ωη, such that sup t∈[0,T] ∥X(t, ·) − Y (t, ·)∥L∞ ≤ Ω � ∥¯̺ − ¯σ∥M + ∥V − U∥L∞(C) + ∥ν − η∥L∞(C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='10) The modulus of continuity Ω in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6 can be explicitly computed as soon as one can invert the integral function GV,η,λ(r) = � r r0 ds ωV (s + λ ωη(s)), r ≥ 0, r0 > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11) naturally brought by the Osgood condition assumed in (O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' In fact, the stability esti- mate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='10) follows by simply differentiating a localized integral distance between the flows with respect to the time variable, and then applying the classical Bihari–LaSalle inequality (see [22, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1] for instance) with Osgood modulus of continuity r → ωV (r + λ ωη(r)), for some specific parameter λ > 0 depending on ∥¯̺∥M and ∥¯σ∥M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6 clearly rephrases as a stability result of the flow of the ODE in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' From the point of view of Control Theory, the stability estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='10) yields a continuous dependence of the (Lagrangian) solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', of the flows induced by the corresponding ODE in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8), in terms of the (non-local) control given by the convolution kernel, as well as of the velocity vector field and of the initial datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proofs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Existence of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2, we need some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We begin with an Aubin–Lions-type Lemma, which is inspired by [15, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1 (Compactness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let (̺n)n∈N ⊂ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M(Rd)−w∗) be such that sup n∈N ∥̺n∥L∞(M) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) Assume that, for each ϕ ∈ C∞ c (Rd), the functions Fn[ϕ]: [0, T] → R, given by Fn[ϕ](t) = � Rd ϕ d̺n(t, ·), t ∈ [0, T], are uniformly equicontinuous on [0, T], that is, ∀ε > 0 ∃δ > 0 : s, t ∈ [0, T], |s − t| < δ =⇒ sup n∈N |Fn[ϕ](s) − Fn[ϕ](t)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2) Then, there exist a subsequence (̺nk)k∈N and ̺ ∈ C([0, T], M(Rd)−w∗) such that lim k→+∞ sup t∈[0,T] ���� � Rd ϕ d̺nk(t, ·) − � Rd ϕ d̺(t, ·) ���� = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3) for all ϕ ∈ C0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let D ⊂ Cc(Rd) be a countable and dense set in C0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' In virtue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2), for each ϕ ∈ D the sequence (Fn[ϕ])n∈N is equibounded and equicontinuous on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' By Ascoli–Arzelà Theorem and a standard diagonal argument, we can find a subsequence (nk)k∈N such that, for each ϕ ∈ D, the sequence (Fnk[ϕ])k∈N is uniformly convergent to some F[ϕ] ∈ C([0, T]), with ∥F[ϕ]∥L∞([0,T]) ≤ ∥ϕ∥L∞ sup n∈N ∥̺n∥L∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4) By construction, the function ϕ �→ F[ϕ](t) is a continuous linear functional on D for each t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Thus, for each fixed t ∈ [0, T], we can extend the map ϕ �→ F[ϕ](t) to a linear and continuous functional on C0(Rd) for which we keep the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' A plain approximation argument readily proves that, for each ϕ ∈ C0(Rd), the map t �→ F[ϕ](t) is continuous on [0, T] and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' By Riesz’s Representation Theorem, for each t ∈ [0, T] there exists a finite Borel measure ̺(t, ·) ∈ M(Rd) such that F[ϕ](t) = � Rd ϕ d̺(t, ·) for all ϕ ∈ C0(Rd), so that ̺ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M(Rd)−w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Finally, in virtue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4), for ϕ ∈ C0(Rd) and ψ ∈ D, we can estimate sup t∈[0,T] |Fnk[ϕ](t) − F[ϕ](t)| ≤ sup t∈[0,T] |Fnk[ψ](t) − F[ψ](t)| + 2 ∥ψ − ϕ∥L∞ sup n∈N ∥̺n∥L∞(M) and the desired (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3) readily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' □ In order to exploit Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1, we need the following mass preservation property for weak solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' INVERSI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' STEFANI Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2 (Mass preservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let V and η be as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' If ̺ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k−w∗) is a weak solution of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial datum ̺ ∈ M+(Rd)k, then ∥̺i(t, ·)∥M = ∥¯̺i∥M (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5) for t ∈ [0, T] and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k} be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' By applying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4) to the test function ϕ(t, x) = α(t) β(x), (t, x) ∈ [0, T] × Rd, where α ∈ C∞ c ([0, T)) and β ∈ C∞ c (Rd), we get � T 0 � Rd � α′β + α V i(t, x, ̺ ∗ ηi) · ∇β � d̺i(t, ·) dt + α(0) � Rd β(x) d¯̺i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Since α ∈ C∞ c ([0, T)) is arbitrary and ̺ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k−w∗), we infer that t �→ � Rd β d̺i(t, ·) ∈ AC1,1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' R) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6) with � Rd β d̺i(t, ·) = � Rd β d¯̺i + � t 0 � Rd V i(s, ·, ̺ ∗ ηi) · ∇β d̺i(s, ·) ds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='7) for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Now let t ∈ [0, T] be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We let (βR)R>0 ⊂ C∞ c (Rd) be such that βR ≥ 0, supp βR ⊂ B2R, βR = 1 on BR, ∥∇βR∥L∞ ≤ 2 R for all R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' By the Monotone Convergence Theorem, we infer that lim R→+∞ � Rd βR d̺i(t, ·) = ∥̺i(t, ·)∥M as well as lim R→+∞ � Rd βR d¯̺i = ∥¯̺i∥M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Since ���� � t 0 � Rd V i(s, ·, ̺ ∗ ηi) · ∇βR d̺i(s, ·) ds ���� ≤ 2 R ∥̺i∥L∞(M) ∥V i∥L∞(C) for all R > 0, we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5) by applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='7) to βR and passing to the limit as R → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' □ We are ready to prove our existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let (ℓε)ε>0 ⊂ C∞ c (Rd+k) and (\uf6beε)ε>0 ∈ C∞ c (Rd) be two families of standard non-negative mollifiers and set V i,j ε (t, ·) = V i,j(t, ·) ∗ ℓε, ηi,j ε = ηi,j(t, ·) ∗ \uf6beε, where in both cases the (component-wise) convolution occur in the spatial variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Since Vε and ηε clearly satisfy the Lipschitz property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='5) for each ε > 0, by [14, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1] there exists a weak solution ̺ε ∈ C([0, T], M+(Rd)k−w∗) of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial datum ¯̺ ∈ M+(Rd)k, so that � T 0 � Rd � ∂tϕ + V i ε (t, ·, ̺ε ∗ ηi ε) · ∇ϕ � d̺i ε(t, ·) dt + � Rd ϕ(0, ·) d¯̺i = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8) for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k and ε > 0 and ϕ ∈ C∞ c ([0, T) × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION 7 Now let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k} be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We claim that (any sequence in) the family (̺i ε)ε>0 satisfies the assumptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Indeed, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2 we get ∥̺i ε(t, ·)∥M = ∥¯̺i∥M (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='9) for all t ∈ [0, T] and ε > 0, from which (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2), we simply argue as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='7), we easily recognize that the time derivative of the function Fε[β](t) = � Rd β(·) d̺i ε(t, ·), t ∈ [0, T], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='10) is bounded by ���� � Rd V i ε (t, x, ̺ε ∗ ηi ε) · ∇β d̺i ε(t, x) ���� ≤ ∥V i∥L∞(C) ∥∇β∥L∞ ∥¯̺i∥M for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' t ∈ [0, T] and for each ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' In particular, the family (Fε[β])ε>0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='10) is equi-Lipschitz and thus satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1, we find a sequence (̺εn)n∈N ⊂ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k−w∗) and ̺ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' M+(Rd)k−w∗) such that lim n→+∞ sup t∈[0,T] ���� � Rd β d̺εn(t, ·) − � Rd β d̺(t, ·) ���� = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11) for all β ∈ C0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To conclude, we just need to prove that ̺ is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1) starting from the initial datum ¯̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We do so by passing to the limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='8) along (εn)n∈N as n → +∞ for each given ϕ ∈ C∞ c ([0, +∞) × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Indeed, on the one side, since lim n→+∞ � Rd ∂tϕ d̺i εn(t, ·) = � Rd ∂tϕ d̺i(t, ·) because of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11) and ���� � Rd ∂tϕ d̺i εn(t, ·) ���� ≤ ∥∂tϕ∥L∞ ∥¯̺i∥M because of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='9), for all t ∈ [0, T], by the Dominated Convergence Theorem we infer that lim n→+∞ � T 0 � Rd ∂tϕ d̺i εn(t, ·) dt = � T 0 � Rd ∂tϕ d̺i(t, ·) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='12) On the other side, since ηi(t, ·) ∈ C0(Rd) in virtue of the assumption (η), we have that ηi εn(t, ·) → ηi(t, ·) in C0(Rd) as n → +∞, so that lim n→+∞ � ̺εn(t, ·) ∗ ηi εn(t, ·) � (x) = lim n→+∞ � Rd ηi εn(t, x − y) d̺εn(t, y) = � Rd ηi(t, x − y) d̺(t, y) = � ̺(t, ·) ∗ ηi(t, ·) � (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='13) for each x ∈ Rd and all t ∈ [0, T] as a weak-strong convergent pair, due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Moreover, again in virtue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='9) and (η), we can estimate ∥̺εn(t, ·) ∗ ηi εn(t, ·)∥ ≤ ∥̺i∥M ∥ηi∥L∞(C) and ��� � ̺εn(t, ·) ∗ ηi εn(t, ·) � (x) − � ̺εn(t, ·) ∗ ηi εn(t, ·) � (y) ��� ≤ � Rd ���ηi εn(t, x − ·) − ηi εn(t, y − ·) ��� d̺εn(t, ·) ≤ ωη(|x − y|) ∥̺i∥M 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' INVERSI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' STEFANI for all n ∈ N and t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' By Arzelà–Ascoli’s Theorem, we thus get that the pointwise convergence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='13) must be uniform on compact sets in Rd, uniformly in t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' An analogous argument relying on the assumption (V ) proves that also V i εn(t, ·) → V i(t, ·) as n → +∞ uniformly on compact sets in Rd, uniformly in t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Again by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11), by weak-strong convergence and by the Dominated Convergence Theorem, we hence get lim n→+∞ � T 0 � Rd V i εn(t, ·, ̺εn ∗ ηi εn) · ∇ϕ d̺i εn(t, ·) dt = � T 0 � Rd V i(t, ·, ̺ ∗ ηi) · ∇ϕ d̺i(t, ·) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='14) Thus, the conclusion follows by combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='12) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Lagrangian stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We deal with the Lagrangian stability of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We begin with the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let t ∈ [0, T] be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Given x, y ∈ Rd and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=', k}, in virtue of assumption (η) and of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2, we can estimate |̺ ∗ ηi(t, x) − ̺ ∗ ηi(t, y)| ≤ k � j=1 � Rd |ηi,j(t, x − z) − ηi,j(t, y − z)| d̺j(t, z) ≤ k � j=1 � Rd ωη(|x − y|) d̺k(t, z) = ∥̺(t, ·)∥M ωη(|x − y|) = ∥¯̺∥M ωη(|x − y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Thus, thanks to assumption (V ), we get that ���V i� t, x, ̺ ∗ ηi(t, x) � − V i� t, y, ̺ ∗ ηi(t, y) ���� ≤ ωV � |x − y| + |̺ ∗ ηi(t, x) − ̺ ∗ ηi(t, y)| � ≤ ωV � |x − y| + ∥¯̺∥M ωη(|x − y|) � and the conclusion immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' □ We conclude our paper with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Let V, U, η, ν, ¯̺, ¯σ, X, Y and ̺, σ be as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Fix ζ ∈ C(Rd) with ζ ≥ 0 and � Rd ζ(x) dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Letting µ ∈ M+(Rd) be defined by µ = |¯̺| + |¯σ| + ζ L d, we consider the quantity Qζ(t) = k � i=1 − � Rd |Xi(t, ·) − Y i(t, ·)| dµ for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Note that t �→ Qζ(t) is well defined and Lipschitz, with Qζ(0) = 0 and |Qζ(s) − Qζ(t)| ≤ k (∥U∥L∞(C) + ∥V ∥L∞(C)) |s − t| for all s, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Therefore, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' we can write Q′ ζ(t) ≤ k � i=1 − � Rd |V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' ̺ ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Xi)) − Ui(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' σ ∗ νi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i))| dµ ≤ k � i=1 (1)i + (2)i + (3)i + (4)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' LAGRANGIAN STABILITY FOR A NON-LOCAL CE SYSTEM UNDER OSGOOD CONDITION 9 where (dropping the variables of X and Y for notational convenience) (1)i = − � Rd |V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' ̺ ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Xi)) − V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' ̺ ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i))| dµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2)i = − � Rd |V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' ̺ ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i)) − V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' σ ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i))| dµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (3)i = − � Rd |V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' σ ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i)) − V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' σ ∗ νi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i))| dµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (4)i = − � Rd |V i(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' σ ∗ νi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i)) − Ui(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' σ ∗ νi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i))| dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' We now estimate each term separately at a given t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='4 and Jensen’s inequality, we can easily estimate the first term as (1)i ≤ − � Rd ωV � |Xi − Y i| + ∥¯̺∥M ωη(|Xi − Y i|) � dµ ≤ ωV � − � Rd |Xi − Y i| dµ + ∥¯̺∥M ωη � − � Rd |Xi − Y i| dµ �� ≤ ωV � Qζ(t) + ∥µ∥M ωη(Qζ(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Concerning the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' since |(̺ − σ) ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' x)| = ���� � Rd ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' x − y) d(X#¯̺(y) − Y#¯σ(y)) ���� ≤ k � j=1 � Rd |ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='j(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' x − Xj) − ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='j(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' x − Y j)| d¯̺j + � Rd |ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='j(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' x − Y j)| d|¯̺j − ¯σj| ≤ k � j=1 � Rd ωη(|Xj − Y j|) d¯̺j + ∥η∥L∞(C)∥¯̺j − ¯σj∥M ≤ � Rd ωη \uf8eb \uf8ed k � j=1 |Xj − Y j| \uf8f6 \uf8f8 d|¯̺| + ∥η∥L∞(C)∥¯̺ − ¯σ∥M for all x ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' again by Jensen’s inequality we get (2)i ≤ − � Rd ωV � |(̺ − σ) ∗ ηi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Y i)| � dµ ≤ ωV �� Rd ωη � k � i=1 |Xi − Y i| � d|¯̺| + ∥η∥L∞(C)∥¯̺ − ¯σ∥M � ≤ ωV � ∥µ∥M ωη(Qζ(t)) + ∥η∥L∞(C)∥¯̺ − ¯σ∥M � ≤ ωV � Qζ(t) + ∥µ∥M ωη(Qζ(t)) � + ωV � ∥η∥L∞(C)∥¯̺ − ¯σ∥M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' The last two terms can be trivially estimated as (3)i ≤ ωV � ∥σ∥L∞(M) ∥η − ν∥L∞(C) � = ωV � ∥¯σ∥M ∥η − ν∥L∞(C) � ≤ ωV � ∥µ∥M ∥η − ν∥L∞(C) � thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='2, and (4)i ≤ ∥V − U∥L∞(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' INVERSI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' STEFANI Putting everything altogether, we conclude that Q′ ζ(t) ≲ ωV � Qζ(t) + λ ωη(Qζ(t)) � + M, where λ = ∥̺∥M + ∥σ∥M + 1 and M = ωV � ∥η∥L∞(C)∥¯̺ − ¯σ∥M � + ωV � λ ∥η − ν∥L∞(C)) � + ∥V − U∥L∞(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' At this point, we just need to recall the Osgood condition assumed in (O) and the integral function in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Indeed, by the classical Bihari–LaSalle inequality (see [22, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='1] for instance), we find a modulus of continuity Ω: [0, +∞) → [0, +∞), only depending on T, ∥¯̺∥M, ∥¯σ∥M, ∥η∥L∞(C), ∥ν∥L∞(C), ωV , ωη, such that sup t∈[0,T] Qζ(t) ≤ Ω � ∥¯̺ − ¯σ∥M + ∥V − U∥L∞(C) + ∥ν − η∥L∞(C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='15) We remark that Ω is independent of ζ, as long as we choose ζ ≥ 0 and ∥ζ∥L1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' To conclude, we choose a family (ζx0,ε)ε>0 of standard mollifiers around x0 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' Since the flows X(t, ·), Y (t, ·) are continuous maps, we deduce that lim ε→0+ Qζx0,ε(t) = |X(t, x0) − Y (t, x0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='16) Thus, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='10) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='it or giorgio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='stefani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFKT4oBgHgl3EQfdy7h/content/2301.11822v1.pdf'} diff --git a/LNFQT4oBgHgl3EQfUDZM/content/tmp_files/2301.13295v1.pdf.txt b/LNFQT4oBgHgl3EQfUDZM/content/tmp_files/2301.13295v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1734a8947f20aaa8344ddb8f29d8bf137d4af4c6 --- /dev/null +++ b/LNFQT4oBgHgl3EQfUDZM/content/tmp_files/2301.13295v1.pdf.txt @@ -0,0 +1,4373 @@ +Quantum Boltzmann Machines +Applications in Quantitative Finance +Cameron Perot1 +Master’s Thesis +submitted to +The Faculty of Mathematics, Computer Science, and Natural Sciences +of RWTH Aachen University +written at +Jülich Supercomputing Centre +Forschungszentrum Jülich +First Examiner: Prof. Dr. Kristel Michielsen2,3 +Second Examiner: Prof. Dr. Holger Rauhut2 +Adviser: Dr. Dennis Willsch3 +July 4, 2022 +1cameron.perot@pm.me +2RWTH Aachen University, D-52056 Aachen, Germany +3Jülich Supercomputing Centre, Institute for Advanced Simulation, Forschungszentrum Jülich, +D-52425 Jülich, Germany +arXiv:2301.13295v1 [quant-ph] 30 Jan 2023 + +Abstract +In this thesis we explore using the D-Wave Advantage 4.1 quantum annealer to sample from +quantum Boltzmann distributions and train quantum Boltzmann machines (QBMs). We +focus on the real-world problem of using QBMs as generative models to produce synthetic +foreign exchange market data and analyze how the results stack up against classical models +based on restricted Boltzmann machines (RBMs). Additionally, we study a small 12-qubit +problem which we use to compare samples obtained from the Advantage 4.1 with theory, +and in the process gain vital insights into how well the Advantage 4.1 can sample quantum +Boltzmann random variables and be used to train QBMs. Through this, we are able to show +that the Advantage 4.1 can sample classical Boltzmann random variables to some extent, +but is limited in its ability to sample from quantum Boltzmann distributions. Our findings +indicate that QBMs trained using the Advantage 4.1 are much noisier than those trained +using simulations and struggle to perform at the same level as classical RBMs. However, +there is the potential for QBMs to outperform classical RBMs if future generation annealers +can generate samples closer to the desired theoretical distributions. +i + +Contents +1 +Introduction +1 +2 +Data Analysis & Preprocessing +3 +2.1 +Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2.1 +Data Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2.2 +Additional Information . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3 +The Classical Restricted Boltzmann Machine +10 +3.1 +Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +3.1.1 +Optimizing an RBM . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3.2 +The Classical Market Generator . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3.2.1 +Models +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.2.2 +Results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.2.3 +Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +4 +The Quantum Boltzmann Machine +20 +4.1 +Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4.1.1 +Optimizing a QBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +4.1.2 +Quantum Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +4.2 +12-Qubit Problem +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +4.2.1 +Sampling From a Quantum Boltzmann Distribution +. . . . . . . . . +26 +4.2.2 +Training Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.2.3 +Simulation-based Model . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.2.4 +D-Wave Advantage 4.1-based Model . . . . . . . . . . . . . . . . . . +31 +4.3 +The Quantum Market Generator . . . . . . . . . . . . . . . . . . . . . . . . +34 +4.3.1 +Setting the Annealer’s Hyperparameters . . . . . . . . . . . . . . . . +34 +4.3.2 +Results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +4.3.3 +Comparison to Gate-Based Models . . . . . . . . . . . . . . . . . . . +37 +4.3.4 +Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +4.4 +Challenges of Using a D-Wave Annealer to Train QBMs . . . . . . . . . . . +41 +4.4.1 +Choosing an Embedding . . . . . . . . . . . . . . . . . . . . . . . . . +41 +4.4.2 +Sampling the Proper Distribution . . . . . . . . . . . . . . . . . . . . +41 +4.4.3 +QPU Limitations and Imperfections +. . . . . . . . . . . . . . . . . . +42 +5 +Conclusion +44 +5.1 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +5.2 +Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +Appendix A Definitions and Methodologies +46 +A.1 Correlation Coefficients +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +A.2 Annualized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +A.3 Learning Rate Decay Schedule +. . . . . . . . . . . . . . . . . . . . . . . . . +47 +ii + +A.4 Autocorrelation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +A.5 Kullback-Leibler Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +A.5.1 +Kullback-Leibler Divergence in Practice . . . . . . . . . . . . . . . . +48 +A.6 Tail Concentration Functions +. . . . . . . . . . . . . . . . . . . . . . . . . . +48 +A.7 Exact Computation of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +A.8 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +Appendix B Restricted Boltzmann Machine +50 +B.1 +Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +B.2 +Log-Likelihood Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +Appendix C Quantum Boltzmann Machine +52 +C.1 +Log-Likelihood Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +C.2 +Log-Likelihood Lower Bound +. . . . . . . . . . . . . . . . . . . . . . . . . . +53 +C.3 +Log-Likelihood Lower Bound Derivative +. . . . . . . . . . . . . . . . . . . . +54 +C.4 +Effective β as a Learnable Parameter . . . . . . . . . . . . . . . . . . . . . . +55 +iii + +Chapter 1 +Introduction +In recent years we have seen the inception of cloud-based quantum computing, with a +number of different providers offering various services. In terms of maturity, the quantum +computing industry as a whole is still in the early stages and there are a lot of obstacles +left to overcome before mainstream adoption. Quantum computing is not only trying to +advance the theory and technology, but also yearning for practical applications in which +quantum computing offers advantages over classical computing. +There are two main branches of quantum computing: universal quantum computing, +i.e., gate-based quantum computing, and adiabatic quantum computing, i.e., quantum +annealing. In our work here we focus on the latter, as current generation devices are slightly +more mature and have much higher numbers of qubits than the former. We discuss the +theory behind quantum annealing later in Section 4.1.2. One such cloud-based quantum +computing service is D-Wave’s Leap platform [1], which allows users to access quantum +annealers and other solvers across the world. +D-Wave is a pioneer in this field, having been researching and developing quantum +annealers since 1999. They revolutionized the field with the release of the world’s first +commercially available quantum annealer in 2011 [2]. Since then, they have released a +new version every 2-3 years, each having more qubits and couplers than the previous. +Their latest version, the D-Wave Advantage, has over 5000 qubits with 15 connections per +qubit [3]. +In this thesis we take a journey into the field of quantum machine learning and explore +the possibilities of using quantum Boltzmann machines (QBMs) as generative models for +real-world financial data. As we will see, there is a deep connection between the quantum +Boltzmann machine and quantum annealing, allowing one to train QBMs using a quantum +annealer. +Risk management is one of the most important components of the financial system, and +in 2008 it failed, leading to the financial crisis which wreaked havoc on economies around +the world. The success of risk management hinges on how accurately the underlying risk +models capture the true behavior of the market. Therefore, it is essential that we continu- +ously strive to find new and innovative ways of modeling that can help us understand the +real risks involved and implement policies to effectively mitigate such risks. +In the globalized economy of today, foreign exchange (forex) fluctuations expose a +number of firms to a lot of risk if not properly mitigated. +Forex markets had a daily +volume of $6.6T in 2019 [4], the majority of which was concentrated in a few major pairs. +In the 2019 paper The Market Generator [5], Kondratyev and Schwarz detail how a classical +restricted Boltzmann machine (RBM) can be used to generate synthetic forex data, and +the advantages it offers over traditional parametric models. We use their work as a basis to +build our classical models upon, which we then use as a reference to compare our quantum +models with. +1 + +In Chapter 2, we start by visualizing the data set in various ways to get an idea how +it is distributed. We further analyze quantitative metrics to get a better understanding of +some of the intricacies of the data set. Finally, we go through and detail how we preprocess +the data set into a model-friendly format. +With the data set in hand, we move to explaining the theory behind the classical +RBM in Chapter 3 and describing some of the difficulties associated with training and +using classical RBMs. We then train several classical models on the data set discussed +in Chapter 2 using different preprocessing methods and compare them with each other +using visualizations and a number of quantitative metrics. +In Chapter 4, we start from the theory of quantum Boltzmann machines, detailing how +they work and their connection to quantum annealing. We study a small 12-qubit problem +which we can simulate, allowing us to compare annealer performance with that of theory, +and gaining key insights into how to train and use QBMs. With those insights, we move +to the final stage of training a model using the data set from Chapter 2, then assessing +the performance versus the classical models from Chapter 3. Additionally, we cover some +of the challenges of using D-Wave quantum annealers to train QBMs in Section 4.4. +Lastly, we summarize our findings in Chapter 5, as well as discuss future directions in +which this research can be expanded. +In addition to the research and results presented here, we also introduce the open source +Python package qbm [6] to make it easier for the community to train and study quantum +Boltzmann machines. All work presented here is reproducible (except for that involving +quantum measurements), and the code is available on GitHub 1. +1https://github.com/cameronperot/qbm-quant-finance +2 + +Chapter 2 +Data Analysis & Preprocessing +2.1 +Data Analysis +Our raw data set consists of the daily open, high, low, and close (OHLC) values for the +time period 1999-01-01 through 2019-12-31 of the following major currency pairs +• EURUSD - Euro € / U.S. Dollar $ +• GBPUSD - British Pound Sterling £ / U.S. Dollar $ +• USDCAD - U.S. Dollar $ / Canadian Dollar $ +• USDJPY - U.S. Dollar $ / Japanese Yen ¥ +obtained from Dukascopy historical data feed [7]. We filter the data set to remove days +with zero volume, as well as NYSE and LSE holidays, resulting in 5165 training samples. +Here we use the notation xopen, xhigh, xlow, and xclose to denote the open, high, low, and +close values of a currency pair on a particular day. +Given that the raw data values are on an absolute basis, we need to convert them to +relative terms in order to be able to compare data from different time periods on a more +equal footing. The natural way to do so is to use the intraday returns +r = xclose − xopen +xopen +. +(2.1) +However, this is not necessarily the best way to approach this. Instead, we opt to use the +log returns +˜r = log(1 + r) = log +�xclose +xopen +� +(2.2) +due to several advantages, such as log-normality and small r approximation [8]. +We begin our analysis by taking a look at the histograms depicted in Fig. 2.1. From +visual examination we see that the log returns are roughly normally distributed with the +statistics given in Table 2.1. +We also visualize the log returns in a violin and box plot in Fig. 2.2 to identify outliers +and see how they are distributed. Two major outliers clearly stand out from the rest: one to +the downside for the GBPUSD pair, and another to the upside for the USDJPY pair. The +former occurred on 2016-06-24, the day the Brexit referendum result was announced [9]. +The latter occurred on 2008-10-28, right in the midst of the financial crisis when people +were talking about the end of the Yen carry trade [10]. In the final training data set, +we remove outliers greater than 10σ from the mean, resulting in only removing the day +corresponding to the Brexit referendum result, which lies 11.1σ below the mean. +3 + +Figure 2.1: Histograms of the log returns data set. +Log Returns Data Set Statistics +Currency Pair +Mean +Standard Deviation +EURUSD +5.15 · 10−5 +6.17 · 10−3 +GBPUSD +−8.49 · 10−6 +5.73 · 10−3 +USDCAD +−5.04 · 10−5 +5.40 · 10−3 +USDJPY +−6.31 · 10−5 +6.32 · 10−3 +Table 2.1: Statistics of the log returns data set. +Next we examine the correlations between the currency pairs to get an idea of the inter- +dependencies between them. We visualize this with scatter plots shown in Fig. 2.3 where we +observe a clear positive correlation between EURUSD/GBPUSD, and clear negative cor- +relations between EURUSD/USDCAD and GBPUSD/USDCAD, where the / is used to +denote the pairs being compared against each other. This is further verified by the Pearson +r, Spearman ρ, and Kendall τ correlation coefficients laid out in Table 2.2. Furthermore, +we find the correlation coefficients to be positive for pairs of the form XUSD/Y USD, and +negative for pairs of the form XUSD/USDY , for X, Y ∈ {EUR, GBP, CAD, JPY}, as +expected. Details on how the correlation coefficients are computed and how to interpret +them can be found in Appendix A.1. +2.2 +Data Preprocessing +The models in the following chapters require the training data to be in the form of bit +vectors, so we must first convert our data set to such a form. Let X ∈ R4×N represent +the training data set of log returns with N samples, where training samples are vectors in +the column space, thus element xij represents the ith currency pair log return for the jth +training sample. +To discretize the data, we rescale and round the entries of X to integer values in +4 + +EURUSD +GBPUSD +100 +100 +80 +80 +60 +60 +40 +40 +D +20 +20 +0 ++0 +-1 +ｏi +2２3 +-3 +-2 +2 +3 +-3 +-2 +Log Return +1e-2 +Log Return +1e-2 +USDCAD +USDJPY +100 +100 +80 +80 +60 +nsi +nsi +Der +40 +Del +40 +20 +20 +0 ++0 +-1 +-1 +-3 +-2 +0 +1 +2 +3 +-3 +-2 +0 +1 +2 +3 +1e-2 +1e-2 +Log Return +Log ReturnFigure 2.2: Violin and box plot of the log returns data set illustrating the distribution of +the outliers. +Figure 2.3: Scatter plots of the log returns data set. +{0, 1, . . . , 2nbits − 1}, represented by the matrix X′ ∈ N4×N with entries +x′ +ij = +� +xij − mink{xik} +maxk{xik} − mink{xik} · (2nbits − 1) +� +, +(2.3) +where ⌊ · ⌉ denotes rounding to the nearest integer. +5 + +1e-2 +6 +4 +XX +XX +2 + Return +0 +Log +-2 +KX& +-6 +X +EURUSD +GBPUSD +USDCAD +USDJPY +Currency Pair1e-2 +1e-2 +1e-2 +4 +4 +4 +2 +2 +2 +GBPUSD +SDCAD +USDJPY +0 +0 +0 +S +G +-2 +2 +2 +¥-2¥0 +2 +4 +-4-20 +2 +4 +-4　-20 +4 +2 +4 +EURUSD +1e-2 +EURUSD +1e-2 +EURUSD +1e-2 +1e-2 +1e-2 +le-2 +4 +4 +4 : +2 +2 +2 +SDCAD +DJPY +USDJPY +0 +0 +0 +S +2 +-2 +2 +-4 +4 +.4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +GBPUSD +1e-2 +GBPUSD +1e-2 +USDCAD +1e-2Correlation Coefficients +Currency Pairs +Pearson +Spearman +Kendall +EURUSD/GBPUSD +0.62 +0.62 +0.44 +EURUSD/USDCAD +-0.44 +-0.41 +-0.29 +EURUSD/USDJPY +-0.26 +-0.30 +-0.21 +GBPUSD/USDCAD +-0.42 +-0.37 +-0.26 +GBPUSD/USDJPY +-0.14 +-0.21 +-0.15 +USDCAD/USDJPY +0.00 +0.06 +0.04 +Table 2.2: Correlation coefficients of the log returns data set. +A new matrix V ∈ {0, 1}4·nbits×N is then created with the columns being the nbits- +length bit vectors corresponding to the binary representation of the entries of the columns +of X′ concatenated together. For example, if x′ = (x′ +1, x′ +2, x′ +3, x′ +4) is a column of X′ and the +function bitvector(x′) takes in an integer x′ and returns an nbits-bit binary representation +bit vector, then the corresponding column in V is +v = +� +��� +bitvector(x′ +1) +bitvector(x′ +2) +bitvector(x′ +3) +bitvector(x′ +4) +� +��� ∈ {0, 1}4·nbits. +(2.4) +For this research we take nbits = 16, giving us a training set V ∈ {0, 1}64×N, thus our +training samples are bit vectors of length 64. The discretization errors associated with this +conversion and data set are on the order of 10−7, well within the desired tolerance for this +purpose. +2.2.1 +Data Transformation +Due to how the data is linearly converted to a discrete form before rounding, it opens up the +possibility of the discretized data being clustered in the mid-range values if large outliers +are present. To mitigate this, we use a transformation to reduce the gap between outliers by +scaling outliers beyond a certain threshold τ using the procedure detailed in Algorithm 1. +We call this the outlier power transformation. +In practice, we take τ = 1 and α = 0.5, thus the standardized data points above +one standard deviation are mapped to their square roots, as illustrated in Fig. 2.4. We +tested a few other combinations of τ and α, but found these values to produce the best +model results out of those we tried; of course this could likely be further optimized. The +effect this transformation has on the model results versus the base dataset can be seen +in Section 3.2.2. This transformation is invertible when ¯x, σx, and δ are saved. +Histograms of the transformed data set are shown in Fig. 2.5, and a violin and box +plot is shown in Fig. 2.6. In these, we observe the appearance of "shoulders" around the +threshold τ = 1 standard deviation, and that the transformed outliers appear much less +extreme, allowing us to better utilize the full range of discrete values. Table 2.3 shows +that the transformation reduces the standard deviations to roughly 78% of their originals +values given in Table 2.1. +2.2.2 +Additional Information +As mentioned in [5], one can use additional binary indicator variables to enrich the training +data set. One such bit of information is the rolling volatility relative to the historical median +(see Appendix A.2 for definition of annualized volatility). If the 3-month rolling volatility +6 + +Algorithm 1 Outlier Power Transformation +1: procedure Transform(x, α, τ) +▷ α is the power, τ is the threshold +2: +N ← length(x) +3: +¯x ← 1 +N +�N +i=1 xi +4: +σx ← +� +1 +N +�N +i=1(xi − ¯x)2 +5: +δ ← τ − τ α +▷ ensures the transformation is bijective +6: +for i in 1 to N do +7: +xi ← (xi − ¯x)/σx +▷ standardize +8: +if xi > τ then +9: +xi ← (|xi|α + δ) · sign(xi) +▷ scale standardized values beyond τ +10: +end if +11: +xi ← xi · σx + ¯x +▷ undo standardization +12: +end for +13: end procedure +Figure 2.4: Transformation defined in Algorithm 1 using τ = 1 and α = 0.5, for the +purpose of reducing large gaps in the discretized data set by scaling outliers above τ +standard deviations. +Transformed Log Returns Data Set Statistics +Currency Pair +Mean +Standard Deviation +EURUSD +5.54 · 10−5 +4.88 · 10−3 +GBPUSD +1.66 · 10−5 +4.48 · 10−3 +USDCAD +−6.42 · 10−5 +4.21 · 10−3 +USDJPY +−4.68 · 10−5 +4.93 · 10−3 +Table 2.3: Statistics of the outlier power-transformed log returns data set. +is below (above) the historical median it is assigned a value of 0 (1) to indicate the low +(high) volatility regime. The 3-month rolling volatilities versus their historical medians are +7 + +1 +0 +-1 +X +-2 +-6 +-4 +-2 +0 +2 +4 +6 +(x - x)/ox OriginalFigure 2.5: Histograms of the outlier power-transformed log returns data set. +Figure 2.6: Violin and box plot of the outlier power-transformed log returns data set +illustrating the distribution of the rescaled outliers. +plotted in Fig. 2.7. +These additional binary indicator variables are then concatenated onto the training +data set and fed to the model to make it more flexible by allowing for the model outputs +to be conditioned on a specific volatility regime. Adding one indicator for each of the +four currency pairs increases the number of rows in our training data set by four, thus the +8 + +EURUSD +GBPUSD +100 +100 +80 +80 +60 +60 +40 +40 +D +20 +20 +0. +0 +-iｏi +23 +23 +-3 +-2 +m- +-2 +1e-2 +Log Return +1e-2 +Log Return +USDCAD +USDJPY +100 +100 +80 +80 +W60 +60 +nsi +nsi +Del +40 +40 +D +20 +20 +0 ++0 +-2 +-2 +-1 +-3 +-1 +0 +1 +2 +3 +-3 +0 +1 +2 +3 +1e-2 +1e-2 +Log Return +Log Returnle-2 +2 +1 + Return +0 +Log +-1 +X +-2 +EURUSD +GBPUSD +USDCAD +USDJPY +Currency Pairvolatility-concatenated data set is in the space {0, 1}68×N. +Figure 2.7: 3-month rolling volatilities of the log returns data set compared with their +historical medians. +9 + +EURUSD +GBPUSD +3 × 10-1 +3× 10-1 +3M Rolling +2 × 10-1 +2 × 10-1 +Volatility +Median +lity +Volatil +10-11 +10- +6 × 10-2 +6 × 10-2 +4 × 10-2 +4 × 10-2 +3 × 10-2 +3 × 10-2 +2000 2004 2008 2012 2016 2020 +2000 2004 2008 2012 2016 2020 +Date +Date +USDCAD +USDIPY +3 × 10-1 +3 × 10-1 +2 × 10-1 +2 × 10-1 +Volatility +latility +10-1 +10-1. +6 × 10-2 +6× 10-2 +4 × 10-2 +4 × 10-2 +3 × 10-2 +3 × 10-2 +2000 2004 2008 2012 2016 2020 +2000 2004 2008 2012 2016 2020 +Date +DateChapter 3 +The Classical Restricted Boltzmann +Machine +3.1 +Theory +The restricted Boltzmann machine (RBM) is an energy-based model defined by the energy +function [11] +E(v, h) = − +nv +� +i=1 +aivi − +nh +� +j=1 +bjhj − +nv +� +i=1 +nh +� +j=1 +viwijhj += −a⊺v − b⊺h − v⊺Wh, +(3.1) +where +• v ∈ {0, 1}nv represents the visible units, with associated bias vector a ∈ Rnv. +• h ∈ {0, 1}nh represents the hidden units, with associated bias vector b ∈ Rnh. +• W ∈ Rnv×nh represents the weights corresponding to the interaction strengths be- +tween visible and hidden units. +It is termed restricted due to the fact that there are no intralayer connections, i.e., +visible units are only connected to hidden units, and vice versa. An example diagram is +depicted in Fig. 3.1. +The probability to find the system in the configuration (v, h) is given by the Boltzmann +distribution (with β = 1/kT = 1) +p(v, h) = 1 +Z e−E(v,h), +(3.2) +with intractable [12] partition function +Z = +� +v,h +e−E(v,h), +(3.3) +where � +v,h denotes the sum over all possible configurations of v and h. +The imposed restrictions on intralayer connections enable us to write the conditional +probabilities of the layers as the product of the individual units’ probabilities 1 (see Ap- +1Here σ(x) is the element-wise logistic sigmoid function and ⊙ denotes element-wise multiplication. +10 + +Figure 3.1: Diagram of a restricted Boltzmann machine with nv visible units and nh hidden +units. +pendix B.1 for derivation) +p(h|v) = +nh +� +j=1 +σ +� +(2h − 1) ⊙ (b + W⊺v) +� +j, +p(v|h) = +nv +� +i=1 +σ +� +(2v − 1) ⊙ (a + Wh) +� +i. +(3.4) +3.1.1 +Optimizing an RBM +Due to the intractability of the partition function, the model cannot be solved exactly in +general, thus we resort to other methods to optimize it such as likelihood maximization +via gradient descent. For data set distribution pdata and parameters θ = (W, a, b), the +log-likelihood is given by +ℓ(θ) = +� +v +pdata(v) log p(v) += +� +v +pdata(v) log +� 1 +Z +� +h +e−E(v,h) +� +, +(3.5) +with gradients (see Appendix B.2 for derivation) +∂wijℓ(θ) = ⟨vihj⟩data − ⟨vihj⟩model, +∂aiℓ(θ) = ⟨vi⟩data − ⟨vi⟩model, +∂bjℓ(θ) = ⟨hj⟩data − ⟨hj⟩model. +(3.6) +The part of the gradient under the data set distribution is referred to as the positive +phase, and the part under the model distribution is referred to as the negative phase. It +is trivial to compute the expectation values in the positive phase, but not so much in the +negative phase because p(v) cannot be sampled directly. +11 + +h1 +h2 +hi +V1 +2 +ViFigure 3.2: Illustration of the n-step Gibbs sampling procedure. +In practice the negative phase expectation values are sampled using a Markov chain +Monte Carlo (MCMC) method. This is done via Gibbs sampling [13], which uses the con- +ditional probabilities p(h|v) and p(v|h). One starts with a visible vector and then samples +the hidden units conditioned on the visible units, followed by sampling the visible units +conditioned on the hidden units, and so forth until the desired thermalization threshold is +reached. The number of steps required to reach thermalization is model dependent and can +be estimated by analyzing the autocorrelations of a sample chain generated by the model. +The algorithm for Gibbs sampling is given in Algorithm 2 and illustrated in Fig. 3.2. The +algorithm is presented in a vectorized format for brevity. +Algorithm 2 Gibbs Sampling +1: procedure Gibbs(v, n, W, a, b) +2: +nv ← length(a) +3: +nh ← length(b) +4: +for k in 1 to n do +5: +r ∼ Uniform(0, 1, nh) +6: +h ← r < σ(b + W⊺v) +▷ σ, < applied element-wise +7: +r ∼ Uniform(0, 1, nv) +8: +v ← r < σ(a + Wh) +▷ σ, < applied element-wise +9: +end for +10: +return v +11: end procedure +The Uniform(a, b, n) function in Algorithm 2 produces a length n vector of uniform +i.i.d. random variables on the interval [a, b), and the < operator acts element-wise with +(true, false) �→ (1, 0). +The standard procedure for training an RBM is called n-step contrastive divergence +(CD-n), with n often taken to be one in practice [13]. The algorithm is detailed in Al- +gorithm 3, where one can see that n corresponds to how many Gibbs sampling steps are +between the positive and negative phase gradients. Applying the algorithm to a mini- +batch is essentially the same except that one divides the learning rate by the size of the +mini-batch to get a mini-batch averaged gradient. +3.2 +The Classical Market Generator +In The Market Generator [5] by Kondratyev and Schwarz, they show how an RBM can be +used as a generative model to produce synthetic market data. Specifically, they study how +it performs on the log returns of forex data for the same currency pairs we use here for the +time period 1999-2019. In this section we use some of the same metrics, as well as a couple +additional ones, so that we can verify our models achieve similar performance to theirs, as +12 + +ho +h2 +hr! +h1 +h2 +hnh +h1 +h2 +h.1 +p(ho /vo) +p(vi|ho) +p(hi/vi) +p(vn|hn-1) +p(v2|h1) +V2 +Unu +V2 +Uny +V1 +02 +Uny +V1 +Jo +Vo +V1 +V2 +VnAlgorithm 3 n-Step Contrastive Divergence (CD-n) +1: procedure CD(v+, n, W, a, b, η) +▷ v+ is a training sample +2: +h+ ← σ(b + W⊺v+) +▷ σ applied element-wise +3: +v− ← Gibbs(v+, n, W, a, b) +4: +h− ← σ(b + W⊺v−) +▷ σ applied element-wise +5: +W ← W + η(v+h⊺ ++ − v−h⊺ +−) +6: +a ← a + η(v+ − v−) +7: +b ← b + η(h+ − h−) +8: +return W, a, b +9: end procedure +well as give us a good reference point to compare our quantum models within Chapter 4. +3.2.1 +Models +We train and analyze four RBM models using variations of the filtered data set from Chap- +ter 2, each with slightly different preprocessing procedures denoted by: +• (B): base data set. +• (X): base data set transformed using Algorithm 1. +• (V): base data set with additional volatility indicators. +• (XV): base data set transformed using Algorithm 1 with additional volatility indica- +tors. +The models here have 64 (68 for ones with volatility indicators) visible units and 30 hidden +units (the same as in [5]) to act as regularized autoencoders. We use a mini-batch size of 10, +and an initial learning rate of 10−3 that decays by a factor of half every 1000 epochs after +epoch 5000 as defined in Appendix A.3, for a total of 104 epochs. We base the models on a +modified version of scikit-learn’s [14] BernoulliRBM class, which we forked 2 to implement +the ability to use a learning rate schedule with the BernoulliRBM class. +One of the drawbacks of the RBM is that it is not easy to track the training progress +for our use case, as the pseudolikelihood metric implemented by the scikit-learn package +is not necessarily a good proxy for our models’ performances. The Kullback-Leibler (KL) +divergence of pmodel from pdata, denoted DKL(pdata ∥ pmodel), is a suitable quantity to +track model performance as it measures the information loss associated with using the +model distribution pmodel to approximate the data set distribution pdata (more information +in Appendix A.5). However, due to the high number of epochs and the thermalization +requirements of samples generated by the RBM, this is not very feasible because generating +samples to compute the KL divergence every epoch significantly increases model training +times. Therefore, we only present the final results of the models. +3.2.2 +Results +Autocorrelations +As mentioned before, the classical RBM sampling method is based on an MCMC algorithm, +and thus samples produced via this method are autocorrelated. Therefore, we first examine +the autocorrelations to see how dependent samples are on the previous, so that we can get +an idea of how many Gibbs steps are needed between samples to consider them statistically +2https://github.com/cameronperot/scikit-learn/ +13 + +independent. We use Gibbs sample chains of length 108 for this analysis. More information +about the autocorrelation function and time can be found in Appendix A.4. +Fig. 3.3 shows the autocorrelation functions for the various models and currency pairs. +It is immediately clear that the autocorrelations fall off much sooner for the models trained +on the transformed data sets for all currency pairs. +This observation is confirmed by +examining the integrated autocorrelation times in Table 3.1. +It is not immediately clear why the transformed data sets lead to such shorter integrated +autocorrelation times, but this is a welcome trend as it means that less sampling steps are +required to reach thermalization. +Figure 3.3: +Autocorrelation functions of the RBM models. +Integrated Autocorrelation Times +Currency Pair +RBM (B) +RBM (X) +RBM (V) +RBM (XV) +EURUSD +295.7 +147.5 +267.4 +129.2 +GBPUSD +307.0 +173.2 +308.9 +121.6 +USDCAD +340.6 +120.0 +258.8 +91.3 +USDJPY +33.9 +46.7 +28.8 +36.7 +Table 3.1: +Integrated autocorrelation times of the RBM models. +The results in the rest of this section are derived from an ensemble of 100 sample sets +consisting of 104 samples each, and 104 Gibbs sampling steps between samples to ensure +thermalization. +14 + +EURUSD +GBPUSD +100 +100 +Autocorrelation +Autocorrelation +10-1 +10- +RBM (B) +RBM (X) +10-3. +10-3. +RBM (V) +RBM (XV) +10-4, +10-4 +100 +101 +102 +103 +100 +101 +104 +102 +103 +104 +Lag +Lag +USDCAD +USDIPY +100 +100 +Autocorrelation +Autocorrelation +10-1 +10-1 +10-2 +10- +2 +10-3. +10-3. +10-4, +10-4. +100 +101 +102 +103 +104 +100 +101 +102 +103 +104 +Lag +LagMarginal Distributions +To get an idea of how well the models perform, we examine the KL divergences of the +marginal distributions of each currency pair in Table 3.2. Here we observe that all models +reproduce the marginal distributions quite well, but the models trained on the trans- +formed data sets perform slightly better, particularly on the USDCAD marginal. +The +performance of the models on the marginal distributions is also visualized with Q-Q plots +in Fig. 3.4. More information on how the KL divergences are computed can be found +in Appendix A.5.1. +DKL(pdata || pmodel) +Currency Pair +RBM (B) +RBM (X) +RBM (V) +RBM (XV) +EURUSD +0.010 ± 0.001 +0.007 ± 0.001 +0.011 ± 0.002 +0.009 ± 0.001 +GBPUSD +0.007 ± 0.001 +0.006 ± 0.001 +0.011 ± 0.001 +0.007 ± 0.001 +USDCAD +0.017 ± 0.002 +0.007 ± 0.001 +0.015 ± 0.002 +0.008 ± 0.001 +USDJPY +0.008 ± 0.001 +0.007 ± 0.001 +0.010 ± 0.001 +0.009 ± 0.001 +Mean +0.010 ± 0.001 +0.007 ± 0.001 +0.011 ± 0.002 +0.008 ± 0.001 +Table 3.2: +KL divergences of the RBM models. The values are shown in the format mean +± one standard deviation from an ensemble of 100 sample sets consisting of 104 samples +each. +Correlations +The distribution is in a sense more than just the sum of its parts. Beyond learning the +marginal distributions, the models should also capture the correlations between the cur- +rency pairs. To verify this, we turn to the correlation coefficients in Table 3.3 to see how +well the models capture the correlations. We find that the models reproduce the structure +of the correlation coefficients reasonably well, with the models trained on the transformed +data sets encoding more of the behavior. +Volatilities +Examining the historical volatilities in Table 3.4 confirms the models can produce synthetic +data with similar volatilities to the training data set, albeit marginally higher in all cases. +Tails +It is extremely important for the models to learn the tail events because these play a +crucial role in financial risk management. The models trained on the transformed data +sets reproduce the lower tails a little better for most currency pairs, but overestimate some +of the upper tails. It is difficult to say overall if one model performs better than another +here, as it really depends on what one wants to do with the generated data. +We also study the tail concentration functions (see Appendix A.6 for definitions and in- +terpretations) between currency pairs in Fig. 3.5. Here we see that all models perform quite +well for the most part except for a few of the extreme regions in the EURUSD/GBPUSD, +EURUSD/USDJPY, and GBPUSD/USDJPY plots. +15 + +Figure 3.4: Log return Q-Q plots of the RBM models for each currency pair. Note that +these plots only use the same number of samples as the size of the training data set (5165), +and thus are not entirely representative of the models’ performances. +Conditional Sampling +For the data sets with additional volatility indicators, we have the ability to condition on +these indicators to sample from a specific volatility regime. This is useful, for example, if +we are trying to generate real-world data that fits the current volatility landscape. +This leads us to look at the conditional volatilities, i.e., seeing how well the models +reproduce the volatilities from the two volatility regimes. Laid out in Table 3.6, we observe +that the samples produced by the RBMs have slightly lower (higher) volatilities in the high +(low) regime, but are overall in good agreement with the data set. +3.2.3 +Summary +The classical RBM results presented in this section are in line with those obtained by +Kondratyev and Schwarz in [5], and the differences can likely be accounted for by the +different data sets used in training (e.g., different sources, different filtering, etc.), model +16 + +RBM (B) +RBM (X) +RBM (V) +RBM (XV) +1e-2 EURUSD +1e-2 EURUSD +1e-2 EURUSD +1e-2 EURUSD +4 +4 - +4 +4 +2 +2 +2 +2 +Model +e +0 +d +a +0 +0 +a +0 +Moc +Mo +Mo +-2 +-2 +-2 +-2 +-4 - +-4-20　 +-4 -20 +-4-2024 +２ +4 +4 +-4 -2　 +２4 +０ +Data Set le-2 +Data Setle-2 +Data Setle-2 +Data Setle-2 +1e-2 GBPUSD +1e-2GBPUSD +1e-2GBPUSD +1e-2GBPUSD +4 - +4 +4 - +4 +2 +2 +2 +2 +e +a +0 +a +0 +a +0 +a +0. +Moc +Mo +Mo +Mo +-2 +-2 +-2 +-2 +-4七 +-4-2024 +-4-2０24 +-4-2０24 +-4-2024 +Data Setle-2 +Data Setle-2 +Data Setle-2 +Data Setle-2 +1e-2USDCAD +1e-2USDCAD +1e-2USDCAD +1e-2USDCAD +4 - +4 +4 - +4 +2 +2 +2 +2 + Model +e +0 +0 +a +0 +a +0 +Mo( +Mo +-2 +-2 +-2 +-2 +-4七 +-4 -2 0 +-4 -20 +-4 -2 +-4 -2 +２ +4 +4 +24 +2 +24 +Data Setle-2 +Data Setle-2 +Data Setle-2 +Data Setle-2 +1e-2 USDJPY +1e-2 USDJPY +1e-2 USDJPY +1e-2 USDJPY +4 +4 +4 +4 +2 +2 +2 +2 +e +e +e +0. +a +0 +a +0. +0. +Mo( +M +M +M +-2 +-2 +-2 +-2 +-4七 +-4-2024 +-4-2024 +-4-2024 +-4-2024 +Data Setle-2 +Data Setle-2 +Data Setle-2 +Data Setle-2Correlation Coefficients +Data Set +RBM (B) +Currency Pairs +Pearson +Spearman +Kendall +Pearson +Spearman +Kendall +EURUSD/GBPUSD +0.62 +0.62 +0.44 +0.48 ± 0.01 +0.53 ± 0.01 +0.38 ± 0.01 +EURUSD/USDCAD +-0.44 +-0.41 +-0.29 +-0.33 ± 0.01 +-0.34 ± 0.01 +-0.24 ± 0.01 +EURUSD/USDJPY +-0.26 +-0.30 +-0.21 +-0.21 ± 0.01 +-0.25 ± 0.01 +-0.17 ± 0.01 +GBPUSD/USDCAD +-0.42 +-0.37 +-0.26 +-0.31 ± 0.01 +-0.33 ± 0.01 +-0.22 ± 0.01 +GBPUSD/USDJPY +-0.14 +-0.21 +-0.15 +-0.15 ± 0.01 +-0.18 ± 0.01 +-0.13 ± 0.01 +USDCAD/USDJPY +0.00 +0.06 +0.04 +0.06 ± 0.01 +0.07 ± 0.01 +0.05 ± 0.01 +RBM (X) +RBM (V) +Currency Pairs +Pearson +Spearman +Kendall +Pearson +Spearman +Kendall +EURUSD/GBPUSD +0.56 ± 0.01 +0.59 ± 0.01 +0.42 ± 0.01 +0.48 ± 0.01 +0.54 ± 0.01 +0.38 ± 0.01 +EURUSD/USDCAD +-0.39 ± 0.01 +-0.39 ± 0.01 +-0.27 ± 0.01 +-0.34 ± 0.01 +-0.36 ± 0.01 +-0.25 ± 0.01 +EURUSD/USDJPY +-0.24 ± 0.01 +-0.29 ± 0.01 +-0.19 ± 0.01 +-0.20 ± 0.01 +-0.23 ± 0.01 +-0.16 ± 0.01 +GBPUSD/USDCAD +-0.36 ± 0.01 +-0.35 ± 0.01 +-0.24 ± 0.01 +-0.30 ± 0.01 +-0.33 ± 0.01 +-0.22 ± 0.01 +GBPUSD/USDJPY +-0.16 ± 0.01 +-0.20 ± 0.01 +-0.13 ± 0.01 +-0.14 ± 0.01 +-0.17 ± 0.01 +-0.12 ± 0.01 +USDCAD/USDJPY +0.05 ± 0.01 +0.07 ± 0.01 +0.05 ± 0.01 +0.05 ± 0.01 +0.07 ± 0.01 +0.05 ± 0.01 +RBM (XV) +Currency Pairs +Pearson +Spearman +Kendall +EURUSD/GBPUSD +0.54 ± 0.01 +0.59 ± 0.01 +0.42 ± 0.01 +EURUSD/USDCAD +-0.39 ± 0.01 +-0.38 ± 0.01 +-0.26 ± 0.01 +EURUSD/USDJPY +-0.22 ± 0.01 +-0.27 ± 0.01 +-0.19 ± 0.01 +GBPUSD/USDCAD +-0.36 ± 0.01 +-0.36 ± 0.01 +-0.24 ± 0.01 +GBPUSD/USDJPY +-0.16 ± 0.01 +-0.20 ± 0.01 +-0.13 ± 0.01 +USDCAD/USDJPY +0.05 ± 0.01 +0.07 ± 0.01 +0.05 ± 0.01 +Table 3.3: Correlation coefficients of the data set vs. samples generated by the RBM +models. The RBM values are shown in the format mean ± one standard deviation from +an ensemble of 100 sample sets consisting of 104 samples each. +Historical Volatilities +Currency Pair +Data Set +RBM (B) +RBM (X) +RBM (V) +RBM (XV) +EURUSD +9.78% +9.98% ± 0.11% +10.10% ± 0.10% +10.18% ± 0.11% +10.27% ± 0.10% +GBPUSD +8.98% +9.34% ± 0.11% +9.38% ± 0.12% +9.55% ± 0.12% +9.53% ± 0.11% +USDCAD +8.56% +8.98% ± 0.13% +9.01% ± 0.11% +9.29% ± 0.13% +9.12% ± 0.12% +USDJPY +10.02% +10.26% ± 0.13% +10.42% ± 0.14% +10.82% ± 0.16% +10.46% ± 0.12% +Table 3.4: Historical volatilities of the data set vs. samples generated by the RBM models. +The RBM values are shown in the format mean ± one standard deviation from an ensemble +of 100 sample sets consisting of 104 samples each. +hyperparameters, and the stochastic nature of the models. This further confirms that the +RBM is performant and can be used to generate synthetic data from distributions with +intricate structures, such as the correlations and volatilities seen here. +Overall it is difficult to say if one of the models performs better than the others, as it +depends on the desired use case, but the models trained on the transformed data sets do +yield lower KL divergence values and capture more of the correlations between currency +pairs. This offers evidence that the results might be able to be further improved through the +use of more advanced data preprocessing methods. We do not investigate these possibilities +any further though, given that this is not the main scope of this thesis. The results in this +section act mainly as a point of reference to compare the quantum models within the next +chapter. +17 + +Lower Tails (1st Percentile) +Currency Pair +Data Set +RBM (B) +RBM (X) +RBM (V) +RBM (XV) +EURUSD +-1.64% +-1.80% ± 0.04% +-1.68% ± 0.05% +-1.90% ± 0.05% +-1.76% ± 0.06% +GBPUSD +-1.47% +-1.59% ± 0.04% +-1.57% ± 0.05% +-1.63% ± 0.05% +-1.66% ± 0.06% +USDCAD +-1.40% +-1.54% ± 0.05% +-1.57% ± 0.06% +-1.59% ± 0.05% +-1.58% ± 0.06% +USDJPY +-1.70% +-2.03% ± 0.07% +-1.92% ± 0.06% +-2.17% ± 0.09% +-1.96% ± 0.06% +Upper Tails (99th Percentile) +Currency Pair +Data Set +RBM (B) +RBM (X) +RBM (V) +RBM (XV) +EURUSD +1.62% +1.59% ± 0.04% +1.84% ± 0.06% +1.70% ± 0.04% +1.81% ± 0.05% +GBPUSD +1.42% +1.45% ± 0.04% +1.54% ± 0.04% +1.53% ± 0.03% +1.57% ± 0.04% +USDCAD +1.51% +1.61% ± 0.04% +1.53% ± 0.05% +1.60% ± 0.05% +1.56% ± 0.05% +USDJPY +1.59% +1.56% ± 0.04% +1.60% ± 0.05% +1.61% ± 0.04% +1.61% ± 0.05% +Table 3.5: Lower and upper tails, i.e., 1st and 99th percentiles, of the data set vs. samples +generated by the RBM models. The RBM values are shown in the format mean ± one +standard deviation from an ensemble of 100 sample sets consisting of 104 samples each. +Conditional Volatilities +Low Regime +High Regime +Currency Pair +Data Set +RBM (V) +RBM (XV) +Data Set +RBM (V) +RBM (XV) +EURUSD +6.72% +7.67% ± 0.23% +7.60% ± 0.24% +13.04% +13.19% ± 0.31% +12.92% ± 0.27% +GBPUSD +6.67% +7.45% ± 0.21% +7.50% ± 0.22% +12.69% +12.13% ± 0.31% +11.61% ± 0.28% +USDCAD +6.05% +6.72% ± 0.22% +6.40% ± 0.21% +12.86% +12.53% ± 0.37% +12.14% ± 0.30% +USDJPY +7.36% +9.01% ± 0.32% +8.76% ± 0.27% +13.15% +12.63% ± 0.38% +12.41% ± 0.31% +Table 3.6: Conditional historical volatilities of the data set vs. samples generated by the +RBM models. The RBM values are shown in the format mean ± one standard deviation +from an ensemble of 100 sample sets consisting of 104 samples each. +18 + +Figure 3.5: Tail concentration functions of the data set vs. samples generated by the RBM +models. +19 + +EURUSD/GBPUSD +EURUSD/USDCAD +0.4 +0.6 +0.3 +0.4 +0.2 +Data Set +RBM (B) +0.2 +RBM (X) +0.1 +RBM (V) +RBM (XV) +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +EURUSD/USDIPY +GBPUSDUSDCAD +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +GBPUSD/USDIPY +USDCAD/USDIPY +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Chapter 4 +The Quantum Boltzmann Machine +4.1 +Theory +The Quantum Boltzmann Machine detailed here is based on the work in Quantum Boltz- +mann Machine by Amin et al. [15]. In this section we use spin eigenvalues +1 and −1 +rather than binary values 0 and 1, respectively, in order to maintain consistency with the +language of quantum mechanics. We start with the n-qubit Hamiltonian +H = − +n +� +i=1 +Γiσx +i − +n +� +i=1 +biσz +i − +n +� +i=1 +n +� +j=i+1 +wijσz +i σz +j , +(4.1) +where +σx +i = I⊗i−1 ⊗ σx ⊗ I⊗n−i, +σz +i = I⊗i−1 ⊗ σz ⊗ I⊗n−i, +(4.2) +with σx and σz being the Pauli x and z matrices, and I being the 2×2 identity matrix. We +denote the first nv qubits as the visible units and the last nh qubits as the hidden units, +thus we have a total of nv + nh = n qubits. +The system’s distribution is modeled by the density matrix +ρ = 1 +Z e−H, +(4.3) +where e−H = �∞ +n=0 +1 +n!(−H)n is the matrix exponential, and Z = tr(e−H) is the partition +function. The probability to observe the system in state |v, h⟩ is given by +p(v, h) = tr(|v, h⟩ ⟨v, h| ρ), +(4.4) +and if we define the projection operator +Λv = |v⟩ ⟨v| ⊗ I⊗nh, +(4.5) +then the marginal probability to measure the visible units in state |v⟩ is given by +p(v) = tr(Λvρ). +(4.6) +Using the probabilities above we can obtain the log-likelihood, which for data set dis- +tribution pdata and parameters θ = (W, a, b) is +ℓ(θ) = +� +v +pdata(v) log tr(Λvρ), +(4.7) +where � +v denotes the sum over all possible configurations of v. +20 + +4.1.1 +Optimizing a QBM +When optimizing a QBM, it is preferable to maximize the lower bound of the log-likelihood +rather than maximizing the log-likelihood itself. The reason for this is that the partial +derivative of the log-likelihood with respect to the parameters has a term which is compu- +tationally expensive to compute, as discussed in Appendix C.1. The lower bound of the +log-likelihood is given by (see Appendix C.2 for derivation) +˜ℓ(θ) = +� +v +pdata(v) log tr(ρv), +(4.8) +where we have what is referred to as the clamped Hamiltonian, which for a given visible +vector v is +Hv = ⟨v|H|v⟩ , +(4.9) +with corresponding clamped density matrix +ρv = 1 +Zv +e−Hv, +(4.10) +and Zv = tr(e−Hv). This is called clamped because the visible qubits are held to the +classical state of the visible vector v. +The associated derivatives with respect to the parameters of the lower bound are given +by (see Appendix C.3 for derivation) +∂wij ˜ℓ(θ) = ⟨σz +i σz +j ⟩data − ⟨σz +i σz +j ⟩model, +∂bi ˜ℓ(θ) = ⟨σz +i ⟩data − ⟨σz +i ⟩model, +(4.11) +where ⟨ · ⟩data is the expectation value with respect to the data set, and ⟨ · ⟩model is the +expectation value with respect to the original density matrix. +If connections are restricted within the hidden layer, then the hidden unit probabili- +ties are independent in the positive phase and can be computed easily, as shown in Ap- +pendix C.3. This leads to positive phase expectation values of +⟨σz +i ⟩data = +� +v +pdata(v)vi, i ∈ Iv, +⟨σz +i ⟩data = +� +v +pdata(v) b′ +i(v) +Di(v) tanh +� +Di(v) +� +, i ∈ Ih, +⟨σz +i σz +j ⟩data = +� +v +pdata(v)vivj, i, j ∈ Iv, +⟨σz +i σz +j ⟩data = +� +v +pdata(v)vi +b′ +j(v) +Dj(v) tanh +� +Dj(v) +� +, i ∈ Iv, j ∈ Ih, +(4.12) +where b′ +i(v) = bi + (W⊺v)i, Di(v) = +� +Γ2 +i + b′ +i(v)2, Iv = {1, . . . , nv} represents the visible +qubit indices, and Ih = {nv + 1, . . . , n} represents the hidden qubit indices. +4.1.2 +Quantum Annealing +Quantum annealing, also known as adiabatic quantum computing, is a branch of quantum +computing that is based on the adiabatic theorem, which in the (translated) words of +Born and Fock [16]: "A physical system remains in its instantaneous eigenstate if a given +perturbation is acting on it slowly enough and if there is a gap between the eigenvalue +21 + +and the rest of the Hamiltonian’s spectrum." This can be achieved by implementing a +Hamiltonian of the form [17] +H(s) = A(s)Hinitial + B(s)Hfinal, +(4.13) +where s ∈ [0, 1]. For a linear anneal schedule s(t) = t/ta, where ta is the annealing time. +Hinitial is the initial Hamiltonian which describes the system at s = 0 and is responsible +for introducing quantum fluctuations. Hfinal is the final Hamiltonian which describes the +system at s = 1 and is responsible for encoding the problem defined by the user. +The functions A(s) and B(s) must be such that they satisfy the relations +A(0) ≫ B(0), +A(1) ≪ B(1). +(4.14) +In essence, a quantum annealer starts in the ground state of the initial Hamiltonian, +then slowly evolves the system over time so that it remains in the instantaneous ground +state. By the time the annealing process is completed, the Hamiltonian is just that of +the problem, and if the system evolved adiabatically, then it should have remained in the +instantaneous ground state. Therefore, when the qubits are measured at the end, they +should correspond to a low energy solution of the final Hamiltonian. +D-Wave Quantum Annealer +D-Wave quantum annealers implement a time-dependent Hamiltonian of the form [18] +H(s) = A(s) +� +− +n +� +i=1 +σx +i +� ++ B(s) +� +n +� +i=1 +hiσz +i + +n +� +i=1 +n +� +j=i+1 +Jijσz +i σz +j +� +. +(4.15) +From this we see the initial Hamiltonian has the ground state where all qubits are aligned +in the x-direction, i.e., |+⟩⊗n, which corresponds to an equal superposition of all possible +states in the computational basis. The final Hamiltonian corresponds to the Ising model +described by the hi and Jij values. +The quantum processing unit (QPU) is made up of superconducting qubits under the +influence of external magnetic fluxes [17] that change the Hamiltonian from the initial to +the final over the duration of the annealing process. These qubits are arranged in a graph +structure similar to that seen in Fig. 4.1. The default anneal schedule for the D-Wave +Advantage 4.1 is shown in Fig. 4.2 +Mapping the QBM to the D-Wave Quantum Annealer +As stated in [15], in order get a quantum annealer to sample from a quantum Boltzmann +distribution, one would need to freeze the evolution at some point s∗ during the annealing +process and then perform the measurements. The authors go on to say that this can be done +in practice using a nonuniform s(t) that anneals slowly in the beginning, then quenches +the system (completes the annealing as fast as possible) at the freeze-out point s∗, if s∗ +is in the quasistatic regime. In an earlier paper [21], Amin showed that the quasistatic +regime begins around 1 µs for the D-Wave 2000Q, so it should not be an issue to reach the +quasistatic regime for annealing times longer than 5 µs. +Because a quantum annealer is a real-world physical device, samples generated with it +have an associated temperature called the effective temperature. To be more specific, the +corresponding density operator is of the form +ρ(s, T) = 1 +Z e−βH(s), +(4.16) +22 + +Figure 4.1: +A lattice with 4 × 4 Pegasus unit cells (P4). The D-Wave Advantage QPU is +based on a lattice with 16 × 16 Pegasus unit cells (P16) [19]. +Figure 4.2: +Default anneal schedule of the D-Wave Advantage 4.1 with linear s(t) = t/ta +and ta = 20 µs [20]. +23 + +10 +A(s(t) +B(s(t) +8 +9 +[ZHO] : +4 +2 +0 +５ +0 +10 +15 +20 +t [μs]where β = 1/kT is the effective inverse temperature. In principle, β is an unknown quantity +and must be determined in order to effectively use the annealer to generate samples from +a quantum Boltzmann distribution. +Comparing the density operator of the QBM in Eq. (4.3) to the one in Eq. (4.16) at +the freeze-out point s∗, we find +Γi = βA(s∗), +bi = −βB(s∗)hi, +wij = −βB(s∗)Jij. +(4.17) +This enables us to map the QBM to the annealer if β can be determined to some reasonable +degree of accuracy. +Learning the Effective Inverse Temperature +There is the possibility to treat β as a learnable parameter rather than having to choose +a value empirically, as detailed by Xu and Oates in [22]. +The method is based on a +log-likelihood maximization approach leading to parameter updates of the form (see Ap- +pendix C.4 for how one arrives at this result) +∆ˆβ = ηˆβ +� +⟨E⟩data − ⟨E⟩model +� +, +(4.18) +where ˆβ = 1/k ˆT is the estimator of the effective inverse temperature, and ηˆβ is the asso- +ciated learning rate. We must note though, that this approach is only valid for classical +Boltzmann distributions, but this fits our current use case as we will see in Section 4.2.1. +D-Wave Ocean SDK +D-Wave offers an easy-to-use Python package called Ocean SDK [23] to interact with their +Leap [1] cloud-based quantum annealing platform, which allows users to access various +quantum annealers and other solvers around the globe. +One of the most important steps in solving a problem using a D-Wave annealer is +finding an embedding, i.e., a mapping of the logical qubits to the physical qubits, and +the SDK offers a heuristic method to do so. If the problem cannot be directly embedded +(1:1 logical:physical qubits), then a cluster of physical qubits called a chain is created to +represent one logical qubit. Chains introduce added complexity into the problem, because +one then needs to tune the chain strength, i.e., the coupling constant between the qubits +in the chains. If the measured values of the qubits in a chain differ, this is called a chain +break, and the system will report back the majority vote of the measured values in the +chain. Therefore, it is best to avoid chains if possible, but they are often a necessary evil +for larger problems due to connectivity limitations. +Samples can be easily generated by the annealer using the sample_ising(h, J) +function which takes in the user-defined hi and Jij values and returns a sample set of +specified size (maximum 104). The returned sample set contains the sampled state vectors +(an array of shape (nsamples, n) with values ±1 corresponding to the qubit measurements), +their energies, and other information about the run. +It must be noted that for the purposes of using a D-Wave annealer for quantum Boltz- +mann sampling, one must disable autoscaling to properly estimate the effective tempera- +ture, as per Eq. (4.17). The sample_ising(h, J) function has the keyword argument +24 + +autoscale=True, which rescales the hi and Jij values by the factor [24] +rautoscale = max +� +max +� +max{hi} +max{hrange}, 0 +� +, max +� +min{hi} +min{hrange}, 0 +� +, +max +� max{Jij} +max{Jrange}, 0 +� +, max +� min{Jij} +min{Jrange}, 0 +�� +. +(4.19) +This is because the main use case of D-Wave annealers is to maximize the probability of +measuring the ground state, thus the problem is rescaled so that the hi and Jij values +fully utilize the allowed range of values, essentially decreasing the effective temperature; +therefore, we set autoscale=False to avoid this. For the Advantage 4.1 system the +allowed value ranges are hrange = [−4, 4] and Jrange = [−1, 1] [25]. +Generating More Robust Statistics +QPUs are not perfect, and sometimes specific qubits or parts of the chip might have readout +biases. To mitigate such issues, one can perform a gauge transformation on the problem. +If we have an n-qubit problem, then we can generate a random vector r ∈ {+1, −1}n which +allows us to change the submission to the solver without actually changing the underlying +problem. This is done by taking +hi → rihi, +Jij → rirjJij, +(s1, . . . , sn) → (r1s1, . . . , rnsn), +(4.20) +and then transforming the results back using the third relation above, where si is the +measured value of qubit i. +Previous Work in This Field +In recent years, a number of researchers have studied using D-Wave quantum annealers to +train Boltzmann machines [22, 26–33]. The most common approach is to train a classical +RBM with quantum assistance, i.e., using the annealer to generate the samples in the +negative phase rather than using Gibbs sampling. Classical RBMs trained with quantum +assistance are a special case of the QBM, i.e., when s∗ = 1 the problem reduces to a +classical RBM because lims→1 Γi = 0 in Eq. (4.17). +One thing that stands out the most about some of the previous research is that very +few discuss embeddings and anneal schedules, which as we will see in the next section are +important for getting the best possible performance out of the annealer. Therefore, we +aim to create a basic framework with which one can use to approach the problem of using +a D-Wave annealer to sample from a (quantum) Boltzmann distribution. +4.2 +12-Qubit Problem +In order to get a better understanding of how the QBM works, we study a small 12-qubit +problem that can be solved exactly. For this purpose we take a QBM with restrictions in +both the visible and hidden layers and train it using the log-likelihood lower bound maxi- +mization approach; we call this a bound-based quantum restricted Boltzmann machine, or +BQRBM for short. We configure the model with 8 visible and 4 hidden units to act as a +regularized autoencoder. +25 + +4.2.1 +Sampling From a Quantum Boltzmann Distribution +Before training the model, we first need to assess the Advantage 4.1’s ability to sample +from quantum Boltzmann distributions. To this end, we randomly generate the values of +hi and Jij from a normal distribution with µ = 0 and σ = 0.1, then use the KL divergence +to compare samples generated by the Advantage 4.1 with theoretical distributions. +Anneal Schedule Format +The A(s) and B(s) values for a D-Wave annealer are fixed and depend on the specific +system [20], but the Ocean SDK allows us to define a nonuniform s(t) using a list of (t, s) +tuples, which then determine the A(s(t)) and B(s(t)) curves. In this section we use what +we call pause-and-quench anneal schedules that +1. start at (t = 0, s = 0), +2. pause the system at (tpause, spause) for a duration of ∆pause, +3. quench the system at (tquench, squench) over a duration of ∆quench. +Thus, the anneal schedules provided to the solver are of the form +[(0, 0), (tpause, spause), (tquench, squench), (tquench + ∆quench, 1)], +(4.21) +where +squench ≡ spause, +tpause = spause · trelative, +tquench = tpause + ∆pause. +(4.22) +An annotated example of a custom pause-and-quench anneal schedule with squench = 0.55, +trelative = 20 µs, and ∆pause = 10 µs is given in Fig. 4.3. +The minimum quench duration ∆quench is a function of squench and is limited by the +system’s fastest anneal rate αquench +∆quench(squench) = 1 − squench +αquench +. +(4.23) +The Advantage 4.1 system allows a maximum of αquench = 2 µs−1 [24]. +Verifying the Distribution +We use the KL divergence DKL(ptheory ∥ psamples) to compare the probabilities of the ener- +gies computed from the samples returned by the Advantage 4.1 with the theoretical energy +distributions for s = 0.01, 0.02, . . . , 1 and T = 10−3, 2, 4, . . . , 200 mK, which we visualize as +heatmaps in Fig. 4.4. More information on how the KL divergences are computed can be +found in Appendix A.5.1, how the density matrix (from which the theoretical distributions +are obtained) is computed in Appendix A.7, and the required constants in Appendix A.8. +In the right heatmap, where squench = 0.55, we observe a narrow band in which the +Advantage 4.1-generated samples closely resemble a quantum Boltzmann distribution, and +in fact the samples approximate multiple distributions depending on the effective temper- +ature. Marshall et al. present similar results using a D-Wave 2000Q in [34], in which +they discuss if the distribution returned by the annealer fits that of a quantum Boltzmann +distribution late in the anneal process when A(s∗)/B(s∗) ≪ 1, then the distribution at s∗ +should be close to a classical Boltzmann distribution, i.e., +e−βH(s∗) ≈ e−βB(s∗)Hfinal. +(4.24) +26 + +Figure 4.3: +Example of a custom pause-and-quench anneal schedule for the D-Wave +Advantage 4.1 [20]. Annotations indicate the points (t, B(s(t))), as well as the periods +over which the annealing is paused and quenched. +This in turn means that not only is there one optimal s∗ and effective temperature which +models the distribution, but rather a set of them corresponding to a family of distributions +for which βB(s∗) is constant. Therefore, this explains the streak pattern in the heatmaps. +Furthermore, we observe that the left heatmap, where squench = 0.25, is quite similar to +the right one where squench = 0.55, but with higher KL divergence values and temperatures. +This indicates that quenching at squench = 0.25 produces samples that are distributed more +as a classical Boltzmann distribution, and that we cannot generate samples from quantum +Boltzmann distributions with s∗ ⪅ 0.45, at least not with the anneal schedules we use here. +It must also be noted that the effective temperature corresponding to the classical +Boltzmann distribution (s∗ = 1) is significantly higher than that of the D-Wave temper- +ature of TDW = 15.4 ± 0.1 mK [1]1. +It is not entirely clear exactly why the effective +temperature of the distribution is so much higher than the device temperature, but in [34] +they give several possible reasons, including the discrepancy between the temperature of +the device and the qubits, fluctuations in the temperature while annealing, and control +errors masquerading as higher temperatures. In principle, higher effective temperatures +are unwanted because they shrink the range of allowed values for the weights and biases +as per Eq. (4.17), but there is not much one can do about this. +From analysis of the heatmaps and the fact that we cannot produce distributions with +s∗ ⪅ 0.45, we conclude that nontrivial dynamics occur while the system is quenching, +i.e., the system cannot quench fast enough. It is difficult to compare directly since the +2000Q is a different system than the Advantage 4.1 we study here, but in [34] they also +allude to the possibility of nontrivial dynamics occurring. The 2000Q allows for quenching +with αquench = 1, which is only a factor of two smaller than that of the Advantage 4.1. +Therefore, if as supposed in [34] that the quench is not fast enough, then likely such a +small difference in how fast the system can be quenched would not drastically change the +1Temperature obtained from the system properties in the Leap interface. +27 + +Squench = 0.55, trelative = 20 μs, Apause = 10 μs +10 +A(s(t) +B(s(t) +8 +(tquench + △quench, B(1) +6 +(tquench, B(Squench)) +[GHz] +4 +(tpause, B(Spause)) +[(0, B(0)) +△quench +0 +0 +5 +10 +15 +20 +[sr] Figure 4.4: +Heatmaps of DKL(ptheory ∥ psamples) comparing the distribution produced by +samples from the D-Wave Advantage 4.1 to a set of theoretical QBM distributions for two +different quench points using embedding 10. The dashed lines represent the optimal values +of B(s)/T = constant, computed by taking the value of T which produces the lowest KL +divergence for each s ≥ 0.5. Data represents an ensemble average over 10 random gauge +sample sets consisting of 104 samples each. +results. +If we take a second to think about it, the qubits are oscillating at a frequency in terms of +gigahertz. This means that a quench duration of a few hundred nanoseconds still allows for +a number of oscillations in the qubits, which is likely enough time for nontrivial dynamics +to take place. It would be interesting to verify via simulation how fast a quench must be +in order to freeze out the distribution at the desired point s∗. +We conclude that we are unable to reliably generate arbitrary quantum Boltzmann +distributed samples using the Advantage 4.1 system. +Therefore, for the remainder of +this thesis we focus on training models with s∗ = 1 using classical Boltzmann distributed +samples generated by the Advantage 4.1, also enabling us to use the aforementioned method +of learning the effective temperature. +Choosing an Embedding +We compare 10 different heuristically generated embeddings based on how well they ap- +proximate the desired distribution. +In this embedding comparison, we use only direct +embeddings (no chains), so the embeddings only differ by the location of the qubits on the +chip, and pause-and-quench anneal schedules with trelative = 20 µs and ∆pause = 0 µs. +It is difficult to compare the heatmaps of all embeddings and quench points due to +the higher dimensionality of the data, so we take the minimum KL divergence over s and +T, and plot it as a function of squench in Fig. 4.5. We immediately see how varied the +results are depending on the embedding and quench point, highlighting the importance of +choosing a good embedding and anneal schedule. +Our findings indicate that embedding 10 is likely a good choice because it produces the +best results at squench = 0.55. The rest of the results in this subsection use embedding 10. +Choosing an Anneal Schedule +With the chosen embedding we want to see if there is a way in which we can alter the +anneal schedule to further reduce the KL divergence. +We start with the same anneal +28 + +trelative = 20 μs, Apause = 0 μs +Squench = 0.25 +Squench = 0.55 +140 +0.20 +140 +0.20 +B(s)/T= 72.4 GHz/K +B(s)/T= 93.5 GHz/K +120 +120 +(sadesd +Tpw = 15.4 ± 0.1 mK +Tpw = 15.4 ± 0.1 mK +0.15 +0.15 +100 +100 +[mK] +80 +[mK] +80 +0.10 +0.10 +60 +60 +人 +40 +40 +0.05 +DKL( +0.05 +DkL( +20 +20 +0. +0. +0.00 +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +s +sFigure 4.5: +Comparison of mins,T +� +DKL(ptheory ∥ psamples) +� +for different embeddings and +squench values. Data represents an ensemble average over 10 random gauge sample sets +consisting of 104 samples each. Shaded regions represent one standard deviation. +schedule formula as before, except we introduce pausing before initiating the quench for +durations ∆pause = 0, 10, 100 µs, as well as the addition of trelative = 100 µs. +Fig. 4.6 illustrates that pausing and longer annealing times have little effect, and that +quenching in the range of squench ∈ [0.55, 0.6] produces the best results. With this in- +formation, we opt to use an anneal schedule with squench = 0.55, trelative = 20 µs, and +∆pause = 0 µs, as it offers a good balance between performance and QPU usage time. +4.2.2 +Training Data +Having verified that the Advantage 4.1 can indeed produce Boltzmann distributed samples +to some degree of accuracy, we proceed with training models using both a simulation +and the Advantage 4.1. +We randomly generate a training data set consisting of 1500 +samples, 1000 from a N(−2, 1) distribution and 500 from a N(3, 1) distribution, visualized +in Fig. 4.7. +4.2.3 +Simulation-based Model +The first step is training a model using a simulation in which the samples are generated +using the probabilities obtained from computing ρ exactly. Here we use a mini-batch size +of 10, s∗ = 1, and an initial learning rate of η = 0.1 with a schedule that exponentially +decays the learning rate every 10 epochs by a factor of 2 beginning at epoch 50 as defined +in Appendix A.3. The learning rate ηˆβ for the parameter ˆβ follows a similar schedule, +except it has a decay period of 20 as opposed to 10, to allow for more range of motion in +the ˆβ parameter later in the training process if the estimate needs to adapt more quickly +to a new effective β. +29 + +trelative = 20 μS, △pause = 0 μs +0.025 +Embedding 1 +Embedding 5 +Embedding 9 +Embedding 2 +Embedding 6 +Embedding 10 +Embedding 3 +Embedding 7 +Average +0.020 +Embedding 4 +Embedding 8 +0.015 +0.010 +S +0.005 +0.000 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +SquenchFigure 4.6: +Comparison of mins,T +� +DKL(ptheory ∥ psamples) +� +for various pause-and-quench +anneal schedules using embedding 10. Data represents an ensemble average over 10 random +gauge sample sets consisting of 104 samples each. Shaded regions represent one standard +deviation. Some of the sample sets with longer annealing times and pause durations contain +less than 104 samples as to satisfy the maximum allowed run time of the D-Wave Advantage +4.1. +Figure 4.7: +Histogram of the training data set used in the 12-qubit problem. +30 + +0.025 +trelative = 20 μS, △pause = 0 μs +trelative = 100 μs, △pause = 0 μs +trelative = 20 μs, Apause = 10 μs +trelative = 100 μs, Apause = 10 μs +trelative = 20 μs, Apause = 100 μs +trelative = 100 μs, Apause = 100 μs +0.020 +0.015 +0.010 +0.005 +0.000 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Squench0.25 +0.20 +0.10 +0.05 +0.00 +4 +-2 +0 +N +4 +6 +xResults +Fig. 4.8 shows the results of training the simulation-based model on the aforementioned +data set. We use the KL divergence DKL(pdata ∥ pmodel) as a way to track the progress of +the training and get a read on how well the model learns the data set distribution, because +minimizing the KL divergence is equivalent to maximizing the log-likelihood [35]. The KL +divergence is computed at the end of every epoch using a sample set of size 104. In the +left plot of Fig. 4.8, we observe a clear trend of the KL divergence being minimized. The +learning curve reaches an optimal value after about 80 epochs, then remains steady for the +next 20 epochs until the end of training. +Figure 4.8: +Training results of the 12-qubit model trained using the simulation. On the +left is the KL divergence DKL(pdata ∥ pmodel) plotted against the epochs; each data point +was generated using 104 samples at the end of every epoch. On the right is the learned +temperature estimator ˆT plotted against the epochs, as well as the effective temperature +that the simulation was configured to generate samples at. +We designed the simulation such that we can set the effective β to any value we desire. +To verify that the model can learn an accurate value for the estimator ˆβ, we configure the +simulation to generate samples at an effective value of β = 0.5 GHz−1 (T ≈ 96 mK) and +initialize the model with a value of ˆβ = 1 GHz−1 ( ˆT ≈ 48 mK). The results in the right +plot of Fig. 4.8 confirm that it is able to learn a value of ˆβ close to the actual effective β. +Overall, the results show that the model can generate samples similar to the training +distribution reasonably well when trained using the simulation, i.e., the best case scenario. +Additionally, we are able to verify that the model can accurately learn an estimate of the +effective temperature. We use the results of this model as a baseline to compare the models +trained using the Advantage 4.1 in the next subsection with. +4.2.4 +D-Wave Advantage 4.1-based Model +Having successfully trained the 12-qubit BQRBM using samples generated via exact sim- +ulation, we move to switching the sample generation part to the Advantage 4.1. We take +the same hyperparameters as the simulation and an anneal schedule using squench = 0.55, +trelative = 20 µs, and ∆pause = 0 µs. We see in Fig. 4.4 that squench = 0.55 has an optimal +temperature of around 90 mK for s∗ = 1, thus we take ˆβ = 0.5 GHz−1 ( ˆT ≈ 96 mK) as +31 + +0.200 +100 +0.175 +90- +0.150 +80 +0.125 +[mK] +0.100 +70 +