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+LINEAR QUADRATIC REGULATION CONTROL FOR FALLING
+LIQUID FILMS
+OSCAR A. 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
+([email protected], [email protected], [email protected]).
+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). Radu Cim-
+peanu and Susana Gomes also acknowledge EPSRC support via grant EP/V051385/1.
+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.
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+

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+IEEE ROBOTICS AND AUTOMATION LETTERS. 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: [email protected])
+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:
+[email protected])
+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.
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+

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: [email protected]
+† E-mail: steff[email protected]
+© 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
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+MNRAS 000, 1–7 (2022)
+

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.
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+

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+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, Oxford, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Girardeau, Relationship between systems of impene-  trable bosons and fermions in one dimension, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' 1, 516 (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  [3] A.' 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=' Gogolin, A.' 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=' Nersesyan, and A.' 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=' Tsvelik,  Bosonization and Strongly Correlated Systems (Cam-  bridge University Press, Cambridge, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Giamarchi, Quantum Physics in One Dimension (Ox-  ford University Press, Oxford, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Cheianov and 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=' Zvonarev, Nonunitary  Spin-Charge Separation in a One-Dimensional Fermion  Gas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' 92, 176401 (2004), arXiv:cond-  mat/0308470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  [6] G.' 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=' Fiete and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Balents, Green’s Function for Mag-  netically Incoherent Interacting Electrons in One Dimen-  sion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content=' 93, 226401 (2004), arXiv:cond-  mat/0403744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dA0T4oBgHgl3EQfOv8t/content/2301.02164v1.pdf'}
+page_content='  [7] G.' 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'}
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@@ -0,0 +1,1048 @@
+arXiv:2301.04448v1  [gr-qc]  11 Jan 2023
+Emergent diffeomorphism invariance in toy models
+Hrvoje Nikoli´c
+Theoretical Physics Division, Rudjer Boˇskovi´c Institute,
+P.O.B. 180, HR-10002 Zagreb, Croatia
+e-mail: [email protected]
+January 12, 2023
+Abstract
+Conceptual difficulties in semiclassical and quantum gravity arise from dif-
+feomorphism invariance of classical general relativity.
+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. 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. A change of intuition may also induce new technical results and,
+hopefully, contribute to better understanding of semiclassical and quantum gravity.
+Acknowledgements
+The author is grateful to T. Juri´c for discussions. This work was supported by the
+Ministry of Science of the Republic of Croatia.
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+20
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+On the Complexity of the Two-Stage Majority Rule*
+Yongjie Yang
+Chair of Economic Theory, Saarland University, Saarb¨ucken, Germany
+[email protected]
+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
+
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+16
+

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+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<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)
+(x)
+(EaEb)e ∪ e
+(9)
++
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+(−1)n(e)
+(x)
+(EaPα)e ∪ e +
+�
+1≤α<β≤d2
+e=(PαPβ)
+(−1)n(e)
+(x)
+(PαPβ)e ∪ e
+(10)
+where n(e) is the number of edges ordered before e.
+Example 2.4. Consider a complete graph G = K6 with six vertices (see the left-hand image in Figure 5) and its
+enhanced state S = (s, c) ∈ C4,2(G) (see the right-hand image in Figure 5).
+The enhanced state S can be written as
+x
+12, 13, 23 | 46 |
+x
+5.
+
+6
+SO YAMAGATA
+1
+2
+3
+4
+5
+6
+x
+1
+x
+1
+2
+3
+4
+5
+6
+Figure 5. Complete graph G = K6 and enhanced state S = (s, c) ∈ C4,2(G)
+In this example, the differential ∂4,2 would be calculated as follows.
+∂4,2 �
+x
+12, 13, 23 | 46 |
+x
+5
+�
+=
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (14) +
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (16) −
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (24) −
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (26)
+−
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (34) −
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (36) −
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (45) +
+�
+x
+12, 13, 23 | 46 |
+x
+5
+�
+∪ (56)
+=
+x
+12, 13, 14, 23, 46 |
+x
+5 +
+x
+12, 13, 16, 23, 46 |
+x
+5 −
+x
+12, 13, 23, 24, 46 |
+x
+5 −
+x
+12, 13, 23, 26, 46 |
+x
+5
+−
+x
+12, 13, 23, 34, 46 |
+x
+5 −
+x
+12, 13, 23, 36, 46 |
+x
+5 −
+x
+12, 13, 23 |
+x
+45, 46 +
+x
+12, 13, 23 |
+x
+46, 56.
+For the later reference let us give a list of descriptions of S ∪ e · f below, where we omit color x for simplicity.
+(I) Ee
+a ∪ f =
+
+Ee
+a | E f
+b ≔
+�
+E1 | . . . | Ee
+a | . . . | E f
+b | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb itself;
+Ee, f
+a
+| ≔
+�
+E1 | . . . | Ee, f
+a
+| . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ee
+a itself;
+Ee
+a | (EbEc) f ≔
+�
+E1 | . . . | Ee
+a | . . . | (EbEc) f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb and Ec;
+�Ee
+aEb
+� f | ≔
+�
+E1 | . . . | �Ee
+aEb
+� f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ee
+a and Eb;
+Ee
+a | (EbPα)f ≔
+�
+E1 | . . . | Ee
+a | . . . | (EbPα)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb and Pα;
+�Ee
+aPα
+� f | ≔
+�
+E1 | . . . | �Ee
+aPα
+� f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ee
+a and Pα;
+Ee
+a |
+�
+PαPβ
+�f ≔
+�
+E1 | . . . | Ee
+a | . . . | Ed1 |
+�
+PαPβ
+� f | P1 | . . . | Pd2
+�
+if f =
+�
+PαPβ
+�
+∈ E(G);
+(II) (EaEb)e ∪ f =
+
+(EaEb)e | E f
+c ≔
+�
+E1 | . . . | (EaEb)e | . . . | E f
+c | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ec itself;
+�
+EaE f
+b
+�e | ≔
+�
+E1 | . . . |
+�
+EaE f
+b
+�e | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb itself
+�
+E f
+aEb
+�e | ≔
+�
+E1 | . . . |
+�
+E f
+aEb
+�e | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea itself
+(EaEb)e, f | ≔
+�
+E1 | . . . | (EaEb)e, f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f(� e) connects Ea and Eb;
+(EaEb)e | (EcEd)f ≔
+�
+E1 | . . . | (EaEb)e | . . . | (EcEd)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ec and Ed;
+(EaEb)e (EaEc) f | ≔
+�
+E1 | . . . | (EaEb)e (EaEc)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea and Ec;
+(EaEb)e (EbEc) f | ≔
+�
+E1 | . . . | (EaEb)e (EbEc)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb and Ec;
+(EaEb)e | (EcPα) f ≔
+�
+E1 | . . . | (EaEb)e | . . . | (EcPα) f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ec and Pα;
+(EaEb)e (EaPα)f | ≔
+�
+E1 | . . . | ((EaEb)e Pα)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea and Pα;
+(EaEb)e (EbPα)f | ≔
+�
+E1 | . . . | ((EaEb)e Pα)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb and Pα;
+(EaEb)e |
+�
+PαPβ
+�f ≔
+�
+E1 | . . . | (EaEb)e | . . . | Ed1 |
+�
+PαPβ
+� f | P1 | . . . | Pd2
+�
+if f =
+�
+PαPβ
+�
+∈ E(G);
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+7
+(III) (EaPα)e ∪ f =
+
+(EaPα)e | E f
+b ≔
+�
+E1 | . . . | (EaPα)e | . . . | E f
+b | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb itself;
+�
+E f
+a Pα
+�e | ≔
+�
+E1 | . . . |
+�
+E f
+aPα
+�e | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea itself;
+(EaPα)e, f | ≔
+�
+E1 | . . . | (EaPα)e, f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects (EaPα)e itself;
+(EaPα)e | (EbEc)f ≔
+�
+E1 | . . . | (EaPα)e | . . . | (EbEc)f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb and Ec;
+(EaPα)e (EaEb) f | ≔
+�
+E1 | . . . | (EaPα)e (EaEb) f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea and Eb;
+(EaPα)e |
+�
+EbPβ
+� f ≔
+�
+E1 | . . . | (EaPα)e | . . . |
+�
+EbPβ
+�f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Eb and Pβ;
+(EaPα)e �
+EaPβ
+� f | ≔
+�
+E1 | . . . | (EaPα)e �
+EaPβ
+� f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea and Pβ;
+(EaPα)e |
+�
+PβPγ
+� f ≔
+�
+E1 | . . . | (EaPα)e | . . . | Ed1 |
+�
+PβPγ
+� f | P1 | . . . | Pd2
+�
+if f =
+�
+PβPγ
+�
+∈ E(G);
+(EaPα)e �
+PαPβ
+�f | ≔
+�
+E1 | . . . | (EaPα)e �
+PαPβ
+�f | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f =
+�
+PαPβ
+�
+∈ E(G);
+(IV)
+�
+PαPβ
+�e ∪ f =
+
+E f
+a |
+�
+PαPβ
+�e ≔
+�
+E1 | . . . | E f
+a | . . . | Ed1 |
+�
+PαPβ
+�e | P1 | . . . | Pd2
+�
+if f connects Ea itself;
+(EaEb) f |
+�
+PαPβ
+�e ≔
+�
+E1 | . . . | (EaEb)f | . . . | Ed1 |
+�
+PαPβ
+�e | P1 | . . . | Pd2
+�
+if f connects Ea and Eb;
+�
+EaPγ
+�f |
+�
+PαPβ
+�e ≔
+�
+E1 | . . . |
+�
+EaPγ
+�f | . . . | Ed1 |
+�
+PαPβ
+�e | P1 | . . . | Pd2
+�
+if f connects Ea and Pγ;
+(EaPα)f �
+PαPβ
+�e | ≔
+�
+E1 | . . . | (EaPα) f �
+PαPβ
+�e | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea and Pα;
+�
+EaPβ
+� f �
+PαPβ
+�e | ≔
+�
+E1 | . . . |
+�
+EaPβ
+� f �
+PαPβ
+�e | . . . | Ed1 | P1 | . . . | Pd2
+�
+if f connects Ea and Pβ;
+�
+PαPβ
+�e |
+�
+PγPδ
+� f ≔
+�
+E1 | . . . | Ed1 |
+�
+PαPβ
+�e |
+�
+PγPδ
+�f | P1 | . . . | Pd2
+�
+if f =
+�
+PγPδ
+�
+∈ E(G);
+�
+PαPβ
+�e �
+PαPγ
+� f | ≔
+�
+E1 | . . . | Ed1 |
+�
+PαPβ
+�e �
+PαPγ
+�f | P1 | . . . | Pd2
+�
+if f =
+�
+PαPγ
+�
+∈ E(G);
+�
+PαPβ
+�e �
+PβPγ
+� f | ≔
+�
+E1 | . . . | Ed1 |
+�
+PαPβ
+�e �
+PβPγ
+�f | P1 | . . . | Pd2
+�
+if f =
+�
+PβPγ
+�
+∈ E(G);
+3. A splitting property of the chromatic homology
+In this section we show a splitting property of the chromatic homology using the combinatorial description of en-
+hanced states introduced in the previous section. The following proposition gives the generators of Ker (∂i, j : Ci, j(G) →
+Ci+1, j(G)) for any connected graph G.
+Proposition 3.1. Let G be a connected graph. The generators of Ker (∂i, j : Ci, j(G) → Ci+1, j(G)) consist of the
+following two types of elements.
+(I) For fixed (i1, . . ., ip, k1, . . ., kq), 1 ≤ i1 < · · · < it < · · · < ip ≤ d1, 1 ≤ k1 < · · · < kt′ < · · · < kq ≤ d2, p + q = j, and
+Eh, h = 1, . . ., d1 with �d1
+h=1 #Eh = i − 1
+(11)
+�
+e∈E(G)\�d1
+h=1 Eh
+(−1)n(e) �
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ e.
+(II) For fixed d1, d2 with d1 + d2 = j and �d1
+h=1 #Eh = i,
+(12)
+x
+E1 | . . . |
+x
+Ed1 |
+x
+P1 | . . . |
+x
+Pd2,
+where each Eh is full of edges, i.e., FEh = Eh holds for h = 1, . . ., d1.
+Proof. (I) Let us show that elements of the form (11) are in Ker ∂i, j. This can be written as
+�
+e∈E(G)\�d1
+h=1 Eh
+(−1)n(e) �
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ e
+(13)
+
+8
+SO YAMAGATA
+=
+�
+1≤a≤d1
+�
+e∈FEa \Ea
+(−1)n(e) (x)
+Ee
+a ∪ e +
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)
+(x)
+(EaEb)e ∪ e
+(14)
++
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+(−1)n(e)
+(x)
+(EaPα)e ∪ e +
+�
+1≤α<β≤d2
+e=(PαPβ)
+(−1)n(e)
+(x)
+(PαPβ)e ∪ e
+(15)
+Let us compute the boundary of each term of (14) and (15). In the rest of this computation, we omit the color x for
+simplicity, which does not affect any of the computations.
+∂i, j
+
+�
+1≤a≤d1
+�
+e∈FEa \Ea
+(−1)n(e) (x)
+Ee
+a ∪ e
+ =
+�
+1≤a≤d1
+�
+e∈FEa \Ea
+(−1)n(e)∂i, j
+� (x)
+Ee
+a ∪ e
+�
+(16)
+=
+�
+1≤a,b≤d1
+a�b
+�
+e∈FEa \Ea
+f∈FEb \Eb
+(−1)n(e)+n(f) �
+Ee
+a | E f
+b
+�
+∪ e · f
+(17)
++
+�
+1≤a≤d1
+�
+e, f∈FEa \Ea
+e� f
+(−1)n(e)+n(f) �
+Ee, f
+a
+|
+�
+∪ e · f
+(18)
++
+�
+1≤a,b,c≤d1
+b<c
+b,c�a
+�
+e∈FEa \Ea
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+Ee
+a | (EbEc) f �
+∪ e · f
+(19)
++
+�
+1≤a,b≤d1
+a�b
+�
+e∈FEa \Ea
+f∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)+n(f) ��Ee
+aEb
+�f |
+�
+∪ e · f
+(20)
++
+�
+1≤a,b≤d1
+a�b
+1≤α≤d2
+�
+e∈FEa \Ea
+f∈FEb ∧Pα\FEα
+(−1)n(e)+n(f) �
+Ee
+a | (EbPα) f�
+∪ e · f
+(21)
++
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa \Ea
+f∈FEa ∧Pα\FEa
+(−1)n(e)+n(f) �
+(Ee
+aPα) f |
+�
+∪ e · f
+(22)
++
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e∈FEa \Ea
+f=(PαPβ)
+(−1)n(e)+n(f) �
+Ee
+a |
+�
+PαPβ
+� f�
+∪ e · f
+(23)
+∂i, j
+
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e) (EaEb)e ∪ e
+ =
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)∂i, j ((EaEb)e ∪ e)
+(24)
+=
+�
+1≤a<b≤d1
+1≤c≤d1
+c�a,b
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEc \Ec
+(−1)n(e)+n(f) �
+(EaEb)e | E f
+c
+�
+∪ e · f
+(25)
++
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb \Eb
+(−1)n(e)+n(f) ��
+EaE f
+b
+�e |
+�
+∪ e · f
+(26)
++
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa \Ea
+(−1)n(e)+n(f) ��
+E f
+aEb
+�e |
+�
+∪ e · f
+(27)
++
+�
+1≤a<b≤d1
+�
+e, f∈FEa ∧FEb \FEa ∪FEb
+e� f
+(−1)n(e)+n(f) �
+(EaEb)e, f |
+�
+∪ e · f
+(28)
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+9
++
+�
+1≤a<b≤d1
+1≤c<d≤d1
+(a,b)�(c,d)
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEc ∧FEd \FEc ∪FEd
+(−1)n(e)+n(f) �
+(EaEb)e | (EcEd)f �
+∪ e · f
+(29)
++
+�
+1≤a<b≤d1
+1≤c≤d1
+c�a,b
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa ∧FEc \FEa ∪FEc
+(−1)n(e)+n(f) �
+(EaEb)e (EaEc) f |
+�
+∪ e · f
+(30)
++
+�
+1≤a<b≤d1
+1≤c≤d1
+c�a,b
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EaEb)e (EbEc) f |
+�
+∪ e · f
+(31)
++
+�
+1≤a<b≤d1
+1≤c≤d1
+c�a,b
+1≤α≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEc ∧Pα\FEc
+(−1)n(e)+n(f) �
+(EaEb)e | (EcPα)f �
+∪ e · f
+(32)
++
+�
+1≤a<b≤d1
+1≤α≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa ∧Pα\FEa
+(−1)n(e)+n(f) �
+(EaEb)e (EaPα)f |
+�
+∪ e · f
+(33)
++
+�
+1≤a<b≤d1
+1≤α≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb ∧Pα\FEb
+(−1)n(e)+n(f) �
+(EaEb)e (EbPα)f |
+�
+∪ e · f
+(34)
++
+�
+1≤a<b≤d1
+1≤α<β≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f=(PαPβ)
+(−1)n(e)+n(f) �
+(EaEb)e |
+�
+PαPβ
+� f�
+∪ e · f
+(35)
+∂i, j
+
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+(−1)n(e) (EaPα)e ∪ e
+
+(36)
+=
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+(−1)n(e)∂i, j ((EaPα)e ∪ e)
+(37)
+=
+�
+1≤a,b≤d1
+a�b
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEb \Eb
+(−1)n(e)+n(f) �
+(EaPα)e | E f
+b
+�
+∪ e · f
+(38)
++
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈Ea∧Pα\FEa
+f∈FEa \Ea
+(−1)n(e)+n(f) ��
+E f
+aPα
+�e |
+�
+∪ e · f
+(39)
++
+�
+1≤a≤d1
+1≤α≤d2
+�
+e, f∈FEa ∧Pα\FEa
+e� f
+(−1)n(e)+n(f) �
+(EaPα)e, f |
+�
+∪ e · f
+(40)
++
+�
+1≤a≤d1
+1≤b<c≤d1
+b,c�a
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EaPα)e | (EbEc)f �
+∪ e · f
+(41)
++
+�
+1≤a,b≤d1
+a�b
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)+n(f) �
+(EaPα)e (EaEb) f |
+�
+∪ e · f
+(42)
++
+�
+1≤a,b≤d1
+a�b
+1≤α,β≤d2
+α�β
+�
+e∈FEa ∧Pα\FEa
+f∈FEb ∧Pβ\FEb
+(−1)n(e)+n(f) �
+(EaPα)e |
+�
+EbPβ
+� f�
+∪ e · f
+(43)
+
+10
+SO YAMAGATA
++
+�
+1≤a≤d1
+1≤α,β≤d2
+α�β
+�
+e∈FEa ∧Pα\FEa
+f∈Ea∧Pβ\FEa
+(−1)n(e)+n(f) �
+(EaPα)e �
+EaPβ
+� f |
+�
+∪ e · f
+(44)
++
+�
+1≤a≤d1
+1≤α≤d2
+1≤β<γ≤d2
+β,γ�α
+�
+e∈FEa ∧Pα\FEa
+f=(PβPγ)
+(−1)n(e)+n(f) �
+(EaPα)e |
+�
+PβPγ
+�f �
+∪ e · f
+(45)
++
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e∈FEa ∧Pα\FEa
+f=(PαPβ)
+(−1)n(e)+n(f) �
+(EaPα)e �
+PαPβ
+�f |
+�
+∪ e · f
+(46)
++
+�
+1≤a≤d1
+1≤β<α≤d2
+�
+e∈FEa ∧Pα\FEa
+f=(PβPα)
+(−1)n(e)+n(f) �
+(EaPα)e �
+PβPα
+�f |
+�
+∪ e · f
+(47)
+∂i, j
+
+�
+1≤α<β≤d2
+e=(PαPβ)
+(−1)(n(e)) �
+PαPβ
+�e ∪ e
+
+(48)
+=
+�
+1≤α<β≤d2
+e=(PαPβ)
+(−1)(n(e))∂i, j ��
+PαPβ
+�e ∪ e
+�
+(49)
+=
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa \Ea
+(−1)n(e)+n(f) �
+E f
+a |
+�
+PαPβ
+�e�
+∪ e · f
+(50)
++
+�
+1≤a<b≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)+n(f) �
+(EaEb)f |
+�
+PαPβ
+�e�
+∪ e · f
+(51)
++
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa ∧Pα\FEa
+(−1)n(e)+n(f) ��
+EaPγ
+� f |
+�
+PαPβ
+�e�
+∪ e · f
+(52)
++
+�
+1≤a≤d1
+1≤α<β≤d2
+1≤γ≤d2
+γ�α,β
+�
+e=(PαPβ)
+f∈FEa ∧Pγ\FEa
+(−1)n(e)+n(f) �
+(EaPα) f �
+PαPβ
+�e |
+�
+∪ e · f
+(53)
++
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa ∧Pβ\FEa
+(−1)n(e)+n(f) ��
+EaPβ
+� f �
+PαPβ
+�e |
+�
+∪ e · f
+(54)
++
+�
+1≤α<β≤d2
+1≤γ<δ≤d2
+(α,β)�(γ,δ)
+e=(PαPβ), f=(PγPδ)
+(−1)n(e)+n(f) ��
+PαPβ
+�e |
+�
+PγPδ
+�f �
+∪ e · f
+(55)
++
+�
+1≤α<β≤d2
+1≤γ≤d2
+γ�α,β
+e=(PαPβ), f=(PαPγ)
+(−1)n(e)+n(f) ��
+PαPβ
+�e �
+PαPγ
+�f |
+�
+∪ e · f
+(56)
++
+�
+1≤α<β≤d2
+1≤γ≤d2
+γ�α,β
+e=(PαPβ), f=(PβPγ)
+(−1)n(e)+n(f) ��
+PαPβ
+�e �
+PβPγ
+� f |
+�
+∪ e · f
+(57)
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+11
+By the following computations we can see that
+∂i, j
+
+�
+e∈E(G)\�d1
+h=1 Eh
+(−1)n(e) �
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ e
+ = 0.
+(17) =
+�
+1≤a<b≤d1
+�
+e∈FEa \Ea
+f∈FEb \Eb
+(−1)n(e)+n(f) �
+Ee
+a | E f
+b
+�
+∪ e · f +
+�
+1<b<a≤d1
+�
+e∈FEa \Ea
+f∈FEb \Eb
+(−1)n(f)+n(e) �
+E f
+b | Ee
+a
+�
+∪ e · f
+= 0.
+(18) =
+�
+1≤a≤d1
+
+�
+e∈FEa \Ea
+�
+f∈FEa \Ea∪{e}
+(−1)n(e)+n(f) �
+Ee, f
+a
+|
+�
+∪ e · f +
+�
+f∈FEa \Ea
+�
+e∈FEa \Ea∪{ f}
+(−1)n(e)+n(f) �
+Ee, f
+a
+|
+�
+∪ f · e
+
+= 0.
+(19) + (25) =
+�
+1≤a<b<c≤d1
+�
+e∈FEa \Ea
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+Ee
+a | (EbEc) f�
+∪ e · f
++
+�
+1≤c<a<b≤d1
+�
+f∈FEc \Ec
+e∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)+n(f) �
+E f
+c | (EaEb)e�
+∪ e · f
++
+�
+1≤b<a<c≤d1
+�
+e∈FEa \Ea
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EbEc) f | Ee
+a
+�
+∪ e · f
++
+�
+1≤a<c<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEc \Ec
+(−1)n(e)+n(f) �
+(EaEb)e | E f
+c
+�
+∪ e · f
++
+�
+1≤b<c<a≤d1
+�
+e∈FEa \Ea
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EbEc) f | Ee
+a
+�
+∪ e · f
++
+�
+1≤a<b<c≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEc \Ec
+(−1)n(e)+n(f) �
+(EaEb)e | E f
+c
+�
+∪ e · f
+= 0
+(20) + (26) + (27) =
+�
+1≤a<b≤d1
+�
+e∈FEa \Ea
+f∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)+n(f) ��Ee
+aEb
+�f |
+�
+∪ e · f
++
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa \Ea
+(−1)n(e)+n(f) ��
+E f
+aEb
+�e |
+�
+∪ e · f
++
+�
+1≤b<a≤d1
+�
+e∈FEa \Ea
+f∈FEa ∧FEb \FEa ∪FEb
+(−1)n(e)+n(f) ��
+Ee
+bEa
+�f |
+�
+∪ e · f
++
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa
+(−1)n(e)+n(f) ��
+EaE f
+b
+�e |
+�
+∪ e · f
+= 0
+
+12
+SO YAMAGATA
+(21) + (38) =
+�
+1≤a,b≤d1
+a�b
+1≤α≤d2
+�
+e∈FEa \Ea
+f∈Eb∧Pα\FEb
+(−1)n(e)+n(f) �
+Ee
+a | (EbPα) f �
+∪ e · f +
+�
+1≤a,b≤d1
+a�b
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEb \Eb
+(−1)n(e)+n(f) �
+E f
+b | (EaPα)e�
+∪ e · f
+= 0
+(22) + (39) =
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa \Ea
+f∈FEa ∧Pα\FEa
+(−1)n(e)+n(f) ��Ee
+aPα
+�f |
+�
+∪ e · f +
+�
+1≤a≤d1
+1≤b≤d2
+�
+e∈Ea∧Pα\FEa
+f∈FEa \Ea
+(−1)n(e)+n(f) ��
+E f
+aPα
+�e |
+�
+∪ e · f
+= 0
+(23) + (50) =
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e∈FEa \Ea
+f=(PαPβ)
+(−1)n(e)+n(f) �
+Ee
+a |
+�
+PαPβ
+�f �
+∪ e · f +
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa \Ea
+(−1)n(e)+n(f) ��
+PαPβ
+�e | E f
+a
+�
+∪ e · f
+= 0
+(28) =
+�
+1≤a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+�
+f∈FEa ∧FEb \FEa ∪FEb ∪{e}
+(−1)n(e)+n(f) �
+(EaEb)e, f |
+�
+∪ e · f
++
+�
+1≤a<b≤d1
+�
+f∈FEa ∧FEb \FEa ∪FEb
+�
+e∈FEa ∧FEb \FEa ∪FEb ∪{ f}
+(−1)n(e)+n(f) �
+(EaEb)e, f |
+�
+∪ f · e
+= 0
+(29) =
+�
+1≤a<b<c<d≤d1
+�
+e∈FEa ∧FEb \Ea∪Eb
+f∈FEc ∧FEd \Ec∪Ed
+(−1)n(e)+n(f) �
+(EaEb)e | (EcEd) f�
+∪ e · f
++
+�
+1≤c<d<a<b≤d1
+�
+e∈FEa ���FEb \Ea∪Eb
+f∈FEc ∧FEd \Ec∪Ed
+(−1)n(e)+n(f) �
+(EcEd) f | (EaEb)e�
+∪ e · f
++
+�
+1≤a<c<b<d≤d1
+�
+e∈FEa ∧FEb \Ea∪Eb
+f∈FEc ∧FEd \Ec∪Ed
+(−1)n(e)+n(f) �
+(EaEb)e | (EcEd)f �
+∪ e · f
++
+�
+1≤c<a<d<b≤d1
+�
+e∈FEa ∧FEb \Ea∪Eb
+f∈FEc ∧FEd \Ec∪Ed
+(−1)n(e)+n(f) �
+(EcEd) f | (EaEb)e�
+∪ e · f
++
+�
+1≤c<a<b<d≤d1
+�
+e∈FEa ∧FEb \Ea∪Eb
+f∈FEc ∧FEd \Ec∪Ed
+(−1)n(e)+n(f) �
+(EaEb)e | (EcEd)f �
+∪ e · f
++
+�
+1≤a<c<d<b≤d1
+�
+e∈FEa ∧FEb \Ea∪Eb
+f∈FEc ∧FEd \Ec∪Ed
+(−1)n(e)+n(f) �
+(EcEd) f | (EaEb)e�
+∪ e · f
+= 0
+(30) =
+�
+1≤a<b<c≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa ∧FEc \FEa ∪FEc
+(−1)n(e)+n(f) �
+(EaEb)e (EaEc)f |
+�
+∪ e · f
++
+�
+1≤a<c<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa ∧FEc \FEa ∪FEc
+(−1)n(e)+n(f) �
+(EaEc) f (EaEb)e |
+�
+∪ f · e
++
+�
+1≤c<a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa ∧FEc \FEa ∪FEc
+(−1)n(e)+n(f) �
+(EcEa) f (EaEb)e |
+�
+∪ f · e
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+13
+= 0
+(31) =
+�
+1≤a<b<c≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EaEb)e (EbEc)f |
+�
+∪ e · f
++
+�
+1≤a<c<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EaEb)e (EcEb)f |
+�
+∪ e · f
++
+�
+1≤c<a<b≤d1
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) �
+(EcEb) f (EaEb)e |
+�
+∪ f · e
+= 0
+(32) + (41) =
+�
+1≤a<b<d1
+1≤c≤d1
+c�a,b
+1≤α≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEc ∧Pα\FEc
+(−1)n(e)+n(f) �
+(EaEb)e | (EcPα)f �
+∪ e · f
++
+�
+1≤a≤d1
+1≤b<c≤d1
+b,c�a
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEb ∧FEc \FEb ∪FEc
+(−1)n(e)+n(f) ��
+(EbEc) f | EaPα
+�e�
+∪ e · f
+= 0
+(33) + (34) + (42) =
+�
+1≤a<b≤d1
+1≤α≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEa ∧Pα\FEa
+(−1)n(e)+n(f) �
+(EaEb)e (EaPα) f |
+�
+∪ e · f
++
+�
+1≤a<b≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEb ∧FEa \FEb ∪FEa
+(−1)n(e)+n(f) �
+(EaEb) f (EaPα)e |
+�
+∪ e · f
++
+�
+1≤a<b≤d1
+1≤α≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f∈FEb ∧Pα\FEb
+(−1)n(e)+n(f) �
+(EaEb)e (EbPα) f |
+�
+∪ e · f
++
+�
+1≤b<a≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEι ∪Ea \FEι ∪FEa
+(−1)n(e)+n(f) �
+(EaEb)f (EbPα)e |
+�
+∪ e · f
+= 0
+(35) + (51) =
+�
+1≤a<b≤d1
+1≤α<β≤d2
+�
+e∈FEa ∧FEb \FEa ∪FEb
+f=(PαPβ)
+�
+(EaEb)e |
+�
+PαPβ
+�f �
+∪ e · f +
+�
+1≤a<b≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa ∧FEb \FEa ∪FEb
+�
+(EaEb) f |
+�
+PαPβ
+�e�
+∪ e · f
+= 0
+(40) =
+�
+1≤a≤d1
+1≤α≤d2
+�
+e∈FEa ∧Pα\FEa
+�
+f∈FEa ∧Pα\FEa ∪{e}
+(−1)n(e)+n(f) �
+(EaPα)e, f |
+�
+∪ e · f
++
+�
+1≤a≤d1
+1≤α≤d2
+�
+f∈FEa ∧Pα\FEa
+�
+e∈FEa ∧Pα\FEa ∪{ f}
+(−1)n(e)+n(f) �
+(EaPα)e, f |
+�
+∪ f · e
+= 0
+
+14
+SO YAMAGATA
+(43) =
+�
+1≤a<b≤d1
+1≤α<β≤d2
+�
+e∈Ea∧Pα\FEa
+f∈Eb∧Pβ\FEb
+(−1)n(e)+n(f) �
+(EaPα)e |
+�
+EbPβ
+�f �
+∪ e · f +
+�
+1≤b<a≤d1
+1≤β<α≤d2
+�
+e∈Ea∧Pα\FEa
+f∈Eb∧Pβ\FEb
+(−1)n(e)+n(f) ��
+EbPβ
+�f | (EaPα)e�
+∪ e · f
++
+�
+1≤a<b≤d1
+1≤β<α≤d2
+�
+e∈Ea∧Pα\FEa
+f∈Eb∧Pβ\FEb
+(−1)n(e)+n(f) �
+(EaPα)e |
+�
+EbPβ
+� f�
+∪ e · f +
+�
+1≤b<a≤d1
+1≤α<β≤d2
+�
+e∈Ea∧Pα\FEa
+f∈Eb∧Pβ\FEb
+(−1)n(e)+n(f) ��
+EbPβ
+� f | (EaPα)e�
+∪ e · f
+= 0
+(44) =
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEa ∧Pβ\FEa
+(−1)n(e)+n(f) �
+(EaPα)e �
+EaPβ
+� f |
+�
+∪ e · f +
+�
+1≤a≤d1
+1≤β<α≤d2
+�
+e∈FEa ∧Pα\FEa
+f∈FEa ∧Pβ\FEa
+(−1)n(e)+n(f) ��
+EaPβ
+�f (EaPα)e |
+�
+∪ e · f
+= 0
+(45) + (52) =
+�
+1≤a≤d1
+1≤α≤d2
+1≤β<γ≤d2
+β,γ�α
+�
+e∈FEa ∧Pα\FEa
+f=(PβPγ)
+(−1)n(e)+n(f) �
+(EaPα)e |
+�
+PβPγ
+� f�
+∪ e · f
++
+�
+1≤a≤d1
+1≤α<β≤d2
+1≤γ≤d2
+γ�α,β
+�
+e∈FEa ∧Pγ\FEa
+f=(PαPβ)
+(−1)n(e)+n(f) ��
+EaPγ
+� f |
+�
+PαPβ
+�e�
+∪ e · f
+= 0
+(46) + (53) =
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e∈FEa ∧Pα\FEa
+f=(PαPβ)
+(−1)n(e)+n(f) �
+(EaPα)e �
+PαPβ
+� f |
+�
+∪ e · f
++
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈Ea∧Pα\FEa
+(−1)n(e)+n(f) �
+(EaPα)f �
+PαPβ
+�e |
+�
+∪ e · f
+= 0
+(47) + (54) =
+�
+1≤a≤d1
+1≤β<α≤d2
+�
+e∈FEa ∧Pα\FEa
+f=(PβPα)
+(−1)n(e)+n(f) �
+(EaPα)e |
+�
+PβPα
+� f�
+∪ e · f
++
+�
+1≤a≤d1
+1≤α<β≤d2
+�
+e=(PαPβ)
+f∈FEa ∧Pβ\FEa
+(−1)n(e)+n(f) ��
+EaPβ
+�f |
+�
+PαPβ
+�e�
+∪ e · f
+= 0
+(55) =
+�
+1≤α<β<γ<δ≤d2
+(−1)n(e)+n(f) ��
+PαPβ
+�e |
+�
+PγPδ
+�f �
+∪ e · f +
+�
+1≤γ<δ<α<β≤d2
+(−1)n(e)+n(f) ��
+PγPδ
+� f |
+�
+PαPβ
+�e�
+∪ e · f
++
+�
+1≤α<γ<β<δ≤d2
+(−1)n(e)+n(f) ��
+PαPβ
+�e |
+�
+PγPδ
+� f �
+∪ e · f +
+�
+1≤γ<α<δ<β≤d2
+(−1)n(e)+n(f) ��
+PγPδ
+� f |
+�
+PαPβ
+�e�
+∪ e · f
++
+�
+1≤γ<α<β<δ≤d2
+(−1)n(e)+n(f) ��
+PγPδ
+�f |
+�
+PαPβ
+�e�
+∪ e · f +
+�
+1≤α<γ<δ<β≤d2
+(−1)n(e)+n(f) ��
+PαPβ
+�e |
+�
+PγPδ
+�f �
+∪ e · f
+= 0
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+15
+(56) + (57) =
+�
+1≤α<β<γ≤d2
+(−1)n(e)+n(f) ��
+PαPβ
+�e �
+PαPγ
+�f |
+�
+∪ e · f +
+�
+1≤α<γ<β≤d2
+(−1)n(e)+n(f) ��
+PαPβ
+�e �
+PαPγ
+�f |
+�
+∪ e · f
++
+�
+1≤α<β<γ≤d2
+(−1)n(e)+n(f) ��
+PαPβ
+�e �
+PβPγ
+�f |
+�
+∪ e · f +
+�
+1≤γ<α<β≤d2
+(−1)n(e)+n(f) ��
+PγPα
+�f |
+�
+PαPβ
+�e�
+∪ e · f
++
+�
+1≤α<γ<β≤d2
+(−1)n(e)+n(f) ��
+PαPγ
+�e �
+PβPγ
+�f |
+�
+∪ e · f +
+�
+1≤γ<α<β≤d2
+(−1)n(e)+n(f) ��
+PγPβ
+� f �
+PαPβ
+�e |
+�
+∪ e · f
+= 0
+(II) Since all components of the element of the form (12) are colored x and each component Eh, h = 1, . . ., d is full
+of edges, only bridges can be added. Thus, the element is obviously that of Ker ∂i, j(G).
+Conversely, let us consider S ∈ Ker ∂i, j. We show that there are only two cases for the S : (I) S is the sum of a
+finite number of enhanced states such that all of them share i − 1 edges and the position of color x in common; (II) all
+components E1, . . ., Ed1, P1 . . . , Pd2 of the S are colored x and they are full of edges; i.e., FEa = Ea holds for all a.
+(I) Given S ∈ Ker ∂i, j, let us find a finite number of enhanced states S h, h = 1, . . ., l such that S = S 1 +· · ·+S l and l
+is minimal as possible. For fixed 1 ≤ i1 < · · · < it < · · · < ip ≤ d1, 1 ≤ k1 < · · · < kt′ < · · · < kq ≤ d2 take the enhanced
+state S e of the form
+S e =
+�
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ e,
+and S of the form
+S =
+�
+e∈E(G)\�d1
+h=1 Eh
+(−1)n(e)S e,
+where �d1
+h=1 #Eh = i−1. In this case, we can see that S ∈ Ker ∂i, j by the former computations. Notice that the set E(G)\
+�d1
+h=1 Eh is the set of all edges which can be added to the enhanced state E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2.
+For the set B = {e ∈ E(G) | e connects two components which are colored x} the enhanced state S ∈ Ker ∂i, j is the sum
+of #
+�
+E(G) \ �d1
+h=1 Eh
+�
+− #B terms. Let us show that the number #
+�
+E(G) \ �d1
+h=1 Eh
+�
+− #B is minimal. Let us assume
+that the minimal number is less than #
+�
+E(G) \ �d1
+h=1 Eh
+�
+−#B, say l with l < #
+�
+E(G) \ �d1
+h=1 Eh
+�
+−#B. Then, there exist
+enhanced states S i1, . . . , S iM−l ∈ {S 1, . . ., S M} such that ∂i, j(S 1 + · · ·+ S M) = ∂i, j(S 1 + · · ·+ S M − (S i1 + · · ·+ S iM−l)) = 0,
+in particular, we have �M−l
+p=1 ∂i, j(S ip) = 0, where M = #
+�
+E(G) \ �d1
+h=1 Eh
+�
+− #B. On the other hand, non-vanishing terms
+exist in the boundary �M−l
+p=1 ∂i, j(S ip). Actually, for the set X = {e ∈ E(G) | S ip ∋ e, p = 1, . . ., M − l} the boundary
+∂i, j
+
+�
+1≤a≤d1
+�
+e∈(FEa \Ea)∩X
+(−1)n(e) (x)
+Ee
+a ∪ e
+ =
+�
+1≤a≤d1
+�
+e∈(FEa \Ea)∩X
+(−1)n(e)∂i, j
+� (x)
+Ee
+a ∪ e
+�
+has the terms
+�
+1≤a<b≤d1
+�
+e∈(FEa \Ea)∩X
+f∈FEb \Eb
+(−1)n(e)+n(f) �
+Ee
+a | E f
+b
+�
+∪ e · f +
+�
+1≤a<b≤d1
+�
+e∈(FEa \Ea)
+f∈(FEb \Eb)∩X
+(−1)n(e)+n(f)+1 �
+Ee
+a | E f
+b
+�
+∪ e · f � 0.
+Thus, l = #
+�
+E(G) \ �d1
+h=1 Eh
+�
+− #B is the minimal number of the terms of the sum S = S 1 + · · · + S l such that the
+element S is in Ker ∂i, j if all enhanced sates S h, h = 1, . . ., l share i − 1 edges and the position of color x.
+Now, let us fix f ∈ E(G) \ �d1
+h=1 Eh and consider S e denoted by
+S e =
+�
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ f · e ∈ Ci, j(G), e ∈ E(G) \
+d1
+�
+h=1
+Eh,
+where # �d1
+h=1 Eh = i − 2, and for fixed 1 ≤ i1 < · · · < it < · · · < ip ≤ d1, 1 ≤ k1 < · · · < kt′ < · · · < kq ≤ d2 components
+Eit, Pkt′ are colored x. Let us take an enhanced state T ∈ Ci, j(G) such that for some
+S e′ ∈
+S e
+������� S e =
+�
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ f · e, e ∈ E(G) \
+d1
+�
+h=1
+Eh
+
+
+16
+SO YAMAGATA
+the number of common edges of T and S e is i − 2. Specifically, assume T to be of the form
+T =
+�
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ f ′ · e,
+where f ′(� f) and e(� f ′) ∈ E(G) \ �d1
+h=1 Eh are fixed. Then, any term of ∂i, j(S e′) is of the form
+(−1)n(g) �
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ f · e′ · g,
+where g(� e, f) ∈ E(G) \ �d1
+h=1 Eh, while any term of ∂i, j(T) is of the form
+(−1)n(g) �
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+∪ f ′ · e · g.
+Since f ′ � f, these two cannot be canceled by each other. Thus, if there exists such T in the sum S = S 1 + · · · + S l ∈
+Ker ∂i, j, the sum should be of the form S = �l1
+h=1 S h + �l2
+k=1 Tk such that S h, h = 1, . . ., l1 and Tk, k = 1, . . ., l2 have
+i − 1 common edges and the position of color x respectively, and S h, Tk have i − 2 edges commonly for any h, k,
+whose construction needs to take more than #
+�
+E(G) \ �d1
+h=1 Eh
+�
+elements of Ci, j(G), which results in a contraction of
+the minimality of l.Thus, S 1, . . ., S l should have i − 1 edges in common.
+Let us next show that for S ∈ Ker ∂i, j we cannot choose finite enhanced states S h, h = 1, . . ., l such that S =
+S 1 + · · · + S l and S h, h = 1, . . ., l share i − 1 edges but not the position of color x. We show this fact in a constructive
+way. For arbitrarily fixed S 1 ∈ Ci, j(G), let us assume that there exist enhanced states S h, h = 1, . . ., l such that
+∂i, j(S 1 + S 2 + · · · + S l) = 0 holds. Choose S 1 ∈ Ci, j(G) of the form
+S 1 = . . . | Eτ | . . . |
+x
+Eit1 | . . . | Ea1 | . . . |
+x
+Eit2 | . . . | Ea2 | . . . |
+x
+Pkt′ | . . . | Pα | . . . ,
+where # �d1
+h=1 Eh = i, and for fixed 1 ≤ i1 < · · · < it < · · · < ip ≤ d1, 1 ≤ k1 < · · · < kt′ < · · · < kq ≤ d2 components
+Eit, Pkt′ are colored x. Notice that in the rest of the construction we do not care about the position of label x of the
+components Pkt′ for simplicity.
+Let us consider the term
+x
+�
+EτEit1
+�e of ∂i, j(S 1). To cancel the term we need to add S 2 ∈ Ci, j(G) of the form
+(A) : . . . |
+x
+Eτ | . . . |
+ˆx
+Eit1 | . . . | Ea1 | . . . |
+(x)
+Eit2 | . . . | Ea2 | . . . or
+(B) : . . . | Eτ | . . . |
+x
+Eit1 | . . . | Ea1 | . . . |
+ˆx
+Eit2 | . . . |
+x
+Ea2 | . . . .
+If we take S 2 of the form (A), the boundary ∂i, j(S 2) would have the term
+x
+�
+EτEit1
+�e, so the term is canceled in ∂i, j(S 1 −
+S 2). Then, to delete the term �EτEa1
+�e of ∂i, j(S 1) we need to add an enhanced state S 3 of the form
+(C) : . . . | Eτ | . . . |
+ˆx
+Eit1 | . . . | Ea1 | . . . |
+(x)
+Eit2 | . . . |
+x
+Ea2 | . . . or
+(D) : . . . | Eτ | . . . |
+x
+Eit1 | . . . | Ea1 | . . . |
+ˆx
+Eit2 | . . . |
+x
+Ea2 | . . . .
+If (C) is the case, the term
+�
+Eit1 Ea
+�e appears in ∂i, j(S 1 − S 2 − S 3), and we need to add an enhanced state S 4 ∈ Ci, j(G)
+such that the terms
+x
+�
+Eit1 Ea1
+�e and �EiτEa1
+�e are deleted in ∂i, j(S 1 − S 2 − S 3 − S 4), but such an enhanced state does
+not exist. On the other hand, if (D) is the case, the term
+x
+�
+EτEit1
+�e reappears. Thus, both approaches result in new
+terms–which cannot be deleted by adding new enhanced states– appearing every time we add enhanced states in such
+a way that any term of ∂i, j(S 1) is deleted, and thus, we cannot construct the element of Ker ∂i, j if S 2 is of the form (A).
+Next, let us take S 2 of the form (B). In this case, the boundary ∂i, j(S 2) would have the terms
+x
+�
+EτEit1
+�e and �EτEa1
+�e,
+so the same terms of ∂i, j(S 1) can be canceled in ∂i, j(S 1 − S 2). In a similar way if we take S h, 3 ≤ h ≤ p of the form
+. . . | Eτ | . . . |
+(x)
+Eit1 | . . . | Ea1 | . . . |
+(x)
+Eit2 | . . . |
+(x)
+Ea2 | . . . |
+ˆx
+Eith | . . . |
+x
+Eah−1 | . . .
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+17
+we can delete terms of the form
+x
+�
+EτEith
+�e and �EτEah
+�, h = 1, . . ., p. But again in this construction, the term(s) of the
+form
+(x)
+�EτEim
+�e for some m appears in ∂i, j(S 1 − S 2 − · · ·− S p). The term also cannot be deleted by adding new enhanced
+states.Thus, this approach is also inappropriate for the construction.
+Therefore, enhanced states S h, h = 1, . . ., l of the sum S = S 1 + · · · + S l ∈ Ker ∂i, j hold the property that they have
+i − 1 edges and the position of the color x commonly, and thus we obtain the element of the form of type (I).
+(II) Let us take S ∈ Ker ∂i, j of the form
+S 2 =
+�
+E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2
+�
+.
+A similar observation as the one that we cannot take enhanced states S h, h = 1, . . ., l such that S = S 1+· · ·+S l ∈ Ker ∂i, j
+and S h, h = 1, . . ., l share i − 1 edges but not the position of color x, can be applied if j < d1 + d2. Thus, we cannot
+take a finite number of enhanced states S 1, . . ., S l ∈ Ci, j(G) such that S = S 1 + · · · + S l ∈ Ker ∂i, j.
+Next, let us consider j = d1 + d2. Take an element S ∈ Ci,d1+d2(G) of the form
+x
+E1 | . . . |
+x
+Ed1 |
+x
+P1 | . . . |
+x
+Pd2.
+By the form of ∂i,s+t(S ) the terms (EaEb)e, (EaPα)e,
+�
+PαPβ
+�e are 0. To obtain the element of Ker ∂i,s+t for each
+component Ee
+a, a = 1, . . ., s should be non-existent; equivalently, all components of S are full of edges, i.e., FEa = Ea
+holds for all a.
+□
+Proposition 3.2. Let G be a graph and e be its edge that is not a bridge. Then, for any i, j, the connecting homomor-
+phism γi, j of the following diagram is a 0-map:
+0
+0
+0
+0
+Ker δi−1, j
+Ker di, j
+Ker ∂i, j
+0
+Ci−1, j(G/e)
+Ci, j(G)
+Ci, j(G − e)
+0
+0
+Ci, j(G/e)
+Ci+1, j(G)
+Ci+1, j(G − e)
+0
+coker δi−1, j
+coker di, j
+coker ∂i, j
+0
+0
+0
+0
+αi−1, j
+βi, j
+δi−1, j
+di, j
+∂i, j
+αi, j
+βi+1, j
+γi, j
+
+18
+SO YAMAGATA
+Proof. First, let us take S
+S =
+x
+E1 | . . . |
+x
+Ed1 |
+x
+P1 | . . . |
+x
+Pd2 ∈ Ker ∂i, j,
+where d1+d2 = j, �d1
+h=1 #Eh = i and Eh, h = 1, . . ., d1 are full of edges as components of an enhanced state of Ci, j(G−e)
+except for e as components of an enhanced state of Ci, j(G). Consider S ∪ e ∈ Ci, j(G). By definition βi, j(S ∪ e) = S and
+we have
+di, j(S ∪ e) =
+�
+1≤a≤d1
+�
+e∈FEa \Ea
+(−1)n(e) � x
+E1 | . . . |
+x
+Ee
+a | . . . |
+x
+Ed1 |
+x
+P1 | . . . |
+x
+Pd2
+�
+∪ e.
+Take an element S /e ∈ Ci−1, j(G/e) in such a way that
+S /e =
+�
+1≤a≤d1
+�
+e∈FEa \Ea
+(−1)n(e)
+� x
+E1 | . . . |
+x
+E/e
+a | . . . |
+x
+Ed1 |
+x
+P1 | . . . |
+x
+Pd2
+�
+∪ e,
+where
+x
+E/e
+p is a component defined by contracting e of
+x
+Ee
+p. Then, we have S /e ∈ Ci−1, j(G/e) such that αi−1, j(S /e) = S ∪e.
+By commutativity of the diagram, the element sending the given enhanced state S ∈ Ker ∂i, j by γi, j is 0 ∈ cokerδi−1, j.
+Next, let us take
+S =
+�
+e∈E(G)\�d1
+h=1 Eh
+(−1)n(f)(E1 | . . . |
+x
+Eit | . . . | Ed1 | P1 | . . . |
+x
+Pkt′ | . . . | Pd2) ∪ e ∈ Ker ∂i, j,
+where for fixed 1 ≤ i1 < · · · < it < · · · < ip ≤ d1 and 1 ≤ k1 < · · · < kt′ < · · · < kq ≤ d2 components Ei1, . . ., Eip,
+Pk1, . . . , Pkq, p + q = j are colored x. Consider S ∪ e ∈ Ci, j(G). By definition we have βi, j(S ∪ e) = S . By Proposition
+3.1 the enhanced state S ∪ e is also an element in Ker di, j. Then, again by commutativity of the diagram the element
+S ∈ Ker ∂i, j corresponds to 0 ∈ coker δi−1, j.
+□
+By Proposition 3.2 if e is not a bridge, then we have a short exact sequence
+(58)
+0 → Hi, j(G/e) → Hi+1, j(G) → Hi+1, j(G − e) → 0.
+for i, j.
+Next, consider a map
+ϕi+1, j : Ci+1, j(G − e) → Ci+1, j(G)
+that sends an enhanced state S ∈ Ci+1, j(G − e) to S ′ ∈ Ci+1, j(G) such that S and S ′ are the same form as graphs. Then,
+since e in not a bridge, if S ∈ Ker ∂i+1, j, then S ′ = ϕi, j(S ) ∈ Ker di+1, j. In particular, the maps ϕi, j induce sections
+¯ϕi+1, j : Hi+1, j(G − e) → Hi+1, j(G)
+of the sequence (58) for all i, j, and thus we can see that the sequences split for all i, j. The following is the main
+theorem of the present paper.
+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
+(59)
+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
+(60)
+0 → Hi(G/e) → Hi+1(G) → Hi+1(G − e) → 0
+for i.
+
+A SPLITTING PROPERTY OF THE CHROMATIC HOMOLOGY
+19
+4. Chromatic homology for the complete graph
+As an example, let us compute the chromatic homology of the complete graph Kn with n (n ≥ 4) vertices. Fix a
+vertex v0 ∈ V(Kn) and define the graph Gm as
+G0 = Kn
+Gm+1 = Gm \ e,
+where e ∈ E(Kn) contains v0 as its end vertex. We have the following lemma.
+Lemma 4.1. If m ≤ n − 2, the graph Gm has just one connected component.
+Proof. It suffices to show that to obtain two connected components we need to remove at least n − 1 edges. Let us
+assume Kn = Kn1 ∧ Kn2 for arbitrarily fixed n1, n2(≥ 1) with n1 + n2 = n. Then, to obtain two connected components,
+we need to remove at least # �Kn1 ∧ Kn2 \ E(Kn1) ∪ E(Kn2)� = n1n2 = n1(n − n1) edges.
+Consider
+n1(n − n1) − (n − 1) = (n1 − 1)(n − n1 − 1).
+Since n1 ≥ 1, n2 = n − n1 ≥ 1, the formula obviously must be (n1 − 1)(n − n1 − 1) ≥ 0. Thus, to obtain at least two
+connected components by removing some edges from Kn, the number of removed edges should be at least n − 1. In
+particular, the number of the connected components is always one when we remove any n1 ≤ n − 2 edges.
+□
+By Lemma 4.1 it follows that when m ≤ n − 2, e is not a bridge, and thus we can apply Theorem 3.3 for the graphs
+Gm, 1 ≤ m ≤ n − 2. In particular, since Kn/e = Kn−1, we have the following proposition.
+Proposition 4.2. For 1 ≤ j ≤ n − 2, we have the following split exact sequence
+0 → Hi, j(Kn−1) → Hi+1, j(Gm) → Hi+1, j(Gm+1) → 0
+for i, j.
+By summing up by j we have the split exact sequence
+0 → Hi(Kn−1) → Hi+1(Gm) → Hi+1(Gm+1) → 0
+for i.
+Thus, as a corollary of Proposition 4.2 we obtain a recursive description of the chromatic homology of the complete
+graph.
+Theorem 4.3 (Conjecture 6.8 [9]). For n ≥ 4 the chromatic homology groups of a complete graph Kn are given
+recursively as
+(61)
+Hi(Kn) =
+
+Z{n}
+i = 0
+Hi−1(Kn−1)⊕(n−2) ⊕ Hi(Kn−1){1}
+1 ≤ i ≤ n − 2
+0
+i ≥ n − 1.
+Proof. To begin with, let us consider H0(Kn). By Proposition 3.1 we can see that H0(Kn) ≃ Ker
+�
+∂0 : C0(Kn) → C1(Kn)
+�
+is generated by only
+x
+P1 | . . . |
+x
+P j | . . . |
+x
+Pn. Thus, we have H0(Kn) = Z{n}. The case i ≥ n − 1 is also obvious. By
+Lemma 4.1 if we remove at least n − 1 edges, then there would exist two-component enhanced states which are of the
+form
+(x)
+E1 |
+(x)
+E2 in Ci(Kn). That is, Ci(Kn) contains enhanced states of the form
+(x)
+E and
+(x)
+EE′, and thus we have Ker ∂i, j = 0.
+Finally, let us show the case 1 ≤ i ≤ n − 2. By Proposition 4.2 we can write Hi(Kn) as follows:
+Hi(Kn) = Hi(G0) = Hi−1(Kn−1) ⊕ Hi(G1)
+Hi(G1) = Hi−1(Kn−1) ⊕ Hi(G2)
+Hi(G2) = Hi−1(Kn−1) ⊕ Hi(G3)
+· · · · · · · · ·· · · · · · · · · · · ·· · · · · ·
+Hi(Gn−3) = Hi−1(Kn−1) ⊕ Hi(Gn−2).
+
+20
+SO YAMAGATA
+Remark that since Gn−2 is a graph obtained by adding a pendant edge, which is an edge of G such that one of the end
+vertices has no edges, to the complete graph Kn−1, we have Hi(Gn−2) ≃ Hi(Kn−1){1}, and thus we obtain
+Hi−1(Kn−1)⊕(n−2) ⊕ Hi(Kn−1){1},
+which completes the proof.
+□
+Remark 4.4. Theorem 4.3 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.
+Remark 4.5. It would be interesting to generalize the Theorem 3.3 with the general algebra Z[x]/(xm), i.e., the chro-
+matic homology of graphs with colors in Z[x]/(xm). Such a generalization might lead to further developments in the
+study of chromatic homology, including insights into the conjectures given by Pabiniak et al. [15].
+References
+[1] V. Baranovsky and R. Sazdanovic, Graph homology and graph configuration spaces, Journal of Homotopy and Related Structures, 7, 223-235
+(2012).
+[2] M. B¨okstedt and E. Minuz, Graph cohomologies and rational homotopy type of configuration spaces, arXiv:1904.01452 [math.AT].
+[3] A. Chandler and R. Sazdanovic, A broken circuit model for chromatic homology theories, European Journal of Combinatorics, 104, 103538
+(2022).
+[4] M. Chmutov and S. Chmutov, and Y. Rong, Knight move for chromatic graph cohomology, European Journal of Combinatorics, 29 (1), 311-321
+(2008).
+[5] Z. Dancso and A. Licata, Odd Khovanov homology for hyperplane arrangements, Journal of Algebra, 436 (15), 102-144 (2015).
+[6] A. Hasegawa, Khovanonv homology of graph and quandles, Master’s thesis, Department of Mathematics, Hokkaido University (2020).
+[7] L. Helme-Guizon, J. Przytycki, and Y. Rong, Torsion in graph homology, Fundamenta Mathematicae 190 (1), 139-177 (2006).
+[8] L. Helme-Guizon and Y. Rong, Graph Cohomologies from Arbitrary Algebras, arXiv:0506023 [math.QA].
+[9] L. Helme-Guizon and Y. Rong, A categorification for the chromatic polynomial, Algebraic & Geometric Topology, 5(4), 1365-1388 (2005).
+[10] E. F. Jasso-Hernandez and Y. Rong, A categorification for the Tutte polynomial, Algebraic & Geometric Topology, 6(5), 2031-2049 (2006).
+[11] M. Khovanov, A categorification of the Jones polynomial, Duke Mathematical Journal, 101, 359-426 (2000).
+[12] I. Kriz, On the rational homotopy type of configuration spaces, Annals of Mathematics, 139(2), 227-237 (1994).
+[13] M. Loebl and I. Moffatt, The chromatic polynomial of fatgraphs and its categorification, Advances in Mathematics, 217 (4), 1558-1587 (2008).
+[14] A. M. Lowrance and R. Sazdanovic, Chromatic homology, Khovanov homology, and torsion, Topology and its Applications, 222, 77-99 (2017).
+[15] M. D. Pabiniak, JH. Przytycki, and R. Sazdanovi´c, On the first group of the chromatic cohomology of graphs, Geometriae Dedicata, 140, 19-48
+(2009).
+[16] JH. Przytycki, When the theories meet: Khovanov homology as Hochschild homology of links, Quantum Topology, 1(2), 93-109 (2010).
+[17] R. Sazdanovic and D. Scofield, Patterns in Khovanov link and chromatic graph homology, Journal of Knot Theory and Its Ramifications, 27
+(3), 1840007 (2018).
+[18] R. Sazdanovic and M. Yip, A categorification of the chromatic symmetric function, Journal of Combinatorial Theory, Series A, 154, 218-246
+(2018).
+[19] Z. Zhuang, On the homology theory for the chromatic polynomials, arXiv:2107.03671 [math.GT].
+Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka, 814-0180, Japan.
+Email address: [email protected]
+

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+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
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+(M. Inversi) Department Mathematik und Informatik, Universität Basel, Spiegelgasse 1,
+4051 Basel, Switzerland
+Email address: [email protected]
+(G. Stefani) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265,
+34136 Trieste (TS), Italy
+Email address: [email protected] or [email protected]
+

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+On particle dynamics near the Schwarzschild singularity
+A. Radosz
+Faculty of Basic Problems of Technology (Wroclaw),
+Wroclaw University of Science and Technology, 50-370Wroclaw, Poland∗
+A. V. Toporensky
+Sternberg Astronomical Institute, Lomonosov Moscow
+State University and Kazan Federal University,
+Kremlevskaya 18, Kazan 420008, Russia†
+O. B. Zaslavskii
+Department of Physics and Technology,
+Kharkov V.N. Karazin National University,
+4 Svoboda Square, Kharkov 61022, Ukraine‡
+The problem of the speed of the objects inside the Schwarzschild black hole is
+considered. The general result is that the value of the relative speed of the objects
+following their non-zero angular momentum trajectories, both of geodesic and non-
+geodesic character, when approaching the ultimate singularity, tends to the value of
+speed of light. There is only one exception when both objects move in the same plane
+and have parallel angular momenta. This outcome appears to have a deeper sense: it
+reflects the anisotropic character of the dynamics of interior of this particular black
+hole. The result in question means that near the singularity, collisions of two particles
+lead to an indefinitely large energy in the center of mass frame. Aforementioned
+properties have their counterpart in the phenomenon of an indefinitely large blueshift
+near the singularity.
+PACS numbers: 04.20.-q; 04.20.Cv; 04.70.Bw
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+‡Electronic address: [email protected]
+arXiv:2301.11651v1  [gr-qc]  27 Jan 2023
+
+2
+I.
+INTRODUCTION.
+The studies of the properties of the strong gravitational fields and in particular the
+properties of the black holes (BH) have got in a recent decade a significant theoretical but
+also experimental impact.
+The development of the theoretical interest has been mainly
+related to the BSW-like effect: two-particle collisions undergoing in the vicinity of the BH’s
+horizon would lead to the unbounded energy release [1]. The first ever picture of the BH
+namely, supermassive M-87 BH [2], gravitational waves emission following two BHs merger
+[3] and the intriguing temporarily varying radiation emission of the accretion disk of the
+sources of the strong gravitational field [4] - [7] are the most important recent experimental
+aspects of the presence of the strong gravitational fields.
+In this broad landscape of the BHs associated phenomena, there has been the steadily
+growing interest in the studies of the BH’s interior.
+The progress in the understanding
+(mostly kinematical) phenomena undergoing inside the Schwarzschild’s black hole has been
+based on two features. One is a particular property of the black hole’s horizon: the speed of
+a test particle following geodesic and approaching the horizon tends to that of light, V → 1
+in the static frame. The other one is an anisotropic character BH’s interior: exterior of the
+Schwarzschild black hole is a static and isotropic but its interior, also called T-”sphere” [8] is
+anisotropic and dynamic, whose spatial-like section is a hypercylinder of R1 ×S2 symmetry.
+It is expanding longitudinally, along homogeneity direction R and contracting transversely,
+perpendicularly to this direction in the angular coordinates of the sphere S2 (see e.g. [8], [9]).
+Interplay of these two features has allowed to find an interpretation of variety of seemingly
+contradicting outcomes. One can invoke the case of a test object radially falling towards
+the horizon, whose speed increases to 1 with respect to the static frame, but after crossing
+the horizon the speed turns out to be decreasing to zero [10], as if the test particle being
+hampered inside horizon.
+It is also worth mentioning that this non-monotonic behavior of a 3-velocity is a spe-
+cific property of the observers, static outside and resting inside horizon [10] and does not
+appear in the frame connected with Lemaˆıtre coordinates. Indeed, it was shown that the
+3-velocity with respect to the Lemaˆıtre frame at a horizon can take any value from 0 to
+1, and this velocity decreases in a monotonic way (if it is not equal to zero identically) in
+the Schwarzschild black hole reaching 0 at a singularity (if the angular momentum is zero)
+
+3
+[11]. Thus some kinematical properties of particles falling into a black hole can be more
+easily interpreted in a coordinate system, different from the one offered by Schwarzschild
+coordinates (after mutual exchange of a spatial and temporal coordinates). Another nice
+feature of the frame in question is that hypersurfaces of a constant Lemaˆıtre time are flat,
+so, for example, the proper distance between two points at the same radius is simply the
+difference in their static coordinates r which is useful for visualizing the properties of a free
+fall. The price paid for it consists in loosing some symmetry properties of the space-time
+under study, since the metric in the Lemaˆıtre frame depends upon both spatial and temporal
+coordinates, turning the equations of motion into a rather cumbersome form. In the present
+paper we will deal mostly with the Schwarzschild coordinate frame, however, noticing the
+difference between that one and the Lemaˆıtre frame in some aspects.
+Another apparently self-contradicting example is the observation that the speed of a
+uniformly accelerated test particle (that initially increases) eventually decreases to zero
+value (see e.g. [12]). The cause in both cases is the longitudinal expansion of this particular
+space-time, extremely violent in its final stage, that brings eventually all of the objects
+(moving along homogeneity axis) to the state of relative rest.
+This expansion is also responsible for the for the indefinitely large Doppler redshift near
+the singularity. Namely, the Doppler frequency shift d is defined as the the ratio of the
+frequencies, d = ωr
+ωs recorded by a receiver, ωr and by a sender ωs. An object falling radially
+from the outer space initial position’s r0 records the radiation emitted at r0 as redshifted and
+the redshift d monotonically decreases, reaching value 1/2 on the horizon and then tends to
+zero, when approaching to the ultimate singularity. This may be interpreted as a conclusion
+about the growing darkness inside Schwarzschild black hole.
+This, however, appears to be incorrect if we consider massless particles with non-zero
+angular momentum. It leads to a bright ring around a singularity (see page B-25 of [13]). A
+careful analytical study of red/blue shift for such massless particles have been recently done
+in [14]. Namely, it was shown that if an object following a non-zero angular momentum
+geodesic perceives the zero-angular momentum light ray incoming from the outer space
+sources it turns out to be not red- but blueshifted. Moreover, an object following zero-
+angular momentum trajectory records a non-zero angular momentum light ray as blueshifted;
+in both cases the blueshift is indefinite near the singularity. Therefore, the Doppler redshift
+described for the radially infalling objects recording radial radiation turns out to be an
+
+4
+exception but not the rule. Thus, the interior of the Schwarzschild black hole does not turn
+to be darker, as it has been sometimes believed (see e.g. [15], [16])) but turns out to be
+getting brighter when approaching the singularity. All these results can be rather easily
+explained using the properties of peculiar velocities inside a horizon, as it was shown in [18].
+Motivated by those findings, we are going to make a revision of the seemingly well-
+understood properties of the time-like geodesics inside the Schwarzschild BH. The aim of
+this paper is to reconsider the question of the relative speed of the particles inside the
+horizon in a general case of non-zero angular momentum trajectories of one or both of the
+particles. One finds that such a relative speed may tend to the speed of light in some cir-
+cumstances; then the question of unbounded energy collision will be revised with a rather
+challenging outcome. Finally, we will present an argument that the effects described in this
+paper and those discussed in the context of the blueshift are both direct outcomes of the
+anisotropic character of the dynamics of the interior of Schwarzschild BH. It is also argued
+that although anisotropic dynamics is a common property for all of the BHs interior, the
+critical contraction, as specified below is a particular feature of the Schwarzschild BH that
+causes the indefinite blueshift and, at the same time, leads to the unbounded energy colli-
+sions. Therefore, we are going to discuss in detail the problem: how the critical contraction
+affects the dynamics of the test objects in the vicinity of the ultimate singularity (although
+their final fate is beyond the scope of our considerations). We shall consider the problem of
+the relative speed of the massive test objects. We examine separately two cases: geodesic
+motion and motion under the action of some finite force. In doing so, we assume that, in
+general, particles have non-zero angular momenta.
+The paper is organized as follows. In Sec. II we define the line element and the tetrad for
+resting observer inside Schwarzschild BH and we apply them (Sec. III) for description of the
+kinematics of a freely falling test particle. In Sec. IV the anisotropy of the this space-time is
+described. Particle collisions are analyzed in the following three sections: V - general setup,
+VI - in-plane collisions, VII - different planes collisions. In Secs. VIII-IX the effects of action
+of external force are described. In Sec. X we consider the effect of tidal forces. Discussion
+and final remarks are presented in the final section.
+
+5
+II.
+METRIC, TETRAD
+Let us consider the black hole metric
+ds2 = −fdt2 + dr2
+f
++ r2dω2.
+(1)
+Here, f(r+) = 0, where r+ is the radius of the event horizon. Our main concern is the
+Schwarzschild metric for which f = 1 − r+
+r . We will also discuss very briefly the case of the
+Reissner-Nordstr¨om metric. Inside the horizon, the mutual role of temporal, t and spatial, r,
+coordinates interchanges. We can choose T = −r, y = t, where −r+ ≤ T ≤ 0, −∞ < y < ∞
+[17]. Then
+ds2 = −dT 2
+g
++ gdy2 + T 2dω2,
+(2)
+where g = −f.
+In what follows, it is convenient to use the tetrad attached to a resting observer with
+constant spatial, y, θ, φ coordinates. Such an observer follows a geodesic that has no analogue
+in the outer part of space-time [9]. Namely, in the coordinates (T, y, θ, φ)
+ˆh(0)µ = − 1
+√g(1, 0, 0, 0),
+(3)
+ˆh(1)µ = (0, √g, 0, 0),
+(4)
+ˆh(2)µ = (0, 0, |T| , 0)
+ˆh(3)µ = (0, 0, 0, |T| sin θ).
+(5)
+III.
+MOTION OF A FREE PARTICLE
+Outside the horizon there exists the time-like Killing vector that corresponds to the time
+translation and it leads to the conservation of a particle’s energy. Inside the horizon it be-
+comes a space-like one and this leads to the conservation of the y-component of momentum.
+The angular momentum is conserved everywhere.
+Then, the four-velocity uµ of a particle that moves within the plane θ = π
+2 has the form
+in the coordinate system (2)
+uµ = (P, − p
+g, 0, L
+T 2),
+(6)
+
+6
+where
+P =
+�
+p2 + g(L2
+T 2 + 1),
+(7)
+p = −uy is the specific conserved momentum (the sign minus is chosen to keep the maximum
+similarity with the region outside the horizon), L = uφ is the specific, conserved angular
+momentum.
+The corresponding tetrad components read
+u(a) = uµh(a)
+µ
+= ( P
+√g, − p
+√g, 0, L
+|T|).
+(8)
+Then, one can obtain (see Sec. 8 in [18]) that the three-velocity has components
+V (1) = − p
+P ,
+(9)
+V (3) = L√g
+|T| P .
+(10)
+The absolute value of the velocity V =
+�
+(V (1))2 + (V (3))2,
+V =
+�
+1 − g
+P 2.
+(11)
+The Lorentz gamma factor
+γ =
+1
+√
+1 − V 2 = P
+√g.
+(12)
+For given p and L, the velocity under discussion obeys the condition
+V 2
+1 − V 2 = p2
+g + L2
+T 2.
+(13)
+If the singularity is being approached, T → 0, g → ∞.
+Then, if L ̸= 0, we have
+|V | ≈ 1 − 1
+2
+�T
+L
+�2
+.
+(14)
+In doing so, V (1) → 0, V (3) → ±1.
+If L = 0, V (3) = 0. In the limit under discussion V (1) → 0.
+Thus, in any case V (1) → 0.
+
+7
+IV.
+GEOMETRY AND DYNAMICS
+The above result can be given the following geometric interpretation. The space-time
+described by the line element (2) may be referred to as a T-”sphere” [8]. It has got some
+particular properties: it is non-static, homogeneous, finite in time extent. It has a hyper-
+cylindrical a space-like section V 3 = R1×S2 with no symmetry center, open (−∞ < y < ∞)
+in radial, homogeneity direction R1. This may be regarded as an anisotropic cosmological
+model, expanding longitudinally and contracting transversely in a two-sphere S2 of radius
+|T| (see also [9] - [12]). Expansion along y axis is finally getting extremely violent: all of
+the objects are carried away in such a manner that their ”own” speeds are getting negligible
+- they are finally in a relative rest. And that is the meaning of the first of the results of
+the former section: the resting (in y axis) observer measures the speed of the test object
+travelling along this axis as diminishing to zero, V → 0, as is seen from (13) when L = 0
+and the singularity is approached, so g → ∞.
+If the velocity vector of a test object has got a transverse (to y axis) component, i.e.
+its angular momentum is non-zero, L ̸= 0, it is also carried away transversely due to the
+transverse contraction. This transverse contraction is of critical character: hypercylinder
+V 3 collapses to the line as the radius of the two-sphere tends to zero. All of the massive
+and massless particles are carried away in the following manner. The value of the speed of
+the massive test objects as measured by resting observers goes to that of light, V → 1, and
+the light rays (massless test objects) are perceived by the resting observer as indefinitely
+blueshifted [14].
+Then, the following interesting question arises: what is relative speed of the two observers
+depending on their angular momenta?
+If particles collide, whether their energy in the
+center of mass frame remains finite or grows indefinitely? In particular, it concerns particles
+travelling with (a) parallel, (b) antiparallel angular momenta. These questions are considered
+below.
+
+8
+V.
+PARTICLE COLLISIONS: GENERAL SETUP
+Now, we consider collisions of two particles of masses m1 and m2 and briefly analyze the
+behavior of the energy Ec.m. in the center of mass at the point of collision. By definition,
+E2
+c.m. = −PµP µ,
+(15)
+where P µ = m1uµ
+1 + m2uµ
+2 is the total four-momentum. Then,
+E2
+c.m. = m2
+1 + m2
+2 + 2m1m2γ12,
+(16)
+where w has the meaning of the relative speed, the Lorentz factor of relative motion
+γ12 = −u1µu2µ =
+1
+√
+1 − w2
+(17)
+should not be confused with the individual gamma factor of each particle (12).
+Below, we discuss two cases separately.
+VI.
+PARTICLES MOVE IN THE SAME PLANE
+Then, it follows from (6) and (17) that
+γ12 = P1P2 − p1p2
+g
+− L1L2
+T 2 ,
+(18)
+It is instructive to describe collisions in terms of kinematics characteristics. One can
+define the angle ψ between particles 1 and 2 according to
+cos ψ =
+⃗V1⃗V2
+V1V2
+,
+(19)
+where ⃗V1⃗V2 = V (1)
+1
+V (1)
+2
++ V (3)
+1
+V (3)
+2
+. Then, it follows from (9), (10) that
+cos ψ =
+1
+�
+p2
+1 + g L2
+1
+T 2
+�
+p2
+2 + g L2
+2
+T 2
+(p1p2 + L1L2g
+T 2
+),
+(20)
+γ12 = γ1γ2(1 − cos ψ).
+(21)
+Our main concern is the behavior of γ12 near the singularity.
+For fixed L1, L2, the
+absolute velocity of each particle in the limit when the singularity is approached, can take
+only two values: either V = 0 or V = 1 [18], [20]. Below, we enumerate different sub-cases
+separately depending on the angular momentum of each particle.
+
+9
+A.
+A L1 = 0 = L2.
+Then,
+P =
+�
+p2 + g.
+(22)
+For g → ∞ we have
+γ12 ≈ 1 + w2
+2 ,
+(23)
+where
+w ≈ |p1 − p2|
+√g
+→ 0.
+(24)
+Also,
+V1 → 0, V2 → 0,
+(25)
+cos ψ → sign(p1p2),
+(26)
+If a particle entered the interior of the horizon from its exterior, p > 0. If it entered from
+the left (mirror) region, p < 0. Thus in a physically relevant case when both particles came
+from infinity, ψ → 0.
+B.
+B L1 = 0, L2 = L ̸= 0
+γ12 ≈
+����
+L
+T
+���� → ∞,
+(27)
+w2 ≈ 1 − T 2
+L2 → 1,
+(28)
+V1 → 0, V2 → 1,
+(29)
+cos ψ → 0.
+(30)
+C.
+C L1L2 > 0
+γ12 → L2
+1 + L2
+2
+L1L2
+,
+(31)
+w → |L2
+1 − L2
+2|
+L2
+1 + L2
+2
+< 1,
+(32)
+
+10
+V1 → 1, V2 → 1,
+(33)
+cos ψ ≈ 1 − T 2
+g
+(p1L2 − p2L1)2
+L2
+1L2
+2
+.
+(34)
+Particles move almost parallel to each other near the singularity.
+D.
+D L1L2 < 0
+γ12 ≈ 2|L1L2|
+T 2
+→ ∞,
+(35)
+w ≈ 1 −
+T 4
+4L2
+1L2
+2
+→ 1,
+(36)
+V1 → 1, V2 → 1,
+(37)
+cos ψ → −1.
+(38)
+This means that head-on collision occurs, ψ → π.
+Now, we can summarize the results of the present section.
+L1
+L2
+w
+ψ
+A 0
+0
+0
+0
+B 0
+̸= 0
+1
+π
+2
+C ̸= 0 ̸= 0, L2 parallel to L1
+separated from 1 0
+D ̸= 0 ̸= 0, L2 antiparallel to L1 1
+π
+Table 1. Types of particles collisions near the singularity
+VII.
+PARTICLES MOVE WITHIN DIFFERENT PLANES
+As is well-known, in the case of conserved angular momentum a particle moves within
+a plane. According to above consideration, we can choose this plane to be θ = π
+2 for, say,
+particle 1. However, in general, this is not the case for particle 2, the variable θ will be
+varying in time. Will it significantly affect the results for the relative velocity and Lorentz
+factor γ12 near the singularity? To answer this question, we generalize the results of the
+previous section. Omitting the details of derivation, we give the corresponding formulas
+below. Now,
+uµ = (P, − p
+g, σQ
+T 2 ,
+L
+T 2 sin2 θ),
+(39)
+
+11
+where σ = ±1,
+Q =
+�
+L2
+tot −
+L2
+sin2 θ,
+(40)
+P =
+�
+p2 + g(1 + L2
+tot
+T 2 ),
+(41)
+it is implied that
+Ltot ≥ |L|
+sin θ.
+(42)
+Here, the integral of motion Ltot has the meaning of the total angular momentum of a
+particle, while L is its component corresponding to a variable φ. Then, for V (a) = (V (1),
+V (2), V (3)) one finds
+V ((a) = (− p
+P , σQ√g
+|T| P ,
+L√g
+|T| P sin θ),
+(43)
+eq. (12) is still valid but now with (41). Obviously,
+V⊥ =
+�
+(V (2))2 + (V (3))2 =
+√g
+|T| P Ltot.
+(44)
+We assume that for particle 1 θ = π
+2, Q1 = 0, L1tot = |L1|. Then, in the point of collision
+both particles have the same coordinates, so θ = π
+2 for particle 2 as well. It is convenient to
+introduce an angle α for particle 2. so that L2 = Ltot cos α, where cos α can have any sign.
+Then, in the point of collision we have for particle 2
+uµ = (P, − p
+g, L2tot sin α
+T 2
+, L2tot cos α
+T 2
+).
+(45)
+Eqs. (18), (21) are also valid but in P the quantity Ltot appears instead of L.
+Now,
+cos ψ =
+1
+�
+p2
+1 + g L2
+1
+T 2
+1
+�
+p2
+2 + g L2
+2tot
+T 2
+(p1p2 + L1L2g
+T 2
+),
+(46)
+γ12 = P1P2 − p1p2
+g
+− L1L2
+T 2 .
+(47)
+When a singularity is approached, V (1) → 0 as before, while V⊥ → 1, so V → 1 as well.
+Let us denote the cases A-D depend on the L1, L2 in the manner similar to that in the
+former section. Then, one can see that cases A and B coincide with those from Table 1.
+Indeed, if one of angular momenta is zero, one can choose the equatorial plane for another
+particle to be θ = π
+2, so nothing new happens. Obviously, case D is similar to that from
+Table 1. It remains to check what happens in case C. Then,
+
+12
+cos ψ →
+L2
+L2tot
+= cos α,
+(48)
+ψ = α. Taking into account that V1 → 1 and V2 → 1, we see that according to (21), in case
+C a new possibility arises :
+γ12 ≈ |L1| (L2tot − L2)
+T 2
+,
+(49)
+so γ12 → ∞ in spite of L1L2 > 0. Such a possibility was absent when both particles had been
+moving within the same plane (see Table 1 above). The similar phenomenon for massless
+particles was discussed in [19].
+VIII.
+MOTION UNDER THE ACTION OF FORCE
+Let now some force act on a particle. Then, the equations of motion formally retain their
+form but the quantities p and L cease to be integrals of motion and become the functions
+of time. If there is an acceleration aµ, one finds its tetrad components using (3) - (5) that
+(assuming θ = π
+2)
+a(3) = |T| aφ = aφ
+|T|,
+(50)
+a(y) = √gay = ay
+√g,
+(51)
+a(ˆt) = − aT
+√g = aT
+√g.
+(52)
+If ξµ is the Killing vector, it is easy to notice that
+d
+dτ (ξµuµ) = ξµaµ.
+(53)
+Then,
+dp
+dτ = ay = √ga(y),
+(54)
+dL
+dτ = aφ = |T| a(3),
+(55)
+where we used the same definitions p = −uy and L = uφ as for free particles. Now,
+p2
+g −
+�
+uT�2
+g
++ L2
+T 2 = −1,
+(56)
+
+13
+where
+uT = dT
+dτ =
+�
+p2 + g(1 + L2
+T 2).
+(57)
+It follows from equations of motion that
+dp
+dT = −dp
+dr =
+√ga(y)
+�
+p2 + g(1 + L2
+T 2)
+,
+(58)
+dL
+dT = −dL
+dr =
+|T| a(3)
+�
+p2 + g(1 + L2
+T 2)
+.
+(59)
+It is clear from (58), (59) that p and L remain finite, if a(y) and a(3) are finite.
+This has important consequences for the properties of velocities. In particular, in the
+tetrad (3) - (5) V (1) → 0 and V (3) → ±1, if L ̸= 0 (V (3) = 0 for L = 0). These conclusions
+are valid for any finite E, L[20], so they apply to the case under discussion as well.
+IX.
+WHEN PARTICLE VELOCITY CAN APPROACH THE SPEED OF LIGHT
+It follows from the above consideration that eq. (13) indeed retains its validity, if the
+constants of motion p and L are replaced by their momentary values p(T) and L(T). In
+turn, this has an important consequence. For a finite acceleration, the velocity can reach
+the limiting value V = 1 only in two cases: when approaching the horizon and/or singularity.
+In the first case, the right hand side of (13) diverges due to the first term where g → 0. In
+the second one it does so due to the second term where T → 0.
+All these conclusions are obtained with the assumption that a(i) are finite and hence p and
+L are finite as well. If we relax the requirement of finiteness of a(i), an additional possibility
+opens that V → 1 due to unbounded acceleration and, correspondingly, unbounded p and
+L. Thus there are three possibilities for getting V → 1: (i) horizon, (ii) singularity, (iii)
+infinite acceleration.
+This result is valid for the velocities with respect to the Lemaˆıtre frame as well. Moreover,
+it is valid with respect to a general radially free falling system formed by particles with the
+specific energy e0. It is known that in such a general case the radial component of velocity
+of a particle with specific energy e is given by (see [20])
+V (1) = P0e − Pe0
+e0e − PP0
+(60)
+
+14
+and the angular component is
+V (3) =
+Lf
+r(e0e − PP0),
+(61)
+where
+P0 =
+�
+e2
+0 − f.
+(62)
+Formulae of the present paper for the components of velocity of an individual particle
+can be thought of as a particular case e0 = 0 (corresponding to a resting observer) of
+these general formulae. Assuming that e is finite, we see that V (3) < 1 as it should be
+and V (3) → 1 at singularity. As for the radial component, substituting P and P0 into the
+condition V (1) = 1 and considering finite e we get, after a simple algebra, that f = 0. This
+means that V (1) can take the value 1 at the horizon only, provided e is finite. It is worth
+noting that this result is valid for any spherically symmetric static space-times since we do
+not specify the function f. The coordinate system e0 = 0 considered here becomes singular
+at the horizon itself. However, discussion contained in Sec. VI explains, why V (1) = 1 for
+more general systems is safe from physical point of view (note, that unlike a singularity we
+have equality, not a limit here!).
+X.
+MOTION WITH RESPECT TO A FRAME VERSUS MOTION OF NEARBY
+PARTICLES
+Apart from the behavior of velocity with respect to a fixed frame, another interesting
+question is mutual movement of nearby points. The properties of such a motion can be very
+different from the motion with respect to a fixed frame, even if one of the points considered
+is at rest with respect to the coordinate system in question. The reason is that the velocity
+with respect to a frame is a local entity, while a distance between two points is a space-like
+variable. Indeed, consider the radial motion. The velocity along the leg of a hypercylinder
+in metric (2) is known to decrease and vanish in a singularity [18], [20]. As for the distance
+to a nearby point, it increases as it can be clearly seen from the form of the metrics (1),
+(2). At the singularity the proper distance diverges. The picture is similar to the Big Rip
+cosmological singularity, apart from the fact that the Big Rip is isotropic.
+In popular books, when describing influence of tidal forces to an unhappy observer, falling
+freely into a black hole, authors usually illustrate the text by emotional pictures of an ob-
+
+15
+server ”spaghettified” in the direction toward a singularity. This has no sense in coordinates
+like T, y since (i) they are homogeneous inside a horizon, and (ii) a singularity, being space-
+like and in absolute future for an observer, is not present in any of an observer’s T = const
+slices. By itself, ”spaghettization” does occur but has another meaning: if we make a series
+of snapshots of cross-sections T = const for different T, the object extends more and more
+when T grows approaching T = 0.
+Another picture arises in the Lemaˆıtre coordinates. Singularity is present in the sections
+of constant Lemaˆıtre time, so the direction towards a singularity makes sense. Since for
+two radially separated particles singularity occur at different moments of the proper time
+τ along the trajectory, the separation between two points reaches its finite maximum when
+the ”inner” particle hits a singularity (in this context the word ”hits” is conditional since
+the singularity is space-like, we use it for brevity only).
+It is easy to estimate this maximum for a pure radial motion. Suppose two particles,
+being at rest with respect to the Lemaˆıtre system are separated by some distance. Let a
+particle move with E = m, so it would start its motion from the rest at infinity. Then, it
+is known (see, e.g. eq. 2.3.12 of [22]) that if a particle moves from some r to rf < r, the
+proper time is equal to
+τ(r, rf) = (2/3)r−1/2
+g
+(r3/2 − r3/2
+f ).
+(63)
+In particular, the proper time between a given position r and the singularity r = 0 is
+obtained from (63) if we put there r = 0, so
+τ(r, 0) = (2/3)r−1/2
+g
+r3/2.
+(64)
+It is also worth mentioning that for such a particle the Lemaˆıtre time coincides with the
+proper one (see, e.g. eq. 14 of [18]). Also, the proper distance in this case is equal to the
+difference of the coordinate values of r.
+Let we have two such particles initially separated by the coordinate distance l. We want
+to find the location rf of the ”outer” particle initially located at r+l, at the moment τ when
+the ”inner” particle hits the singularity. Equating τ(r + l, rf) = τ(r, 0), assuming small l
+and expanding the right hand side with respect to l/r we get
+rf = [(3/2)l]2/3r1/3,
+(65)
+which gives for the ratio
+rf/l = (3/2)2/3(r/l)1/3.
+(66)
+
+16
+Thus small absolute displacements remain small. However, relative displacement may be
+arbitrary large.
+As a trilling example we can consider the following situation: suppose that different parts
+of human body (l ∼ 1m) start to move geodesically after tidal acceleration gt exceeds the
+free fall acceleration at the surface of Earth (gE ∼ 10m/s2 ∼ 10−16m−1 in natural units
+c = 1). For small l, gt ≈ rgl/r3 (see, e.g. page B-20 of [13]). Thus free fall begins at
+r = (rgl/gE)1/3. Using these data we can estimate
+rf/l ∼ 60r1/9
+g ,
+(67)
+where rg is expressed in meters.
+This indeed indicates ”spagettization” - the size in r-
+direction enlarges from about 100 times for a stellar mass black hole to about 1000 times
+for a supermassive (109 solar masses) one (note, that the dependence upon black hole mass
+is rather weak).
+For the motion in the angular direction the situation is quite opposite. Suppose we have
+a particle with a zero angular momentum, so it falls along φ = 0 line, and a nearby particle
+does so with some small but non-zero L. We know that V (3) of the second particle tends to 1
+when the singularity is approached. Does this mean that the proper distance between these
+two particles increase rapidly? The answer is ”no” as the direct dependence φ(r) in the
+Schwarzschild metric shows (Fig.1). In this picture we plot V (3) of a particle with L = m2
+inside a horizon. It tends to 1 near a singularity. In the same plot we show the distance
+from the line φ = 0 to this particle (we assume that this particle crosses the line φ = 0
+at the horizon) which is equal to rφ. This distance first increases due to non-zero L (as it
+would be in a flat space also), then it starts to decrease despite growing velocity V (3). The
+contraction in the angular direction overcomes, and the distance in the φ direction appears
+to be always smaller than it would be without gravity.
+This picture is qualitatively the same in the Lemaˆıtre coordinates as well.
+The only
+difference is that V (3) in static coordinate always vanishes at a horizon (this is a counterpart
+of the statement that radial velocity is always 1 at a horizon), while the analog of this value
+with respect to the Lemaˆıtre system can take any value from 0 to 1.
+As for the angle φ itself, it reaches a finite value at singularity. This value grows with
+growing L, tending to π for L → ∞ (see eq.21of [12]).
+
+17
+FIG. 1: The angular component V (3) of velocity of a particle with L = m2 inside a horizon (green)
+and distance to the particle in angular direction from the radius φ = 0 crossed by this particle at
+a horizon (blue). The unit for V (3) is c, the unit for the distance is rg.
+
+0.8-
+0.6
+0.4-
+0.2-
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+60
+1.0
+118
+XI.
+DISCUSSION AND CONCLUSIONS
+In this paper we have considered dynamical phenomena in the vicinity of the singularity of
+the Schwarzschild space-time. In this case one can regard horizon’s interior as an anisotropic,
+dynamical V 4 space-time with a hypercylinder V 3 = R1 × S2 space-like sections. There is
+a longitudinal, R1-expansion and transversal, S2-contraction. Due to extremely violent R1-
+expansion in its final stage one could expect the asymptotic state of mutual rest of all the
+particles moving along y-direction (see e.g
+[12]). This picture has been completed by a
+Doppler’s blueshift, for the case of transverse component of trajectories: a light-like, non-
+zero angular momentum geodesics have been recorded blueshifted [14]. We have verified
+here the kinematics of the test particles following time-like, non-zero angular momentum
+trajectories of both geodesic and non-geodesic character.
+If a test particle moves along
+an arbitrary non-zero angular momentum trajectory, then its speed as measured by resting
+observers, those with constant spatial coordinates, approaches that of light, w → 1 as T → 0.
+Previously, it was found that this is valid for geodesic trajectories [18], [20]. Now, we showed
+that this is valid for an arbitrary finite force.
+If, instead of one particle, we take the two particles following non-zero angular momenta
+trajectories, their relative velocity w → 1 with only one exceptional case. It occurs if both
+particles move in the same plane and have parallel angular momenta; then the value of their
+relative speed w is smaller than that of light, w < 1. This also happens if both particles
+have zero angular momenta. Otherwise, non-zero angular momentum of a test particle is a
+necessary and sufficient condition for w → 1.
+It should be pointed out that there exists the reason, common for both the indefinite
+blueshift for the class of non-zero angular momentum light-like geodesics and indefinite
+tendency of the relative speed of the particles following their non-zero angular momentum
+trajectories to the speed of light when approaching the ultimate singularity T → 0 of
+Schwarzschild BH’s interior. This effect is caused by a contraction in the course of highly
+anisotropic dynamics of space-time. Indeed, when approaching T → 0, [8] the hypercylinder
+is critically contracting, i.e. the radius |T| of the two-sphere, diminishes to the zero value,
+T → 0. This critical contraction carries all of the objects, massive and massless, in such a
+way that the light recorded by a resting or moving along y-axis observer turns out to be
+indefinitely blueshifted and the speed of a test particle as measured by resting or moving
+
+19
+along y axis observer tends indefinitely to the speed of light, w → 1. When two colliding
+massive particles follow their non-zero angular momenta trajectories, then in general they
+experience head-on collision and their relative speed approaches that of light. (For motion
+within the same plane and the same directions of the angular momenta the effect is moderate:
+the relative speed of the colliding particles is found to be smaller than the speed of light,
+w < 1. This is an analogy of the finding in [14] where for motion within the same plane and
+the same directions of the angular momenta of the observer and the light a finite blueshift
+is found).
+The result w → 1 may be regarded as a center of mass energy collision tending to infinity.
+This interpretation provides a particular perspective. All of the variety of the BSW effect,
+unbounded energy collisions in the vicinity of the black hole horizons, outer or inner, have
+lead to the conclusion about arbitrary large limit which, however, is not reached in any
+particular collision, so an infinite limit cannot be realized.
+This is called a principle of
+kinematic censorship [21]. Meanwhile, in the case under discussion this principle is violated
+when T → 0 (r → 0). This is probably quite natural since in the singularity itself all known
+laws of physics can be violated and geometry as such ceases to exist.
+Since exact vanishing of angular momentum and exact coinciding of planes of motion for
+two particles represent zero-measure set of initial conditions and cannot be exactly satisfied
+in any realistic physical situation, we can conclude that tending w to 1 at a singularity
+is unavoidable. Correspondingly, indefinite growth of Ec.m. is general feature for particle
+collisions near the singularity.
+It is also shown for spherically symmetric space-times of a quite general form that a
+particle velocity can approach the speed of light only in three cases: (i) on the horizon, (ii)
+in the singularity, (iii) when a proper acceleration diverges.
+XII.
+ACKNOWLEDGEMENT
+The work of AT is supported by the Program of Competitive Growth of Kazan Federal
+University and by the Interdisciplinary Scientific and Educational School of Moscow Uni-
+versity in Fundamental and Applied Space Research. O. Z. thanks H. V. Ovcharenko for
+
+20
+useful discussion.
+[1] M. Ba˜nados, J. Silk and S.M. West, Kerr black holes as particle accelerators to arbitrarily
+high energy, Phys. Rev. Lett. 103, 111102 (2009). arXiv:0909.0169
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+neutron stars, Bull. Am. Astron. Soc. 51, 352 (2019). arXiv:1903.06844
+[5] P.C. Fragile, S.M. Etheridge, P. Anninos, B. Mishra and W. Klu´zniak, Relativistic viscous
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+[6] D. Farrah et al., Stellar and black hole assembly in z < 0.3 infrared-luminous mergers: inter-
+mittent starbursts versus super-Eddington accretion, Mon. Not. Roy. Astron. Soc. 513, 4770
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+Quantum contribution to luminosity of quasars,
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+holes or how a uniformly accelerated particle may slow down. Eur. Phys. J.C 79, 876 (2019)
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+[13] E.Taylor and J.Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison
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+[20] A. V. Toporensky and O. B. Zaslavskii, General radially moving references frames in the black
+hole background, arXiv:2210.03670.
+[21] Yu. V. Pavlov and O. B. Zaslavskii, Kinematic censorship as a constraint on allowed scenarios
+of high energy particle collisions, Grav. Cosmol. 25, 390 (2019). [arXiv:1805.07649].
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+

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+Reading and Reasoning over Chart Images for Evidence-based
+Automated Fact-Checking
+Mubashara Akhtar, Oana Cocarascu and Elena Simperl
+Department of Informatics, King’s College London
+{mubashara.akhtar,oana.cocarascu,elena.simperl}@kcl.ac.uk
+Abstract
+Evidence data for automated fact-checking
+(AFC) can be in multiple modalities such as
+text, tables, images, audio, or video. While
+there is increasing interest in using images for
+AFC, previous works mostly focus on detect-
+ing manipulated or fake images. We propose
+a novel task, chart-based fact-checking, and
+introduce ChartBERT as the first model for
+AFC against chart evidence. ChartBERT lever-
+ages textual, structural and visual information
+of charts to determine the veracity of textual
+claims.
+For evaluation, we create ChartFC,
+a new dataset of 15, 886 charts. We system-
+atically evaluate 75 different vision-language
+(VL) baselines and show that ChartBERT out-
+performs VL models, achieving 63.8% accu-
+racy. Our results suggest that the task is com-
+plex yet feasible, with many challenges ahead.
+1
+Introduction
+Charts are often used to present data in news ar-
+ticles, reports, scientific publications, and across
+social media posts (Lo et al., 2022; Zhang et al.,
+2021). For example, in recent years, charts have
+been widely used to guide policymakers in decid-
+ing health policies and to communicate COVID
+information with the general public; a popular ex-
+ample is the coronavirus dashboard by Johns Hop-
+kins University,1 which was integrated in several
+websites (Perkel, 2020).
+Misinformation can spread through charts in var-
+ious ways. Previous works in data visualization
+have discussed how misleading chart design can
+cause misinformation (Lo et al., 2022). However, a
+more subtle form of misinformation occurs during
+chart interpretation (e.g. through invalid compar-
+isons, framing correlation as causation, or spread-
+ing of misleading claims). To identify these mis-
+information types not only the stand-alone chart
+but the chart together with its message need to be
+1https://coronavirus.jhu.edu/map.html
+Claim: Both Thane Baker and Nate Cartmell were ranked
+last.
+Evidence:
+Label: Supports
+Figure 1: An example from the ChartFC dataset where
+the claim is supported by the evidence chart.
+considered jointly (Lo et al., 2022). In this work,
+we focus on verifying whether charts support or
+refute claims about them.
+There has been substantial progress in automated
+fact-checking (AFC) in recent years, with a fo-
+cus on verifying claims against text (Wang, 2017;
+Thorne et al., 2018; Schuster et al., 2021; Thorne
+et al., 2021; Diggelmann et al., 2020), table (Aly
+et al., 2021; Diggelmann et al., 2020; Chen et al.,
+2020a; Akhtar et al., 2022), and image (Yao et al.,
+2022; Zlatkova et al., 2019; Qu et al., 2022) evi-
+dence. Previous work has widely ignored claim
+verification against chart images. There are sev-
+eral challenges related to chart fact-checking as
+opposed to other evidence modalities: the struc-
+tural information, text in charts, and location of
+text need to be considered jointly for chart under-
+standing. Text plays a key role and is used, for
+example, as bar labels, chart titles, or in legends
+to explain the use of colors. Moreover, verifying
+claims against charts requires different reasoning
+arXiv:2301.11843v1  [cs.CL]  27 Jan 2023
+
+usain bolt
+1
+andy stanfield
+2
+carl lewis
+2
+shawn crawford
+2
+athlete
+don quarrie
+5
+pietro mennea
+5
+charlie paddock
+7
+frankie fredericks
+7
+nate cartmell
+9
+thanebaker
+6
+0
+2
+4
+6
+8
+ranktypes, e.g. retrieving values, finding extremes, or
+calculating a sum.
+To address these challenges, we propose the
+chart fact-checking task where, given a text claim
+and a chart, the goal is to classify if it supports or
+refutes the claim. We introduce ChartBERT as the
+first model for AFC against chart evidence com-
+prising (i) an OCR-based reading component to
+extract text and structural information from chart
+images; (ii) a sequence generation component to
+process the extracted information; and (iii) an en-
+coding component that extends the BERT archi-
+tecture (Devlin et al., 2019) with three additional
+structural embeddings to jointly learn textual and
+structural representations of chart images.
+Moreover, we release ChartFC as the first bench-
+mark for chart-based AFC, created using TabFact
+(Chen et al., 2020a) as a seed dataset. Our dataset
+contains 15.9k human-written claims and bars of
+different colors, orientations, and backgrounds (see
+Figure 1 for an example). Our highest-performing
+ChartBERT model achieves 63.8% accuracy on
+ChartFC. We compare ChartBERT to 75 vision-
+language (VL) baselines, combining five vision
+encoders, three language encoders, and five fu-
+sion methods. The best-performing VL model is
+a transformer-based (Vaswani et al., 2017), dual
+encoder architecture that uses a simple, yet effec-
+tive fusion block: concatenation and gated recur-
+rent units (GRUs) (Bahdanau et al., 2015). Our
+results suggest that state-of-the-art VL approaches
+struggle with the proposed task, calling for more
+research.
+Our contributions are as follows: 1) we pro-
+pose the chart fact-checking task and build Chart-
+BERT as the first chart fact-checking model; 2)
+we introduce ChartFC, the first dataset for AFC
+with chart evidence; 3) we systematically evalu-
+ate state-of-the-art language/vision encoders and
+fusion methods on the proposed task, highlighting
+challenges and providing an analysis of common
+reasoning types that contribute to failures.2
+2
+Related Work
+2.1
+Verifying Claims against Evidence
+Evidence-based fact-checking aims to predict
+claims’ veracity given evidence data. While many
+datasets focus on text (Thorne et al., 2018; Kotonya
+and Toni, 2020; Schuster et al., 2021; Wang, 2017)
+2The ChartFC dataset, trained models, and our code are
+available at github/link/to/chartfc.com.
+and table evidence (Chen et al., 2020a; Gupta et al.,
+2020; Aly et al., 2021; Wang et al., 2021a; Akhtar
+et al., 2022), human fact-checkers use a wider range
+of modalities for verification (Nakov et al., 2021b;
+Alam et al., 2021). They consult experts and extract
+information from databases, text, tables, graphics,
+and audio/video material from numerous sources.3
+Charts influence how messages are perceived
+(Pandey et al., 2014). For example, Lee et al. (2021)
+use the term “counter-visualization” to describe
+data visualizations by the anti-vaccination commu-
+nities in the US who created charts from publicly
+available data and interpreted them in a way that
+challenged the narrative of the pandemic, leading
+to disinformation.
+2.2
+Automated Fact-Checking with Images
+Given that claims and evidence can be conveyed
+through different modalities, interest in AFC with
+images has increased recently (Nakov et al., 2021a;
+Cao et al., 2020; Alam et al., 2021; Yao et al., 2022;
+Sharma et al., 2022). Previous tasks focus mainly
+on detecting manipulated or fake images rather than
+on evidence-based claim verification (Blaier et al.,
+2021; Kiela et al., 2020; Alam et al., 2021; Sharma
+et al., 2022; Abdali, 2022). Whilst manipulated or
+fake images can be detected using the image only,
+claim verification requires understanding the claim
+and evidence jointly.
+2.3
+Chart Images in Other NLP Tasks
+Two tasks related to chart fact-checking are
+chart question answering and chart summarization.
+Given a chart image, the summarization task re-
+quires to generate a summary of the chart in natural
+language text (Kantharaj et al., 2022; Tan et al.,
+2022). For question answering (chartQA) the an-
+swer to natural language questions is extracted
+from chart images. However, different to claim
+verification, questions typically provide strong in-
+dicators for the correct answers. Existing chartQA
+datasets are either small (Kim et al., 2020) or
+comprise automatically-generated, template-based
+questions (Chaudhry et al., 2020; Kahou et al.,
+2018; Kafle et al., 2018).
+3
+ChartBERT Model
+We introduce ChartBERT, a first BERT-based chart
+fact-checking model. Our model consists of (i) a
+3https://ballotpedia.org/The_methodologies_of_
+fact-checking
+
+Figure 2: The ChartBERT architecture.
+reading component which extracts text and struc-
+tural information from charts (Section 3.2); (ii) a
+component for generating textual sequences from
+the information previously extracted (Section 3.3);
+and (iii) a BERT-based encoder with additional
+structural embeddings for the text extracted from
+charts (Section 3.4). The model architecture is
+shown in Figure 2.
+3.1
+Task Formulation
+Following previous AFC work (Chen et al., 2020a;
+Aly et al., 2021; Thorne et al., 2018; Wang et al.,
+2021b), we view chart fact-checking as a classifi-
+cation task where, given a natural language claim
+and a piece of evidence (i.e. the chart image), the
+goal is to decide if the evidence supports or refutes
+the claim. We use support/refute as labels for claim
+classification instead of true/false as we only as-
+sess the claim veracity given the provided evidence
+rather than claiming universal statements.
+Each ChartFC sample i = (ci, imgi, yi) com-
+prises a natural language claim ci, a chart image
+imgi (see Figure 1 for an example), and a label
+yi ∈ {supports, refutes}.
+3.2
+Reading Text from Chart Images
+Given an image imgi, the reading component ex-
+tracts text and structural information. First, we
+detect text regions in the chart using a Tesser-
+act OCR model (Kay, 2007).
+Specifically, for
+each image, the model extracts n text regions
+Ti = {t1, t2, ..., tn}nj=1, where each region tj con-
+sists of textj, a sequence of m tokens, and a rect-
+angular bounding box bj that surrounds the text
+region in the chart. The bounding box is a tuple
+bj = (xj, yj, wj, hj) where xj and yj are the pixel
+coordinates of the top left point of the box, and wj
+and hj represent the width and height of the box in
+pixels. Thus, for each image imgi we obtain the
+following output oi:
+oi = fR(imgi) = {(textj, xj, yj, wj, hj)}nj=1
+3.3
+Text Sequence Generation
+Next, we process the reading component’s output
+into a text sequence si consisting of m tokens:
+si = fSeqGen(oi) = [s1, s2, ...sm]
+We compare two approaches as follows.
+Concatenation: The concatenation method pro-
+cesses the text regions (i.e. tj ∈ Ti) based on their
+coordinates xj and yj so that texts that are close in
+the chart are also close in the generated sequence.
+The chart text is concatenated into one sequence
+and tokens that belong to different text regions are
+separated using a [; ] token. Thus, for the chart Fig-
+ure 1 we obtain a text sequence starting with “usain
+bolt ; 1 ; andy stanfield ; 2 ; [...].”
+Template: We use the structural information (i.e.
+x, y, wj, hj) to fill templates and generate text se-
+quences. We evaluate three templates (an example
+for each template, extracted from Figure 1, is pro-
+vided in brackets):
+tmp1: entry [num] : [lx] is [textx]; [ly] is [texty]
+(entry one: athlete is usain bolt ; rank is 1);
+tmp2: “row [num] : [lx] is [textx]; [ly] is [texty]”
+(“row 0: athlete is usain bolt ; rank is 1”);
+tmp3: “[lx] is [textx] when [ly] is [texty]”
+(“athlete is usain bolt when rank is 1”).
+The placeholder [lx] is replaced with the x-axis
+label from the chart (e.g. “rank” in Figure 1). Simi-
+larly, the y-axis label (e.g. “athlete”) replaces [ly].
+Based on the coordinates, we classify a bounding
+boxes that contain axes labels (i.e. the boxes with
+the largest y coordinates).
+A counter starting from one replaces [num] and
+numbers the bars in the chart. We fill [texty] and
+and [textx] with text regions detected as bar labels
+and axis ticks given their positions.
+3.4
+Encoding and Classification
+ChartBERT captures the structure of charts through
+three learned embeddings: the x coordinate embed-
+ding which captures the horizontal location of the
+text in the chart, the y coordinate embedding which
+captures the vertical location, and the label embed-
+ding which takes value 1 if the text region is part
+
+CHART EVIDENCE
+usain bolt
+andy stanfield
+carl lewis
+shawn crawford
+athlete
+don quarrie
+5
+pietro mennea
+charlie paddock
+CLAIM
+frankie fredericks
+nate cartmell
+9
+BothThaneBaker
+thane baker
+6
+and Nate Cartmell
+0
+2
+8
+were ranked last.
+rank
+(1) Reading
+Component
+(2) Seq-
+Text: [entry
+(3) BERT-
+one: athlete
+(4)
+Generation
+based
+is usain
+Classifier
+Component
+bolt...]
+encoderFigure 3:
+ChartBERT input representation with the text extracted from the chart and concatenated following
+the approach in Section 3.3. We include additional structural embeddings (i.e. x and y coordinates and label
+embeddings) to the BERT input embeddings (i.e. token, segment and position embeddings).
+of the x-axis label (lx), 2 if the text region is part
+of the for y-axis label (ly) and 0 otherwise.
+Figure 3 shows an example of the encoder with
+the structural embeddings. We concatenate claim
+ci and sequence si, separate them with a [SEP] to-
+ken, add [CLS] as the first input token, and feed the
+resulting vector as input to ChartBERT which gen-
+erates 768-dimensional representations hi ∈ R768.
+Finally, we pass hi through a fully connected layer
+and determine the predicted label using sigmoid.
+ChartBERT uses binary cross entropy to minimize
+loss on the training set.
+inpi = (ci, si, {xj, yj, lxj, lyj}nj=1)
+hi = fEncoder(inpi)
+pi = σ(fFC(hi))
+4
+Evaluation
+For evaluation, we first create a new dataset,
+ChartFC. We compare ChartBERT with several
+VL baselines, each comprising three components:
+a vision encoder, a language encoder, and a fusion
+block to obtain joint representations. We evaluate
+the dataset size and potential biases, discuss results
+obtained with ChartBERT and the baselines, and
+analyse reasoning types the models fail on.
+4.1
+ChartFC Dataset
+This section provides an overview of the ChartFC
+dataset and its creation process. Each dataset entry
+comprises a natural language claim, a chart image,
+and a label ∈ {supports, refutes}.
+4.1.1
+The TabFact Dataset
+We use TabFact (Chen et al., 2020a) as a seed
+dataset. TabFact is a table fact-checking dataset
+of natural language claims and tables extracted
+from Wikipedia as evidence, where the veracity
+of the claim is decided based on the accompanying
+table. Claims were written and evaluated by hu-
+man crowdworkers with at least 95% approval rates
+for prior tasks and more than 500 accepted HITs
+on Amazon Mechanical Turk. The inter-annotator
+agreement for the claim verification task is Fleiss
+κ = 0.75.
+4.1.2
+Creation Pipeline
+Figure 4 shows the dataset creation process.4 Start-
+ing with 117, 784 claims and 16, 000 Wikipedia
+tables from TabFact, we first generate sub-tables.
+To link the claim text to table columns, we (i) lem-
+matize and tokenize the claim and the table con-
+tent, (ii) link claim tokens to column headers and
+cells using string matching and heuristic rules, and
+(iii) decide if a claim token is linked to multiple
+columns using the minimum Levenshtein distance
+4Figure 7 in the Appendix A illustrates the pipeline.
+
+0
+Xc
+0
+Xusain
+Xbolt 
+0
+X1
+0
+Xandy
+Xstanf
+0
+...
+Xathlet
+Xrank
+0
+X-coordinate emb.
+0
+Ye
+0
+Yusain
+Ybolt
+0
+Y1
+0
+Yandy
+Ystanf
+0
+...
+Vathlete
+Yrank
+0
+y-coordinate emb.
+0
+0
+0
+0
+0
+0
+0
+0
+0
+2
+label embedding
+0
+0
+...
+0
+E(cs)
+Eelaim
+E[sEP) 
+Eusain
+Ebolt
+E;
+E1
+E,
+Eandy
+Estanf
+E'
+Eathlete
+Erank
+...
+E[SEP]
+BERT input emb.
+[CLS] 
+claim
+[SEP]
+usain
+bolt
+1
+;
+andy
+stanf
+athlete
+rank
+[SEP]
+concatenated text
+;
+..
+usain bolt
+Both Thane
+andy stanfield
+2
+Baker and Nate
+carl lewis
+Cartmell were
+shawn crawford
+ranked last.
+lete
+don quarrie
+pietro mennea
+charlie paddock
+frankie fredericks 
+nate cartmell
+thane baker
+2
+rankFigure 4: Dataset creation process.
+Train
+Valid
+Test
+Sum
+Support
+7,048
+896
+885
+8,829
+Refute
+5,654
+697
+706
+7,057
+Sum
+12,702
+1,593
+1,591
+15,886
+Table 1: Class distribution across dataset split.
+(Levenshtein, 1966), and finally, (iv) filter sub-
+tables with a maximum of twenty rows and two
+linked columns. This results in a total of 15, 886
+pairs of claims and sub-tables.
+Finally, we generate charts using the Python li-
+braries seaborn and matplotlib. The charts vary
+across the dimensions (i) orientation (horizontal,
+vertical); (ii) bar colors (green, blue, pink); and
+(iii) background (no/white grid lines, white/gray
+background color). We show an example in Fig-
+ure 1. We partition the dataset into training, val-
+idation, and test sets using 8:1:1 ratio and show
+statistics in Table 1.
+4.1.3
+Dataset Evaluation
+To assess the data quality, we apply human and
+automated evaluation. We evaluate the sub-table
+generation step (step 2 in Figure 4) by checking
+the verifiability of claims against the extracted sub-
+tables with TableBERT (Chen et al., 2020a). We
+obtain 69.3% accuracy on our test set, comparable
+to 65.1% accuracy reported by Chen et al. (2020a)
+on their test set.
+For human validation, we extract 100 random
+dataset entries and manually evaluate the claims
+against sub-tables and charts. Of the 100 claims, 92
+were successfully verifiable against their sub-tables
+and chart images, six claims were not verifiable
+because a relevant column was missing in the sub-
+table, and two claims were already mislabelled in
+the TabFact dataset.
+4.1.4
+Chart Reasoning Types
+We label 100 random test samples with chart rea-
+soning types, using a taxonomy of common reason-
+ing types humans apply while interacting with data
+visualisations (Amar et al., 2005). We find seven
+Figure 5:
+Number of chart reasoning types found in
+100 dataset entries.
+reasoning types present in our data: retrieve value,
+filter, comparison, compute derived value, find ex-
+tremum, determine range, and find anomalies.5 On
+average, we find 1.4 different types per claim with
+most claims including either one or two different
+reasoning types (see Figure 5). The reasoning type
+retrieve value, which requires extracting a value
+from the chart image given certain criteria, occurs
+most frequently (51%), followed by find extremum,
+i.e. highest or lowest values in the chart, and fil-
+ter, which occur in approximately a quarter of all
+labelled claims. More complex types such as com-
+pute derived value or extracting all values in a given
+range are less frequent.
+4.2
+Vision-Language Baselines
+We evaluate our task with several VL baselines,
+which jointly use claim text and visual information
+from images for claim verification. We also assess
+the top-3 VL baselines with OCR-extracted chart
+text as additional input. Each baseline consists of
+a language encoder, a vision encoder, and a fu-
+sion component to obtain joint representations. We
+systematically evaluate various state-of-the-art en-
+coders and fusion techniques: we use shallow (e.g.
+BERT Embedder (Chen et al., 2020b)) and deep
+encoders (e.g. DenseNet (Huang et al., 2017)), as
+well as model-agnostic (e.g. concatenation) and
+model-based (e.g. transformer layers) fusion meth-
+ods.
+Language encoders: Given a claim ci, we use
+a language encoder to obtain a feature vector:
+htext
+i
+= fLangEncoder(ci)
+We experiment with three language encoders:
+BERT Embedder: Following Chen et al. (2020b),
+we tokenize the claim text into sub-words. For each
+token, we add the word and position embeddings to
+5We describe the chart reasoning types in detail and give
+examples in Appendix B.
+
+Starting point:
+(1) Link claim to
+(2) Generate
+TabFact dataset
+table columns
+sub-tables
+(4) Evaluate
+(3) Create charts
+1
+山Three reasoning types
+Two reasoning types
+33
+63
+Onereasoningtypeobtain the text representation which we then pass
+through a normalization (Ba et al., 2016) layer.
+LSTM: We encode the text with 32-dimensional
+word embeddings and pass them through two
+LSTMs (Hochreiter and Schmidhuber, 1997) with
+768-dimensional hidden states in each layer. We
+use the hidden states of the second layer as text
+representations.
+BERT: We use a twelve-layer BERT encoder, ini-
+tialized with weights from a pretrained BERT-base
+model.
+Vision encoders: We use a vision encoder to
+extract representations for the chart images:
+himg
+i
+= fV isEncoder(imgi)
+We evaluate five vision encoders:
+Fully connected layer: We use a fully connected
+layer to extract 768-dimensional representations
+per image himg
+i
+∈ R768.
+AlexNet: Using AlexNet (Krizhevsky et al., 2012),
+for each image, we obtain a representation vector
+himg
+i
+∈ R1024 by extracting the model output after
+the third max pooling layer.
+ResNet: We use ResNet-152 (He et al., 2016) to
+obtain 2048-dimensional image representations by
+extracting the model output before the two final
+layers of ResNet-152, i.e. before the average pool-
+ing layer.
+DenseNet: We use a DenseNet (DN) (Huang et al.,
+2017) comprising three dense blocks, with 6, 12,
+and 24 layers, respectively. We extract and concate-
+nate the output of the first and third dense block
+as low- and high-level feature vectors: himg
+i
+=
+fconcat(fDN[block1](imgi); fDN[block3](imgi)).
+Vision Transformer (ViT): We split images into
+sequences of n 16x16 patches before using them as
+input to a pretrained base-ViT model (Dosovitskiy
+et al., 2021).6 We extract the hidden states from
+the model’s final layer and use them as image rep-
+resentations, resulting in 768-dimensional vectors
+for each patch: himg
+i
+= [h ∈ R768]n.
+Fusion methods: We then fuse the text and im-
+age representations:
+hjoint
+i
+= fFusion(himg
+i
+; htext
+i
+)
+We experiment with five fusion methods:
+Concatenation and multiplication: Concatena-
+tion and multiplication are common baseline ap-
+proaches for multimodal fusion (Baltrušaitis et al.,
+6https://huggingface.co/google/
+vit-base-patch16-224
+2018). We reshape the text and image representa-
+tions and either (i) concatenate both vectors, or (ii)
+perform element-wise multiplication.
+Concatenation with GRUs: Inspired by Kafle
+et al. (2020), we concatenate the text and image rep-
+resentations and pass the resulting vector through
+m 1x1 convolutional layers and two GRUs. The
+first GRU takes the input in a forward direction,
+while the second GRU processes the input vector
+in a backwards direction to incorporate contextual
+information:
+hconcat
+i
+= fconv(fconcat{himg
+i
+; htext
+i
+})
+hjoint
+i
+= fconcat{f−−−→
+GRU(hconcat
+i
+); f←−−−
+GRU(hconcat
+i
+)}
+Multimodal Compact Bilinear Pooling (MCB):
+MCB is an efficient and popular baseline for multi-
+modal fusion (Fukui et al., 2016). The text and im-
+age representations are each projected to a higher
+dimensional space using the projection function
+Count Sketch (Charikar et al., 2004). The outer
+product of the projected vectors is then calculated
+in Fast Fourier Transform space to obtain a joint
+representation for both modalities and thus reduce
+the amount of learnable parameters during model
+training.
+Transformer layers: Given the recent popularity
+of transformer layers used for joining text and vi-
+sual representations (Tan and Bansal, 2019; Chen
+et al., 2020b; Yang et al., 2021), we use a three-
+layer transformer to get cross-modal embeddings.
+The representation hjoint
+i
+is passed through two
+fully-connected layers and sigmoid to obtain the
+classification. We use binary cross entropy loss
+and stratified sampling in each training batch to
+minimize the loss on the training set.
+4.3
+Experimental Setup
+We perform hyper-parameter search on the valida-
+tion set and select the best-performing combination
+from the following values: {8, 16, 32} for batch
+size, {1e−3, 7e−4, 5e−5, 5e−6, 5e−7} for learning
+rate, {1, ..., 50} for training epochs with early stop-
+ping. We also experimented with different learning
+rates for the language and vision encoders. Ulti-
+mately, we used one learning rate for the entire VL
+model as the modality-specific learning rates did
+not provide any performance gains.7
+7The hyper-parameters for each VL baseline can be found
+in our GitHub repo.
+
+SeqGen
+Val Acc
+Val F1
+Test Acc
+Test F1
+concat.
+59.2
+55.1
+60.6
+57.0
+temp. tmp1
+62.4
+59.1
+63.3
+61.0
+temp. tmp2
+62.0
+59.4
+61.9
+58.7
+temp. tmp3
+62.1
+59.7
+63.8
+61.1
+Table 2:
+Results for ChartBERT with differ-
+ent sequence generation (SeqGen) approaches: con-
+catenation and template.
+V-Encoder
+Fusion
+no OCR
+text concat
+ViT
+concat GRU
+59.8
+60.5
+ResNet
+mult
+60.1
+61.3
+ResNet
+concat
+59.8
+62.7
+Table 3: Test accuracy of top-3 VL baselines: without
+(no OCR) chart text and chart text concatenated. All
+models use BERT as language encoder.
+We run all experiments on a single NVIDIA
+Tesla V100 GPU with 32GB RAM. We measure
+model performance with prediction accuracy and
+(macro) F1 on the test dataset.
+4.4
+Results & Discussion
+How does ChartBERT perform on the task?
+How do different approaches for sequence gen-
+eration influence model performance?
+Table 2 gives an overview of the results obtained
+by ChartBERT. The best ChartBERT variant yields
+63.8% test accuracy and processes chart text into
+text sequences using the template tmp3. Com-
+pared to the concatenation approach, using tmp3
+increases the accuracy by +3.2%.
+Interestingly, the choice of template design im-
+pacts the model performance only slightly. While
+template tmp3 might seem more “natural” to hu-
+mans, it does not yield much higher performance
+compared to tmp2.
+How do VL baselines perform on ChartFC?
+How does the selection of encoder or fusion
+method impact model performance?
+In contrast to many state-of-the-art VL ap-
+proaches that use simple vision encoders and
+attention-based fusion (Chen et al., 2020b; Kim
+et al., 2021; Xia et al., 2021), the three best-
+performing VL models on ChartFC use BERT as
+language encoder, ViT or ResNet to obtain image
+representations, and either concatenation, multipli-
+cation, or concatenation with GRUs as a fusion
+method. Using only the claim and chart as input
+(i.e. without the OCR-extracted chart text), the
+highest test accuracy we obtain is 60.1% with the
+model consisting of BERT, ResNet, and multiplica-
+tion fusion (see Table 3).
+Regarding the language encoder,8 models that
+use BERT perform best, irrespectively of the vi-
+sion encoder and fusion method: the best LSTM-
+based model achieves 56.1% test accuracy and the
+best model with BERT embedder yields 56.5% ac-
+curacy, both lower than the best BERT-based VL
+model with 60.1% accuracy. In contrast, we obtain
+similar accuracy scores across different vision en-
+coder: for example, replacing ResNet in Table 3
+row two with a fully connected layer reduces the
+accuracy slightly by 0.6% to 59.7%. The choice
+of fusion method does not impact performance
+strongly: while using multiplication mostly outper-
+forms other methods by a small margin, no fusion
+method stands out across all vision and language
+encoders. We also evaluate the chartQA model
+PReFIL (Kafle et al., 2020), which uses LSTM as
+language encoder, DenseNet for image representa-
+tions, and concatenation with GRUs for fusion, and
+obtain on ChartFC a low test accuracy of 55.6%.
+How does OCR-extracted chart text influence
+performance of VL models?
+In addition to claim text and chart images used
+in VL baselines, we also include the text extracted
+from the charts through OCR as input (see Sections
+Sections 3.2 and 3.3 for details). Table 3 shows that
+using the concatenated chart text as input improves
+accuracy compared to the models that do no use
+the chart text (e.g. from 59.8% to 62.7%). The
+highest accuracy 62.7% is obtained with the BERT-
+ResNet-concatenation baseline.
+Do models fail on particular chart reasoning
+types?
+We evaluate the best VL baseline, consisting of
+BERT, ViT, and concatenation with GRUs, on the
+chart reasoning types present in ChartFC and de-
+scribed in Section 4.1.4. We find that the model
+performs best on the reasoning types retrieve value,
+filter, and finding extremum, while struggling partic-
+ularly with compute derived values. Figure 6 shows
+that the model classifies correctly 65% (i.e. 33 out
+of 51) of claims that require retrieval and 61% of
+claims that require filtering. However, only 50%
+of comparison claims and 38% of claims required
+to compute derived values are correctly predicted.
+These results are in line with previous works that
+discuss limitations of state-of-the-art models in
+tasks requiring numerical reasoning capabilities
+(Thawani et al., 2021).
+8The complete set of results obtained with different en-
+coders and fusion methods can be found in Tables 5, 6, and 7
+in the Appendix.
+
+Figure 6: Chart reasoning types: total count and cor-
+rect predictions of manually annotated test samples.
+Training Samples
+Test Accuracy
+127 (1%)
+51.6
+3,175 (25%)
+57.0
+6,351 (50%)
+57.1
+9,526 (75%)
+58.0
+12,702 (100%)
+59.8
+Table 4: Performance of VL baseline (BERT, ViT, and
+concatenation with GRUs) with different training set
+sizes.
+Is the dataset size sufficient for our proposed
+task? Do ChartFC claims contain biases?
+We evaluate the size of the dataset by training
+our VL baseline (i.e. using BERT, ViT, and con-
+catenation with GRUs) on various subsets of the
+training data as shown in Table 4 and report the
+accuracy on the test set. The performance on the
+test set improves as the number of training sam-
+ples increases. While the performance gain is high
+when increasing the training set from 1% to 25%
+(51.6% accuracy compared to 57%), the difference
+in accuracy between the baseline trained on half
+of the training data and the entire training data is
+only 2.6%, indicating that our training set has a
+reasonable size.
+We also train a claim-only BERT model to deter-
+mine whether claims contain biases that allow the
+model to correctly predict the label while ignoring
+the evidence charts. Trained on the claim text only,
+the model achieves 52% accuracy on the test set,
+compared to ChartBERT’s accuracy of (63.8%).
+We conclude that the claim text itself is not suffi-
+cient for correct classification.
+What are the dis-/advantages of an automated
+dataset pipeline for chart fact-checking?
+We automatically create ChartFC using a table
+fact-checking dataset as seed by identifying sub-
+tables relevant to the claims and then building the
+charts. ChartFC includes common stylistic varia-
+tions: bars of different colors, horizontal/vertical
+orientations, different backgrounds (light/dark, grid
+lines/no grid lines). While natural charts come with
+large stylistic variation, using them results in re-
+duced control over task complexity and dataset.
+In future work, we plan to explore two alterna-
+tive dataset creation pipelines: first, automated
+pipelines for other charts types to extend the cur-
+rent dataset, and second, a pipeline with natural
+charts where we would create claims for charts.
+Using natural charts would require a multi-step
+annotation process: selecting and separating charts
+from other images (Vougiouklis et al., 2020); writ-
+ing claims which support/refute them; evaluating
+the claims to check for correctness, typos, etc. We
+would require annotators with proficiency in inter-
+preting charts, and with basic mathematical and
+language skills to create claims with different rea-
+soning types (see Figure 5).
+5
+Conclusion and Future work
+We propose the chart fact-checking task and intro-
+duce ChartBERT, a novel model for fact-checking
+claims against chart images comprising three main
+components: a reading component, a sequence
+generation component, and an encoder that ex-
+tends BERT’s encoder with structural embeddings.
+We also introduce ChartFC as the first dataset for
+fact-checking against chart images, consisting of
+15, 886 claims and chart images.
+ChartBERT
+achieves
+63.8%
+accuracy
+on
+ChartFC. We systematically evaluate 75 different
+VL baselines, using various language encoders, vi-
+sion encoders, and fusion methods. The highest-
+performing VL baseline uses BERT as language
+encoder, ResNet to extract image representations,
+and concatenation to obtain joint representations
+for both modalities. The model achieves 62.7%
+test accuracy. Our results indicate that chart fact-
+checking, which requires extracting and reasoning
+over text and structural information from charts, is
+a challenging task for future research on AFC and
+VL methods.
+
+51
+50
+40
+33
+30
+total
+correct
+23
+24
+20
+20
+16
+14
+14
+10
+10
+6
+4
+Chart Reasoning TypeLimitations
+The TabFact dataset (Chen et al., 2020a) has been
+a valuable resource for creating ChartFC. However,
+using it as (the sole) seed dataset has limitations.
+ChartFC consists of bar charts only; indeed,
+given the claims and tables found in TabFact, the
+bar chart was deemed the most appropriate chart
+type. Various types of charts exist (e.g. pie charts,
+line charts) and their effectiveness in different data
+contexts and tasks has been investigated in the lit-
+erature. For example, Saket et al. (2019) evaluated
+the effectiveness of chart types using crowdsourc-
+ing experiments across the chart reasoning types
+we discussed in Section 4.1.4. In the context of
+small datasets, i.e. up to 34 rows and two columns
+which is similar to our setting, Saket et al. (2019)
+found bar charts to be the most accurate visualiza-
+tion type for the given chart reasoning types. In
+addition to bar charts, other types of charts used as
+evidence for fact-checking tasks ought to be inves-
+tigated. Behrisch et al. (2018) studied visualization
+methods for different data types (i.e. multi- and
+high-dimensional data, relational data, geo-spatial
+data, sequential and temporal data, and text data).
+For example, they found that scatter plots were ap-
+propriate visualization types for queries regarding
+data distribution (e.g. correlations and clusters),
+while line charts were more appropriate for queries
+about temporal aspects of data. To extend ChartFC
+with other chart types, we require more diverse
+data types (e.g. sequential and temporal data) and
+appropriate claims.
+Moreover, ChartFC claims are restricted to En-
+glish, whereas misinformation is commonly spread
+in different languages. Future work is necessary to
+address the limited availability of non-English fact-
+checking datasets and to contribute to the efforts
+done in this space (Gupta and Srikumar, 2021).
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+China. Association for Computational Linguistics.
+A
+Dataset Pipeline
+In Figure 7, we give an example of the dataset cre-
+ation pipeline. Starting with the claim and initial
+TabFact table, we first filter columns required to de-
+cide the claims veracity label: “age at appointment”
+and “prior occupation”. This sub-table is used to
+create the evidence chart (bottom right).
+B
+Chart Reasoning Types
+We label 100 random test set samples with chart
+reasoning types. Next, we briefly describe each
+type, for more details we refer to the taxonomy by
+Amar et al. (2005):
+• Retrieve Value: Given some conditions, re-
+trieve a single value from the chart image.
+• Filter: Find all data points in the chart that
+fulfill some specified conditions.
+• Compute Derived Value: Calculate an aggre-
+gated value (e.g. average or count) using data
+points extracted from the chart.
+• Find Extremum: Extract the top-n data points
+given some conditions.
+• Determine Range: Based on some conditions,
+find a span of values such that all extracted
+data points fulfil the conditions.
+• Find Anomalies: Find any anomalies in a spec-
+ified set of data points.
+• Compare: Compare the values of different
+data points to each other.
+
+Figure 7: Example for dataset creation process.
+Figure 8:
+Encoders and fusion methods used in VL
+baselines.
+C
+VL Baselines
+Figure 8 provides an overview of all encoders and
+fusion methods we use in our evaluation.
+Table 5, 6, and 7 provide an overview of all VL
+baselines we evaluated on ChartFC.
+
+Claim: There are four people who were appointed at secretary at the age of 50
+1. Initial table
+romanised name
+chinese name
+ageatappointment
+portfolio
+prior occupation
+0
+donald tsang yam - kuen
+曾槿
+58 
+chief secretary for administration (cs)
+chief secretary foradministration (cs)
+1
+anthony leung kam - chung
+梁锦松
+50
+financial secretary (fs)
+financial secretary (fs)
+2
+elsie leung oi - see
+梁愛詩
+63
+secretary for justice (si)
+secretary for justice (si)
+3
+joseph wong wing - ping
+王永平
+54
+secretary for civil service
+secretary for civil service
+henry tang ying - yen
+唐英年
+50
+secretary for commerce,industry and technology
+chairman , federation of hong kong industries
+chief secretary for administration (cs)
+58
+financial secretary (fs)
+ 50
+2. Subtable
+3. Chart
+secretary for justice (sj)
+9
+ageat appointment
+prior occupation
+secretary for civil service
+ 54
+0
+58
+chief secretary for administration (cs)
+prior occupation
+chairman, federation of hong kong industries
+50
+1
+50
+financial secretary (fs)
+secretary for financial services
+50
+s9
+secretary for justice (sj)
+chief financial officer . pccw
+50
+3
+ 54 
+secretary for civil service
+md of greater china , ch2 m hill
+51
+4
+50
+chairman , federation of hong kong industries
+chairman , arts development council
+ 52 
+..
+secretary for constitutional affairs
+58
+vice - chancellor , chinese university
+ 57
+secretary for health and welfare
+56 
+20
+40
+60
+age at appointmentVision encoders
+Language encoders
+Fusion methods
+FC layer
+concatenation
+BERT
+concatenation
+AlexNet
+Embedder
++ GRUs
+ResNet
+LSTM
+multiplication
+DenseNet
+BERT
+MCB Pooling
+Transformer
+ViT
+layersLang Encoder
+Vis Encoder
+Fusion
+Val Acc
+Val F1
+Test Acc
+Test F1
+BERT Emb
+FC
+concatenation
+56.7
+37.8
+55.6
+36.6
+BERT Emb
+FC
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+FC
+multiplication
+56.6
+52.8
+56.5
+52.3
+BERT Emb
+FC
+MCB
+56.2
+36.1
+55.6
+35.7
+BERT Emb
+FC
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+AlexNet
+concatenation
+56.5
+40.2
+55.1
+38.1
+BERT Emb
+AlexNet
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+AlexNet
+multiplication
+57.0
+41.4
+55.9
+39.9
+BERT Emb
+AlexNet
+MCB
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+AlexNet
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ResNet 152
+concatenation
+56.5
+45.4
+56.2
+45.5
+BERT Emb
+ResNet 152
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ResNet 152
+multiplication
+56.6
+38.3
+56.3
+38.8
+BERT Emb
+ResNet 152
+MCB
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ResNet 152
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+DenseNet (6, 12, 24)
+concatenation
+56.5
+43.7
+54.0
+40.7
+BERT Emb
+DenseNet (6, 12, 24)
+concatenation, biGRU
+56.6
+45.3
+54.1
+42.2
+BERT Emb
+DenseNet (6, 12, 24)
+multiplication
+56.5
+37.1
+55.6
+36.4
+BERT Emb
+DenseNet (6, 12, 24)
+MCB
+56.2
+36.1
+55.6
+35.7
+BERT Emb
+DenseNet (6, 12, 24)
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ViT
+concatenation
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ViT
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ViT
+multiplication
+57.1
+42.1
+54.8
+37.6
+BERT Emb
+ViT
+MCB
+56.2
+36.0
+55.6
+35.7
+BERT Emb
+ViT
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+Table 5: VL baselines using BERT embedder for text encoding, different vision encoders, and fusion methods
+Lang Encoder
+Vis Encoder
+Fusion
+Val Acc
+Val F1
+Test Acc
+Test F1
+LSTM
+FC
+concatenation
+56.6
+36.9
+55.5
+35.8
+LSTM
+FC
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+LSTM
+FC
+multiplication
+56.2
+36.0
+55.6
+35.7
+LSTM
+FC
+MCB
+56.2
+36.0
+55.6
+35.7
+LSTM
+FC
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+LSTM
+AlexNet
+concatenation
+56.3
+39.6
+56.1
+39.8
+LSTM
+AlexNet
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+LSTM
+AlexNet
+multiplication
+56.2
+36.0
+55.6
+35.7
+LSTM
+AlexNet
+MCB
+56.2
+36.0
+55.6
+35.7
+LSTM
+AlexNet
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+LSTM
+ResNet 152
+concatenation
+56.2
+36.0
+55.6
+35.7
+LSTM
+ResNet 152
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+LSTM
+ResNet 152
+multiplication
+56.2
+36.0
+55.6
+35.7
+LSTM
+ResNet 152
+MCB
+56.4
+36.3
+56.0
+35.9
+LSTM
+ResNet 152
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+LSTM
+DenseNet (6, 12, 24)
+concatenation
+56.2
+36.0
+55.6
+35.7
+LSTM
+DenseNet (6, 12, 24)
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+LSTM
+DenseNet (6, 12, 24)
+multiplication
+56.2
+36.0
+55.6
+35.7
+LSTM
+DenseNet (6, 12, 24)
+MCB
+56.2
+36.0
+55.6
+35.7
+LSTM
+DenseNet (6, 12, 24)
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+LSTM
+ViT
+concatenation
+56.2
+36.0
+55.6
+35.7
+LSTM
+ViT
+concatenation, biGRU
+56.2
+36.0
+55.6
+35.7
+LSTM
+ViT
+multiplication
+56.2
+36.0
+55.6
+35.7
+LSTM
+ViT
+MCB
+56.3
+36.7
+55.7
+36.5
+LSTM
+ViT
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+Table 6: VL baselines with LSTM as language encoder, different vision encoders, and fusion methods
+
+Lang Encoder
+Vis Encoder
+Fusion
+Val Acc
+Val F1
+Test Acc
+Test F1
+BERT
+FC
+concatenation
+59.3
+50.7
+59.6
+51.0
+BERT
+FC
+concatenation, biGRU
+58.8
+51.1
+58.5
+50.2
+BERT
+FC
+multiplication
+59.4
+54.5
+59.7
+54.9
+BERT
+FC
+MCB
+59.7
+49.6
+59.1
+49.3
+BERT
+FC
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+BERT
+AlexNet
+concatenation
+59.5
+47.9
+59.1
+47.6
+BERT
+AlexNet
+concatenation, biGRU
+59.2
+48.2
+58.0
+47.0
+BERT
+AlexNet
+multiplication
+59.0
+56.2
+59.6
+57.0
+BERT
+AlexNet
+MCB
+58.8
+45.2
+57.4
+43.9
+BERT
+AlexNet
+Transformer layers
+57.6
+50.8
+59.5
+52.6
+BERT
+ResNet 152
+concatenation
+59.8
+50.9
+59.8
+50.8
+BERT
+ResNet 152
+concatenation, biGRU
+59.1
+47.0
+58.8
+46.7
+BERT
+ResNet 152
+multiplication
+59.3
+52.2
+60.1
+53.6
+BERT
+ResNet 152
+MCB
+58.2
+47.0
+58.7
+48.9
+BERT
+ResNet 152
+Transformer layers
+56.2
+36.0
+55.6
+35.7
+BERT
+DenseNet (6, 12, 24)
+concatenation
+59.1
+51.4
+59.1
+52.4
+BERT
+DenseNet (6, 12, 24)
+concatenation, biGRU
+60.2
+53.0
+59.0
+51.0
+BERT
+DenseNet (6, 12, 24)
+multiplication
+59.4
+49.2
+58.7
+48.7
+BERT
+DenseNet (6, 12, 24)
+MCB
+59.9
+49.6
+58.8
+48.6
+BERT
+DenseNet (6, 12, 24)
+Transformer layers
+58.7
+48.0
+58.1
+46.8
+BERT
+ViT
+concatenation
+56.2
+36.0
+55.6
+35.7
+BERT
+ViT
+concatenation, biGRU
+59.0
+51.2
+59.8
+51.7
+BERT
+ViT
+multiplication
+58.0
+42.7
+56.6
+41.1
+BERT
+ViT
+MCB
+59.2
+49.5
+59.2
+49.6
+BERT
+ViT
+Transformer layers
+57.1
+40.8
+55.9
+39.1
+Table 7: VL baselines with BERT as language encoder, different vision encoders, and fusion methods
+

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+ 
+1
+Contrasting Analog and Digital Resistive Switching 
+Memory 
+Characteristics 
+in 
+Solution-Processed 
+Copper (I) Thiocyanate and Its Polymer Electrolyte 
+Based Memristive Devices  
+Rajesh Deb1, Saumya R. Mohapatra*1, Manjula G. Nair2, and Ujjal Das3  
+1Solid State Ionics Laboratory, Department of Physics, National Institute of Technology Silchar, 
+Silchar-788010, Assam, India 
+2Department of Physics, Indian Institute of Technology, Patna, Bihar, India, 801106 
+3Quantum Materials and Device Unit, Institute of Nano Science and Technology, Mohali-
+140306, Punjab, India 
+*Email:[email protected] 
+ 
+KEYWORDS:  Memristive devices, analog and digital memory, synaptic plasticity, Schottky 
+emission, charge trapping-detrapping, electrochemical metallization cell (ECM). 
+ 
+ 
+
+ 
+2
+ABSTRACT 
+   Usually, resistive switching (RS) devices show digital RS memory (sharp SET and RESET 
+process), which is most suitable for digital data storage applications. Some RS devices also 
+manifest ideal memristive behavior or analog memory characteristics (gradual change in 
+resistance states). The analog RS properties of memristive devices widen their application 
+domain to a much broader field of neuromorphic computing. The tunability of memristive 
+devices to digital or analog memory applications greatly depends upon the switching medium. In 
+this work, we report a comparative study on RS properties of two kinds of memristive devices 
+based upon copper (I) thiocyanate (CuSCN) and a solid polymer electrolyte (SPE) made up of 
+CuSCN as ionic moieties in polyethylene oxide (PEO). The device (ITO/CuSCN/Cu), prepared 
+by spin-coating CuSCN layer between ITO and copper electrode, shows simultaneous analog 
+and digital RS characteristics. The RS property of the device is tunable by varying the thickness 
+of the CuSCN layer. The current-voltage characteristics reveal that devices prepared at 3000 rpm 
+(thicker) during the spin-coating show only digital bipolar RS memory. In comparison, the 
+devices deposited at 4000 rpm (thinner) show both analog and digital RS memory. The 
+conduction mechanism responsible for RS behavior in CuSCN-based devices is Schottky 
+emission mediated charge trapping and de-trapping at the interfacial states. Contrastingly, when 
+the same CuSCN is used as the electrolyte in SPE film, the device only shows bipolar digital 
+non-volatile memory characteristics. The RS behavior is due to the electrochemical metallization 
+(ECM) mechanism. The ON and OFF states are achieved by the formation and rupture of copper 
+filaments due to the redox reactions at the interface.  
+1. INTRODUCTION 
+
+ 
+3
+   With the advent and adaptation of new technologies such as the internet of things (IoT), 
+artificial intelligence (AI), and neuromorphic computing, etc., in many walks of our life, society 
+is increasingly becoming more data-driven. To cope with the surge of demand in storing vast 
+amounts of data and processing it faster, the storage media, i.e., digital memory technology, is 
+facing the challenge of improving memory density and switching speed. So far, memory density 
+has increased in tandem with Moore's law. Hence, the cell size has been continuously 
+downscaled and may already hit the physical limit (i.e., 4F2 design protocol where F is the 
+minimum feature size in lithography).1-3 Similarly, the slow switching speed of the memory cell 
+creates a bottleneck between the memory and the processor in the von-Neumann architecture 
+based platforms, where they are segregated as different units.3-5 Hence, the overhauling of digital 
+memory technology is eminent and rightly happening, as evident from the recent developments 
+of in-memory-computing or near-data-processing device architectures.5-8 In this context, the 
+'resistive switching memory' (RSM) brings a new prospective with versatile memory 
+characteristics. The RSM devices, also known as memristive devices, show voltage modulated 
+resistance states, and their memory characteristics can be broadly classified as digital and analog 
+types. Digital RSM devices have features of sharp SET and RESET processes that can be 
+observed in the bipolar or unipolar mode of operations. The early studies on RSM are primarily 
+focused on observing digital non-volatile memory operations, which is useful in data storage. On 
+the other hand, the analog RSM shows a gradual change of resistance states as the voltage is 
+scanned. It represents ideal memristive behavior i.e., showing pinched hysteresis in the current-
+voltage characteristics.  
+   Recent reports of RS devices show digital RS memory characteristics can be repurposed to 
+observe analog RS memory, and vice-versa, simply by maneuvering the biasing conditions.9-11 
+
+ 
+4
+Such versatility in the memory characteristics of memristive devices widens their application 
+potential from conventional digital storage to the implementation of neuromorphic computing 
+hardware and mimicking biological synaptic memory. Further, memristive devices also see many 
+specialized niche applications with novel device architecture and by on-boarding them with other 
+micro/nanodevices. Lately, memristor-based sensors (memsensors), where along with the 
+electrical bias, other external stimuli also strongly influence the resistance state of the devices, 
+are explored for various sensing applications.12-13 Similarly, the self-powered memristive device 
+is a new trend where hybrid device architectures are designed with nanogenerators, thin film 
+solar cells, etc.14-17  
+   Memristive devices endowed with such varied functionalities and multiple memory 
+characteristics owe much to the switching medium. This work focuses on copper thiocyanate 
+(CuSCN) as a promising switching medium. CuSCN is already reasonably known as an excellent 
+electronic and optoelectronic material having applications in a wide range of devices such as 
+perovskite photovoltaics, organic LED, deep-UV photodetectors, thin film transistors, and 
+radiofrequency Schottky diode, etc.18-22 Attributes such as chemical stability, optical 
+transparency (98% in the visible range), good hole mobility (0.01–0.1 cm2 V−1 s−1) and suitable 
+energy band alignments to support hole transport are some of the reasons make it very popular as 
+a hole transport layer (HTL) in these devices.23 CuSCN is a P-type semiconductor from the 
+family of pseudohalides and, due to the interbonding of layers, can form a coordination polymer. 
+Moreover, CuSCN is solution processable with solvents such as diethyl sulfide (DES) and 
+NH4OH, producing large-area thin films without cracks and pin-holes.24 Hence, it has a huge 
+scope for low-cost solution-processable flexible and transient electronic applications. But, 
+despite these advantages, CuSCN as the primary or sole switching material in memristive 
+
+ 
+5
+devices is less explored. Initial studies on CuSCN-based RS memristive devices used a 
+composite of CuSCN or copper-dopped CuSCN solid electrolytes showing digital RS 
+characteristics.25-26 In both cases, the deposition method of CuSCN was rigorous and 
+complicated, involving either thermal evaporation of KSCN on the copper surface or dipping in 
+the copper electrode in a solution of NaSCN. Another work involving CuSCN was reported by 
+B. Cheng et al. 27 They showed bipolar RS behavior in a multilayer RS medium made up of 
+CuSCN/PMMA/ZnO that worked as a p-i-n heterojunction diode. More recently, W. Chen et al. 
+reported negative differential resistance and bipolar resistive switching memory in symmetric 
+ITO/CuSCN/ITO devices where the CuSCN layer was electrodeposited.28 Here, in this study, we 
+fabricated two types of memristive devices with the switching medium as (i). CuSCN and (ii). 
+CuSCN-based solid polymer electrolyte (Cu-SPE). Both the switching materials are solution-
+processed and spin-coated. The Cu-SPE is comprised of CuSCN dissolved in polyethylene oxide 
+(PEO). The CuSCN and Cu-SPE films are pretty different regarding their electrical properties. 
+The CuSCN layer, though often considered a solid electrolyte, its ionic moieties are not mobile. 
+It is purely an electronic conductor through holes.29 But in Cu-SPE, CuSCN remains dissolved 
+with separate and mobile ionic species (Cu+ and SCN–). Hence, it remains predominantly an 
+ionic conductor with a high ion-transport number.32 Our first device with ITO/CuSCN/Cu 
+stacking shows both digital and analog-type resistive switching properties with the ability of 
+synaptic plasticity. In contrast, the devices with ITO/Cu-SPE/Cu structure show only digital RS 
+memory based on the electrochemical metallization (ECM) mechanism.  
+2. EXPERIMENTAL SECTION 
+2.1 Fabrication of CuSCN based memristive cells 
+
+ 
+6
+   Copper (I) thiocyanate (CuSCN) and Polyethylene Terephthalate (PET) substrates with 
+Indium-tin Oxide (ITO) layer pre-deposited were brought from Sigma Aldrich. The substrate 
+was cut into pieces of 18 ×18 mm2 areas. The solution of CuSCN was prepared using diethyl 
+sulfide as the solvent and spin-coated onto the ITO layer of the PET substrate. Devices were 
+prepared with three different thicknesses of CuSCN layer by spin-coating at the speed of 3000, 
+4000, and 5000 rpm for 60 secs. The prepared films were then vacuum-dried at 60 0C for 5 hours 
+to remove the solvent. Finally, a 40 nm thick circular copper (Cu) top electrode of diameter (100 
+µm) was deposited using a stainless steel shadow mask with a thermal evaporation method. So, a 
+vertical two-terminal ITO/CuSCN/Cu device was formed, where ITO and Cu act as bottom and 
+top electrodes, respectively.  
+2.2 Preparation of PEO-CuSCN Solid Polymer Electrolyte (SPE) 
+   Poly(ethylene oxide) (PEO) is known to solubilize many alkali and alkaline earth metals salts 
+due to the large dipole moment on the ether oxygen.30 Copper (I) thiocyanate as a pseudohalide 
+is also expected to dissolve in PEO. Following the earlier reports, we prepared solid polymer 
+electrolytes (SPEs) using PEO and CuSCN.31 – 32 PEO of molecular weight 6×105 was purchased 
+from Sigma Aldrich. For the preparation of the electrolytic solution, 0.5g of PEO was initially 
+dissolved in 20 ml of acetonitrile under constant stirring for 3 hours. Then CuSCN was added to 
+the above solution as weight fraction (x wt.%) of PEO where x =  0.25, 0.5, 1, 2, 3, and 5. The 
+mixtures were stirred for another 21 hours to get a homogeneous solution. 
+2.3 Fabrication of Cu-SPE based Memristive Cells 
+   The polymer electrolyte solution containing both PEO and CuSCN was spin-coated onto the 
+ITO layer of the PET substrate. The spin-coating was carried out at 1000 rpm for 10 sec, 
+
+ 
+7
+followed by at 3000 rpm for 120 sec. The prepared films were then dried in a vacuum oven at 60 
+0C for 5 hours to remove any left-out solvent. The thickness of the copper-ion conductive Cu-
+SPE film is estimated to be ~ 250 nm from the cross-sectional scanning electron microscope 
+(SEM) image, as shown in supporting information (SI) Figure S5. Finally, a 40 nm thick circular 
+copper (Cu) top electrode of diameter (~100 µm) was deposited, as mentioned in section 2.1. 
+Hence, the desired device is a vertical stack of ITO/Cu-SPE/Cu. 
+2.4 Characterization of Materials and Devices 
+   The structural properties and absorption spectra of the CuSCN film were characterized using 
+X-ray diffraction (XRD) and UV-visible spectroscopy. The film's chemical structure and surface 
+topography were studied using Raman spectra and atomic force microscopy (AFM). The Cu-SPE 
+films were characterized by using XRD and FTIR. XRD study shows two broad peaks at 19.1° 
+and 23.3° due to the PEO, as presented in SI Figure S3a. No Bragg peaks corresponding to 
+CuSCN were observed in the electrolyte films. This confirms CuSCN is dissolved and ionic 
+species are separated in the polymer electrolyte. SI Figure S3b shows the FTIR spectra of the 
+Cu-SPE films with varying CuSCN concentrations. The characteristic vibrational modes of PEO 
+and CuSCN are identified in the figure. 
+     The electrical characterization of the CuSCN and Cu-SPE film-based memristive device were 
+studied by using Keithley 4200-Semiconductor Characterization System (SCS). The PET 
+substrate containing the vertical cells was placed on a probe station, and contact was made with 
+the electrodes using the tungsten tips. 
+3. RESULT AND DISCUSSION 
+3.1 Material Characterization  
+
+ 
+8
+   The XRD spectra of as-received CuSCN powder and thin films spin-coated at 4000 rpm are 
+shown in Figure 1a. As indexed, the powder CuSCN is observed to be in the β phase, which has 
+a hexagonal (rhombohedral) structure.33-34 After making films from the solution, the CuSCN 
+though still remains in the β phase, becomes semicrystalline with a considerable amorphous 
+fraction. The Bragg peaks observed at 16.27⁰ and 27.4⁰ are significantly broadened, as shown in 
+the inset of Figure 1a. For the thin film, the average crystallite size for (003) and (101) peaks 
+observed at 16.27⁰ and 27.4⁰ found to be ~ 9.33 nm and ~ 4.47 nm, respectively, as calculated 
+using the Debye-Scherer formula. The UV-visible absorption spectra of the CuSCN powder and 
+thin films deposited on ITO are presented in Figure 1b in the wavelength range of 200-800 nm. 
+Above 350 nm (mainly in the visible region), no significant absorption is observed for the 
+powder CuSCN. While the thin film of CuSCN shows some absorption in the visible region may 
+be due to the defects or trap states created in the film.35 The absorption spectra of the CuSCN 
+powder and the thin film show a peak at ~ 297 nm, a characteristic of CuSCN.35 The optical band 
+gap energy was calculated from the Tauc plot, as shown in the inset of Figure 1b. The band gap 
+energy of the CuSCN powder and thin film spin-coated at 4000 rpm were found to be 3.6 eV and 
+3.4 eV, respectively, which agrees well with the reported values.36  
+
+ 
+Figure 1. (a) XRD pattern of CuSCN powder and thin film (inset) deposited over ITO coated PET substrate, (b) 
+UV-visible absorption spectra of CuSCN powder and thin film deposited over 
+represents the determination of band gap energy 
+of CuSCN powder and thin film deposited 
+peak of ITO, (d) AFM topography image of CuSCN thin film deposited over ITO coat
+   The Raman spectra in the frequency region of 50
+film are shown in Figure 1c. I
+and 244 cm-1 belong to the stretching vibration of Cu
+at 430 cm-1 and 876 cm-1 corresponds to 
+at 746 cm-1 is for C-S stretching. In the high
+2173 cm-1 corresponding to the C
+(a) XRD pattern of CuSCN powder and thin film (inset) deposited over ITO coated PET substrate, (b) 
+visible absorption spectra of CuSCN powder and thin film deposited over a glass substrate. Inset of (b) 
+represents the determination of band gap energy of CuSCN powder and thin film from Tauc plot
+of CuSCN powder and thin film deposited over ITO coated PET substrate. The symbol star (*) represents the Raman 
+peak of ITO, (d) AFM topography image of CuSCN thin film deposited over ITO coated PET substrate. 
+The Raman spectra in the frequency region of 50-3000 cm-1 of the CuSCN powder
+In the powder sample, the low-frequency modes at around 205 cm
+belong to the stretching vibration of Cu-S and Cu-N, respectively. 
+corresponds to the bending vibration of S-C≡N, 
+S stretching. In the high-frequency region, CuSCN shows a single peak at 
+corresponding to the C≡N stretching.24, 37 The thin film of CuSCN layer exhibit
+9
+ 
+(a) XRD pattern of CuSCN powder and thin film (inset) deposited over ITO coated PET substrate, (b) 
+glass substrate. Inset of (b) 
+er and thin film from Tauc plot, (c) Raman spectra 
+ver ITO coated PET substrate. The symbol star (*) represents the Raman 
+ed PET substrate.  
+of the CuSCN powder and thin 
+frequency modes at around 205 cm-1 
+N, respectively. The Raman shift 
+ while the Raman shift 
+frequency region, CuSCN shows a single peak at ~ 
+The thin film of CuSCN layer exhibits the 
+
+(003)
+(101)
+Intensity (a.u.)
+(003)
+(101)
+(a.u.)
+Intensity (
+(104)
+10
+20 30
+2
+(900)
+(110)
+(600
+(012)
+(015)
+(01
+人
+10
+20
+30
+40
+50
+20 (Degrees)
+- Thin film
+*—ITO peak
+a
+Intensi
+Cu-N
+v(Cu-S)
+S(SCN)
+人
+500
+1000
+1500
+Raman shift (c(110)
+=2000
+100bPowder2
+260
+70
+80
+300
+400
+500WayelengthC297 nmPowder2500-1owder'hin tilm-Thin filmowderThin Film36e13
+4Energy(ew600
+700
+800(nm)「55504535302520102 um 
+10
+peaks of S-C≡N bending vibration, C-S stretching mode, and C≡N stretching mode located at 
+430 cm-1, 746 cm-1, and 2173 cm-1, respectively. These Raman peaks are characteristics of the β-
+phase of CuSCN. The Raman peaks indicated by the star mark (*) in the thin film correspond to 
+the ITO layer on the PET substrate.38 The surface topography image of CuSCN film deposited at 
+4000 rpm on ITO-coated PET substrate is presented in Figure 1d. The root mean square surface 
+roughness measured over an area of 5µm × 5µm is ~ 10 nm with CuSCN grains on the surface. 
+3.2 Memory Characteristics of CuSCN based memristive device 
+   The schematic diagram of the ITO/CuSCN/Cu devices as prepared by spin-coating of the 
+CuSCN solution is presented in the inset of Figure 2a. Particularly for the device prepared at 
+4000 rpm while spin-coating, the current-voltage characteristics is shown in Figure 2a. The 
+devices were biased over a small voltage range (0.8 V to – 0.8 V). As the voltage sweeps from 
+0V to 0.8 V, the current gradually rises, and the device attends some low resistance state (LRS). 
+Again, by reverse basing (0 → – 0.8V → 0), the device steadily returns to the high resistance 
+state. This I-V characteristic represents the ideal memristive behavior, i.e., pinched hysteresis 
+loop with zero crossing.39 Such resistive switching memory characteristic is also known as the 
+analog RS memory. The device shows very stable analog RS behavior over repeated cycles. 
+Figure 2b shows the semilog plot of current vs. voltage for the first ten cycles of the voltage 
+scan, where the analog RS behavior is almost reproducible without any significant deviation. The 
+OFF-ON resistance ratio measured at a read voltage of 0.2V is 3.2 and 3.1 for the 1st and 10th 
+cycle, respectively. 
+
+ 
+Figure 2. (a) Linear current-voltage (
+sweeping the voltage from 0 to +0.8V to 
+of (a) shows the schematic device structure, (b) r
+complete cycles by sweeping the voltage from 0 to +0.8V to 
+   As elucidated in the introduction, the gradual change in resistance or conductance state 
+observed in the analog RS has 
+neuromorphic computation. To 
+plotted I-V characteristics by sweeping the voltage for three different ranges 
+±1.0V. In all these three sweeping cycles, the current gradually 
+forward biasing direction ( 0 
+when biasing is reversed ( 0 
+voltage of 0.2V have resistance 
+±0.5V, ±0.8V, and ±1.0V, respectively. It is note
+the sweeping cycle, the lower 
+the forward direction. Hence as a consequence, in 
+attained with less resistance as 
+ltage (I-V) characteristic of ITO/CuSCN/Cu memristive
+the voltage from 0 to +0.8V to – 0.8V to 0. The bias sweeping direction is indicated by black arrows. Inset 
+chematic device structure, (b) represents the semi-logarithmic plot of the memrist
+the voltage from 0 to +0.8V to – 0.8V to 0 for 10 continuous voltage sweep cycles.
+As elucidated in the introduction, the gradual change in resistance or conductance state 
+erved in the analog RS has an advantage over the digital RS memory for 
+neuromorphic computation. To further reveal the analog RS properties in this direction, we 
+characteristics by sweeping the voltage for three different ranges 
+±1.0V. In all these three sweeping cycles, the current gradually SETs to some LRS in the 
+forward biasing direction ( 0 → 0.5V, 0.8V or 1.0V → 0 ) and RESETs to HRS 
+when biasing is reversed ( 0 → – 0.5V, – 0.8V or – 1.0V → 0 ). The LRSs obtained at a read 
+voltage of 0.2V have resistance 110 kΩ, 43 kΩ, and 13 kΩ for voltage sweeping in the range of  
+respectively. It is noteworthy that with increasing the voltage range of 
+the sweeping cycle, the lower LRSs (or higher conductance states) are achieved while biasing in 
+the forward direction. Hence as a consequence, in the reverse biasing direction
+with less resistance as the voltage sweep range widens from ±0.5V to ±1.0V.
+11
+ 
+tive device of first cycle by 
+indicated by black arrows. Inset 
+plot of the memristive device in 
+0.8V to 0 for 10 continuous voltage sweep cycles. 
+As elucidated in the introduction, the gradual change in resistance or conductance state 
+advantage over the digital RS memory for implementing 
+the analog RS properties in this direction, we 
+characteristics by sweeping the voltage for three different ranges ±0.5V, ±0.8V, and 
+s to some LRS in the 
+s to HRS progressively 
+. The LRSs obtained at a read 
+for voltage sweeping in the range of   
+worthy that with increasing the voltage range of 
+are achieved while biasing in 
+reverse biasing direction also, HRSs are 
+voltage sweep range widens from ±0.5V to ±1.0V.  
+
+Cu
+2.0x10-4
+1.5x10-4
+oCu
+CuSCN
+OC
+NO
+1.0x10
+Current (A)
+ITO
+5.0x10-5
+PET
+0.0
+-5.0x10-5
+-1.0x10-4
+-0.8
+-0.4
+0.0
+VoltagE-00
+-0.4OA 
+12
+Another testimony of analog RS memory is depicted in Figure 3b. We successively bias the 
+device in forwarding biasing direction with a sequence of 0 → 0.8V → 0 up to five consecutive 
+cycles. After each cycle, the device achieved a new LRS state with lower resistance. Then in the 
+reverse direction also, the device was biased with a sweeping sequence of 0 → – 0.8V → 0 
+consecutively for five cycles. The RESET process is also gradual, with an incremental rise in the 
+resistance value of the HRS as the sweeping cycle varies from 6 to 10. This type of continuous 
+and gradual change in the LRS and HRS states with bias modulating is the signature feature of 
+analog RS memory.  
+   To better represent the change in resistance states with voltage sweeps, Figure 3a and Figure 
+3b are reproduced in Figure 3c and Figure 3d, respectively, as the time evolution of current, as 
+voltage sweeps. Figure 3c shows the steady rise in current as the range of sweeping voltage 
+increases. Hence, multiple LRS and HRS can be achieved with different biasing voltages. In 
+Figure 3d, repetitive forward biasing cycles produce incrementally higher conductance states, 
+whereas the reverse biasing cycles produce progressively lower conductance states. These I-V 
+characteristics relate to the synaptic functions, such as potentiation and depression of the 
+synaptic weight.9, 40-41 Hence, this is suggestive that the analog RS memory prepared with 
+CuSCN has the propensity to show synaptic behavior and can be employed to implement 
+neuromorphic computing. It is also observed in Figure 4a that the memory window (resistance 
+ratio) increases with increase in the sweeping range of biasing voltage. Similarly, the memory 
+window decreases with increase in cycle number in consecutive forward and reverse biasing 
+cycles as shown in Figure 4b. In the forward biasing cycles, the current is self-limiting and may 
+attain saturation after some cycles. In the reverse biasing cycles, the HRS state's resistance 
+approaches the resistance of pristine cells as the number of cycles increases.  
+
+ 
+Figure 3. (a) Current-voltage characteristics of the 
+±0.5V (0V → 0.5V → 0V → – 0.5V
+ITO/CuSCN/Cu memristive device, where 1, 2, 3, 4
+voltage sweeps (0V → 0.8V → 0V) and 6, 7, 8, 9
+voltage sweeps (0V → – 0.8V → 0V) , (c) 
+and  (d) Temporal evolution of current 
+   The memristive device ITO/CuSCN/Cu can also show digital RS memory after a forming stage 
+which occurs just above 1.0 V.  But in successive cycles
+voltages ~ 0.4V and ~ – 0.25
+ON resistance ratio observed for 40 cycles as ~10
+showed digital and analog RS memories together.
+analog RS memory makes the CuSCN
+voltage characteristics of the ITO/CuSCN/Cu memristive device in cycles b
+0.5V → 0V), 0V to ±0.8V and 0V to ±1V. (b) Current
+, where 1, 2, 3, 4 and 5 represents the I-V curves under consecutive five positive 
+0V) and 6, 7, 8, 9 and10 represents I-V curves under consecutive five negative 
+0V) , (c) Temporal evolution of current when voltage sweeps with widening range 
+Temporal evolution of current for five consecutive positive and negative voltage sweeps
+The memristive device ITO/CuSCN/Cu can also show digital RS memory after a forming stage 
+just above 1.0 V.  But in successive cycles, SET and RESET occur at
+0.25V, respectively, as shown in Figure 5a. Figure
+ON resistance ratio observed for 40 cycles as ~105. Very few reports on
+nd analog RS memories together.9-11The simultaneous observation of digital and 
+analog RS memory makes the CuSCN-based devices more versatile. To gain more insight into 
+13
+ 
+device in cycles between 0V to 
+0V), 0V to ±0.8V and 0V to ±1V. (b) Current-voltage characteristics of 
+curves under consecutive five positive 
+curves under consecutive five negative 
+evolution of current when voltage sweeps with widening range 
+positive and negative voltage sweeps. 
+The memristive device ITO/CuSCN/Cu can also show digital RS memory after a forming stage 
+SET and RESET occur at very low 
+Figure 5b shows the OFF-
+on memristive systems 
+The simultaneous observation of digital and 
+based devices more versatile. To gain more insight into 
+
+1E-4.
+(a)
+1E-5
+1E-6
+1E-7-
+1E-8
+-0.8
+-0.4
+0.0
+Voltage (V
+1.0
+Voltage
+Current
+0.5
+0.0
+-0.5
+(c)
+-1.0-
+0
+100
+200
+300
+Time (s)F1E-63... 0.5 V0.8 V-10
+1F7.-0.4
+0.8
+-0.8
+-0.4
+0.0Voltag1E-4A
+）
+←100.0
+00F-50.0
+-0.4-0.8--100.0400
+500
+600
+0
+100
+200
+300Time5210.4
+0.8e (V)F1000 CurrentF50.0CF-100.0400
+500
+600(s)e 
+14
+the RS behavior of CuSCN-based devices, we carried out the electrical characterization of 
+ITO/CuSCN/Cu devices with different CuSCN layer thicknesses prepared at 3000 and 5000 rpm 
+of spin-coating. The devices prepared at 5000 rpm were found to be mostly short-circuited and 
+hence discarded from further investigation. The devices prepared at 3000 rpm don't show any 
+analog RS behavior, as shown in supporting information Figure S1a. However, it shows stable 
+bipolar digital RS behavior when swept between 0 and ±2V (SI Figure S1b). Compared to the 
+devices prepared at 4000 rpm, the SET voltage is higher. The observation of both analog and 
+digital RS in devices prepared at 4000 rpm and the absence of analog RS switching in devices 
+prepared at 3000 rpm suggests that the thickness of the CuSCN layer is the crucial factor in 
+observing the ideal memristive behavior. To ascertain whether the resistive switching 
+phenomena is due to the electrochemical metallization (ECM) (by copper ion diffusion from the 
+top Cu electrode) or trapping of charge carriers at the interface, we prepared the devices with a 
+gold top electrode (ITO/CuSCN/Au). The thickness of the CuSCN layer is maintained the same 
+by spin-coating it at 4000 rpm. The device shows both the analog and digital RS memory (SI 
+Figure S2a,b), similar to the ITO/CuSCN/Cu device. This suggests that the RS phenomenon is 
+due to charge trapping and de-trapping at the interface. For such interfacial RS behavior, high 
+electronic conductivity (~ 0.01 S m-1) of the CuSCN layer owing to the hole transport plays a 
+pivotal role.42 For materials like MXene (Ti3C2) and 1T-MoS2 nanosheets, the role of electronic 
+conductivity (due to electron or hole) in observing RS is already established.10, 39 
+
+ 
+Figure 4. (a) Plot of resistance ratio 
+consecutive positive and negative voltage 
+   Although we are ruling out the possibility of 
+the role of Copper ion (Cu+) diffusion
+voltage range over which the RS is observed in these devices
+electrode). Unlike the case of the copper top electrode
+switching for lower voltage sweep (< 1.0V) 
+RESET voltages of digital RS memory in gold
+based devices. So, we believe the RS memory behavior in ITO/CuSCN/Cu device
+carrier trapping and de-trapping modulated by the copper ion diffusion from the top copper 
+electrode. Such role of ionic modulation and coupling of ionic and electronic currents in 
+governing the resistive switching behavior is already testified 
+(a) Plot of resistance ratio for different voltage sweeping cycles, (b) Plot of resistance ratio with 
+voltage sweeping. 
+Although we are ruling out the possibility of an ECM type of mechanism for RS in CuSCN
+diffusion can't be completely played down if we are looking at the 
+voltage range over which the RS is observed in these devices (with Au and Cu as 
+case of the copper top electrode, we could not observe the analog 
+switching for lower voltage sweep (< 1.0V) with the gold top electrode. Similarly, the SET and 
+of digital RS memory in gold-based devices are much higher than in copper
+based devices. So, we believe the RS memory behavior in ITO/CuSCN/Cu device
+trapping modulated by the copper ion diffusion from the top copper 
+Such role of ionic modulation and coupling of ionic and electronic currents in 
+governing the resistive switching behavior is already testified in other memristive sys
+15
+, (b) Plot of resistance ratio with 
+ECM type of mechanism for RS in CuSCN, 
+yed down if we are looking at the 
+(with Au and Cu as a top 
+, we could not observe the analog 
+. Similarly, the SET and 
+based devices are much higher than in copper-
+based devices. So, we believe the RS memory behavior in ITO/CuSCN/Cu devices is charge 
+trapping modulated by the copper ion diffusion from the top copper 
+Such role of ionic modulation and coupling of ionic and electronic currents in 
+in other memristive systems.43-45  
+
+ 
+Figure 5. (a) Bipolar I-V characteristics of 
+resistances with switching cycle. 
+3.3 Current conduction mechanism in CuSCN based memristive devices
+   To elucidate further the current conduction mechanism in the ITO/CuSCN/Cu devices, the 
+analog I-V characteristics for the first positive voltage scan
+as shown in Figure 6a. It is observed that in the low biasing regi
+of the linearly fitted line is nearly one
+charge carriers in the CuSCN layer. The non
+region 0.27 V < V < 0.5 V is plotted as ln(
+is well fitted by the linear relationship
+Schottky emission in the particular voltage region. The slope of the curve in th
+0.5V < V< 0.8 V is 2.96. It suggests that the current conduction in 
+the trap-assisted space-charge
+in the HRS, the current conduction is 
+and trap assisted-SCLC. After the 
+slope is 1.02 for the entire biasin
+characteristics of ITO/CuSCN/Cu memristive device, (b) Plot of ON and OFF state 
+3.3 Current conduction mechanism in CuSCN based memristive devices
+To elucidate further the current conduction mechanism in the ITO/CuSCN/Cu devices, the 
+characteristics for the first positive voltage scan (Figure 3b) is plotted in log
+. It is observed that in the low biasing region, i.e., 0 < V < 0.27 V, the slope 
+is nearly one, indicating the Ohmic conduction due to the intrinsic 
+CuSCN layer. The non-linear part of the curve of Figure
+0.5 V is plotted as ln(I) vs √𝑉 as shown in the inset of 
+linear relationship, confirming that the current conduction is governed by 
+chottky emission in the particular voltage region. The slope of the curve in th
+. It suggests that the current conduction in the higher voltage region is
+charge-limited conduction (SCLC), i.e., I α Vm where m > 2.
+in the HRS, the current conduction is Ohmic initially and then followed by S
+SCLC. After the SET process, when sweeping back to zero from 0.8 V, the 
+slope is 1.02 for the entire biasing region, indicating the Ohmic type conduction agai
+16
+ 
+, (b) Plot of ON and OFF state 
+3.3 Current conduction mechanism in CuSCN based memristive devices 
+To elucidate further the current conduction mechanism in the ITO/CuSCN/Cu devices, the 
+) is plotted in log-log scale 
+0 < V < 0.27 V, the slope 
+hmic conduction due to the intrinsic 
+Figure 6a in the voltage 
+the inset of Figure 6a. The curve 
+current conduction is governed by 
+chottky emission in the particular voltage region. The slope of the curve in the biasing region 
+higher voltage region is 
+where m > 2.9, 46-48 Thus, 
+initially and then followed by Schottky emission 
+when sweeping back to zero from 0.8 V, the 
+the Ohmic type conduction again in the 
+
+1E-5
+(A)
+RESET
+Current
+1E-7
+1st
+1E-9
+10th
+20th
+30th
+40th
+1E-1
+-0.50
+-0.25
+0.00
+Voltage1'ON·ROFFS0.25
+0.50
+0
+10Switc>10°20
+30
+40hing cycleSHT 
+LRS. In the negative sweeping direction, the current conduction 
+Ohmic, Schottky emission, and SCLC while
+Figure 6. (a) log-log plot of the analog 
+device. The inset of (a) represents the ln(
+Plot of Schottky barrier height (ɸB
+digital I-V for the first SET process in the 
+linear fitting in the voltage region 0.20 V < V < 0.31 V.
+   CuSCN is a p-type semiconductor with work function (
+= 4.4 eV) and ITO (ΦITO ~ 4.8 to 5.2 eV
+respectively, at both the interfaces
+the Schottky contacts get modified
+negative sweeping direction, the current conduction follows the same sequence of 
+and SCLC while sweeping the bias from 0 V to 
+log plot of the analog I-V for the first positive voltage scan of the ITO/CuSCN/Cu memristive 
+nset of (a) represents the ln(I) vs √𝑉 plot with linear fitting in the voltage region 0.27 V< V < 0.5 V, (b) 
+B) with consecutive positive and negative voltage scan,  (c) log
+for the first SET process in the ITO/CuSCN/Cu memristive device, (d) ln (I) vs 
+linear fitting in the voltage region 0.20 V < V < 0.31 V. 
+type semiconductor with work function (ΦCuSCN ~ 5 eV).4
+~ 4.8 to 5.2 eV), CuSCN makes Schottky 
+at both the interfaces.50-52 So, during positive and negative voltage sweeping cycles, 
+the Schottky contacts get modified, influencing the charge flow to and from the device. At the
+17
+ws the same sequence of 
+sweeping the bias from 0 V to – 0.8V.  
+ITO/CuSCN/Cu memristive 
+plot with linear fitting in the voltage region 0.27 V< V < 0.5 V, (b) 
+) with consecutive positive and negative voltage scan,  (c) log-log plot of the 
+) vs √𝑉 plot of Fig. (c) with 
+49 Hence, with Cu (ΦCu 
+), CuSCN makes Schottky and Ohmic contacts, 
+positive and negative voltage sweeping cycles, 
+the charge flow to and from the device. At the 
+
+1E-4
+Positive voltage
+1E-5
+3
+1.01
+1E-7-
+slope :
+(a)
+1E-3
+0.01
+Voltage
+-O-Digital SET
+1E-51
+slope = 1.0
+Currel
+C
+HR
+1E-9.
+slope
+1E-3
+0.01
+Volta!Schottkvemision0.008e1.0-B0.50 -(h)0.55 0.60 0.65 0.700.58-
++Bca
+--18.4-DSLNO Pos0.92
+-18.8-0.1
+0.45Negitivevoltageative voltage11
+T
+10
+10ion1T0.50
+0.55eT0.54 - 
+18
+Cu/CuSCN interface, the Schottky barrier height can be calculated by using the Richardson-
+Schottky equation. The Richardson-Schottky equation of the I-V characteristics is given by 
+ 
+ 
+ 
+𝐼 = 𝐴𝐴∗𝑇�𝑒���������� ����
+⁄
+� ��
+⁄
+�  
+ 
+                                     (1) 
+where, 𝐴 = 𝜋(50 µ𝑚)� = cell contact area, 𝐴∗ =
+�����∗ ��
+ℎ�
+ = Richardson constant, T = 
+temperature, q = electric charge, 𝜑� = Schottky barrier height, E = electric field, 𝜀� = dielectric 
+constant of material and k = Boltzmann constant.9 Taking the natural logarithm of equation (1) 
+and replacing E by V/d, where d is the distance between the Cu top and ITO bottom electrode, 
+we get 
+ 
+     𝑙𝑛(𝐼) = �𝑙𝑛(𝐴𝐴∗𝑇�) −
+�
+�� 𝜑�� +
+�
+�� �𝑞 4𝜋𝜀�
+⁄
+√𝑉       
+                                  (2) 
+By taking the value of A* = 119.56 A/K2cm2 (𝑚�∗ = 𝑚�), the Schottky barrier height (𝜑�) is 
+calculated from the intercept of 𝑙𝑛(𝐼) vs √𝑉. We find that the Schottky barrier decreases under 
+successive positive voltage scans and increases under subsequent repetitive negative voltage 
+scans, as shown in Figure 6b. Further, from the slope of  𝑙𝑛(𝐼) vs √𝑉 plot, the ideality factor is 
+calculated to be ≈ 10, much greater than the ideality factor of an ideal diode (n =1). The higher 
+ideality factor explains the inhomogeneity in the Schottky barrier that arises due to traps or 
+interfacial states at the Cu/CuSCN interface.53 Quite a similar conduction mechanism is observed 
+for digital RS memory, as shown in Figure 6c. The log-log plot of current-voltage characteristics 
+during the SET process shows linear fit in the low voltage region of HRS with slope ⁓ 0.92. In 
+the high voltage region, i.e., 0.21 V to 0.39 V,  the current conduction mechanism follows 
+Schottky emission, as shown in Figure 6d, followed by trap-assisted SCLC. In the LRS, the slope 
+is well-fitted by Ohmic conduction. 
+
+ 
+19
+   Based on the conduction mechanism discussed, the energy band diagram is schematically 
+presented in Figure 7a-d. In the ITO/CuSCN/Cu memristive device, the Cu/CuSCN and 
+ITO/CuSCN interfaces make Schottky and Ohmic contacts (Figure 7b). When a positive bias is 
+applied to the Cu top electrode, band bending occurs, and as a result, Schottky barrier height 
+reduces. Hence, the current flow gets easier, and the trap states get filled. This makes the device 
+ON. On the other hand, in the reverse polarity of the bias, the Schottky barrier height increases at 
+the Cu/CuSCN interface, and the current flow reduces. The filled trap states release the charge 
+carriers at the Ohmic contact made at the ITO/CuSCN interface. Hence, charge trapping and de-
+trapping at the interfacial states is the main reason behind analog RS behavior in ITO/CuSCN/Cu 
+memristive devices. The electrochemical oxidation of the Cu electrode and influx of Cu+ ions 
+into the CuSCN medium during the forward biasing conditions can also strongly influence the 
+switching mechanism. Particularly in the digital RS memory, the ON state is due to the 
+conductive filament formed as a result of both filled trap states and the reduction of Cu+ ions to 
+metallic copper-dendrites grown at the ITO/CuSCN interface. The conductive filament may not 
+be entirely due to charge trapping or the copper dendrites connecting both interfaces. Instead, it 
+can be the result of both phenomena. A percolative conductive path may be formed between 
+filled interfacial trap states with patches of copper-dendrites in the CuSCN layer. Hence, the off 
+state may be attained by partial dissolution of the conductive filament due to the offloading of 
+charge carriers from the trap-states and oxidation of copper-dendrites.  
+
+ 
+ 
+Figure 7. (a) Energy band diagram of the 
+during the transition from OFF state to ON state (SET process), and (d) ON state to OFF state (RESET process).
+3.4 Current-voltage characteristics in ITO/Cu
+   Cu-SPE is the electrolyte medium containing CuSCN in the ionized form as Cu
+embedded in the PEO matrix, 
+spectroscopic characterization of the Cu
+information (Figure S3), confirming the dissolution of CuSCN in the PEO.
+CuSCN concentration in Cu-
+(a) Energy band diagram of the ITO/CuSCN/Cu memristive device before contact, (b) after contact, (c) 
+OFF state to ON state (SET process), and (d) ON state to OFF state (RESET process).
+voltage characteristics in ITO/Cu-SPE/Cu memristive device
+SPE is the electrolyte medium containing CuSCN in the ionized form as Cu
+ as shown in the schematic diagram in Figure 8
+spectroscopic characterization of the Cu-SPE film is already presented in the supporting 
+confirming the dissolution of CuSCN in the PEO.
+-SPE for better RS characteristics, the ITO/Cu
+20
+ 
+before contact, (b) after contact, (c) 
+OFF state to ON state (SET process), and (d) ON state to OFF state (RESET process). 
+SPE/Cu memristive device  
+SPE is the electrolyte medium containing CuSCN in the ionized form as Cu+ and SCN– ions 
+Figure 8a. The structural and 
+SPE film is already presented in the supporting 
+confirming the dissolution of CuSCN in the PEO. To optimize the 
+ITO/Cu-SPE/Cu cells were 
+
+Before contac
+(a)
+Vacuum
+CBM, -1.8 e
+dcu
+-4.4eV
+Cu
+ΦcuscN= -5 el
+Er,CusCN
+VBM.-5.4 6 -4.8e(c)
+0
+0
+Cu
+0
++VElectronHole+
+Aftar *tuCu(b]EuTO 
+prepared with Cu-SPE containing 
+characteristic of ITO/Cu-SPE/Cu cells 
+5.0%) are shown in the SI Figure
+one with 5 wt.% of CuSCN. However, the RS behavior is 
+for 0.25 and 0.5 wt.%. Whereas above 
+positive and negative biases with increasing concentration
+ionic moieties (Cu+ and SCN
+Hence, the depolarizing field created due to the space
+dominating in Cu-SPE film containing higher 
+RESET voltages increase in ITO/Cu
+5wt.%, the device could not be SET even 
+commonly observed in polymer electrolyte
+characterization of ITO/Cu-SPE/Cu devices, we fixed the CuSCN concentration in Cu
+1.0 wt.%, which shows very stable and repetitive bipolar digital RS behavior
+Figure 8. (a) Schematic diagram of the 
+ITO/Cu-SPE/Cu memristive device 
+SPE containing x wt.% of CuSCN where x = 0.25, 0.5, 1, 2, 3
+SPE/Cu cells containing varying CuSCN concentration
+Figure S6. All the devices show bipolar digital RS behavior except the 
+one with 5 wt.% of CuSCN. However, the RS behavior is unstable and not 
+for 0.25 and 0.5 wt.%. Whereas above 1 wt.%, the SET, and RESET processes occur at higher 
+with increasing concentrations of CuSCN. The concentration of the 
+and SCN–) increase along with CuSCN concentration in the Cu
+field created due to the space-charge polarization 
+SPE film containing higher concentrations of CuSCN. Due to this, SET and 
+ITO/Cu-SPE/Cu devices with higher CuSCN concentration
+5wt.%, the device could not be SET even if the applied bias is increased up to 10.0V. This is 
+commonly observed in polymer electrolyte-based ECM-type memristive devices.
+SPE/Cu devices, we fixed the CuSCN concentration in Cu
+which shows very stable and repetitive bipolar digital RS behavior
+(a) Schematic diagram of the ITO/Cu-SPE/Cu memristive device, (b) Bipolar 
+SPE/Cu memristive device for 1 wt% CuSCN. 
+21
+= 0.25, 0.5, 1, 2, 3, and 5. The I-V 
+CuSCN concentrations (0.25% to 
+All the devices show bipolar digital RS behavior except the 
+not repeatedly observed 
+and RESET processes occur at higher 
+The concentration of the 
+uSCN concentration in the Cu-SPE film. 
+charge polarization becomes more 
+of CuSCN. Due to this, SET and 
+CuSCN concentration, and at 
+the applied bias is increased up to 10.0V. This is 
+type memristive devices.54 For further 
+SPE/Cu devices, we fixed the CuSCN concentration in Cu-SPE as 
+which shows very stable and repetitive bipolar digital RS behavior.  
+ 
+device, (b) Bipolar I-V characteristics of the 
+
+(a)
+Cu
+SPE
+ITO
+PET二-1E-7.-2
+Wolt.fh1E-31st6041RESET 
+   Figure 8b represents the I-V
+film with 1 wt% of CuSCN for 60 consecutive voltage sweep cycles. In the first sweep cycle, the 
+device was SET at a higher voltage of 1.24V, called th
+In the second sweep cycle, the device
+shows digital RS memory with no significant
+to cycle.  
+Figure 9. (a) Distribution of the SET and RESET voltage of 
+shows the ON and OFF state resistances with switching cycles
+read voltage (c) Multi-level resistive switching observed in the ITO/Cu
+concentration of 1 wt% (d) Current-
+V characteristics of ITO/Cu-SPE/Cu devices containing
+1 wt% of CuSCN for 60 consecutive voltage sweep cycles. In the first sweep cycle, the 
+voltage of 1.24V, called the forming process, and RESET at 
+In the second sweep cycle, the device was SET at 0.96V and RESET at 
+with no significant variation of SET and RESET voltages from cycle 
+(a) Distribution of the SET and RESET voltage of ITO/Cu-SPE/Cu memristive device
+resistances with switching cycles (b) Retention test of the ON and OFF states at 30 mV 
+level resistive switching observed in the ITO/Cu-SPE/Cu memristive device with CuSCN 
+-voltage characteristics of the device up to 15 days. 
+22
+SPE/Cu devices containing the Cu-SPE 
+1 wt% of CuSCN for 60 consecutive voltage sweep cycles. In the first sweep cycle, the 
+and RESET at –1.16V. 
+was SET at 0.96V and RESET at – 0.86V. The I-V plot 
+variation of SET and RESET voltages from cycle 
+ 
+SPE/Cu memristive device. The inset of (a) 
+Retention test of the ON and OFF states at 30 mV 
+SPE/Cu memristive device with CuSCN 
+
+60
+50
+ Cycles
+1010
+40
+108
+30
+R
+ON
+>1
+20
+OFF
+10°
+10
+0
+10 20 30
+0
+(a)
+Switchir
+-1.0
+-0.5
+0.0
+Voltage
+1E-31
+-CC=1mA
+-CC=0.1mA
+1E-5
+-CC= 0.01 mA
+A
+Current
+1E-7
+1E-9.
+(c)
+1E-11
+-2
+-1
+0
+VoltageE
+Read @ 30 my5E+
+Du05
+10Tir1E-4-CC=0.1mARESET1E-610.1mA-1mA2
+-2
+-11E-4FV oltHRSTDC100
+1000
+10000ne (s)0Day-0-Dav 150
+1
+201E-6F 
+23
+   Further, Figure 9a represents the SET and RESET voltages distribution of ITO/Cu-SPE/Cu 
+cells during 60 consecutive voltage sweep cycles. The average SET and RESET voltages are 
+0.80 V and –0.78 V. The ITO/Cu-SPE/Cu cells exhibit good cyclic behavior, as shown in the 
+inset of Figure 9a. The ON and OFF states are reproducibly achieved for the SET and RESET 
+cycles. The OFF-ON state resistance ratio is more than 105, maintained throughout all the 
+measuring cycles. Also, retention characteristic was measured in each particular state under 
+continuous 30 mV read voltage, as shown in Figure 9b. Both ON and OFF states are stable up to 
+the measured time duration, i.e., 104 sec. The Cu-SPE based cells also show multi-level bipolar 
+resistance switching, as presented in Figure 9c. Three resistance states were achieved by keeping the 
+compliance at 0.01 mA, 0.1 mA, and 1.0 mA. As the compliance current increases from 0.01 mA to 1 
+mA, the SET and RESET voltage increase. The ON state resistances are in the order of 10 kΩ, 2 kΩ, and 
+500 Ω for 0.01 mA, 0.1 mA, and 1.0 mA compliance, respectively. This shows that with increasing 
+compliance current, the ON-state resistance decreases steadily. The I-V measurements of the device 
+for up to 15 days with an interval of 7 days are shown in Figure 9d. The device showed reliable 
+and reproducible resistive switching behavior over the period without any deterioration of the 
+switching characteristics as an effect of storing or aging. 
+3.5 Current conduction mechanism in Cu-SPE based memristive devices 
+   Further, to understand the role of the top Cu electrode and its electrochemical effect on the 
+switching mechanism, a memristive device was prepared with Au inert electrode by replacing the 
+copper. In the ITO/Cu-SPE/Au memristive device, no switching is observed up to the applied 
+bias of ±5 V, as shown in SI Figure S7. The current remains very low (~ 1.23 µA) even at 5V. 
+This suggests that the Cu top electrode has a definite role in the resistive switching behavior of 
+ITO/Cu-SPE/Cu memristive devices. To investigate the switching mechanism in Cu-SPE, the I-V 
+
+ 
+characteristic of one switching cycle was plotted in a log
+the SET process. The current conduction mechanism in the bias region 0.1V < V < 0.78V is 
+observed to be linear with a slope of 1.46
+for ion-hopping as given in equation (3). 
+                𝐼 = 2𝑧𝑒𝑐𝑎𝑣
+where, c is the concentration of mobile cations, 
+frequency factor. For low electric fields 
+equation (4) with linear dependence of 
+                                          𝐼 =
+This linear behavior infers that
+medium to form copper filaments on applying a positive bias to Cu top electrode.
+   In the LRS, the slope is found to be 1, which means that the current conduction mechanism 
+was dominated by Ohmic conduction as 
+Figure 10. Linear fitting of the I-V
+concentration of 1 wt% in ITO/Cu-SPE/Cu device
+characteristic of one switching cycle was plotted in a log-log scale, as shown in 
+the SET process. The current conduction mechanism in the bias region 0.1V < V < 0.78V is 
+a slope of 1.46. This can be understood using Mott
+hopping as given in equation (3). The Mott-Gurney equation is given by
+𝑧𝑒𝑐𝑎𝑣 𝑒𝑥𝑝 �
+���
+��� 𝑠𝑖𝑛ℎ �
+����
+��� �                                     
+is the concentration of mobile cations, a is the jump distance of ions
+For low electric fields �𝐸 <
+��
+����, the equation (3) can be modified as shown in 
+with linear dependence of I on E similar to Ohmic conduction.
+=
+(��)���
+��
+𝑎�𝑣 𝑒𝑥𝑝 �
+���
+���                                                                             
+This linear behavior infers that the Cu+ ions drifted toward the ITO electrode in the electrolyte 
+filaments on applying a positive bias to Cu top electrode.
+In the LRS, the slope is found to be 1, which means that the current conduction mechanism 
+was dominated by Ohmic conduction as copper filaments are formed in the
+V curve for the (a) SET and (b) RESET process using log
+SPE/Cu device. 
+24
+as shown in Figure 10a for 
+the SET process. The current conduction mechanism in the bias region 0.1V < V < 0.78V is 
+ng Mott-Gurney equation 
+s given by55 
+                                                                       (3) 
+is the jump distance of ions, and v is the 
+can be modified as shown in 
+similar to Ohmic conduction. 
+                                                     (4)  
+ions drifted toward the ITO electrode in the electrolyte 
+filaments on applying a positive bias to Cu top electrode. 
+In the LRS, the slope is found to be 1, which means that the current conduction mechanism 
+in the Cu-SPE film  
+ 
+curve for the (a) SET and (b) RESET process using log-log plot for CuSCN 
+
+ 
+25
+between Cu and ITO electrodes. During the RESET process in the LRS, from 0 to RESET 
+voltage (– 0.84V), the slope is 1 indicating the Ohmic type conduction. In the HRS from – 0.84V 
+< V < – 0.16V, the current conduction mechanism follows space charge limited current (SCLC) 
+with a slope of 2.54.45 This could be due to the emergence of space charge in the electrolyte 
+medium immediately after the breakdown of the filament. As the voltage decreases further, the 
+current follows linearly with the voltage due to the ion-hopping in the electrolyte medium, as 
+explained by equation (4). These studies confirm the switching mechanism to be ECM or 
+conductive-bridge random access memory (CBRAM) type, as observed in other polymer-
+electrolyte systems with electrochemically active electrodes.56,57 The formation and dissolution 
+of copper filament are responsible for the ON and OFF states, and are controlled by the 
+electrochemical redox reaction at the interfaces.  
+     Moreover, no analog RS is observed in these ECM-type memory devices based on Cu-SPE 
+film. In fact, analog RS is less often observed in ECM-type memory devices and more rarely so 
+in polymer-electrolyte based ECM devices.41,58 If the switching layer is electronically insulator, 
+sharp SET and RESET process occurs, leading to digital RS. But when the switching layer is 
+semiconducting (due to electron or hole conduction) along with allowing ion transport, the 
+chances of analog RS arises.59   
+4. CONCLUSIONS 
+   In conclusion, we prepared and studied two types of memristive devices based on CuSCN. The 
+switching media in these two devices are CuSCN and an SPE where CuSCN remains dissolved 
+in the PEO matrix (Cu-SPE). While the CuSCN is known to be a p-type semiconductor, the Cu-
+SPE is an ionic conductor and electronically insulator. The first device consisted of a solution-
+processed CuSCN layer as the switching medium. The RS behavior is strongly dependent on the 
+
+ 
+26
+thickness of the CuSCN layer. Devices with thinner CuSCN layer (prepared at 4000 rpm during 
+spin-coating) show both analog and digital RS memory. The switching mechanism is due to the 
+trapping and de-trapping of charge carriers at the trap or interfacial states modulated by the 
+Schottky emission. The copper-ion diffusion from the top Copper electrode also influences the 
+RS characteristics. The second device, made up of the Cu-SPE layer, shows only digital RS 
+memory. The devices offer good cyclability and retention of ON and OFF states. Multi-level RS 
+switching is also possible, as demonstrated by achieving discrete low resistance states (LRSs) by 
+enforcing different current compliances. However, analog RS is not observed in Cu-SPE-based 
+devices. The switching mechanism in the Cu-SPE based devices is ECM type, where the 
+formation and dissolution of copper filament are responsible for resistive switching behavior. 
+Nevertheless, CuSCN and its electrolytic system show versatile and contrasting RS memory 
+characteristics. Hence, they have immense potential for developing low-cost solution-
+processable nanoelectronics for digital non-volatile memory and neuromorphic computing 
+applications.  
+ 
+ACKNOWLEDGMENTS  
+The authors gratefully acknowledge the fund received from the Department of Science and Technology, 
+Government of India, through the DST-FIST project (SR/FST/PSI-212/2016(C)). The authors also 
+sincerely thank the help received from MRC, MNIT Jaipur in RAMAN and AFM measurements. 
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+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+31
+Supporting information 
+Contrasting Analog and Digital Resistive Switching 
+Memory 
+Characteristics 
+in 
+Solution-Processed 
+Copper (I) Thiocyanate and Its Polymer Electrolyte 
+Based Memristive Devices  
+Rajesh Deb1, Saumya R. Mohapatra*1, Manjula G. Nair2, and Ujjal Das3  
+1Solid State Ionics Laboratory, Department of Physics, National Institute of Technology Silchar, 
+Silchar-788010, Assam, India 
+2Department of Physics, Indian Institute of Technology, Patna, Bihar, India, 801106 
+3Quantum Materials and Device Unit, Institute of Nano Science and Technology, Mohali-
+140306, Punjab, India 
+*Email:[email protected] 
+
+ 
+Figure S1. (a) Current-voltage characteristics of the Cu/CuSCN/ITO memristive device spin
+Inset of (a) represents the semi-logarithimic plot. 
+coated at 3000 rpm. 
+Figure S2. (a) Current-voltage character
+curves under consecutive three positive voltage sweeps (0V
+consecutive three negative voltage sweeps (0V 
+as observed from the I-V characteristics in the 
+Structural and Optical characterization
+The X-ray diffraction pattern of the copper ion conductive 
+CUSCN (Cu-SPE) concentration o
+voltage characteristics of the Cu/CuSCN/ITO memristive device spin
+logarithimic plot. (b) Bipolar I-V curves of Cu/CuSCN/ITO memristive cell spin 
+ 
+voltage characteristics of Au/CuSCN/ITO memristive device, where 1, 2, 3 represents the 
+curves under consecutive three positive voltage sweeps (0V → 2V → 0V) and 4, 5, 6 represents 
+ee negative voltage sweeps (0V → – 2V → 0V), respectively (b) Bipolar 
+characteristics in the Au/CuSCN/ITO memristive device. 
+Optical characterization of Cu-SPE 
+ray diffraction pattern of the copper ion conductive solid polymer electrolyte films with 
+concentration of 0.25% to 5% is shown in Figure S3(a)
+32
+ 
+voltage characteristics of the Cu/CuSCN/ITO memristive device spin-coated at 3000 rpm. 
+curves of Cu/CuSCN/ITO memristive cell spin 
+ 
+, where 1, 2, 3 represents the I-V 
+0V) and 4, 5, 6 represents I-V curves under 
+Bipolar digital resistive switching 
+polymer electrolyte films with 
+(a). The pattern shows 
+
+(a)
+2x10°-
+1x10-5.
+(v)
+Current 
+0
+C
+-1x10s
+-2x10~
+-0.8
+-0.4
+0.0
+VoltagC1E-6RESET1E-6
+C1E-811E-8:-0.8-0.4
+0.0
+0.4
+0.8Yoltage (V)1E-4
+A)
+1E-5
+Current
+4
+C 1E-6
+5
+6
+1E-7 :
+Au/CuSCN/ITO
+-2
+-1
+0
+Voltage0.4
+0.8
+-2
+-1
+0e
+Vol1E-3 RESET1E-4=n3aE1
+2
+-2
+-1Vol(b)SET1E-41JiForming-1st4: 20th 30thot1
+2
+3tage (SET0
+2tage (V) 
+two dominant peaks of PEO at 19.1° and 23.3°, which corresponds to (120) and (112) planes 
+respectively.1,2 It is observed that there is an increase in the intensity of (120) peaks with the rise 
+in the salt concentration. This suggests that salt concentration has a strong influe
+crystallization process of the PEO. 
+in the electrolyte film, even at 5 wt.% of the CuSCN concentration. This indicates that no 
+uncomplexed CuSCN is present in the electrolyte film. 
+plotted as transmittance vs. wave
+the characteristic vibrational modes of PEO in the wavenumber region 800
+at 841cm-1 and 946 cm-1 are due to CH
+C-O stretching. The sharp band at 1094 cm
+relatively small bands at 1279 cm
+wagging, and CH2 deformation, respectively. The absorption band located at 2880 cm
+assigned to the C-H asymmetric stretching mode
+was visible for higher concentrations of copper (I) thiocyanate salt.
+Figure S3. (a) X-ray diffraction pattern of Cu
+region 580-3500 cm-1. 
+PEO at 19.1° and 23.3°, which corresponds to (120) and (112) planes 
+observed that there is an increase in the intensity of (120) peaks with the rise 
+in the salt concentration. This suggests that salt concentration has a strong influe
+crystallization process of the PEO. Further, no Bragg peak corresponding to CuSCN is observed 
+in the electrolyte film, even at 5 wt.% of the CuSCN concentration. This indicates that no 
+uncomplexed CuSCN is present in the electrolyte film. The FTIR spectra of the Cu
+plotted as transmittance vs. wavenumber (580-3500 cm-1) is shown in Figure
+vibrational modes of PEO in the wavenumber region 800
+are due to CH2 asymmetric rocking motion with some contribution of 
+O stretching. The sharp band at 1094 cm-1 is assigned to the C-O-C stretching mode. 
+relatively small bands at 1279 cm-1, 1343 cm-1, and 1463 cm-1 are because of  CH
+deformation, respectively. The absorption band located at 2880 cm
+H asymmetric stretching mode.3,4 The C-N stretching band found at 2168 cm
+was visible for higher concentrations of copper (I) thiocyanate salt. 
+ray diffraction pattern of Cu-SPE films, (b) FTIR spectra of Cu-SPE film
+33
+PEO at 19.1° and 23.3°, which corresponds to (120) and (112) planes 
+observed that there is an increase in the intensity of (120) peaks with the rise 
+in the salt concentration. This suggests that salt concentration has a strong influence on the 
+o Bragg peak corresponding to CuSCN is observed 
+in the electrolyte film, even at 5 wt.% of the CuSCN concentration. This indicates that no 
+IR spectra of the Cu-SPE film 
+shown in Figure S3(b). It confirms 
+vibrational modes of PEO in the wavenumber region 800-3000 cm-1. The bands 
+asymmetric rocking motion with some contribution of 
+C stretching mode. The 
+are because of  CH2 twisting, CH2 
+deformation, respectively. The absorption band located at 2880 cm-1 is 
+N stretching band found at 2168 cm-1 
+ 
+SPE film in the wavenumber 
+
+(120)
+(112)
+(n'r)
+20
+40
+60
+20 (Degrees3102%1%0.57080
+100
+1000
+1500
+2000WayenumlS50/23%SPE
+a27010%0.5%0.25%14CuSCN2500
+3000
+3500er (cm*)8
+1223
+4
+--50620/ 
+X-ray Photoelectron Spectroscopy (XPS) of 
+The chemical and electronic state of atoms in the electrolyte film was investigated by X
+photoelectron spectroscopy (XPS). The XPS full scan spectrum as shown in Fig. 10 (a), indicates 
+the presence of  Cu, S, C, N, and O in the film. The high
+N 1s, O 1s, and Cu 2p are shown in Fig. S4 (b)
+peak observed at binding energy (BE) of 162.2 eV is due to either C
+peaks at 163.8 eV and 168.8 eV correspond to S
+level spectra is dominated by a peak 
+peak at 284.4 eV is due to the C
+Figure S4. (a) XPS survey spectrum of the Cu
+S 2p, C 1s, N 1s, O 1s and Cu 2p regions of the 
+CuSCN at 398.6 eV, and a small peak at 400.3 eV is due to the N
+spectrum of the electrolyte film consists of one peak at 532.3 eV, assigned as C
+ray Photoelectron Spectroscopy (XPS) of Cu-SPE  
+The chemical and electronic state of atoms in the electrolyte film was investigated by X
+photoelectron spectroscopy (XPS). The XPS full scan spectrum as shown in Fig. 10 (a), indicates 
+the presence of  Cu, S, C, N, and O in the film. The high-resolution spectrums of the S 2p, C 1s, 
+N 1s, O 1s, and Cu 2p are shown in Fig. S4 (b)-(f). In the S 2p core level spectra, the component 
+peak observed at binding energy (BE) of 162.2 eV is due to either C-S or Cu
+peaks at 163.8 eV and 168.8 eV correspond to S-C≡N and S-O environments.  The C 1s core 
+level spectra is dominated by a peak at 285.9 eV, which corresponds to CuSCN (S
+peak at 284.4 eV is due to the C-H environments of PEO. N 1s spectra show a dominant peak of
+(a) XPS survey spectrum of the Cu-SPE film for 1 wt% of CuSCN. (b)-(f) High resolution spectra of the 
+S 2p, C 1s, N 1s, O 1s and Cu 2p regions of the Cu-SPE film. 
+CuSCN at 398.6 eV, and a small peak at 400.3 eV is due to the N-H environment. The O 1s 
+rum of the electrolyte film consists of one peak at 532.3 eV, assigned as C
+34
+The chemical and electronic state of atoms in the electrolyte film was investigated by X-ray 
+photoelectron spectroscopy (XPS). The XPS full scan spectrum as shown in Fig. 10 (a), indicates 
+resolution spectrums of the S 2p, C 1s, 
+2p core level spectra, the component 
+S or Cu-S. The other two 
+O environments.  The C 1s core 
+at 285.9 eV, which corresponds to CuSCN (S-C≡N), and the 
+H environments of PEO. N 1s spectra show a dominant peak of 
+ 
+(f) High resolution spectra of the 
+H environment. The O 1s 
+rum of the electrolyte film consists of one peak at 532.3 eV, assigned as C-O-C. From the 
+
+(a)
+0.1s
+Cu: 0.28
+S: 0.11%
+C 1s
+N: 0.48°
+C: 67.97
+Intensity (a.u.)
+0: 31.17
+Cu 2p
+Z
+0
+200
+400
+600
+008
+1000
+120
+Binding energy
+(ev)03.门12S(p)
+S-C=N
+N
+Intensity (a.u.)
+N-H
+396
+398
+400
+Binding energy (eV)160
+108
+282T0D、54DA87
+280
+288 
+high-resolution spectra of the Cu 2p core levels, the peaks at 932 eV and 951.8 eV correspond to 
+the prominent peaks of Cu 2p
+due to the presence of Cu+ in the electrolyte film
+thiocyanate separated into Cu+
+ 
+Figure S5. Cross-sectional SEM image of 
+resolution spectra of the Cu 2p core levels, the peaks at 932 eV and 951.8 eV correspond to 
+the prominent peaks of Cu 2p3/2 and Cu 2p1/2, respectively. The component peak at 933.1 eV is 
+in the electrolyte film.5-10 These sources indicate that the copper (I) 
++ and SCN− ions in the polymer electrolyte. 
+sectional SEM image of Cu-SPE layer deposited over ITO-coated PET substrate
+ 
+35
+resolution spectra of the Cu 2p core levels, the peaks at 932 eV and 951.8 eV correspond to 
+nent peak at 933.1 eV is 
+These sources indicate that the copper (I) 
+ 
+coated PET substrate at 3000 rpm. 
+
+TTO350 nm个PET100nmISCN230 nm
+PEO+Cu 
+Figure S6. I-V plots of the ITO/Cu-
+(c) 1 wt% (d) 2 wt% (e) 3 wt% and 
+Figure S7. Current
+ 
+-SPE/Cu memory device with CuSCN concentration of 
+ (f) 5 wt%. 
+ 
+Current-voltage (I-V) characteristics of Au/SPE/ITO memory device.
+36
+ 
+with CuSCN concentration of (a) 0.25 wt% (b) 0.5 wt% 
+ 
+characteristics of Au/SPE/ITO memory device. 
+
+1E-31
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+-2
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+1E-6
+1E-7
+Current 
+1E-8
+C
+1E-9
+1E-10 -
+Au/s
+LERESET1E-6
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+1E-314
+-2
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+2Voltage )SET
+HETSET1 wt%1
+1Yoltage5 wt%RES?1E-74 
+37
+REFERENCES 
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+ 
+

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+Ananke: A Python Package For Causal Inference Using
+Graphical Models
+Jaron J. R. Lee†,1
+[email protected]
+Rohit Bhattacharya†,2
+[email protected]
+Razieh Nabi3
+[email protected]
+Ilya Shpitser1
+[email protected]
+† Equal contribution
+1Department of Computer Science, Johns Hopkins University
+2Department of Computer Science, Williams College
+3Department of Biostatistics and Bioinformatics, Emory University
+Abstract
+We implement Ananke: an object-oriented Python package for causal inference with
+graphical models. At the top of our inheritance structure is an easily extensible Graph class
+that provides an interface to several broadly useful graph-based algorithms and methods
+for visualization.
+We use best practices of object-oriented programming to implement
+subclasses of the Graph superclass that correspond to types of causal graphs that are
+popular in the current literature. This includes directed acyclic graphs for modeling causally
+sufficient systems, acyclic directed mixed graphs for modeling unmeasured confounding, and
+chain graphs for modeling data dependence and interference. Within these subclasses, we
+implement specialized algorithms for common statistical and causal modeling tasks, such as
+separation criteria for reading conditional independence, nonparametric identification, and
+parametric and semiparametric estimation of model parameters. Here, we present a broad
+overview of the package and example usage for a problem with unmeasured confounding.
+Up to date documentation is available at https://ananke.readthedocs.io/en/latest/.
+Keywords:
+causal graphical models, causal identification, semiparametric estimation
+1. Introduction
+Causal inference is a pipeline comprised of many steps – specification of a causal model,
+identification of the desired causal parameter under assumptions of this model, estimation
+of the parameter from data based on the identifying functional, and robustness checks via
+sensitivity analysis and uncertainty quantification. Any of these steps may be complicated
+by unmeasured confounding, data dependence, and missing data. In Ananke, we implement
+methods that span all of these steps, including nonparametric identification and semipara-
+metric estimation strategies, to provide analysts a unifying interface that allows them to
+set up end-to-end pipelines that exhibit robustness to the aforementioned complications.
+In particular, we adopt an object-oriented paradigm to implement graph-based causal
+inference methods. We build an inheritance structure spanning causal graphical models
+that use any combination of directed (→), bidirected (↔), and undirected (–) edges. We
+hope that due to its object-oriented nature and easily accessible Python implementation
+©2022 Lee, Bhattacharya, Nabi, and Shpitser.
+License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/.
+arXiv:2301.11477v1  [stat.ME]  27 Jan 2023
+
+Lee, Bhattacharya, Nabi, and Shpitser
+Ananke will improve the accessibility of many graph-based causal inference methods, and
+allow interested users to easily extend and build on its current infrastructure.
+Related work: The doWhy package and DAGitty aim to provide a unifying interface
+for distinct steps in the causal inference pipeline. However, their estimation capabilities are
+largely limited to settings without unmeasured confounders or selection bias. In the case of
+doWhy, the causal graph interface is still under active development with plans to interface
+with Ananke rather than build one from scratch (personal communication with developers.)
+Other existing packages emphasize a single step in the pipeline. TETRAD and its Python port
+causal-learn (Scheines et al., 1998), pcalg (Kalisch et al., 2012), and cdt (Kalainathan
+et al., 2020) focus on graph representation and model selection; causaleffect focuses
+on nonparametric identification; npcausal (Kennedy, 2021), zEpid (Zevich, 2018), tmle3
+(Coyle, 2021), and DoubleML (Bach et al., 2022) focus on semiparametric estimation. Other
+standalone packages exist as appendices to papers, and in certain cases we reimplement these
+algorithms in Ananke, e.g., the maximum likelihood algorithms in Evans (2013) and Drton
+et al. (2009). The principle advantage of Ananke over peers is that it offers a unified and
+easily extended interface for causal inference in a single package, with an active community.
+2. Overview of Ananke’s Graph Inheritance Structure
+Graph
+Segregated Graph
+Intrinsic Graph
+Acyclic Directed Mixed Graph
+Chain Graph
+Directed Acyclic Graph
+Bidirected Graph
+Undirected Graph
+Figure 1: Ananke’s graph inheritance structure.
+An overview of graphical models in Ananke
+and their inheritance structure is shown in
+Fig. 1.
+The Graph class currently supports
+the creation of graphs G
+= (V, D, B, U),
+where V denotes a set of vertices and D, B, U
+denote sets of directed (→), bidirected (↔),
+and undirected (–) respectively. Within this
+class we implement methods and algorithms
+that are broadly applicable to any subclass:
+simple methods involving addition and dele-
+tion of edges, finding a subgraph GS com-
+prised of only vertices in S ⊆ V (and associ-
+ated edges), and computing genealogical sets of a vertex, such as its ancestors, descendants,
+and siblings. We also implement a lightweight draw method using a Python interface to
+graphviz (Ellson et al., 2001; Hagberg et al., 2022) for visualizing any instance of the class
+or subclasses of it – all figures in this paper are produced using this functionality. The
+rest of the inheritance structure is based on the types of edges each graph class contains.
+At the lowest levels are graphs only containing a single edge type: Directed acyclic graphs
+(DAGs) (only → edges) are the most popular type of causal graph (Robins, 1986; Spirtes
+et al., 2000; Pearl, 2009); Bidirected graphs (only ↔ edges) are used to represent marginal
+correlations and are popular in genomics (Chaudhuri et al., 2007; Cox and Wermuth, 2014);
+Undirected graphs (only – edges) can be used to encode feedback relationships (Lauritzen,
+1996). Next, we have graphs containing a mixture of edges types: Acyclic directed mixed
+graphs (ADMGs) model systems with causal influence (via → edges) and correlation due
+to unmeasured confounding (via ↔ edges) (Wright, 1921; Verma and Pearl, 1990); Chain
+graphs model causal influence (via → edges) as well as non-iid phenomena such as conta-
+2
+
+Ananke
+gion, feedback, and symmetric relationships (via – edges) (Lauritzen and Richardson, 2002;
+Ogburn et al., 2020; Bhattacharya et al., 2019a); Segregated graphs consisting of all three
+kinds of edges are capable of modeling all three mechanisms discussed above (Shpitser,
+2015). We note that intrinsic graphs shown in the hierarchy of Fig. 1 are not causal graph-
+ical models, but rather a graphical representation created by us to efficiently compute all
+statistical kernels required to parameterize a hidden variable causal model – a necessary
+step for estimation discussed in Section 3. This illustrates additional use cases of our graph
+inheritance structure for intermediate tasks. As another example, we use our chain graph
+implementation to encode equivalence classes of causal DAGs – different models that im-
+ply the same restrictions on the observed data distribution – known as Complete Partially
+Directed Acyclic Graphs (CPDAGs). This allows Ananke to easily interface with or ex-
+tend causal discovery algorithms that output such objects, e.g., implementations of greedy
+equivalence search or the PC algorithm in the causal-learn package (Zhang et al., 2022).
+3. Data Analysis in Ananke
+To illustrate usage of Ananke we step through a hypothetical analysis for assessing the
+effect of smoking on diabetes using a teaching dataset derived1 from the Framingham Heart
+Study (Kannel and Gordon, 1968). We start by encoding substantive assumptions using an
+ADMG shown in Fig. 2 along with the Ananke commands used to create and visualize it.
+age
+smoke
+diabetes
+bp
+>>> vertices = ["age", "smoke", "bp", "diabetes"]
+>>> di_edges = [("age", "smoke"), ("smoke", "bp"),
+>>>
+("bp", "diabetes"), ("age", "diabetes")]
+>>> bi_edges = [("smoke", "diabetes")]
+>>> fdoor = graphs.ADMG(vertices, di_edges, bi_edges)
+>>> fdoor.draw("LR")
+Figure 2: Front-door model visualized using built-in capability.
+An ADMG can imply certain testable independence statements (amongst other general
+constraints) that can be read via m-separation (Richardson, 2003). For example, we can
+verify that age is m-separated from bp given smoke implying age ⊥⊥ bp | smoke.
+>>> fdoor.m_separated("age", "bp", ["smoke"])
+True
+An analyst may verify whether the data supports such assumptions using any standard
+conditional independence test. Assuming Fig. 2 is correct, the next step is to apply identifi-
+cation theory to determine whether the desired causal effect can be expressed as a function
+of observed data. Applying Ananke’s implementation of a sound and complete algorithm
+for identification in presence of unmeasured confounding (Richardson et al., 2017) gives:
+>>> treatments = ["smoke"]; outcomes = ["diabetes"]
+>>> id_fdoor = identification.OneLineID(graph=fdoor, treatments=treatments, outcomes=outcomes)
+>>> print('Identifed =', id_fdoor.id(), '; Functional =', id_fdoor.functional())
+Identifed = True
+Functional = Σagebp φdiabetessmokebp(p(V);G) φsmokediabetesage(p(V);G) φsmokebpage(p(V);G)
+1. The teaching extract of the Framingham Heart Study can be requested from https://biolincc.nhlbi.
+nih.gov/teaching/.
+3
+
+Lee, Bhattacharya, Nabi, and Shpitser
+That is, the counterfactual distribution p(diabetes(smoke)), and hence the effect, is indeed
+identified. Interpreting the output based on Richardson et al. (2017) gives
+p(diabetes(smoke)) =
+�
+age,bp
+p(bp | smoke)p(age)
+� �
+smoke′
+p(diabetes | smoke′, bp, age)p(smoke′ | age)
+�
+.
+While the focus of this analysis is on identification under unmeasured confounding, Ananke
+also implements identification algorithms for missing data (Bhattacharya et al., 2019b; Nabi
+et al., 2020), selection bias, and data fusion (Lee et al., 2020; Lee and Shpitser, 2020). After
+identification, we may choose from a variety of estimation strategies offered in Ananke.
+3.1 Linear Gaussian Structural Equation Models
+One possible choice is to assume a linear structural equation model with correlated errors
+(Wright, 1934). In Ananke, we implement the iterative algorithm described in Drton et al.
+(2009) to obtain maximum likelihood estimates for all edge coefficients; causal effects are
+then computed via path analysis. Applying this to standardized Framingham data gives:
+Age
+Smoke
+-0.19
+Diabetes
+0.11
+BP
+-0.05
+-0.02
+0.08
+>>> lsm = LinearGaussianSEM(front_door)
+>>> lsm = lsm.fit(df_cont)
+>>> lsm.draw(direction="LR")
+>>> lsm.total_effect(["smoke"], ["diabetes"])
+ACE: -0.004
+3.2 M¨obius Parameterization for Discrete Data
+An alternative is to use the M¨obius parameterization of the observed data likelihood, which
+assumes all observed variables are discrete (Evans and Richardson, 2012). We implement
+a coordinate descent algorithm to compute maximum likelihood estimates for the M¨obius
+parameters (which may be variationally dependent in general.) Using binary versions of
+variables in the dataset, we obtain the following result for the average causal effect.
+>>> bnm = binary_nested.BinaryNestedModel(front_door)
+>>> bnm = bnm.fit(X=binary_nested.process_data(df_bin), tol=1e-12 )
+>>> pY1_A0 = bnm.estimate(treatment_dict={"smoke": 0}, outcome_dict={"diabetes": 1})
+>>> pY1_A1 = bnm.estimate(treatment_dict={"smoke": 1}, outcome_dict={"diabetes": 1})
+>>> print("ACE: ", pY1_A1 - pY1_A0)
+ACE: 0.004
+3.3 Semiparametric Estimation of Causal Effects
+If the effect is identified, Ananke lists several semiparametric estimation strategies, proposed
+by Bhattacharya et al. (2022), and suggests the best one according to semiparametric effi-
+ciency theory. The implementation only requires specification of the ADMG, treatment, and
+outcome. Using Ananke’s suggestion of efficient augmented primal IPW estimator gives:
+>>> ace_obj = CausalEffect(graph=front_door, treatment='smoke', outcome='diabetes')
+>>> ace = ace_obj.compute_effect(df_bin, "eff-apipw"); print("ACE: ", ace)
+ACE: -0.002
+Acknowledgments
+4
+
+Ananke
+We thank Preethi Prakash and Ranjani Srinivasan for contributions to Ananke, and Carson
+Kurtz for assisting R.B. in testing code.
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+7
+

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+page_content='edu  † Equal contribution  1Department of Computer Science, Johns Hopkins University  2Department of Computer Science, Williams College  3Department of Biostatistics and Bioinformatics, Emory University  Abstract  We implement Ananke: an object-oriented Python package for causal inference with  graphical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content='  We use best practices of object-oriented programming to implement  subclasses of the Graph superclass that correspond to types of causal graphs that are  popular in the current literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content=' Here, we present a broad  overview of the package and example usage for a problem with unmeasured confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Up to date documentation is available at https://ananke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='io/en/latest/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Keywords:  causal graphical models, causal identification, semiparametric estimation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Introduction  Causal inference is a pipeline comprised of many steps – specification of a causal model,  identification of the desired causal parameter under assumptions of this model, estimation  of the parameter from data based on the identifying functional, and robustness checks via  sensitivity analysis and uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Any of these steps may be complicated  by unmeasured confounding, data dependence, and missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' In Ananke, we implement  methods that span all of these steps, including nonparametric identification and semipara-  metric estimation strategies, to provide analysts a unifying interface that allows them to  set up end-to-end pipelines that exhibit robustness to the aforementioned complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  In particular, we adopt an object-oriented paradigm to implement graph-based causal  inference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' We build an inheritance structure spanning causal graphical models  that use any combination of directed (→), bidirected (↔), and undirected (–) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' We  hope that due to its object-oriented nature and easily accessible Python implementation  ©2022 Lee, Bhattacharya, Nabi, and Shpitser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  License: CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='0, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='11477v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='ME]  27 Jan 2023  Lee, Bhattacharya, Nabi, and Shpitser  Ananke will improve the accessibility of many graph-based causal inference methods, and  allow interested users to easily extend and build on its current infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Related work: The doWhy package and DAGitty aim to provide a unifying interface  for distinct steps in the causal inference pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' However, their estimation capabilities are  largely limited to settings without unmeasured confounders or selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' In the case of  doWhy, the causal graph interface is still under active development with plans to interface  with Ananke rather than build one from scratch (personal communication with developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=')  Other existing packages emphasize a single step in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' TETRAD and its Python port  causal-learn (Scheines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 1998), pcalg (Kalisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2012), and cdt (Kalainathan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2020) focus on graph representation and model selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' causaleffect focuses  on nonparametric identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' npcausal (Kennedy, 2021), zEpid (Zevich, 2018), tmle3  (Coyle, 2021), and DoubleML (Bach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2022) focus on semiparametric estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Other  standalone packages exist as appendices to papers, and in certain cases we reimplement these  algorithms in Ananke, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', the maximum likelihood algorithms in Evans (2013) and Drton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' The principle advantage of Ananke over peers is that it offers a unified and  easily extended interface for causal inference in a single package, with an active community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Overview of Ananke’s Graph Inheritance Structure  Graph  Segregated Graph  Intrinsic Graph  Acyclic Directed Mixed Graph  Chain Graph  Directed Acyclic Graph  Bidirected Graph  Undirected Graph  Figure 1: Ananke’s graph inheritance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  An overview of graphical models in Ananke  and their inheritance structure is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  The Graph class currently supports  the creation of graphs G  = (V, D, B, U),  where V denotes a set of vertices and D, B, U  denote sets of directed (→), bidirected (↔),  and undirected (–) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Within this  class we implement methods and algorithms  that are broadly applicable to any subclass:  simple methods involving addition and dele-  tion of edges, finding a subgraph GS com-  prised of only vertices in S ⊆ V (and associ-  ated edges), and computing genealogical sets of a vertex, such as its ancestors, descendants,  and siblings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' We also implement a lightweight draw method using a Python interface to  graphviz (Ellson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Hagberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2022) for visualizing any instance of the class  or subclasses of it – all figures in this paper are produced using this functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' The  rest of the inheritance structure is based on the types of edges each graph class contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  At the lowest levels are graphs only containing a single edge type: Directed acyclic graphs  (DAGs) (only → edges) are the most popular type of causal graph (Robins, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Spirtes  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Pearl, 2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Bidirected graphs (only ↔ edges) are used to represent marginal  correlations and are popular in genomics (Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Cox and Wermuth, 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Undirected graphs (only – edges) can be used to encode feedback relationships (Lauritzen,  1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Next, we have graphs containing a mixture of edges types: Acyclic directed mixed  graphs (ADMGs) model systems with causal influence (via → edges) and correlation due  to unmeasured confounding (via ↔ edges) (Wright, 1921;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Verma and Pearl, 1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Chain  graphs model causal influence (via → edges) as well as non-iid phenomena such as conta-  2  Ananke  gion, feedback, and symmetric relationships (via – edges) (Lauritzen and Richardson, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Ogburn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Segregated graphs consisting of all three  kinds of edges are capable of modeling all three mechanisms discussed above (Shpitser,  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' We note that intrinsic graphs shown in the hierarchy of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' 1 are not causal graph-  ical models, but rather a graphical representation created by us to efficiently compute all  statistical kernels required to parameterize a hidden variable causal model – a necessary  step for estimation discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' This illustrates additional use cases of our graph  inheritance structure for intermediate tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' As another example, we use our chain graph  implementation to encode equivalence classes of causal DAGs – different models that im-  ply the same restrictions on the observed data distribution – known as Complete Partially  Directed Acyclic Graphs (CPDAGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' This allows Ananke to easily interface with or ex-  tend causal discovery algorithms that output such objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', implementations of greedy  equivalence search or the PC algorithm in the causal-learn package (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Data Analysis in Ananke  To illustrate usage of Ananke we step through a hypothetical analysis for assessing the  effect of smoking on diabetes using a teaching dataset derived1 from the Framingham Heart  Study (Kannel and Gordon, 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' We start by encoding substantive assumptions using an  ADMG shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' 2 along with the Ananke commands used to create and visualize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  age  smoke  diabetes  bp  >>> vertices = ["age", "smoke", "bp", "diabetes"]  >>> di_edges = [("age", "smoke"), ("smoke", "bp"),  >>>  ("bp", "diabetes"), ("age", "diabetes")]  >>> bi_edges = [("smoke", "diabetes")]  >>> fdoor = graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='ADMG(vertices, di_edges, bi_edges)  >>> fdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='draw("LR")  Figure 2: Front-door model visualized using built-in capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  An ADMG can imply certain testable independence statements (amongst other general  constraints) that can be read via m-separation (Richardson, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' For example, we can  verify that age is m-separated from bp given smoke implying age ⊥⊥ bp | smoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  >>> fdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='m_separated("age", "bp", ["smoke"])  True  An analyst may verify whether the data supports such assumptions using any standard  conditional independence test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Assuming Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' 2 is correct, the next step is to apply identifi-  cation theory to determine whether the desired causal effect can be expressed as a function  of observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Applying Ananke’s implementation of a sound and complete algorithm  for identification in presence of unmeasured confounding (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2017) gives:  >>> treatments = ["smoke"];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' outcomes = ["diabetes"]  >>> id_fdoor = identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content="OneLineID(graph=fdoor, treatments=treatments, outcomes=outcomes)  >>> print('Identifed =', id_fdoor." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content="id(), ';" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=" Functional =', id_fdoor." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='functional())  Identifed = True  Functional = Σagebp φdiabetessmokebp(p(V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='G) φsmokediabetesage(p(V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='G) φsmokebpage(p(V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='G)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' The teaching extract of the Framingham Heart Study can be requested from https://biolincc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='nhlbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='gov/teaching/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  3  Lee, Bhattacharya, Nabi, and Shpitser  That is, the counterfactual distribution p(diabetes(smoke)), and hence the effect, is indeed  identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Interpreting the output based on Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' (2017) gives  p(diabetes(smoke)) =  �  age,bp  p(bp | smoke)p(age)  � �  smoke′  p(diabetes | smoke′, bp, age)p(smoke′ | age)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  While the focus of this analysis is on identification under unmeasured confounding, Ananke  also implements identification algorithms for missing data (Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Nabi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2020), selection bias, and data fusion (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Lee and Shpitser, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' After  identification, we may choose from a variety of estimation strategies offered in Ananke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='1 Linear Gaussian Structural Equation Models  One possible choice is to assume a linear structural equation model with correlated errors  (Wright, 1934).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' In Ananke, we implement the iterative algorithm described in Drton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  (2009) to obtain maximum likelihood estimates for all edge coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' causal effects are  then computed via path analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Applying this to standardized Framingham data gives:  Age  Smoke  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='19  Diabetes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='11  BP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='08  >>> lsm = LinearGaussianSEM(front_door)  >>> lsm = lsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='fit(df_cont)  >>> lsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='draw(direction="LR")  >>> lsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='total_effect(["smoke"], ["diabetes"])  ACE: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='004  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='2 M¨obius Parameterization for Discrete Data  An alternative is to use the M¨obius parameterization of the observed data likelihood, which  assumes all observed variables are discrete (Evans and Richardson, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' We implement  a coordinate descent algorithm to compute maximum likelihood estimates for the M¨obius  parameters (which may be variationally dependent in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=') Using binary versions of  variables in the dataset, we obtain the following result for the average causal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  >>> bnm = binary_nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='BinaryNestedModel(front_door)  >>> bnm = bnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='fit(X=binary_nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='process_data(df_bin), tol=1e-12 )  >>> pY1_A0 = bnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='estimate(treatment_dict={"smoke": 0}, outcome_dict={"diabetes": 1})  >>> pY1_A1 = bnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='estimate(treatment_dict={"smoke": 1}, outcome_dict={"diabetes": 1})  >>> print("ACE: ", pY1_A1 - pY1_A0)  ACE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='004  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='3 Semiparametric Estimation of Causal Effects  If the effect is identified, Ananke lists several semiparametric estimation strategies, proposed  by Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' (2022), and suggests the best one according to semiparametric effi-  ciency theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' The implementation only requires specification of the ADMG, treatment, and  outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=" Using Ananke’s suggestion of efficient augmented primal IPW estimator gives:  >>> ace_obj = CausalEffect(graph=front_door, treatment='smoke', outcome='diabetes')  >>> ace = ace_obj." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='compute_effect(df_bin, "eff-apipw");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' print("ACE: ", ace)  ACE: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='002  Acknowledgments  4  Ananke  We thank Preethi Prakash and Ranjani Srinivasan for contributions to Ananke, and Carson  Kurtz for assisting R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' in testing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  References  Philipp Bach, Victor Chernozhukov, Malte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Kurz, and Martin Spindler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' DoubleML –  An object-oriented implementation of double machine learning in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Journal of  Machine Learning Research, 23(53):1–6, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  URL http://jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='org/papers/v23/  21-0862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Rohit Bhattacharya, Daniel Malinsky, and Ilya Shpitser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Causal inference under interfer-  ence and network uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' In Proceedings of the 35th Conference on Uncertainty in  Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' AUAI Press, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Rohit Bhattacharya, Razieh Nabi, Ilya Shpitser, and James M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Robins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Identification in  missing data models represented by directed acyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' In Proceedings of the 35th  Conference on Uncertainty in Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' AUAI Press, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Rohit Bhattacharya, Razieh Nabi, and Ilya Shpitser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Semiparametric inference for causal  effects in graphical models with hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Journal of Machine Learning Research,  23:1–76, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Sanjay Chaudhuri, Mathias Drton, and Thomas S Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Estimation of a covariance  matrix with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Biometrika, 94(1):199–216, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  David Roxbee Cox and Nanny Wermuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Multivariate dependencies: Models, analysis and  interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Chapman and Hall/CRC, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Jeremy R Coyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' tmle3: The extensible TMLE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content='  URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content=' Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Computing maximum like-  lihood estimates in recursive linear models with correlated errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Journal of Machine  Learning Research, 10(10), 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  John Ellson, Emden Gansner, Lefteris Koutsofios, Stephen C North, and Gordon Wood-  hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Graphviz—open source graph drawing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' In International Symposium on Graph  Drawing, pages 483–484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Springer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Robin J Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' ADMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content='htm, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='  Robin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Evans and Thomas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Maximum likelihood fitting of acyclic directed  mixed graphs to binary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content='org/abs/1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content='  Aric Hagberg, Dan Schult, and Manos Renieris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Pygraphviz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+page_content='  5  Lee, Bhattacharya, Nabi, and Shpitser  Diviyan Kalainathan, Olivier Goudet, and Ritik Dutta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Causal discovery toolbox: Uncov-  ering causal relationships in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
+page_content=' Journal of Machine Learning Research, 21(37):1–5,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFJT4oBgHgl3EQfNSxs/content/2301.11477v1.pdf'}
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+Scene-centric vs. Object-centric Image-Text
+Cross-modal Retrieval: A Reproducibility Study
+Mariya Hendriksen1[0000−0003−0314−2955], Svitlana
+Vakulenko2[0000−0002−5278−8886]⋆, Ernst Kuiper3[0000−0002−8075−4894], and Maarten
+de Rijke4[0000−0002−1086−0202]
+1 AIRLab, University of Amsterdam, The Netherlands
+[email protected]
+2 Amazon, Spain
+[email protected]
+3 Bol.com, The Netherlands
+[email protected]
+4 University of Amsterdam, The Netherlands
+[email protected]
+Abstract. Most approaches to cross-modal retrieval (CMR) focus either on ob-
+ject-centric datasets, meaning that each document depicts or describes a single
+object, or on scene-centric datasets, meaning that each image depicts or describes
+a complex scene that involves multiple objects and relations between them. We
+posit that a robust CMR model should generalize well across both dataset types.
+Despite recent advances in CMR, the reproducibility of the results and their gen-
+eralizability across different dataset types has not been studied before. We address
+this gap and focus on the reproducibility of the state-of-the-art CMR results when
+evaluated on object-centric and scene-centric datasets. We select two state-of-the-
+art CMR models with different architectures: (i) CLIP; and (ii) X-VLM. Addi-
+tionally, we select two scene-centric datasets, and three object-centric datasets,
+and determine the relative performance of the selected models on these datasets.
+We focus on reproducibility, replicability, and generalizability of the outcomes of
+previously published CMR experiments. We discover that the experiments are not
+fully reproducible and replicable. Besides, the relative performance results par-
+tially generalize across object-centric and scene-centric datasets. On top of that,
+the scores obtained on object-centric datasets are much lower than the scores ob-
+tained on scene-centric datasets. For reproducibility and transparency we make
+our source code and the trained models publicly available.
+1
+Introduction
+Cross-modal retrieval (CMR) is the task of finding relevant items across different modal-
+ities. For example, given an image, find a text or vice versa. The main challenge in CMR
+is known as the heterogeneity gap [5, 22]. Since items from different modalities have
+different data types, the similarity between them cannot be measured directly. There-
+fore, the majority of CMR methods published to date attempt to bridge this gap by
+⋆ Research conducted while the author was at the University of Amsterdam.
+arXiv:2301.05174v1  [cs.IR]  12 Jan 2023
+
+2
+M. Hendriksen et al.
+learning a latent representation space, where the similarity between items from differ-
+ent modalities can be measured [57].
+In this work, we specifically focus on image-text CMR, which uses textual and
+visual data. The retrieval task is performed on image-text pairs. In each image-text pair,
+the text (often referred to as caption) describes the corresponding image it is aligned
+with. For image-text CMR we use either an image or a text as a query [57]. Hence,
+the CMR task that we address in this paper consists of two subtasks: (i) text-to-image
+retrieval: given a text that describes an image, retrieve all the images that match this
+description; and (ii) image-to-text retrieval: given an image, retrieve all texts that can
+be used to describe this image.
+Scene-centric vs. object-centric vs. datasets. Existing image datasets can be grouped
+into scene-centric and object-centric datasets [48, 62]. The two types of dataset are
+typically used for different tasks, viz. the tasks of scene and object understanding, re-
+spectively. They differ in important ways that are of interest to us when evaluating
+performance and generalization abilities of CMR models.
+Scene-centric images depict complex scenes that typically feature multiple objects
+and relations between them. These datasets contain image-text pairs, where, in each
+pair, an image depicts a complex scene of objects and the corresponding text describes
+the whole scene, often focusing on relations and activities.
+Images in object-centric image datasets are usually focused on a single object of
+interest that they primarily depict. This object is often positioned close to the center
+of an image with other objects, optionally, in the background. Object-centric datasets
+contain image-text pairs, where, in each pair, an image depicts an object of interest and
+the corresponding text describes the depicted object and its (fine-grained) attributes.
+To illustrate the differences between the two dataset types in CMR, we consider
+the examples provided in Fig. 1 with an object-centric image-caption pair (left) and a
+scene-centric image-caption pair (right). Note how the pairs differ considerably in terms
+of the visual style and the content of the caption. The pair on the left focuses on a single
+object (“pants”) and describes its fine-grained visual attributes (“multicolor,” “boho,”
+“batic”). The pair on the right captures a scene describing multiple objects (“seagulls,”
+“pier,” “people”) and relations between them (“sitting,” “watching”).
+Research goals. We focus on (traditional) CMR methods that extract features from each
+modality and learn a common representation space. Recent years have seen extensive
+experimentation with such CMR methods, mostly organized into two groups: (i) con-
+trastive experiments on object-centric datasets [17], and (ii) contrastive experiments
+on scene-centric datasets [35]. In this paper, we consider representative state-of-the-art
+CMR methods from both groups. We select two pre-trained models which demonstrate
+state-of-the-art performance on CMR task and evaluate them in a zero-shot setting. In
+line with designs used in prior reproducibility work on CMR [3] we select two models
+for the study. Following the ACM terminology [1], we focus on reproducibility (differ-
+ent team, same experimental setup) and replicability (different team, different exper-
+imental setup) of previously reported results. And following Voorhees [55], we focus
+on relative (a.k.a. comparative) performance results. In addition, for the reproducibility
+experiment, we consider the absolute difference between the reported scores and the
+reproduced scores.
+
+Scene-centric vs. Object-centric Cross-modal Retrieval
+3
+Multicolor boho batic pants
+Seagulls sitting on the ledge of a pier 
+with people watching
+Fig. 1: An object-centric (left) and a scene-centric (right) image-text pair. Sources:
+Fashion200k (left); MS COCO (right).
+We address the following research questions: (RQ1) Are published relative perfor-
+mance results on CMR reproducible? This question matters because it allows us to
+confirm the validity of reported results. We show that the relative performance results
+are not fully reproducible. Specifically, the results are reproducible for one dataset, but
+not for the other dataset).
+We then shift to replicability and examine whether lessons learned on scene-centric
+datasets transfer to object-centric datasets: (RQ2) To what extent are the published rel-
+ative performance results replicable? That is, we investigate the validity of the reported
+results when evaluated in a different setup. We find that relative performance results are
+partially replicable, using other datasets.
+After investigating the reproducibility and replicability of the results, we consider
+the generalizability of the results. We contrastively evaluate the results on object-centric
+and scene-centric datasets: (RQ3) Do relative performance results for state-of-the-art
+CMR methods generalize from scene-centric datasets to object-centric datasets? We
+discover that the relative performance results only partially generalize across the two
+dataset types.
+Main contributions. Our main contributions are: (i) We are one of the first to con-
+sider reproducibility in the context of CMR and reproduce scene-centric CMR experi-
+ments from two papers [44, 61] and find that the results are only partially reproducible.
+(ii) We perform a replicability study and examine whether relative performance differ-
+ences reported for CMR methods generalize from scene-centric datasets to object-cen-
+tric datasets. (iii) We investigate the generalizability of obtained results and analyze the
+effectiveness of pre-training on scene-centric datasets for improving the performance
+of CMR on object-centric datasets, and vice versa. And, finally, (iv) to facilitate the
+reproducibility of our work, we provide the code and the pre-trained models used in our
+experiments.5
+5 https://github.com/mariyahendriksen/ecir23-object-centric-vs-s
+cene-centric-CMR
+
+4
+M. Hendriksen et al.
+2
+Related Work
+Cross-modal retrieval. CMR methods attempt to construct a multimodal representa-
+tion space, where the similarity of concepts from different modalities can be measured.
+Some of the earliest approaches in CMR utilised canonical correlation analysis [15, 26].
+They were followed by a dual encoder architecture equiped with a recurrent and a con-
+volutional component, a hinge loss [12, 58] and hard-negative mining [11]. Later on,
+several attention-based architectures were introduced such as architectures with dual
+attention [39], stacked cross-attention [31], bidirectional focal attention [36].
+Another line of work proposed to use transformer encoders [54] for CMR task [38],
+and adapted the BERT model [8] as a backbone [13, 67]. Some other researchers worked
+on improving CMR via modality-specific graphs [56], or image and text generation
+modules [16].
+There is also more domain-specific work that focused on CMR in fashion [14, 28–
+30], e-commerce [19, 20], cultural heritage [49] and cooking [56].
+In contrast to the majority of prior work on the topic, we focus on the reproducibil-
+ity, replicability, and generalizability of CMR methods. In particular, we explore the
+state-of-the-art models designed for the CMR task by examining their performance on
+scene-centric and object-centric datasets.
+Scene-centric and object-centric datasets. The majority of prior work related to object-
+centric and scene-centric datasets focuses on computer vision tasks such as object
+recognition, object classification, and scene recognition. Herranz et al. [21] investi-
+gated biases in a CNN when trained on scene-centric versus object-centric datasets and
+evaluated on the task of object classification.
+In the context of object detection, prior work focused on combining feature repre-
+sentations learned from object-centric and scene-centric datasets to improve the perfor-
+mance when detecting small objects [48], and using object-centric images to improve
+the detection of objects that do not appear frequently in complex scenes [62]. Finally,
+for the task of scene recognition, Zhou et al. [66] explored the quality of feature rep-
+resentations learned from both scene-centric and object-centric datasets and applied to
+the task of scene recognition.
+Unlike prior work on the topic, in this paper, we focus on both scene-centric and
+object-centric datasets for evaluation on CMR task. In particular, we explore how state-
+of-the-art (SOTA) CMR models perform on object-centric and scene-centric datasets.
+Reproducibility in cross-modal retrieval. To the best of our knowledge, despite the
+popularity of the CMR task, there are very few papers that focus on reproducibility
+of research in CMR. Some rare (recent) examples include [3], where the authors sur-
+vey metric learning losses used in computer vision and explore their applicability for
+CMR. Rao et al. [45] analyze contributing factors that affect the performance of the
+state-of-the-art CMR models. However, all prior work focuses on exploring model per-
+formance only on two popular scene-centric datasets: Microsoft COCO (MS COCO)
+and Flickr30k.
+In contrast, in this work, we take advantage of the diversity of the CMR datasets and
+specifically focus on examining how the state-of-the-art CMR models perform across
+different dataset types: scene-centric and object-centric datasets.
+
+Scene-centric vs. Object-centric Cross-modal Retrieval
+5
+3
+Task Definition
+We follow the same notation as in previous work [4, 53, 65]. An image-caption cross-
+modal dataset consists of a set of images I and texts T where the images and texts are
+aligned as image-text pairs: D = {(x1
+I, x1
+T ), ..., (xn
+I, xn
+T )}.
+The cross-modal retrieval (CMR) task is defined analogous to the standard informa-
+tion retrieval task: given a query q and a set of m candidates Ωq = {x1, . . . , xm} we
+aim to rank all the candidates w.r.t. their relevance to the query q. In CMR, the query
+can be either a text qT or an image qI: q ∈ {qT , qI}. Similarly, the set of candidate
+items can be either visual Iq ⊂ I, or textual Tq ⊂ T data: Ω ∈ {Iq, Tq}.
+The CMR task is performed across modalities, therefore, if the query is a text then
+the set of candidates are images, and vice versa. Hence, the task comprises effectively
+two subtasks: (i) text-to-image retrieval: given a textual query qT and a set of candidate
+images Ω ⊂ I, we aim to rank all instances in the set of candidate items Ω w.r.t. their
+relevance to the query qT ; (ii) image-to-text retrieval: given an image as a query qI
+and a set of candidate texts Ω ⊂ T , we aim to rank all instances in the set of candidate
+items Ω w.r.t. their relevance to the query qI.
+4
+Methods
+In this section, we give an overview of the models included in the study, of the models
+which were excluded, and provide justification for it. All the approaches we focus on
+belong to the traditional CMR framework and comprise two stages. First, we extract
+textual and visual features. The features are typically extracted with a textual encoder
+and a visual encoder. Next, we learn a latent representation space where the similarity
+of items from different modalities can be measured directly.
+4.1
+Methods included for comparison
+We focus on CMR in zero-shot setting, hence, we only consider pre-trained models.
+Therefore, we focus on the models that are released for public use. Besides, as explained
+in Section 1, we follow prior reproducibility work to inform our experimental choices
+regarding the number of models. Given the above-mentioned requirements, we selected
+two methods that demonstrate state-of-the-art performance on the CMR task: CLIP and
+X-VLM.
+Contrastive Language-Image Pretraining (CLIP) [44]. This model is a dual encoder
+that comprises an image encoder, and a text encoder. The model was pre-trained in a
+contrastive manner using a symmetric loss function. It is trained on 400 million image-
+caption pairs scraped from the internet. The text encoder is a transformer [54] with
+modification from [43]. For the image encoder, the authors present two architectures.
+The first one is based on ResNet [18] and it is represented in five variants in total.
+The first two options are ResNet-50, ResNet-101; the last three options are variants of
+ResNet scaled up in the style of EfficientNet [51] The second image encoder architec-
+ture is a Vision Transofrmer (ViT) [9]. It is presented in three variants: ViT-B/32, a
+ViT-B/16, and a ViT-L/14. The CMR results reported in the original paper are obtained
+
+6
+M. Hendriksen et al.
+with a model configuration where vision transformer ViT-L/14 is used as an image en-
+coder, and the text transformer is a text encoder. Hence, we use this configuration in our
+experiments.
+X-VLM [61]. This model consists of three encoders: an image encoder, a text encoder,
+and a cross-modal encoder. The image and text encoder take an image and text as inputs
+and output their visual and textual representations. The cross-modal encoder fuses the
+output of the image encoder and the output of the text encoder. The fusion is done via
+a cross-attention mechanism. For CMR task, the model is fine-tuned via a contrastive
+learning loss and a matching loss. All encoders are transformer-based. The image en-
+coder is a ViT initialised with Swin Transformerbase [37]. Both the text encoder and the
+cross-modal encoder are initialised using different layers of BERT [8]: the text encoder
+is initialized using the first six layers, whereas the cross-modal encoder is initialised
+using the last six layers.
+4.2
+Methods excluded from comparison
+While selecting the models for the experiments, we considered other architectures with
+promising performance on the MS COCO and the Flickr30k datasets. Below, we outline
+the architectures we considered and explain why they were not included.
+Several models such as Visual N-Grams [32], Unicoder-VL [33], ViLT-B/32 [25],
+UNITER [6] were excluded because they were consistently outperformed by CLIP
+on the MS COCO and Flickr30k datasets by large margins. Besides, we excluded
+ImageBERT [42] because it was outperformed by CLIP on the MS COCO dataset.
+ALIGN [23], ALBEF [34], VinVL [64], METER [10] were not included because X-
+VLM consistently outperformed them. UNITER [6] was beaten by both CLIP and X-
+VLM. We did not include other well-performing models such as ALIGN [23], Flamin-
+go [2], CoCa [60] because the pre-trained models were not publicly available.
+5
+Experimental Setup
+In this section, we discuss our experimental design including the choice of datasets,
+subtasks, metrics, and implementation details.
+5.1
+Datasets
+We run experiments on two scene-centric and three object-centric datasets. Below, we
+discuss each of the datasets in more detail.
+Scene-centric datasets. We experiment with two scene-centric datasets: (i) Microsoft
+COCO (MS COCO) [35] contains 123,287 images depicting regular scenes from ev-
+eryday life with multiple objects placed in their natural contexts. There are 91 different
+object types such as “person”, “bicycle”, “apple”. (ii) Flickr30k contains 31,783 im-
+ages of regular scenes from everyday life, activities, and events. For both scene-centric
+datasets, we use the splits provided in [24]. The MS COCO dataset is split into 113,287
+images for training, 5,000 for testing and 5,000 for validation; the Flickr30k dataset has
+
+Scene-centric vs. Object-centric Cross-modal Retrieval
+7
+29,783 images for training, 1,000 for testing and 1,000 for validation. In both datasets,
+every image was annotated with five captions using Amazon Mechanical Turk. Besides,
+we select one caption per image randomly, and use the test set for our experiments.
+Object-centric datasets. We consider three object-centric datasets in our experiments:
+(i) Caltech-UCSD Birds 200 (CUB-200) [59] contains 11,788 images of 200 birds
+species. Each image is annotated with a fine-grained caption from [46]. We selected
+one caption per image randomly. Each caption is at least 10 words long and does
+not contain any information about the birds’ species or actions. (ii) Fashion200k con-
+tains 209,544 images that depict various fashion items in five product categories (dress,
+top, pant, skirt, jacket) and their corresponding descriptions. (iii) Amazon Berkley Ob-
+jects (ABO) [7] contains 147,702 product listings associated with 398,212 images. This
+dataset was derived from Amazon.com product listings. We selected one image per list-
+ing and used the associated product description as its caption. The majority of images
+depict a single product on a white background. The product is located in the center of
+the image and takes at least 85% of the image area. For all object-centric datasets, we
+use the splits provided by the dataset authors and use the test split for our experiments.
+5.2
+Subtasks
+Our goal is to assess and compare the performance of the CMR methods (described in
+Section 5) across the object-centric and scene-centric datasets described in the previous
+subsection. We design an experimental setup that takes into account two CMR subtasks
+and two dataset types. It can be summarized using a tree with branches that correspond
+to different configurations (see Fig. 2). We explain how we cover the branches of this
+tree in the next subsection.
+The tree starts with a root (“Image-text CMR” with label 0) that has sixteen de-
+scendants, in total. The root node has two children corresponding to the two image-text
+CMR subtasks: a text-to-image retrieval (node 1) and image-to-text retrieval (node 2).
+Since we want to evaluate each of these subtasks on both object-centric and scene-
+centric datasets, nodes 1 and 2 also have two children each, i.e., the nodes {3, 4, 5, 6}.
+Finally, every object-centric node has three children: CUB-200, Fashion200k, and ABO
+datasets {7, 8, 9, 12, 13, 14}; and every scene-centric node has two children: MS COCO
+and Flickr30k datasets {10, 11, 15, 16}.
+5.3
+Experiments
+To answer the research questions introduced in Section 1, we conduct two experiments.
+In all the experiments, we use CLIP and X-VLM models in a zero-shot setting. Fol-
+lowing [55], we focus on relative performance results. In each experiment, we consider
+different subtrees from Fig. 2. Following [25, 32, 33, 44, 61], we use Recall@K where
+K = {1, 5, 10} to evaluate the model performance in all our experiments. In addition,
+following [50, 52, 63], we calculate the sum of recalls (rsum) for text-to-image, and
+image-to-text retrieval tasks as well as the total sum of recalls for both tasks.
+For text-to-image retrieval, we first obtain representations for all the candidate im-
+ages by passing them through the image encoder of the model. Then we pass each
+
+8
+M. Hendriksen et al.
+CUB-200
+Fashion200k
+ABO
+MS COCO
+Flickr30k
+Image-text 
+CMR
+Image-to-text 
+retrieval
+Object-centric
+Scene-centric
+Text-to-image 
+retrieval
+Object-centric
+Scene-centric
+CUB-200
+Fashion200k
+ABO
+MS COCO
+Flickr30k
+1
+3
+0
+2
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+Fig. 2: Our experimental design for evaluating CMR methods across object-centric and
+scene-centric datasets. The blue colour indicates parts of the tree used in Experiment 1,
+the green color indicates parts of the tree used in Experiment 2, and the red color indi-
+cates parts used in all experiments. (Best viewed in color.)
+textual query through the text encoder of the model and retrieve the top-k candidates
+ranked by cosine similarity w.r.t. the query.
+For image-to-text retrieval, we do the reverse, using the texts as candidates and im-
+ages as queries. More specifically, we start by obtaining representations of the candidate
+captions by passing them through the text encoder. Afterwards, for each of the visual
+queries, we pass the query through the image encoder and retrieve top-k candidates
+ranked by cosine similarity w.r.t. the query.
+In Experiment 1 we evaluate the reproducibility of the CMR results reported in the
+original publications (RQ1). Both models we consider (CLIP and X-VLM) were origi-
+nally evaluated on two scene-centric datasets, viz. MS COCO and Flickr30k. Therefore,
+for our reproducibility study, we also evaluate these models on these two datasets. We
+evaluate both text-to-image and image-to-text retrieval. That is, we focus on the two
+sub-trees 0←1←4←{10, 11} and 0←2←6←{15, 16} (the red and blue parts of the
+tree) from Fig. 2. In addition to relative performance results, we consider absolute dif-
+ferences between the reported scores and the reproduced scores. Following Petrov and
+Macdonald [41], we assume that the score is reproduced if we obtain a score value equal
+to the reported score given a relative tolerance of ±5%.
+In Experiment 2 we focus on the replicability of the reported results on object-
+centric datasets (RQ2). Thus, we evaluate CLIP and X-VLM on the CUB-200, Fash-
+ion200k, and ABO datasets. This experiment covers the subtrees 0←1←3←{7, 8, 9}
+and 0←2 ←5←{12, 13, 14} (the red and green parts of the tree) in Fig. 2.
+After obtaining the results from Experiment 1 and 2, we examine the generaliz-
+ability of the obtained scores (RQ3). We do so by comparing the relative performance
+results the models achieve on the object-centric versus scene-centric datasets. More
+specifically, we compare the relative performance of CLIP and X-VLM on CUB-200,
+Fashion200k, ABO with their relative performance on MS COCO and Flickr30k. Thus,
+this experiment captures the complete tree in Fig. 2.
+
+Scene-centric vs. Object-centric Cross-modal Retrieval
+9
+Table 1: Results of Experiment 1 (reproducibility study), using the MS COCO and
+Flickr30k datasets. “Orig.” indicates the scores from the original publications. “Repr.”
+indicates the scores that we obtained.
+Text-to-image
+Image-to-text
+Rsum
+Model
+R@1 R@5 R@10 R@1 R@5 R@10
+t2i
+i2t
+total
+MS COCO (5k)
+Orig.
+CLIP [44]
+37.80 62.40 72.20
+58.40 81.50 88.10
+172.40 228.00 400.40
+X-VLM [61] 55.60 82.70 90.00
+70.80 92.10 96.50
+228.30 259.40 487.70
+Repr.
+CLIP
+21.59 40.22 49.80
+24.36 44.13 53.41
+111.61 121.90 233.51
+X-VLM
+42.79 67.61 67.64
+64.60 84.48 84.50
+178.04 233.58 411.62
+Flickr30k (1k)
+Orig.
+CLIP [44]
+68.70 90.60 95.20
+88.00 98.70 99.40
+254.50 286.10 540.60
+X-VLM [61] 71.90 93.30 96.40
+85.30 97.80 99.60
+261.60 282.70 544.30
+Repr.
+CLIP
+74.95 93.09 96.15
+77.02 94.18 96.84
+264.19 268.04 532.23
+X-VLM
+37.82 82.36 82.48
+63.30 91.10 91.10
+202.66 245.50 448.16
+6
+Results
+We focus on the reproducibility (different team, same setup) and replicability (different
+team, different setup) of the CMR experiments reported in the original papers devoted
+to CLIP [44] and X-VLM [61]. To organize our result presentation, we refer to the tree
+in Fig. 2. We traverse the tree bottom up, from the leaves to the root.
+6.1
+RQ1: Reproducibility
+To address RQ1, we report on the outcomes of Experiment 1. We investigate to what
+extent the CMR results reported in the original papers devoted to CLIP [44] and X-
+VLM [61] are reproducible. Given that both methods were originally evaluated on two
+scene-centric datasets, viz. MS COCO and Flickr30k, we evaluate the models on the
+text-to-image and image-to-text tasks on these two datasets. Therefore, we focus on the
+two blue sub-trees 0←1←4←{10, 11} and 0←2←6←{15, 16} from Fig. 2.
+Results. The results of Experiment 1 are shown in Table 1. We recall the scores obtained
+in the original papers [44, 61] (“Orig.”) and the scores that we obtained (“Repr.”), on
+the MS COCO and Flickr30k datasets. Across the board, the scores that we obtained
+(the “reproduced scores”) tend to be lower than the scores obtained in the original pub-
+lications (the “original scores”).
+On the MS COCO dataset, X-VLM consistently outperforms CLIP, both in the orig-
+inal publications and in our setup, for both the text-to-image and the image-to-text tasks.
+Moreover, this holds for all R@n metrics, and, hence, for the Rsum metrics. Interest-
+ingly, the relative gains that we obtain tend to be larger than the ones obtained in the
+original publications. For example, our biggest relative difference is for the image-to-
+text task in terms of the R@1 metric: according to the scores reported in [44, 61],
+
+10
+M. Hendriksen et al.
+X-VLM outperforms CLIP by 21%, whereas in our experiments the relative gain is
+165%.
+On average, the original CLIP scores are as much as ∼70% higher than the repro-
+duced scores; the original scores for X-VLM are ∼20% higher than the reproduced
+ones. When considering the absolute differences between the original scores and the
+reproduced scores and assuming a relative tolerance of ±5%, we see that, on the MS
+COCO dataset, the scores are not reproducible for both models.
+On the Flickr30k dataset, we see a different pattern. For the text-to-image task, the
+original results indicate that X-VLM consistently outperforms CLIP, on all R@n met-
+rics, but according to our results, the relative order is consistently reversed. For the
+image-to-text task, we obtained mixed outcomes: for R@1 and R@5, the original order
+(CLIP outperforms X-VLM) is confirmed, but for R@10 the order is swapped. Accord-
+ing to our experimental results, however, CLIP consistently outperforms X-VLM on all
+tasks, and on all R@n metrics (and hence also on the Rsum metrics).
+On the Flickr30k dataset, the CLIP scores are reproduced on the text-to-image and
+image-to-text retrieval tasks when the model is evaluated on R@5 and R@10. On the
+text-to-image task, the reproduced R@5 score is 2.7% higher than the original score;
+the reproduced R@10 score is 1% higher than the original score. For the image-to-
+text retrieval task, the reproduced R@5 score is 4% lower than the original score; the
+reproduced R@10 score is 2% lower than the original score.
+Answer to RQ1. In the case of the CLIP model, the obtained absolute scores were
+reproducible only on the Flickr30k dataset for the text-to-image and the image-to-text
+tasks when evaluated on R@5 and R@10. For X-VLM, we did not find the absolute
+scores obtained when evaluating the model on the MS COCO and Flickr20k datasets to
+be reproducible, neither for the text-to-image nor the image-to-text tasks.
+The relative outcomes on the MS COCO dataset could be reproduced, for all tasks
+and metrics, whereas on the Flickr30k dataset they could only partially be reproduced,
+that is, only for the image-to-text task on the R@1 and R@5 metrics; for the text-
+to-image task, X-VLM outperforms CLIP according to the original scores, but CLIP
+outperforms X-VLM according to our reproduced scores.
+Upshot. As explained in Section 4, in this paper we focus on CMR in a zero-shot set-
+ting. This implies that the differences that we observed between the original scores and
+the reproduced scores must be due to differences in text and image data (pre-)processing
+and loading. We, therefore, recommend that the future work includes (as much as is
+practically possible) tools and scripts used in these stages of the experiment with the
+publication of its implementations.
+6.2
+RQ2: Replicability
+To answer RQ2, we replicate the originally reported text-to-image and image-to-text
+retrieval experiments in a different setup, i.e., by evaluating CLIP and X-VLM using
+object-centric datasets instead of scene-centric datasets. Thus, we evaluate CLIP and X-
+VLM on the CUB-200, Fashion200k, and ABO datasets and focus on the green subtrees
+0←1←3←{7, 8, 9} and 0←2←5←{12, 13, 14} from Fig. 2.
+Results. The results of Experiment 2 (aimed at answering RQ2) can be found in Ta-
+
+Scene-centric vs. Object-centric Cross-modal Retrieval
+11
+Table 2: Results of Experiment 2 (replicability study), using the CUB-200, Fash-
+ion200k, and ABO datasets.
+Text-to-image
+Image-to-text
+Rsum
+Model
+R@1 R@5 R@10 R@1 R@5 R@10
+t2i
+i2t
+total
+CUB-200
+CLIP
+0.71
+2.38
+4.42
+1.23
+3.40
+5.48
+7.51 10.11 17.62
+X-VLM
+0.70
+2.28
+2.45
+1.16
+2.35
+2.45
+5.43
+5.96 11.39
+Fashion200k
+CLIP
+3.05
+8.56 12.85
+3.43
+9.82 14.56
+24.46 27.81 52.27
+X-VLM
+2.80
+6.62
+6.70
+1.84
+3.96
+4.04
+16.12 09.84 25.96
+ABO
+CLIP
+6.25 13.90 18.50
+7.99 18.96 25.57
+38.65 52.52 91.17
+X-VLM
+3.10
+6.48
+6.56
+3.20
+7.42
+7.50
+16.14 18.12 34.26
+ble 2. On the CUB-200 dataset, CLIP consistently outperforms X-VLM. The biggest
+relative increase is 124% for image-to-text in terms of R@10, while the smallest rel-
+ative increase is 1% for text-to-image in terms of R@1. Overall, on the text-to-image
+retrieval task, CLIP outperforms X-VLM by 38%, and on the image-to-text retrieval
+task, the relative gain is 70%.
+On Fashion200k, CLIP outperforms X-VLM, too. The smallest relative increase
+is 9% for text-to-image in terms of R@1, the biggest relative increase is 260% for
+image-to-text in terms of R@10. In general, on the text-to-image retrieval task, CLIP
+outperforms X-VLM by 52%; on the image-to-text retrieval task, the relative gain is
+83%.
+Finally, on the ABO dataset, CLIP outperforms X-VLM again. The smallest rela-
+tive increase is 101% for text-to-image in terms of R@1, the biggest relative increase
+is 241% for image-to-text again in terms of R@10. In general, on the text-to-image re-
+trieval task, CLIP outperforms X-VLM by 139%; on the image-to-text retrieval task, the
+relative gain is 190%. All in all, CLIP outperforms X-VLM on all three scene-centric
+datasets. The overall relative gain on CUB-200 dataset is 55%, on Fashion200k dataset
+– 101%. The biggest relative gain of 166% is obtained on the ABO dataset.
+Answer to RQ2. The outcome of Experiment 2 is clear. The original relative perfor-
+mance results obtained on the MS COCO and Flickr30k (Table 1) are only partially
+replicable to the CUB-200, Fashion200k, and ABO datasets. On the latter datasets
+CLIP consistently outperforms X-VLM by a large margin, whereas the original scores
+obtained on the former datasets indicate that X-VLM mostly outperforms CLIP.
+Upshot. We hypothesize that the failure to replicate the relative results originally re-
+ported for scene-centric datasets (viz. X-VLM outperforms CLIP) is due to CLIP being
+pre-trained on more and more diverse image data. We, therefore, recommend that future
+work aimed at developing large-scale CMR models quantifies and reports the diversity
+of the training data used.
+
+12
+M. Hendriksen et al.
+6.3
+RQ3: Generalizability
+To answer RQ3, we compare the relative performance of the selected models on object-
+centric and scene-centric data. Thus, we compare the relative performance of CLIP and
+X-VLM on CUB-200, Fashion200k, ABO with their relative performance on MS COCO
+and Flickr30k. We focus on the complete tree from Fig. 2.
+Results. The results of our experiments on the scene-centric datasets are in Table 1; the
+results that we obtained on the object-centric datasets are in Table 2. On object-centric
+datasets, CLIP consistently outperforms X-VLM. However, the situation with scene-
+centric results is partially the opposite. There, X-VLM outperforms CLIP on the MS
+COCO dataset.
+Answer to RQ3. Hence, we answer RQ3 by stating that the relative performance results
+for CLIP and X-VLM that we obtained in our experiments only partially generalize
+from scene-centric to object-centric datasets. The MS COCO dataset is the odd one
+out.6
+Upshot. Given the observed differences in relative performance results for CLIP and X-
+VLM on scene-centric vs. object-centric datasets, we recommend that CMR be trained
+in both scene-centric and object-centric datasets to help improve the generalizability of
+experimental outcomes.
+7
+Discussion & Conclusions
+We have examined two SOTA image-text CMR methods, CLIP and X-VLM, by con-
+trasting their performance on two scene-centric datasets (MS COCO and Flicrk30k)
+and three object-centric datasets (CUB-200, Fashion200k, ABO) in a zero-shot setting.
+We focused on the reproducibility of the CMR results reported in the original pub-
+lications when evaluated on the selected scene-centric datasets. The reported scores
+were not reproducible for X-VLM when evaluated on the MS COCO and the Flickr30k
+datasets. For CLIP, we were able to reproduce the scores on the Flickr30k dataset when
+evaluated using R@5 and R@10. Conversely, the relative results were reproducible
+on the MS COCO dataset, for all metrics and tasks, and partially reproducible on the
+Flickr30k dataset only for image-to-text task when evaluated on R@1 and R@5. We
+also examined the replicability of the CMR results using three object-centric datasets.
+We discovered that the relative results are replicable when we compare the relative per-
+formance on the object-centric datasets with the relative scores on the Flickr30k dataset.
+However, for the MS COCO dataset, the relative outcomes were not replicable. And, fi-
+nally, we explored the generalizability of the obtained results by comparing the models’
+performance on scene-centric vs. object-centric datasets. We observed that the absolute
+scores obtained when evaluating models on object-centric datasets are much lower than
+the scores obtained on scene-centric datasets.
+Our findings demonstrate that the reproducibility of CMR methods on scene-centric
+6 On the GitHub repository for CLIP, several issues have been posted related to the performance
+of CLIP on the MS COCO dataset. See, e.g., https://github.com/openai/CLIP/i
+ssues/115.
+
+Scene-centric vs. Object-centric Cross-modal Retrieval
+13
+datasets is an open problem. Besides, we show that while the majority of CMR methods
+are evaluated on the MS COCO and the Flickr30k datasets, the object-centric datasets
+represent a challenging and relatively unexplored set of benchmarks.
+A limitation of our work is the relatively small number of scene-centric and object-
+centric datasets used for the evaluation of the models. Another limitation is that we
+only considered CMR in a zero-shot setting, ignoring, e.g., few-shot scenarios; this
+limitation did, however, come with the important advantage of reducing the number of
+experimental design decisions to be made for contrastive experiments.
+A promising direction for future work is to include further datasets when contrasting
+the performance of CMR models, both scene-centric and object-centric. In particular, it
+would be interesting to investigate the models’ performance on datasets, e.g., Concep-
+tual Captions [47], the Flower [40], and the Cars [27] datasets. A natural step after that
+would be to consider few-shot scenarios.
+Acknowledgements. We thank Paul Groth, Andrew Yates, Thong Nguyen, and Maurits
+Bleeker for helpful discussions and feedback.
+This research was supported by Ahold Delhaize, and the Hybrid Intelligence Center,
+a 10-year program funded by the Dutch Ministry of Education, Culture and Science
+through the Netherlands Organisation for Scientific Research, https://hybrid-i
+ntelligence-centre.nl.
+All content represents the opinion of the authors, which is not necessarily shared or
+endorsed by their respective employers and/or sponsors.
+
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VdE4T4oBgHgl3EQfnQ2t/content/tmp_files/2301.05175v1.pdf.txt

@@ -0,0 +1,1722 @@
+Scene-Aware 3D Multi-Human Motion Capture from a Single Camera
+D. C. Luvizon, M. Habermann, V. Golyanik, A. Kortylewski, C. Theobalt
+MPI Informatics, Saarland Informatics Campus, Germany
+Input: Single-view RGB Video
+Output: Estimated Humans and Scene
+Our
+Method
+Front
+Top
+ Multi-person
+Applications
+CG Characters Retargeting
+Front
+Top
+Depth-scale aware
+Figure 1. Our approach estimates absolute 3D positions of multiple humans in a scene, body shape and articulation in a globally and
+temporally coherent manner from a single monocular RGB video. It achieves higher 3D reconstruction accuracy than competing methods,
+allows motion re-targeting in the 3D space, and works exceptionally well even for in-the-wild videos.
+Abstract
+In this work, we consider the problem of estimating
+the 3D position of multiple humans in a scene as well as
+their body shape and articulation from a single RGB video
+recorded with a static camera.
+In contrast to expensive
+marker-based or multi-view systems, our lightweight setup
+is ideal for private users as it enables an affordable 3D mo-
+tion capture that is easy to install and does not require ex-
+pert knowledge. To deal with this challenging setting, we
+leverage recent advances in computer vision using large-
+scale pre-trained models for a variety of modalities, in-
+cluding 2D body joints, joint angles, normalized disparity
+maps, and human segmentation masks. Thus, we introduce
+the first non-linear optimization-based approach that jointly
+solves for the absolute 3D position of each human, their ar-
+ticulated pose, their individual shapes as well as the scale
+of the scene.
+In particular, we estimate the scene depth
+and person unique scale from normalized disparity predic-
+tions using the 2D body joints and joint angles. Given the
+per-frame scene depth, we reconstruct a point-cloud of the
+static scene in 3D space. Finally, given the per-frame 3D
+estimates of the humans and scene point-cloud, we perform
+a space-time coherent optimization over the video to ensure
+temporal, spatial and physical plausibility. We evaluate our
+method on established multi-person 3D human pose bench-
+marks where we consistently outperform previous methods
+and we qualitatively demonstrate that our method is ro-
+bust to in-the-wild conditions including challenging scenes
+with people of different sizes. Code: https://github.
+com/dluvizon/scene-aware-3d-multi-human
+1. Introduction
+Estimating the absolute 3D position, body shape, and ar-
+ticulation of multiple people in a scene is a fundamental
+research problem that has many applications in game devel-
+opment, VR/AR, and HCI. Years of research went into de-
+veloping sophisticated and expensive setups such as multi-
+view systems, motion capture suits, and manually or semi-
+automatically denoising of the tracked motions to then, for
+example, animate CG characters with these captured mo-
+tions. However, one ideally would like to obtain such an
+absolute scene understanding from a capture setup that is
+easy to install, affordable, and that does not require expert
+knowledge, i.e. a single RGB camera. Such a lightweight
+setup would enable 3D motion capture for private users, e.g.
+avatar control via the smartphone, but it can also be applied
+for post production in the movie industry where, for exam-
+ple, one person should be replaced by another in a 3D con-
+sistent manner. At the same time, it has to be stated that
+1
+arXiv:2301.05175v1  [cs.CV]  12 Jan 2023
+
+performing motion capture given such limited data is ex-
+ceptionally more difficult compared to multi-view systems.
+The major challenges for such a monocular setting, where
+only a single static video of the entire scene with moving
+persons is given, are the inherent depth ambiguity and oc-
+clusions, among many others.
+Therefore, recent monocular approaches focus on a sin-
+gle human [33, 40] or even assume an actor template is
+given [13,14,61]. Recently, some works started to research
+the multi-person setting, but they either only learn a rela-
+tive depth ordering of people in the scene [20] that is not
+3D consistent over time or they directly predict absolute
+depth, which is prone to overfit to the settings shown in the
+training data [37]. Most of those works leverage recent ad-
+vances in Computer Vision and take as input several types
+of regressed data modalities obtained from models trained
+on large-scale data. This involves 1) 2D body joints [4,11],
+2) joint angles [51], 3) normalized disparity maps [27, 41],
+and 4) human segmentation masks [8]. Interestingly, none
+of those works jointly considers all of those modalities.
+To this end, this work investigates how each of those
+modalities can benefit the task of multi-person absolute 3D
+pose and shape estimation. A particular challenge, however,
+is that each individual modality has, of course, advantages,
+but also disadvantages. While 2D and 3D keypoint detec-
+tions can help to infer the local 3D pose of a single person,
+they cannot ensure 3D consistency across humans and the
+scene. Joint angle estimates can be directly used to drive
+CG characters, but they are usually less accurate than the
+3D keypoint detectors due to error accumulation along the
+kinematic chain. Normalized disparity maps provide global
+reasoning of the entire scene as well as the humans in terms
+of its scale-normalized depth, but they cannot provide abso-
+lute depth and scale of the scene. Finally, human segmen-
+tation masks can provide close to pixel-perfect and identity
+preserving segmentations of humans in the scene, but they
+lack a 3D understanding.
+Now, to unite all the advantages of each of the modali-
+ties while compensating for their potential limitations, we
+propose the first optimization-based approach that jointly
+recovers the absolute 3D position of all humans in the im-
+ages, their articulated pose, their individual shapes, as well
+as the scale of the scene from a single video recorded with a
+static camera; see Fig. 1. In particular, we propose a novel
+energy formulation, which infers the absolute scene depth
+and the person unique scale from scale-normalized dispar-
+ity predictions by using the 2D and joint angle estimates of
+the humans in the scene as a prior. Once the per-frame ab-
+solute depth is known, we reconstruct a dense point cloud of
+the static scene in absolute 3D space by segmenting out the
+humans using the predicted segmentations and aggregating
+per-frame depth over time. Finally, we perform a coherent
+space-time optimization over the entire sequence to ensure
+temporal and spatial consistency as well as physical plausi-
+bility leveraging the aggregated scene estimate and the joint
+angle predictions. Note that in each of those steps, the com-
+bination of different data modalities is leveraged through
+our method and only this specific approach achieves the de-
+sired result in the considered setting, as extensively shown
+in our results. In summary, our primary technical contribu-
+tions are as follows:
+• The first monocular approach for multi-person abso-
+lute pose and unique scale estimation that jointly esti-
+mates multiple human poses and the 3D scene by com-
+bining data modalities in a novel optimization frame-
+work.
+• A human body prior to disambiguate the scale of the
+scene, which allows us to perform a coherent space-
+time reasoning of the human motion in absolute space.
+• We show that the estimated 3D human bodies can be
+refined in 3D space and time by filtering body move-
+ments in 3D coordinates and by penalizing implausible
+poses w.r.t. the estimated scene, resulting in a more co-
+herent final prediction.
+Since our approach estimates joint angles, global positions
+and scale, the recovered 3D human poses can be directly
+applied to CG characters enabling exciting applications as
+shown in Section 4. Moreover, we demonstrate that the joint
+reasoning of the human body shape, pose, and the dense
+scene over the entire video sequence improves state of the
+art in terms of 3D localization, scene and person scale, as
+well as body pose compared to prior work, both, quanti-
+tatively and qualitatively.
+Finally, we show that several
+downstream applications can be directly derived from our
+method, like monocular human motion capture and avatar
+control.
+2. Related Work
+3D human motion capture is an active research area, and
+many works have been proposed in the past [6, 23, 31, 32,
+36, 49, 50, 53, 55, 70]. Since we target a monocular setting,
+we do not review multi-view- and depth-based methods. In-
+stead, we review previous works that are most related to our
+method.
+2.1. 3D Human Pose Estimation
+2.1.1
+Single Person Pose Estimation
+Estimating the human body pose in 3D from a single image
+is a challenging problem that has been successfully handled
+by learning a human body prior from MoCap data [19]. To
+simplify the problem, previous methods usually predict 3D
+coordinates relative to the root joint, assuming a normalized
+human body size [33] and a fixed bounding box around the
+2
+
+person in 3D space [36, 40]. However, when multiple peo-
+ple are interacting with the environment, normalized and
+root-relative predictions are not enough to disambiguate the
+position and scale of individual persons in the scene. In
+addition, directly estimating the 3D joint coordinates could
+result in implausible poses, which is a problem that can be
+mitigated by estimating joint angles instead [71].
+Several works focus on estimating the full human mesh
+deformation from videos [13,14,61], assuming that the ac-
+tor mesh is provided in advance. Other works for single
+human estimation [22,24,39] rely on SMPL [30] as a proxy
+shape. Reconstructing shape proxies along with sparse 3D
+skeletons is desirable in many scenarios (e.g., they can be
+used for body parts segmentation). Moreover, SMPL serves
+as a statistical prior on human body shapes and enables ad-
+ditional supervisory terms such as human silhouette over-
+lays in 2D, which can result in higher accuracy [39].
+2.1.2
+Multiple Person Pose Estimation
+Estimating positions of each person w.r.t. the others is cru-
+cial in multi-human pose estimation. Nonetheless, most of
+the existing multi-person methods are by design perform-
+ing root-relative predictions [1, 45, 46, 51]. Several tech-
+niques predict translations of each person in the camera
+reference frame. They either optimize the translation by
+projecting and fitting the estimated 3D poses into the im-
+age plane [9, 34, 66] or by directly regressing the distance
+of the root joint to the camera with a deep neural net-
+work [28, 37, 56, 69]. The first case can be more robust
+to different camera setups, but is limited by the unknown
+height of each person in the scene. The second strategy is
+highly dependent on the training data and may not gener-
+alize to camera configurations not present in the training.
+Others explore human priors [26] to estimate a global tra-
+jectory [64], but still fail to recover the body size.
+Recent methods performing human depth estimation are
+focused on penalizing depth ordering of multiple humans.
+For instance, Jiang et al. [20] uses instance segmentation
+masks to penalize depth inversion and Sun et al. [52] pro-
+poses to infer the depth of each person based on an imagi-
+nary bird’s-eye-view representation and to estimate the per-
+son age as a proxy for the scale. Other approaches pre-
+dict the relative depth among multiple persons by inferring
+some scene properties. A possible scene simplification is to
+assume a parametric planar floor, in such a way that each
+prediction can be positioned to respect a plausible human-
+floor contact [54, 65]. The common limitation of such ap-
+proaches is the dependency on a simplified floor represen-
+tation, which is often not the case in real applications. Con-
+trarily, we estimate a scene point cloud that can represent a
+arbitrary ground floor.
+The works from Jiang et al. [20] and Ugrinovic et al. [54]
+are the most closely related to ours. Similarly to the for-
+mer, we also render the estimated human models into the
+image plane to provide additional supervision in the depth
+dimension, and, related to the latter, we also disambiguate
+body size and depth for each person by constraining pre-
+dictions with an estimated scene geometry. But differently
+from [20], that does not take the scene into account, and
+from [54], that relies on a simplified scene representation
+and operates in a single frame, our method represents the
+scene as a frustum point cloud and performs optimization
+over the entire video sequence. In our work, we also rely
+on a human body proxy model [30] to estimate joint angles
+and we propose a new formulation to optimize the position
+of the humans and the scene in a joint optimization process.
+Therefore, our model improves the prediction of human po-
+sitions by relying on an estimated proxy scene geometry
+that does not depend on a simplified parametric model.
+2.2. Scene-aware Motion Capture
+Predicting and understanding how humans interact in
+3D has recently gained a lot of attention. Several current
+methods focus on positioning humans in a pre-scanned 3D
+scene [12, 15, 18] and on simultaneous estimation of hu-
+man poses and objects humans interact with [7, 59, 62]. A
+different setup assumes an RGB-D sensor [68] or a mov-
+ing camera [16, 25, 29, 67] that facilitates estimating the
+scene geometry. Recent methods integrate physics-based
+constraints into monocular 3D human motion capture and
+mitigate foot-floor penetration and other severe artefacts
+[47,48]. Yu et al. [63] also support composite scenes in the
+parcours and sports scenarios. Although there is a growing
+interest in investigating the interactions of humans and ob-
+jects [2, 10], 3D motion capture of multiple humans with
+environmental awareness from a single monocular camera
+remains underexplored.
+Determining the absolute human scale in 3D is an ill-
+posed and challenging task. Bieler et al. [3] estimate the
+height of a single person from monocular videos by observ-
+ing jumping people. Dabral et al. [10] require an interaction
+with an object undergoing a free flight to resolve the abso-
+lute scene scale. Both methods assume motion influenced
+by the universal law of gravity near the surface of Earth,
+which allows them to relate the time spent in the air or the
+form of the observed trajectory with absolute distances in
+the metric units.
+The downside is that jumping humans
+or flying objects are restrictive assumptions. In contrast,
+we use a human body and 3D scene priors in 3D multi-
+human motion estimation and do not make strong assump-
+tions about the observed human motions.
+3. Method
+The goal of our method is to estimate the absolute 3D
+position of each human in the scene, i.e., up to a unique and
+3
+
+Input: Video Sequence
+Disparity
+Map
+2D Pose
+SMPL 
+Parameters
+Instance
+Segmentation
+Image Modality Regression and Matching (Section 3.1)
+Pre-processing and Matching
+Scene Scale and Depth Disambiguation (Section 3.2)
+Output: Humans and Scene
+Aggregation
+(median)
+Optimization Framework
+Space-time Coherent Pose Optimization (Section 3.3)
+Eq. 3
+SMPL
+Estimates
+Disparity
+Maps
+Depth
+Maps
+Segmentation
+and 2D Pose
+Renderer
+Background Scene
+Point Cloud
+For each
+person
+Background
+Masks
+3D Humans and
+Scene Estimates
+in a Unique Scale
+Figure 2. Overview of our method. For each frame in a monocular RGB video, we first estimate a normalized disparity map, 2D human
+poses, SMPL model parameters, and segmentation masks. These predictions are matched and tracked across frames to obtain per-person
+associations (blue box). The multi-modal estimates are then fed into our optimization framework. The first part of our optimization
+process estimates per-frame human models in global position and the scene geometry (yellow box). In the second part, the per-frame scene
+predictions are aggregated into a single point cloud representation and the human predictions are refined in a space-time coherent manner
+over the full video (red box). The yellow arrows indicate the energy terms minimized by our method. The output of our method is the
+absolute 3D positions of each human in the scene, their shape and pose as well as the scene scale. ⊙ is the Hadamard product.
+global scale, their proxy shape and pose, as well as the scene
+scale solely from a monocular RGB video recorded with
+a static camera for which we know the intrinsics. To this
+end, we propose a unified approach that, for the first time,
+leverages all available data modalities, including 2D joint
+detections, regressed SMPL parameters, estimated dispar-
+ity maps, and human segmentations. As illustrated in Fig-
+ure 2, our method is divided into two stages. The first stage,
+i.e. Image Modality Regression and Matching (Section 3.1),
+extracts per-frame estimates and aggregates human-related
+predictions to individuals throughout the video sequence.
+The second stage, i.e. the proposed Optimization Frame-
+work, estimates the person and per-frame scene scale, the
+global 3D position of each person in the scene, as well as
+the refined articulated body pose in the form of joint angles
+per frame.
+The optimization framework is further subdivided into
+two parts. The Scene Scale and Depth Disambiguation part
+(Section 3.2) recovers a consistent and absolute 3D scene
+depth per frame, the human scales, and their absolute 3D
+position and body pose by jointly reasoning about multi-
+ple humans and the scene. The second part, referred to as
+Space-time Coherent Pose Optimization (Section 3.3), re-
+fines the pose and position of the estimated humans in a
+space-time coherent formulation, i.e. we enforce over the
+entire sequence the estimated poses to be temporally sta-
+ble and physically plausible. For this, we leverage a rough
+scene geometry estimation, which is obtained by aggregat-
+ing the absolute depth maps also estimated by our method.
+This final part significantly reduces artifacts, such as foot
+sliding, human-scene intersections, and jitter. Before we
+explain our method in more detail, we introduce relevant
+notations.
+Notations.
+The input of our framework is a video se-
+quence It, with t ∈ {1, . . . , T}, where T is the number of
+frames. We leverage the skinned multi-person linear model
+(SMPL) [30] to represent the humans in the scene. SMPL is
+a differentiable parametric human model that takes as input
+the pose parameters θ ∈ R72, corresponding to the axis-
+angles of 24 body joints and the global body rotation, and
+PCA shape parameters β ∈ R10, and produces a skinned
+human mesh
+fsmpl(θ, β) = V,
+(1)
+where V are the posed and shaped vertices of the human
+body; for more details we refer to their paper [30]. The
+mesh vertices regressed by SMPL can also be used to es-
+timate a sparse 3D pose as J (V), where J (·) is a linear
+regressor parameterized by a matrix W ∈ RJ×6890, and J
+4
+
+Adenotes the total number of joints.
+To account for translations in 3D space, we further add a
+translation Γt,n ∈ R3 to the SMPL representation, where n
+is the person index. Furthermore, the 3D human pose mod-
+els are overwhelmingly biased towards adult body sizes.
+Thus, we explicitly model the person scale by sn ∈ R+
+and our final human mesh can be defined as
+˜Vt,n = snVt,n + Γt,n.
+(2)
+This human mesh for person n at time t is then fully deter-
+mined by the parameters θt,n, Γt,n, βn, and sn, which we
+aim to recover in the following. Important to note is that the
+person scale sn and shape βn are unique for each person
+and consistent across the entire video sequence.
+3.1. Input Modality Regression and Matching
+To solve this underconstrained and challenging problem,
+our idea is to unite the strength of all data modalities, which
+recent state-of-the-art Computer Vision methods provide, in
+a single algorithm. More precisely, we leverage data-driven
+priors in the form of four off-the-shelf methods for each
+frame of the input video sequence, as shown in Figure 2.
+First, we obtain normalized disparity maps ˆdt from
+the state-of-the-art DPT model [41], which are then post-
+processed to enhance sharpness [58]. Note that these maps
+only encode relative and normalized depth and they are not
+consistent across frames, which becomes visible in the form
+of depth jitter.
+Second, 2D pose tracking is obtained by AlphaPose [11],
+which coherently detects and tracks 2D joint positions
+ˆP2d
+t,n ∈ RJ×2 in image space and over time. Although this
+method is very robust due to training on large scale data, it
+falls short in predicting 3D.
+Third, we predict the body shape βt,n and joint angles
+ˆθt,n for each person in each frame using ROMP [51]. Since
+ROMP predicts varying shapes for a single person across
+time, we average the predictions over the entire sequence to
+obtain a temporally consistent body shape. Thus, the ver-
+tices (Equation 2) are now only a function of the pose θt,n,
+translation Γt,n, and scale sn, which will be important in
+the next section. Moreover, to match the 2D AlphaPose
+and the SMPL detections, we leverage ROMPs projection
+model, compute the average Euclidean distance in image
+space, and pair detections with the lowest distance based on
+the Hungarian matching. It is worth mentioning that ROMP
+cannot account for out-of-distribution body sizes, e.g. small
+kids, neither it can predict the absolute 3D position of the
+humans with respect to the scene.
+Fourth, we also leverage human segmentation masks,
+referred to as Ωt,n ∈ RH×W , which are obtained from
+Mask2Former [8]. Similarly, if we consider all the remain-
+ing pixels for frame t that do not belong to a person mask,
+we can also obtain a per-frame background segmentation
+mask Bt ∈ RH×W . To ensure that the 2D AlphaPose de-
+tections, the SMPL detections, and the foreground masks
+have a consistent person ID, we read the pixel values of
+the segmented masks at the 2D joint detections for each de-
+tected skeleton and apply a max-voting to retrieve the ID of
+the person.
+In summary, the inputs to our algorithm now are:
+• ˆdt: Normalized disparity maps
+• ˆP2d
+t,n: 2D joint predictions
+• ˆθt,n, ˆβn: Pose angle and shape estimates
+• Ωt,n, Bt: Human and background segmentations
+Note that none of these predictions individually or by a triv-
+ial combination is discriminative enough to fully describe
+the entire scene, i.e. absolute 3D position, pose, and scale
+of the humans in the scene. Next, we demonstrate how our
+proposed method solves this problem.
+3.2. Scene Scale and Depth Disambiguation
+In the first part or our optimization process we focus on
+jointly obtaining the joint angles θt,n, shape parameters βn,
+global translation Γt,n, and scale sn of each person. Impor-
+tantly, this step is performed jointly for the entire sequence,
+where the global reference is in the static camera. How-
+ever, estimating the height of a person and the scale given
+only a single RGB video is, by itself, an ill-posed problem
+as variations in scale can be compensated by a translation
+along the depth and vice versa. As a result, infinitely many
+scale/translation combinations can lead to the same 2D im-
+age projections.
+So far, we only considered individual humans without
+looking at the surrounding scene, although the scene itself
+can provide an important prior that helps to solve the above
+problem. Therefore, we leverage recent advances in monoc-
+ular depth estimation [41], which regress per-pixel normal-
+ized disparity maps ˆdt. It encodes the relative depth of each
+person in the scene, but obtaining the absolute depth val-
+ues solely from ˆdt is also an ill-posed problem, and fur-
+ther these predictions are not consistent across frames. The
+question remains, how the absolute scene depth or equiva-
+lently the human scales and translations can be recovered.
+Our idea is to set the two entities, i.e., the scene and the
+humans, into a relation such that they constrain each other
+in an absolute 3D space. While the humans can already
+be represented in absolute space by means of their global
+translation Γt,n and scale sn, we also require a per-frame
+conversion of temporally inconsistent normalized disparity
+maps to absolute depth maps, which can be defined as
+˜Dt =
+zfar,tznear,t
+ˆdt(zfar,t − znear,t) + znear,t
+(3)
+5
+
+where znear,t and zfar,t are the near and far depth values,
+respectively. Intuitively, this operation shifts and scales the
+normalized disparity maps to convert them to absolute depth
+values. Importantly, these near and far values are optimized
+per-frame to compensate for the temporal inconsistencies in
+the disparity maps.
+Once both humans and the scene can be represented in
+absolute 3D space, we now relate them to each other by
+jointly solving for κt,n ∈ {znear,t, zfar,t, θt,n, βn, Γt,n, sn}
+by minimizing the energy
+arg min
+∀t∈{1,...,T },∀n∈{1,...,N}:κt,n
+EI,
+with
+(4)
+EI = Edepth + E2d + Esmpl + Ereg,
+(5)
+which is jointly optimized over the entire sequence. In par-
+ticular, our energy is composed of a depth term Edepth, a
+2D image evidence term E2d, a joint angle and shape term
+Esmpl, and additional regularization terms Ereg. In the fol-
+lowing, we explain each term in more detail.
+3.2.1
+Depth Consistency Energy
+Most importantly, to ensure a coherent depth between the
+scene and all humans in the scene, we propose a depth con-
+sistency energy
+Edepth = λdepth
+�
+t,n
+�
+M(Ψd( ˜Vt,n)) − M( ˜Dt)
+�2
+,
+(6)
+M(D) =
+� Ωt,n
+|Ωt,n| log(D),
+(7)
+where |Ω| denotes the number of foreground pixels, M(·)
+computes the average of the log-depth in the foreground,
+Ψd(·) is a differentiable rasterizer [43] that projects and
+converts a 3D mesh into a depth map in the image plane,
+and λdepth is a hyperparameter. The vertices ˜Vt,n refer to
+the estimated SMPL models of each person in global space,
+which are a function of the variables θt,n, βn, Γt,n, and sn
+(Equation 2). The rasterized human depths are then com-
+pared to the estimated absolute depth map ˜Dt of the scene
+(Equation 3), which are a function of the variables znear,t
+and zfar,t. Thus, this energy jointly optimizes the human
+and the scene parameters. However, since both sides of the
+penalty term contain free variables, this energy alone would
+not disambiguate the problem.
+3.2.2
+Image Projection Energy
+We introduce an additional data term, which further con-
+strains the human-related variables by enforcing the 3D
+bodies to project accurately into the image plane. More pre-
+cisely, the data term
+E2d = Ejoints + Esilhouette
+(8)
+penalizes the error between the projected 3D body joints
+J ( ˜Vt,n) of the optimized SMPL models and the respective
+2D body joints ˆP2d
+t,n regressed by AlphaPose with
+Ejoints =
+�
+t,n
+���Π(J ( ˜Vt,n)) − ˆP2d
+t,n
+���
+2
+2,
+(9)
+where Π(·) is the perspective camera projection operator.
+The right term of (8) penalizes the discrepancy between the
+SMPL silhouette and the instance segmentation masks:
+Esilhouette = λsilhouette
+|Ω|
+�
+t,n
+σt,n
+���Ψs( ˜Vt,n) − Ωt,n
+���
+2
+2,
+(10)
+where Ψs(·) is a differentiable renderer [43] that projects
+and converts a 3D mesh into a silhouette image and σt,n is
+a visibility mask, so vertices hidden by other humans are
+not penalized.
+3.2.3
+Joint Angle and Shape Energy
+Since (8) only constrains the parameters in 2D image space,
+we further add an additional data term that ensures that
+the optimized SMPL parameters are close the prediction of
+ROMP:
+Esmpl = λsmpl
+�
+t,n
+���θt,n − ˆθt,n
+���
+1 +
+���βn − ˆβn
+���
+1 . (11)
+Here, ∥·∥1 denotes the L1 norm.
+3.2.4
+Temporal and Human Priors
+To further constrain the scale and position of a person, we
+leverage priors on the human body size and on the temporal
+information. This is achieved by our regularization term
+Ereg = Escale + Espeed.
+(12)
+For the scale term Escale, our assumptions are two-fold:
+i) The scale of a person should not deviate too much from
+the standard person size, i.e., the standard SMPL size when
+sn = 1, and ii) the average scale of multiple people in the
+scene should remain close to one. This dual assumption is
+enforced by
+Escale = λscale
+�
+n
+(sn − 1)2 +
+��
+n
+(sn − 1)
+�2
+,
+(13)
+where the first term accounts for the individual person scale
+and the second term accounts for the average scale of mul-
+tiple persons.
+In addition to the person scale, we also introduce an un-
+derlying assumption that locomotion is rather smooth over
+6
+
+Figure 3. Per-frame estimations of our method considering the first
+optimization part (Section 3.2) only. From left to right: Estimated
+depth map, frontal view of the scene and estimated humans, and
+top view. Note how the persons’ absolute 3D location, articulated
+pose and shape as well as the scene scale can be recovered from
+a single input image, even with people of different sizes (bottom
+row).
+time based on the physical limits of the human body, so we
+penalize large movements of the root joint by our energy
+Espeed = λspeed
+�
+t,n
+∥Γt,n − Γt−1,n∥2
+2 .
+(14)
+In the optimization process described above, the per
+frame human parameters and the absolute scene depth
+are obtained by means of the optimized human ∀t
+∈
+{1, ..., T}, ∀n ∈ {1, ..., N} : θt,n, βn, Γt,n, sn, and scene
+znear,t, zfar,t parameters.
+Figure 3 shows our estimated
+scene and humans for example frames. Note that the es-
+timated depth looks plausible, humans and the scene are
+coherent with each other, and the reprojection of humans
+into the input view looks accurate.
+3.3. Space-time Coherent Pose Optimisation
+Since we obtained absolute and per-frame human mod-
+els and scene estimations, both information can be used to-
+gether to further refine the human poses in a spatially and
+temporally coherent manner.
+Therefore, in the last part
+of our optimization method, we refine the estimated poses
+in 3D by enforcing physical plausibility between humans
+and the estimated scene, as well as by applying a temporal
+smoothness term. More precisely, we extend (4) by includ-
+ing a new energy term EII:
+arg min
+∀t∈{1,...,T },∀n∈{1,...,N}:κt,n
+EI + EII,
+with
+(15)
+EII = Econtact + Eslip + Etemporal.
+(16)
+For implementing Econtact and Eslip, we leverage the esti-
+mated scene geometry as a reference for enforcing foot con-
+tact and penalizing foot slipping. In the following, we first
+explain how the per-frame depth maps are aggregated into
+a static 3D scene representation, then we present the energy
+terms of EII in more detail.
+3.3.1
+Scene Point Cloud Estimation
+Our method relies on humans as anchors in the scene, i.e.,
+the estimated geometry around the humans tends to be co-
+herent. However, mainly due to occlusions, the estimated
+per-frame absolute depth values are not yet temporally con-
+sistent for the whole scene. To obtain a static representa-
+tion of the background, we rely on the segmentation masks
+to aggregate the depth values in the background from each
+frame into a single depth map.
+This static depth map
+representation is obtained by computing the per-pixel me-
+dian for the entire video sequence, which is a metric ro-
+bust to outlier depth values. We also experimented with
+more sophisticated aggregation strategies, such as aggre-
+gating values near the human anchors weighted by a Gaus-
+sian distribution—since the human positions are stable—
+but this strategy was significantly more expensive and re-
+sulted in marginal improvements.
+At the end of this ag-
+gregation process, we obtain a single depth map ˆD of the
+scene, which can be then converted to a point cloud repre-
+sentation P ∈ RHW ×3 in absolute 3D space.
+3.3.2
+Improving Physical Plausibility of Estimated
+Motions
+Recently, a series of works highlighted the importance of
+physics awareness in monocular single person motion cap-
+ture [25,44,47,48] with assumptions about the camera and
+floor plane positions. Inspired by them and the fact that we
+obtain a coherent and unique scale estimation of the scene,
+we propose to model in our energy formulation the phys-
+ical interaction between the humans and the environment.
+Here, the first term penalizes ”floating” characters, i.e., hu-
+mans that are not in contact with the ground, and the second
+term penalizes foot sliding, i.e., a foot that is in contact with
+the ground should not move.
+More precisely, given the scene point cloud P and the
+estimated human meshes ˜Vt,n, floating characters are pe-
+nalized by
+Econtact = λcontact
+�
+t,n
+ζ
+����min( ˜Vy+
+t,n − P)
+���
+1
+�
+(17)
+where ˜Vy+
+t,n ∈ R1×3 is the vertex of person n at time t with
+lower Y coordinate, considering that the Y -axis is the grav-
+itational axis for our coordinate frame. In other words, the
+term Econtact minimizes the distance between the lower ver-
+tex ˜Vy+ of each prediction and its respective closest point
+in the scene point cloud. Here, ζ(·) is a robust thresholding
+function, which only considers distances below 20cm.
+The term
+Eslip = λslip
+�
+t,n
+ζ
+���∆( ˜V y+
+t,n )
+���
+1
+(18)
+7
+
+penalizes the movement of this lowest vertex in the time
+domain (∆) when it is in contact with the scene. By ap-
+plying those energy terms, we can now enforce that the hu-
+mans interact more physically accurate with respect to the
+3D scene.
+3.3.3
+Temporally Stable Pose
+Furthermore, since the joint and absolute position optimized
+by EI can still contain smaller jitter, we propose a temporal
+stability term
+Etemporal = λtemporal
+�
+t,n
+���∆t( ˜Vt,n) − ∆t( ¯Vt,n)
+���
+2
+,
+(19)
+based on the 1C filter [5], where ∆t(Vt,n) = Vt,n −
+Vt−1,n is the temporal variation of the human mesh ver-
+tices and ¯Vt,n are the estimated SMPL vertices after tem-
+poral filtering [5]. This term allows us to obtain temporally
+more stable poses with significantly less jitter.
+4. Experiments
+In this section, we present an empirical evaluation of our
+method. We first briefly describe the datasets and metrics
+used in our experiments in Sections 4.1 and 4.2, followed
+by the implementation details in Section 4.3. Next, we com-
+pare our approach with the most related works to ours in
+Section 4.4. In Section 4.5, we perform a thorough abla-
+tion study of the main components of our method and show
+additional qualitative results in Section 4.6.
+4.1. Datasets
+MuPoTs-3D [35] is a test dataset composed of 20 video
+sequences with multiple people, including different types of
+cameras in indoor and outdoor environments. We followed
+the evaluation protocol from [35] in our experiments. This
+dataset is especially challenging due to the large amount
+of interactions between humans and the various types of
+scenes. Ground-truth 3D pose annotations are provided in
+absolute coordinates.
+CMU Panoptic [21] is a dataset recorded in the Panoptic
+Studio with multiple people. As in preliminary work [20,
+65], we use this dataset for evaluation considering the
+sequences haggling1, ultimatum1, and pizza1,
+which are performed by several adults.
+In addition to the previous datasets, we also evaluated
+our method quantitatively on Internet videos considering
+challenging cases with multiple people of different sizes,
+including adults and children.
+4.2. Metrics
+MRPE and AP. We quantitatively evaluate the predic-
+tion of the absolute 3D location of a human using the widely
+adopted mean root position error (MRPE), in millimeters,
+and the average precision of the human root joint (AProot
+25 )
+[37], considering the standard threshold of 25 cm.
+3DPCK. The quality of the articulated 3D pose prediction
+is measured using root-relative 3DPCK [33], with the stan-
+dard threshold of 15 cm. The 3DPCK metric enables mea-
+suring the correctness of the pose, independently of the pre-
+diction of the absolute 3D location of the human.
+MPJPE. For a fair comparison with previous methods,
+we also report root-relative mean per-joint position error
+(MPJPE) in the CMU Panoptic dataset.
+Jitter.
+Finally, since we are targeting high-quality tem-
+poral predictions in 3D coordinates, we also evaluate the
+amount of jitter of our estimations, which is a critical in-
+dicator for many downstream applications. For this eval-
+uation, we adapted the temporal smoothness error esmooth
+from [47] to evaluate the jitter in 3D coordinates.
+4.3. Implementation Details
+Our method is implemented in PyTorch [38] using Py-
+Torch3D [43] for the rasterization (6) and silhouette render-
+ing (10). The camera intrinsics are used in the 3D joint pro-
+jection (9), rasterization (6), and rendering (10) parts, and
+can be obtained from video metadata if not given. We ap-
+ply the RMSprop [17] optimizer with the parameters α and
+momentum set to 0.5 and 0.9, respectively, for all experi-
+ments. In the optimization process, we initially minimize
+the first part (4) only for 30 iterations, then perform the full
+optimization (15) for more 200 iterations. We use a learn-
+ing rate initially set to 0.01 and exponentially decaying with
+factor 0.99.
+The weights λ(.) were empirically defined to
+balance the magnitude of the individual energy terms, and
+fixed in the method in all experiments, except when men-
+tioned otherwise (ablation in Section 4.5).
+The values
+were defined as λdepth = λspeed = 0.05, λsilhouette = 0.1,
+λsmpl = λtemporal = 0.002, λscale = 0.0001, λcontact =
+0.001, and λslip = 0.01.
+For numerical stability, we con-
+strain the variables sn, znear,t, and zfar,t to be non-zero and
+positive. Both human and background segmentation masks
+were post-processed with morphological erosion and dila-
+tion filters of size 3×3 and 5×5, respectively. For the sake
+of GPU memory efficiency, we use mini batches of ten im-
+ages in the depth and silhouette losses. Our experiments run
+on a workstation with one Nvidia Titan V GPU with 12 GB
+of memory.
+4.4. Comparison with Previous Methods
+In Table 1, we compare our method to the most related
+prior work. We compare our method for human localiza-
+tion considering MRPE and AProot
+25
+metrics with the meth-
+ods that are capable of providing such predictions. We use
+two protocols to evaluate the quality of the 3D pose. First,
+we compare against the global 3D pose without any nor-
+8
+
+Table 1. Comparison of our method with previous approaches on
+MuPoTs-3D in the MRPE (lower is better), AProot
+25 , and 3DPCK
+metrics (higher is better), considering the global 3D pose and the
+normalized (univ) ground truths. Our approach is superior to all
+compared methods on the absolute metrics (MRPE, AProot
+25
+and
+3DPCK3d), i.e., the most expressive ones for 3D human motion
+capture. “†” evaluated on samples with IK only; “∗” evaluated on
+root-relative predictions without IK; “‡” results only possible with
+an additional 2D fitting stage, implemented as our baseline.
+Method
+Char.
+control
+MRPE ↓
+AProot
+25
+3DPCK3d
+3DPCKuniv
+LCR-Net [45]
+
+–
+–
+–
+53.8
+LCR-Net++ [46]
+
+–
+–
+–
+70.6
+3DMPPE [37]
+
+–
+31.0
+–
+81.8
+SMAP [69]
+
+–
+45.5
+–
+80.3
+XNect∗ [34]
+
+–
+–
+64.1
+71.9
+XNect† [34]
+
+639
+31.6
+56.5
+60.1
+CRMH [20]
+
+–
+–
+–
+69.1
+BEV [52]
+
+–
+–
+–
+70.2
+Baseline (ROMP+2D fitting)
+
+331‡
+45.4‡
+68.2‡
+71.8
+Ours
+
+266
+62.3
+74.9
+78.9
+malization, which is a fairer protocol for our method, since
+we are capable of estimating the person scale (denoted by
+3DPCK3d). In the second case, we compare against the uni-
+versal 3D pose, which has all bone lengths normalized to a
+standard size, as described in [35] (denoted as 3DPCKuniv).
+For this universal protocol, in our method, we assume per-
+son scale sn equals to one for all predictions. Note how our
+method outperforms all prior work by a wide margin at 3D
+localization and also performs better at estimating the ar-
+ticulated pose compared to all other methods that allow for
+character control.
+As a baseline, we evaluate ROMP [51]
+predictions with an additional stage for fitting estimated
+SMPL models to AlphaPose 2D body joint detections, since
+this is the closest setup to our method without including our
+new energy functions. For this, we assume a unitary person
+scale (w.r.t. the SMPL neutral model) and optimize only
+the global translation in 3D of each person.
+In a similar
+manner, XNect [34] estimates the global position by fitting
+the predicted 3D poses into 2D body joints, assuming a uni-
+versal and normalized human body size. The inverse kine-
+matics (IK) stage from XNect allows this global estimation,
+however, since the optimized 3D human pose differs from
+the preliminary estimated pose, the accuracy after IK drops
+significantly. In summary, we observe that our approach
+outperforms previous methods for human position estima-
+tion by a significant margin, improving the average preci-
+sion of the root joint from 45.4% to 62.3%. Our method
+also outperforms all other approaches for human pose esti-
+mation that are capable of driving a virtual character.
+In Table 2, we compare our method with other ap-
+proaches on the CMU Panoptic dataset. This dataset is spe-
+cially challenging because in many sequences the persons
+are only partially visible, either due to occlusions, or be-
+Table 2. Comparison of our method with previous approaches
+on the CMU Panoptic dataset for 3D pose estimation. Results
+reported in millimeters. Camera views capturing only the upper
+body parts were not used in our evaluation. † evaluated in all the
+sequences. Best results are bold on the standard sequences and
+underlined on the full-body visible sequences.
+Metric
+Method
+Haggling
+Ultimatum
+Pizza
+Avg.
+MPJPE
+CRMH [20]†
+129.6
+153.0
+156.7
+146.4
+BEV [52]†
+90.7
+113.1
+125.2
+109.6
+Baseline
+93.6
+133.8
+145.9
+124.4
+Ours
+84.5
+108.9
+133.2
+108.9
+MRPE
+Baseline
+235.2
+269.6
+356.4
+287.0
+Ours
+213.7
+208.0
+229.7
+217.1
+cause the camera is capturing only the upper body part of
+the actors. Even in this challenging scenario, our method
+performs on par with the recent BEV [52] method, which
+was trained on the CMU Panoptic dataset and, therefore,
+performs better in the cases of partial body visibility then
+our optimization approach. In order to evaluate the perfor-
+mance of our method on the more practical scenario of cam-
+eras recording the full body of the persons, we removed the
+few sequences capturing only the upper body parts. In this
+setup, we largely improve over other methods and over our
+baseline, as can be seen by the underlined numbers in Ta-
+ble 2.
+For many downstream applications, such as gaming and
+character control, jitter is a severe artifact that hinders us-
+ability. Therefore, we also evaluated our method by report-
+ing the temporal smoothness error esmooth in 3D coordi-
+nates. The results from our method, as well as from previ-
+ous work in the literature related to ours, are shown in Ta-
+ble 3. In this experiment, we compared our approach with
+two methods from the literature, showing a significant im-
+provement in reducing the jitter artifact. Furthermore, we
+also evaluated the contribution of different components of
+our method. For instance, the temporal energy term in our
+approach has a critical effect in reducing jitter. In addition,
+the contact and slip terms also contribute in a small propor-
+tion but consistently to all metrics, regardless the presence
+or absence of the temporal energy. When all terms are in-
+cluded, our approach is very stable, with an average jitter
+error below 1cm.
+4.5. Ablation Study
+In this section, we perform additional evaluations of
+the different components of our method. The results on
+MuPoTs-3D are shown in Tables 4 and 5. First, we evaluate
+the influence of the energy terms of the first part of our op-
+timization framework. The energy term Edepth provides es-
+sential information to disambiguate depth and scale, which
+contributes to improving the position estimation. The flex-
+9
+
+Ground-truth
+Ours
+Baseline (ROMP + 2D fitting)
+XNect
+Figure 4. Comparisons of predictions from our method with other approaches. Compared to XNect and our baseline, our method is the
+only one that is able to estimate the person scale. Therefore, it predicts human positions in a more coherent way even for people of smaller
+height. 3D human poses are shown in the image plane (left) and top view (right). The ground-truth pose is not available for all the subjects
+in the dataset. Digital zoom is recommended.
+Table 3. Comparison of our method on MuPoTs-3D with previous
+approaches on temporal smoothness error esmooth, that measures
+the amount of jitter in the predictions in millimeters. We also
+report the MRPE and 3DPCK3d metrics for completeness. Our
+method has a drastically lower jitter in the prediction compared to
+previous multi-person motion capture approaches.
+Method
+Jitter ↓
+MRPE ↓
+3DPCK3d ↑
+XNect [34]
+136.4
+639
+56.5
+ROMP [51]
+59.6
+331
+68.2
+Ours (EI only)
+17.5
+281
+73.5
+Ours (EI + Econtact)
+17.6
+276
+73.7
+Ours (EI + Econtact + Eslip)
+17.1
+273
+73.8
+Ours (EI + Etemporal)
+7.8
+272
+74.8
+Ours (EI + Econtact + Eslip + Etemporal)
+7.5
+266
+74.9
+ibility provided by the person scale factor can be detrimen-
+tal to the overall accuracy of the method if no constraints
+are imposed on it. This can be seen in the second row of
+Table 4, without Escale.
+By constraining our predictions
+to remain close to the original estimates from ROMP, our
+method enforces the final estimates to be valid and prevents
+them from collapsing, as shown in the results without Esmpl.
+Finally, Espeed is relevant for reducing jitter and the silhou-
+ette term provides beneficial contributions to all the metrics.
+With all the energy terms, our method is stable and precise
+in estimating 3D position and pose.
+Since our method
+relies on off-the-shelf predictors as input, we also provide
+a concise evaluation considering two different 2D pose and
+three different depth estimation models from the recent lit-
+erature. The results in Table 5 show that the influence of the
+depth estimation models is relatively small; however, the
+best performing model is the most recent transformer archi-
+tecture, which suggests that our approach directly benefits
+from improved monocular depth estimations. Regarding 2D
+pose estimation, HRNet [57] performed worse than Alpha-
+Pose, since HRNet relies on person detection as a first step,
+Table 4. Ablation study for different energy terms. Without the
+proposed depth and scale terms, the global position in 3D cannot
+be precisely recovered, i.e., AProot
+25
+drops from 62.3 to 47.4% and
+to 22.2%, respectively. The SMPL term is critical for enforcing
+valid estimates, and the speed term contributes to reducing the jit-
+ter. The silhouette term provides consistent improvements in all
+the metrics.
+Experiment
+Jitter ↓
+MRPE ↓
+AProot
+25
+↑
+3DPCK3d ↑
+w/o Edepth
+7.8
+284
+47.4
+75.5
+w/o Escale
+7.7
+541
+22.2
+68.9
+w/o Esmpl
+8.0
+674
+11.5
+56.3
+w/o Espeed
+8.9
+269
+63.6
+74.8
+w/o Esilhouette
+7.6
+270
+62.0
+74.7
+Ours (full)
+7.5
+266
+62.3
+74.9
+Table 5. Our results considering different models for 2D pose and
+monocular depth estimation. We observe that the human posi-
+tion estimation from our method benefits directly from advances in
+the monocular depth estimation when comparing MiDaS v2.1 [42]
+and DPT-Large [41].
+2D Pose Model
+Depth Model
+MRPE ↓
+AProot
+25
+↑
+3DPCK3d ↑
+AlphaPose
+MiDaS v2.1
+278
+55.8
+75.7
+AlphaPose
+DPT-Hybrid
+276
+60.8
+75.0
+AlphaPose
+DPT-Large
+266
+62.3
+74.9
+HRNet
+DPT-Large
+304
+54.9
+72.7
+which makes it susceptible to detection failures.
+4.6. Qualitative Results
+Figure 4 provides additional qualitative results with pre-
+dictions from our method in 3D coordinates, alongside
+the ground truth pose.
+We compare our method with
+XNect [34] and ROMP [51]. We can see that predictions
+from ROMP do often not correspond to the correct posi-
+tion of the humans in the scene, since it is not able to esti-
+mate the correct person scale. For XNect, we can observe
+10
+
+Input image
+Ours
+Baseline (ROMP + 2D fitting)
+Figure 5. 3D Human poses estimated by our method from Internet videos. The baseline method can correctly localise the persons in the
+image plane, but fails drastically in positioning the characters in 3D. Note from our method the correct character order along the depth
+channel and the correctly estimated scale for each person. Digital zoom is recommended.
+BEV
+GLAMR
+Ours
+Input image
+Figure 6. Our results compared to BEV [52] and GLAMR [64] on
+a scene with people of different sizes.
+that it also fails to recover the correct scale of the person,
+which can be observed from the top view. On the other
+hand, our approach can predict a 3D pose that corresponds
+to the ground truth human annotation and is coherently po-
+sitioned in 3D coordinates. We also compare our method
+with GLAMR [64] and BEV [52] in Figure 6. GLAMR
+fails to track all the persons in the scene and BEV fails to
+predict coherent human positions. More qualitative com-
+parisons are in the supplementary video.
+Our method has the advantage of jointly estimating the
+humans and the scene point cloud, which can be further
+used to impose physical constrains in the estimated humans
+over time. The effect of these constraints can be visually
+seen in Figure 7, where we show a sequence of a person
+standing on the floor. In the top row, where no physical
+w/o physics term
+with physics term
+frame t
+frame t+1
+Figure 7. The effect of the physical constrains imposed by the es-
+timated geometry in our predictions. The results without Econtact
+and Eslip (top) contain more foot sliding artifacts than our results
+with physical constrains (bottom).
+constraints were applied, we can observe that the right foot
+oscillates drastically from one frame to another. When the
+physical constraints are applied (the bottom row), this arti-
+fact is drastically reduced, and the right foot stays still in
+contact with the ground.
+Since our method does not require any specific train-
+ing procedure and rely on multiple predictions from models
+trained on a large corpus of data, our approach automati-
+cally generalizes well for in-the-wild and Internet videos, as
+can be seen in Figure 5 and can be directly used to drive vir-
+tual characters from monocular RGB videos; see Figure 8.
+11
+
+XXInput image
+Retarged character
+Figure 8. Our method can be directly used to drive virtual charac-
+ters or animate avatars in augmented reality applications (bottom
+row) from monocular RGB videos. Note the correct character or-
+der along the depth channel. Thanks to our physical plausibility
+constraints, barely any foot-floor penetrations or foot sliding are
+observed in the animations; see the video.
+5. Discussion
+Our method achieves low reconstruction errors, because
+it can successfully leverage multi-modal inputs to disam-
+biguate the relative depths between humans and human
+scales better than previous works. Moreover, our results
+evince significantly less jitter and foot-floor penetrations
+than the evaluated baselines for multi-human 3D pose esti-
+mation and the ablative study confirms that all components
+of the method contribute to the final accuracy. We have
+demonstrated that the recovered 3D human motions can be
+applied for virtual character animation, as one potential ap-
+plication among the many others.
+Limitations and Possible Extensions.
+Although our
+method outperforms competing methods and makes a step
+forward in monocular multi-human 3D motion capture, it
+has several limitations caused by the severe ill-posedness of
+our monocular setting. All these limitations open possibili-
+ties for future extensions and follow-up works as described
+in the following.
+First, our approach relies on multiple inputs from pre-
+trained models (depth maps and 2D body joints) and, there-
+fore, could also be negatively affected by the output of
+those methods; for example if the estimated depth maps
+contain significant artefacts (e.g., when obtained on our-of-
+distribution environments). On the other hand, this implies
+that the performance of our approach has the potential to
+keep increasing in the future with the progress in related
+fields (cf. Table 5).
+Our method also requires that people are entirely visi-
+ble in most of the frames and move in the scene. Other-
+wise, the setting becomes degenerate, and we do not get
+enough cues for accurate reconstruction. Even though we
+mitigate artefacts that appear as violations of physical laws
+by geometric terms, some minor ones of this type remain.
+Further improvements can be attained by methods explic-
+itly modelling physical laws as in single-human 3D motion
+capture [47,48,60].
+Moreover, while the static camera assumption is practi-
+cal, it is also very challenging, and a moving camera could
+provide additional 3D reconstruction cues. Finally, the pro-
+posed approach is an optimization method that can effi-
+ciently process an entire video sequence and extract rele-
+vant information about the scene from all frames globally.
+However, due to this characteristic, the method in its current
+version does not allow real-time applications.
+6. Conclusion
+We present a new holistic approach for multi-human 3D
+motion capture from a single static monocular RGB cam-
+era. Our core statement—that the synergy between multi-
+modal inputs and priors can significantly boost the 3D re-
+construction accuracy in this challenging setting—is con-
+firmed by extensive experiments in which we set a new state
+of the art on commonly used benchmarks. Moreover, as
+expected, we confirm that the constraints from the scene
+point clouds steadily boost the accuracy of the final 3D
+poses.
+Qualitatively, our reconstructions evince substan-
+tially fewer artefacts (such as jitter and foot-floor penetra-
+tions), enabling exciting downstream applications such as
+motion re-targeting for virtual characters. We believe that
+the proposed holistic approach for multi-human 3D motion
+capture can be extended in many useful ways, and we will
+be excited to see follow-ups.
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+

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@@ -0,0 +1,886 @@
+Graph Laplacian for Semi-Supervised Learning
+Or Streicher and Guy Gilboa
+Technion - Israel Institute of Technology, Haifa 3200003, Israel
+[email protected], [email protected]
+Abstract. Semi-supervised learning is highly useful in common scenar-
+ios where labeled data is scarce but unlabeled data is abundant. The
+graph (or nonlocal) Laplacian is a fundamental smoothing operator for
+solving various learning tasks. For unsupervised clustering, a spectral em-
+bedding is often used, based on graph-Laplacian eigenvectors. For semi-
+supervised problems, the common approach is to solve a constrained
+optimization problem, regularized by a Dirichlet energy, based on the
+graph-Laplacian. However, as supervision decreases, Dirichlet optimiza-
+tion becomes suboptimal. We therefore would like to obtain a smooth
+transition between unsupervised clustering and low-supervised graph-
+based classification.
+In this paper, we propose a new type of graph-Laplacian which is adapted
+for Semi-Supervised Learning (SSL) problems. It is based on both density
+and contrastive measures and allows the encoding of the labeled data di-
+rectly in the operator. Thus, we can perform successfully semi-supervised
+learning using spectral clustering. The benefits of our approach are illus-
+trated for several SSL problems.
+Keywords: Graph Representation · Semi-Supervise Learning · Nonlocal
+Laplacian · Spectral Clustering
+1
+Introduction
+Labeling information is a major challenge in modern learning techniques, which
+in many cases can be a long and expensive process. A possible solution to this
+problem is Semi-Supervised Learning (SSL). SSL methods can be thought of as
+the halfway between supervised and unsupervised learning. It uses large amounts
+of unlabeled data and a limited amount of labeled data, to improve the learn-
+ing model. SSL techniques are usually used when one cannot employ supervised
+learning algorithms. The use of supervised learning when limited labels are avail-
+able may result in a lack of generalization and overfitting. Those limited known
+labels, however, can significantly improve performance compared to unsuper-
+vised learning algorithms [11]. Intuitively, the purpose of SSL is to generalize
+the known labels to the unlabeled samples by an appropriate smoothing opera-
+tor. The graph-Laplacian has shown to be highly effective for this purpose.
+In this paper, we focus on Graph-based methods which are well-studied clas-
+sical techniques. We note that the insights presented here can be further used by
+deep learning methods with spectral-graph modules, as in [18], [6], [21], [1]. In
+arXiv:2301.04956v1  [cs.CV]  12 Jan 2023
+
+2
+Or Streicher, Guy Gilboa
+classical graph methods, a weighted graph is constructed based on the affinities
+between data instances. These affinities are usually computed using a metric
+of the features representing each instance. The vast majority of graph-based
+learning methods use the graph-Laplacian as the smoothing operator for gener-
+alization. More advanced nonlinear methods use p-Laplacian operators [8], [5].
+In this study we limit the scope to the linear case (or quadratic energy), not-
+ing that our proposed operators can be further generalized. Data processing
+based on the Laplacian has shown to be effective for a wide range of problems
+including clustering [4] [16], [22], classification [9], segmentation [19], dimension-
+ality reduction [2], [7], [17] and more. For the SSL setting, most graph-based
+learning techniques use the properties of the graph-Laplacian operator to define
+an optimization problem. The labeled information can be inserted as problem
+constraints, see e.g. [12], [3], [9], [14], [13], [20].
+In this paper, we propose a different approach to consider the labeled in-
+formation, by inserting it into the affinity measure that defines the connectivity
+between the nodes of the graph. We first examine the work of [20] on the weighted
+nonlocal Laplacian. In this work, the density of the labeled data is essentially
+increased. This improves the solution of the constrained optimization problem.
+We found that it has a marginal effect on the spectral embedding. Based on
+contrastive arguments, we propose to increase connections between labeled and
+unlabeled data and to increase (remove) connections between labeled data of
+the same (different) clusters. This yields a considerably improved spectral em-
+bedding.
+Our proposed method retains the following main qualities: 1) Interpolation
+between unsupervised and semi-supervised learning. The suggested ap-
+proach enables learning for a changing range of labeled information. 2) Low-
+label regime performance. The proposed method was found most advanta-
+geous in the low-label regime, compared to competitive techniques. 3) Different
+analysis tools. Wide variety of analysis tools can be used for solving SSL prob-
+lems, including spectral and functional analysis. We illustrate the advantages of
+using this new definition on toy examples and on real data sets.
+2
+Setting and Notation
+Let X = {xi}n
+i=1 be a finite set of instances in Rd. These instances are represented
+as nodes on an undirected weighted graph G = (V, E, W), where V is the vertices
+set, E is the edges set and W is the adjacency matrix. The adjacency matrix is
+symmetric and is usually defined by a distance measure between the nodes. For
+example, a common choice is a Gaussian kernel with Euclidean distance,
+Wij = exp
+�
+−||xi − xj||2
+2
+2σ2
+�
+,
+(1)
+where σ is a soft-threshold parameter.
+
+Graph Laplacian for Semi-Supervised Learning
+3
+The degree matrix D is a diagonal matrix where Dii is the degree of the i-th
+vertex, i.e.,
+Dii =
+�
+j
+Wij.
+(2)
+The graph-Laplacian operator is defined by,
+L := D − W.
+(3)
+The graph-Laplacian is a symmetric, positive semi-definite matrix, i.e., ∀f ∈
+Rn , f T Lf ≥ 0. For each vector f ∈ Rn it holds that
+Lf ∈ Rn , Lf(j) =
+n
+�
+i=1
+Wij(fj − fi),
+(4)
+f T Lf =
+n
+�
+i=1
+n
+�
+j=1
+Wij(fi − fj)2.
+(5)
+The eigenvalues of L are real and sorted in ascending order λ1 ≤ λ2 ≤ ... ≤ λn.
+The corresponding eigenvectors form an orthogonal basis, denoted by u1, u2..., un.
+The sample xi can be represented in the spectral embedding space as the ith
+row of the matrix U =
+�u1 · · · uK
+�
+∈ Rn×K, denoted as ϕi. More formally, the
+spectral embedding of an instance xi can be formulated as
+xi �−→ ϕi = [u1(i), u2(i), ..., uK(i)] ∈ RK,
+(6)
+where in most cases K ≪ d.
+For the SSL setting, let us define a discrete function f ∈ Rn over X and
+S ⊂ X, such that |S| = m, m ≤ n, be a subset of X on which the values of f
+are known, i.e., f(x) = g(x), ∀x ∈ S, for a given function g. The purpose of the
+SSL model is to find the values of f of all data-points in X constrained by the
+values of the set S.
+The main SSL problem this work focuses on is clustering. To evaluate the
+clustering performance we examined two common measures. The first one is
+Normalized mutual information (NMI) which is defined as,
+NMI(c, ˆc) =
+I(c, ˆc)
+max{H(c), H(ˆc)},
+(7)
+where I(c, ˆc) is the mutual information between the true labels c and the clus-
+tering result ˆc and H(·) denotes entropy. The second measure is Unsupervised
+Clustering Accuracy (ACC) which is defined as,
+ACC(c, ˆc) = 1
+n max
+π∈Π
+n
+�
+i=1
+1{ci = π(ˆci)},
+(8)
+where Π is the set of possible permutations of the clustering results. To choose
+the optimal permutation π we used the Kuhn-Munkres algorithm [15]. Both indi-
+cators are in the range [0, 1], where high values indicate a better correspondence
+between the clustering result and the true labels.
+
+4
+Or Streicher, Guy Gilboa
+3
+Graph-Laplacian for SSL
+3.1
+Motivation
+A common approach to solve SSL problems, based on the graph-Laplacian (GL),
+is to solve the Dirichlet problem
+min
+f
+1
+2
+�
+xi,xj∈X
+Wij(f(xi) − f(xj))2,
+(9)
+s.t. f(x) = g(x), x ∈ S.
+The solution admits
+�
+xj∈X
+Wij(f(xi) − f(xj)) = 0, xi ∈ X \ S
+(10)
+f(x) = g(x), x ∈ S.
+Note that Eq. (10) can be represented in matrix form. First, we define a con-
+straints vector b ∈ Rn and a mask M ∈ Rn×n such that
+bi =
+�
+g(xi)
+if xi ∈ S
+0
+if xi ∈ X \ S , Mij =
+�
+�
+�
+�
+�
+1
+Lii
+if xi ∈ S and i = j
+0
+if xi ∈ S and i ̸= j
+1
+if xi ∈ X \ S, ∀j
+,
+(11)
+where Lii is the i-th element of the diagonal of L. Now, Eq. (10) can be intro-
+duced in matrix from by
+(M ◦ L)f = b,
+(12)
+where ◦ denotes element-wise multiplication.
+A main problem with GL, as shown in [20], is that for a low sample rate
+of the labeled set, |S|/|X|, the solution is not continuous at the sample points.
+Thus the GL solution does not interpolate well the constraint values. In [20] the
+authors suggested solving the discontinuity problem by using Weighted Nonlocal
+Laplacian (WNLL), assigning a greater weight to the labeled set S compared to
+the unlabeled set X \ S. Formally, the optimization problem of WNLL is given
+by
+min
+f
+�
+xi∈X\S
+�
+xj∈X
+Wij(f(xi) − f(xj))2 + µ
+�
+xi∈S
+�
+xj∈X
+Wij(f(xi) − f(xj))2
+(13)
+s.t. f(x) = g(x), x ∈ S,
+where µ is a regularization parameter. It was suggested to set
+µ = |X|/|S|,
+(14)
+
+Graph Laplacian for Semi-Supervised Learning
+5
+the inverse of the sample rate. This can be interpreted as increasing the density
+(or measure) of the labeled instances. The solution of Eq. (13) is given by solving
+the following linear system,
+�
+xj∈X
+(Wij +Wji)(f(xi)−f(xj))+(µ−1)
+�
+xj∈S
+Wji(f(xi)−f(xj)) = 0, xi ∈ X \S
+(15)
+f(x) = g(x), x ∈ S.
+Similarly to Eq. (12), one can define Eq. (15) in matrix form. Let us introduce
+the linear system as follows,
+�
+xj∈X
+(Wij+Wji)(f(xi)−f(xj))+(µ−1)
+�
+xj∈X
+W labeled
+ij
+(f(xi)−f(xj)) = 0, xi ∈ X\S
+(16)
+f(x) = g(x), x ∈ S,
+such that,
+W labeled
+ij
+=
+�
+Wij
+xi ∈ X \ S, xj ∈ S or xi ∈ S, xj ∈ X \ S
+0
+otherwise
+(17)
+or equivalently,
+�
+xj∈X
+�
+Wij + Wji + (µ − 1)W labeled
+ij
+�
+(f(xi) − f(xj)) = 0, xi ∈ X \ S
+(18)
+f(x) = g(x), x ∈ S.
+Now we can define,
+�
+xj∈X
+[WW NLL]ji(f(xi) − f(xj)) = 0, xi ∈ X \ S
+(19)
+f(x) = g(x), x ∈ S,
+where
+WW NLL = 2W + (µ − 1)W labeled.
+(20)
+Based on WW NLL one can define LW NLL, Eq. (3), such that Eq. (15) is equiv-
+alent to
+(M ◦ LW NLL)f = b.
+(21)
+Inspired by Eq. (20), we would like to define an affinity matrix that takes
+into account the known labels, such that it also distinguishes between labeled
+samples from the same and from different clusters.
+
+6
+Or Streicher, Guy Gilboa
+3.2
+Semi-Supervised Laplacian Definition
+The classical definition of the graph-Laplacian, Eq. (3), is based on the data fea-
+tures in an unsupervised manner. In this section, we introduce a novel definition
+of the graph affinity matrix for SSL problems. That means, the affinity measure
+takes into account not only the feature vectors but also the known information
+about the labels of a subset S of V . The proposed definition is intended to im-
+prove performance for SSL clustering problems. For K clusters, we denote by Sk
+the set of labeled nodes belonging to the k-th cluster, such that S = ∪K
+k=1Sk.
+We suggest the following affinity measure
+WSSL = 2W + αW labeled,
+(22)
+where W is the known unsupervised affinity matrix, α is a scalar parameter, we
+set
+α = |X|/|S| − 1,
+(23)
+and W labeled is defined as follows,
+W labeled
+ij
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+max(W)
+xi, xj ∈ Sk , ∀k ∈ {1, ..K}
+− 2
+αWij
+xi ∈ Sk, xj ∈ Sl , ∀k, l ∈ {1, ..K|k ̸= l}
+Wij
+xi ∈ S, xj ∈ X \ S or xi ∈ X \ S, xj ∈ S
+0
+xi, xj ∈ X \ S
+.
+(24)
+It can be observed that according to this definition, the connection between
+labeled nodes belonging to the same cluster is given the highest weight (max(W)
+is the maximum value of the unsupervised affinity matrix). This strong affinity
+ensures the nodes are well connected inducing high smoothness of the spectral
+solution at these regions. Connections between labeled nodes of different clusters
+are disconnected. This increases the separation of these nodes, avoiding unnec-
+essary regularity between nodes belonging to separate clusters. In addition, as
+in Eq. (17), we reinforce edges between labeled nodes and unlabeled nodes. Now
+we can define the SSL graph-Laplacian as follows,
+LSSL = DSSL − WSSL,
+(25)
+where WSSL is defined in Eq. (22) and DSSL is its associated degree matrix (see
+Eq. (2)). In a similar manner to Eq. (12), one can solve the following problem
+(M ◦ LSSL)f = b.
+(26)
+4
+Analysis of LSSL
+In this section, we analyze the characteristics of LSSL for different scenarios.
+First, we analyze the influence of each of the components in Eq. (24) on the
+spectral embedding. Let us define the following affinity matrices,
+W 1
+ij =
+�
+max(W)
+xi, xj ∈ Sk , ∀k ∈ {1, ..K}
+0
+otherwise
+(27)
+
+Graph Laplacian for Semi-Supervised Learning
+7
+W 2
+ij =
+�
+− 2
+αWij
+xi ∈ Sk, xj ∈ Sl , ∀k, l ∈ {1, ..K|k ̸= l}
+0
+otherwise
+(28)
+W 3
+ij =
+�
+Wij
+xi ∈ S, xj ∈ X \ S or xi ∈ X \ S, xj ∈ S
+0
+otherwise
+(29)
+Note that W 1 and W 2 can be interpreted as Contrastive Affinities, following
+insights of contrastive learning [10]. The contrastive paradigm aims at creating
+an embedding where instances of the same cluster are very close (the role of
+W 1), whereas instances of different clusters are distinctly separated (the role of
+W 2). On the other hand, W 3 can be thought of as Density Affinity. Its purpose
+is to increase the density of the graph in the vicinity of labeled nodes.
+We would like to use those affinity matrices instead of W labeled in Eq. (22)
+and examine the resulting spectral representation based on the graph-Laplacians
+{Li
+SSL}3
+i=1 defined by {W i
+SSL}3
+i=1 such that
+W i
+SSL = 2W + αW i.
+(30)
+The spectral embedding is examined for the 3-Moons dataset containing 900
+nodes, of which 30 are labeled, as can be seen in Fig. 1.
+(a)
+(b)
+Fig. 1: SSL Laplacian Illustration dataset. The 3-Moons dataset containing
+900 nodes appears in Fig. 1a. The labeled nodes are shown in Fig. 1b.
+We examine the spectral representation of the data, Eq. (6), spanned by
+the first two non-trivial leading eigenvectors of the unsupervised Laplacian L,
+{Li
+SSL}3
+i=1 and LSSL. The spectral embedding for each case is shown in Fig. 2.
+We can observe that the spectral representation obtained by LSSL produces the
+clearest division into clusters. Nodes of the same cluster (”moon”) are grouped
+together, whereas nodes of different clusters are further apart. An interesting
+finding in this experiment is that the main effect on the spectral embedding is
+caused by the contrastive affinities, especially of W 1. We will see in the experi-
+mental part this trend is valid also for more complex data. Indeed, the Laplacian
+based on W3, which is equivalent to LW NLL, has virtually no contribution to
+the spectral embedding.
+
+8
+Or Streicher, Guy Gilboa
+(a) L
+(b) L1
+SSL
+(c) L2
+SSL
+(d) L3
+SSL
+(e) LSSL
+Fig. 2: Spectral Embedding Illustration. The spectral embedding obtained
+on the 3-Moons dataset for L, {Li
+SSL}3
+i=1 and LSSL.
+Next, we would like to examine spectral processing compared to constrained
+optimization, both based on LSSL. The performance of both approaches is tested
+over the 2-Moons dataset, which includes 1000 instances. Each node of the graph
+is defined by its Euclidean position. We analyze the spectral properties of the
+graph and the solution to the Dirichlet interpolation problem. The results are
+examined for different labeled sets S. In the spectral case, we find for each node of
+the graph its corresponding value according to the first non-trivial eigenvector
+of the graph-Laplacian, defined using WW NLL or WSSL. In order to find the
+solution for the Dirichlet problem, we set the value 1 over the labeled nodes of
+the first moon and −1 over the labeled nodes of the second moon. Then we find
+the solution to the interpolation problem using L, LW NLL or LSSL. To make a
+division into clusters, we perform K-Means over the resulting solution for each
+case. The obtained results are shown in Fig. 3.
+Analyzing the results, it can be observed that when the amount of labeled
+samples is extremely small, the Dirichlet problem may not generalize that well
+the labels to the unlabeled data. This is especially true when the labels are not
+located near the cluster centers, as can be seen for the sets S2 and S3 (Fig. 3
+bottom two rows). For the Dirichlet problem, the same performance is achieved
+for LW NLL and for LSSL. This is also valid in larger data sets. In more complex
+scenarios the advantages of using the above Laplacians are clear, compared to
+standard L. In this toy example, the differences are minor.
+Intermediate conclusions for these toy examples are that in some cases spec-
+tral analysis of the data is preferred. In addition, the suggested definition of
+LSSL allows us to get good performance for SSL problems both in the spectral
+case and for solving the Dirichlet problem. The reason for this is that LSSL
+includes the contrastive information, which is essential mainly for the spectral
+case, and the density information which is more significant in the optimization
+case. We will now test this in a more comprehensive manner.
+
+0.04
+0.02
+0.00
+-0.02
+-0.04
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+U10.02
+0.00
+-0.02
+-0.04
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+u10.04
+0.02
+0.00
+-0.02
+-0.04
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+U10.04
+0.02
+0.00
+-0.02
+-0.04
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+U10.03
+0.02
+0.01
+0.00
+-0.02
+-0.03
+-0.04
+-0.05
+-0.04
+-0.02
+0.00
+0.02
+0.04
+uiGraph Laplacian for Semi-Supervised Learning
+9
+S
+Spectral
+Spectral
+Dirichlet
+Dirichlet
+Dirichlet
+LW NLL
+LSSL
+L
+LW NLL
+LSSL
+S0
+(0.42,0.80)
+(0.97,1.00)
+(0.94,0.99)
+(0.96,0.99)
+(0.96,0.99)
+S1
+(0.43,0.81)
+(0.96,1.00)
+(0.96,1.00)
+(0.96,1.00)
+(0.96,1.00)
+S2
+(0.42,0.81)
+(0.42,0.81)
+(0.37,0.77)
+(0.37,0.77)
+(0.37,0.77)
+S3
+(0.43,0.81)
+(0.43,0.81)
+(0.24,0.76)
+(0.24,0.76)
+(0.24,0.76)
+Fig. 3: SSL solutions of the 2 Moons dataset. The first column shows different
+labeled sets S. The 2nd and 3rd columns show the spectral clustering result for LW NLL
+and LSSL, respectively. The 4th, 5th and 6th columns show the Dirichlet problem
+result for L, LW NLL and LSSL, respectively. The clustering measures (NMI, ACC) are
+presented below each figure. Spectral LSSL performs well in all configurations.
+5
+Experimental SSL Clustering Results
+In this section, we examine the performance of the different definitions of the
+graph-Laplacian for the clustering problem. To perform clustering in the semi-
+supervised case, we examine two different methods. The first one is Spectral
+Clustering which is based on the division of the data into clusters by performing
+K-Means over the spectral embedding of the data, Eq. (6). The second method
+is based on Dirichlet-form Clustering. To adapt the Dirichlet interpolation
+problem to multiple clusters, we use the algorithm suggested in [20].
+5.1
+2-Moons Clustering
+In this section, we present the statistical clustering performance for the 2-Moon
+dataset when the graph includes 500 nodes. We perform two experiments. In
+the first one, shown in Fig. 4, we examine the effect of changing the standard
+deviation of the noise of the data (that is, the deviation of each point from the
+position on the semicircle that defines the moon). In this case, 10 labeled nodes
+from each class are randomly defined. In the second experiment, we examine
+the effect of changing the size of the labeled set |S| (for fixed noise standard
+deviation set to 0.1). The results of this experiment are summarized in Fig. 5. In
+both experiments, white Gaussian noise is used. The experiments show statistics
+of 100 trials, where a bold line represents the mean value (of NMI or ACC) and
+the lighter regions around each line depict the standard deviation of the measure.
+
+10
+Or Streicher, Guy Gilboa
+The main conclusions from those experiments are that in the spectral case
+the results obtained for LSSL are much better compared to the other Lapla-
+cians, where LW NLL performance degenerates to the unsupervised case. For the
+Dirichlet problem, the performance of LSSL and LW NLL is similar and better
+than using the standard L, especially for the difficult cases, where the noise is
+significant and the amount of labeled information is small. We can conclude that
+the definition of LSSL allows to get the best clustering performance when using
+the Dirichlet problem and especially for spectral analysis of the graph.
+(a)
+(b)
+(c)
+(d)
+Fig. 4: 2 Moons clustering for different noise parameter. NMI and ACC
+measures over 100 different labeled set samples for different noise standard de-
+viation. Figs. 4a-4b are for Spectral Clustering. Figs. 4c-4d are for Dirichlet
+Clustering.
+(a)
+(b)
+(c)
+(d)
+Fig. 5: 2 Moons clustering for different labeled set size. NMI and ACC
+measures over 100 different labeled set samples for different labeled set sizes.
+Figs. 5a-5b are for Spectral Clustering. Figs. 5c-5d are for Dirichlet Clustering.
+5.2
+MNIST and F-MNIST
+Now we examine the clustering performance over the MNIST and Fashion-
+MNIST datasets. Both of these well-known datasets include 28 × 28 gray-scale
+images. For each dataset, we define a graph using the test set which includes
+10, 000 images. The obtained clustering performance, for different size of labeled
+sets, is shown in Fig. 6.
+
+1.0
+0.9
+0.8
+0.7
+NMI
+0.6
+0.5
+L
+0.4
+LWNLL
+0.3
+LsSL
+0.06
+0.08
+0.10
+0.12
+0.14
+Noise std.1.00
+0.95
+? 0.90
+AC
+0.85
+L
+LWNLL
+0.80
+LsSL
+0.06
+0.08
+0.10
+0.12
+0.14
+Noise std.1.0
+0.9
+NMI
+0.8
+0.7
+L
+LWNLL
+0.6
+LsSL
+0.06
+0.08
+0.10
+0.12
+0.14
+Noise std.1.00
+0.98
+0.96
+ACC
+0.94
+0.92
+L
+LWNLL
+0.90
+LsSL
+0.06
+0.08
+0.10
+0.12
+0.14
+Noise std.1.0
+0.9
+0.8
+0.7
+LWNLL
+NMI
+0.6
+LsSL
+0.5
+0.4
+0.3
+0
+20
+40
+60
+80
+100
+ISI1.00
+0.95
+0.90
+LWNLL
+AC
+LSSL
+0.85
+0.80
+0
+20
+40
+60
+80
+100
+[S1.0
+0.9
+0.8
+0.6
+LWNLL
+0.5
+LsSL
+20
+40
+60
+80
+100
+[S]1.000
+0.975
+0.950
+0.925
+cC
+0.900
+0.875
+7
+0.850
+LWNLL
+0.825
+LsSL
+20
+40
+60
+80
+100
+ISIGraph Laplacian for Semi-Supervised Learning
+11
+Fig. 6: MNIST and F-MIST Clustering performance. The mean and stan-
+dard deviation of NMI and ACC of 10 different experiments over the 10,000
+samples of MNIST (Top row) and F-MNIST (bottom row) test sets, for different
+size of a labeled subset |S|.
+It can be observed that using LSSL yields better performance, both for solv-
+ing the Dirichlet problem and for spectral analysis of the graph. In addition, for
+small labeled set |S|, the performance obtained for spectral clustering is better.
+6
+Conclusions
+In this paper, we propose a new definition for the graph-Laplacian designed
+to improve performance for SSL problems. The novel SSL Laplacian, which
+incorporates both contrastive and density affinities, yields improved spectral
+clustering and can be used also in constrained optimization problems. The pro-
+posed operator allows smooth interpolating between the unsupervised and the
+semi-supervised cases. The advantages are most prominent for an extremely low
+amount of labels or noisy data. In this work we have considered only the linear
+case, however, p-Laplacians may also be modified in a similar manner.
+References
+1. Aviles-Rivero, A.I., Sellars, P., Schönlieb, C.B., Papadakis, N.: Graphxcovid: ex-
+plainable deep graph diffusion pseudo-labelling for identifying covid-19 on chest
+x-rays. Pattern Recognition 122, 108274 (2022) 1
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+Dirichlet(L)
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+Dirichlet(LssL)
+Spectral(L)
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+Spectral(LssL)
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+CC
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+Dirichlet(LssL)
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+Spectral(L)
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+Spectral(LssL)
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+

YNE4T4oBgHgl3EQfNwyH/content/tmp_files/2301.04959v1.pdf.txt

@@ -0,0 +1,1367 @@
+Under consideration for publication in J. Fluid Mech.
+1
+Banner appropriate to article type will appear here in typeset article
+Spatial evolution of the turbulent/turbulent
+interface geometry in a cylinder wake
+Jiangang Chen1 and Oliver R. H. Buxton1†
+1Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
+(Received xx; revised xx; accepted xx)
+This study aims to examine the spatial evolution of the geometrical features of the turbu-
+lent/turbulent interface (TTI) in a cylinder wake. The wake is exposed to various turbulent
+backgrounds in which the turbulence intensity and the integral length scale are independently
+varied and comparisons to a turbulent/non-turbulent interface (TNTI) are drawn. The
+turbulent wake was marked with a high-Schmidt-number (𝑆𝑐) scalar and a planar laser
+induced fluorescence (PLIF) experiment was carried out to capture the interface between the
+wake and the ambient flow from 𝑥/𝑑 = 5 to 40 where 𝑥 is the streamwise coordinate from
+the centre of the cylinder and 𝑑 is the cylinder’s diameter. It is found that the TTI generally
+spreads faster toward the ambient flow than the TNTI. A transition region of the interfaces’
+spreading is found at 𝑥/𝑑 ≈ 15, after which the interfaces propagate at a slower rate than
+previously (upstream) and the mean interface positions of both TNTI and TTI scale with the
+local wake half-width. The location of both the TNTI and TTI have non-Gaussian probability
+density functions (PDFs) in the near wake because of the influence of the large-scale coherent
+motions present within the flow. Further downstream, after the large-scale coherent motions
+have dissipated, the TNTI position PDF does become Gaussian. For the first time we explore
+the spatial variation of the “roughness” of the TTI, quantified via the fractal dimension,
+from near field to far field. The length scale in the background flow has a profound effect
+on the TTI fractal dimension in the near wake, whilst the turbulence intensity only becomes
+important for the fractal dimension farther downstream.
+1. Introduction
+All turbulent flows embedded within a non-turbulent background are observed to spread out
+into their environment. The spreading of turbulence into previously irrotational fluid depends,
+in the first instance, on viscous diffusion of vorticity across a well-defined thin layer which
+bounds the turbulent region and separates it from the outer, non-turbulent regions (Townsend
+1976). This convoluted thin layer, usually referred to as a turbulent/non-turbulent interface
+(TNTI), was first examined in detail by Corrsin & Kistler (1955), and extensive studies on
+the dynamical and geometrical features of TNTIs in various turbulent shear flows have been
+conducted ever since (see the review of da Silva et al. 2014). However, numerous situations for
+turbulent industrial and environmental flows have a turbulent background; a typical example
+of which is the wake of a wind turbine developing in the atmospheric turbulent boundary layer
+or the turbulent wake of other upstream wind turbines (e.g. Porté-Agel et al. 2020). In contrast
+† Email address for correspondence: [email protected]
+Abstract must not spill onto p.2
+arXiv:2301.04959v1  [physics.flu-dyn]  12 Jan 2023
+
+2
+to the extensive studies of TNTIs, our knowledge of the interface between flow regions with
+different levels of turbulence intensity, hereinafter referred to as a turbulent/turbulent interface
+(TTI), remains limited, notwithstanding its prevalence in the physical world. In the a recent
+study of Kankanwadi & Buxton (2020) the entrainment across a TTI between a turbulent
+cylinder-wake and a grid-generated turbulent background was experimentally examined. The
+cylinder’s wake was marked with a fluorescent dye of high Schmidt number (𝑆𝑐) such that
+molecular diffusion occurred at a vanishingly small length scale. By examining the velocity
+field in the vicinity of the scalar-marked interface it was revealed that a clear interface
+existed between the wake and the turbulent ambient fluid, independently of the artificially-
+introduced scalar. In particular, a jump in vorticity magnitude over a short distance was
+reported, resembling the vorticity jump across a TNTI. Both the intensity and the integral
+length scale of the background turbulence were independently varied and it was shown that
+in this far-wake region the turbulence intensity was the important parameter in determining
+the geometry of the TTI, characterised by its tortuosity and fractal dimension.
+In their subsequent study of the flow physics governing the behaviour of the TTI, namely
+consideration of the various terms of the enstrophy transport equation, Kankanwadi & Buxton
+(2022b) found the magnitude of the viscous diffusion term is insignificant when compared
+to that of the inertial vorticity stretching term acting at the outermost boundary of the TTI.
+These results imply that viscous diffusion is of little importance to the entrainment process
+across a TTI which contrasts to the scenario of the TNTI in which viscous diffusion is the
+dominant process by which the irrotational fluid acquires vorticity in the so-called viscous
+superlayer (e.g. Corrsin & Kistler 1955; da Silva et al. 2014). Kankanwadi & Buxton (2022b)
+also demonstrated that the vorticity in the vicinity of the TTI is “organised” in such a way
+on the wake side of the TTI that it exploits the enhanced strain rates in the interface-normal
+direction, previously reported for TNTIs (e.g. Buxton et al. 2019; Cimarelli et al. 2015),
+thereby enhancing vorticity stretching/enstrophy production and yielding the enstrophy jump
+across the TTI.
+In spite of these dynamical differences between the TTI and TNTI, their geometries
+both display a common hierarchy of self-similar structures which can be described through
+fractal analysis. The fractal nature of the interface geometry, which renders a much larger
+surface area of the interface than otherwise, is essential to correctly modelling the turbulent
+entrainment rate (e.g. Sreenivasan et al. 1989; Zhou & Vassilicos 2017). Kohan & Gaskin
+(2022) investigated the effect of the background turbulence intensity on the geometry of the
+TTI of an axisymmetric jet and compared it with a TNTI. They found that the turbulence in
+the ambient flow can further stretch and corrugate the interface and thus result in a larger
+fractal dimension of the TTI than the TNTI in results that corroborated those of Kankanwadi
+& Buxton (2020). It is noted that their investigation was carried out in the far field of the jet
+(25 diameters downstream of the orifice) where the coherent motions of the jet have dwindled
+(Tennekes & Lumley 1972; Gordeyev & Thomas 2000). In such a situation, the turbulence
+intensity in the background flow is the dominant parameter in modifying the behaviour of
+the TTI, whilst the size of the energetic eddies in the background flow, characterized by the
+integral length scale, is of less relevance (e.g. Kankanwadi & Buxton 2020).
+However, when it comes to the flow region where the coherent motions prevail, the scenario
+is quite different. It has been reported that the entrainment becomes dominated by large-scale
+engulfment of background fluid under the influence of the coherent motions (e.g. Yule 1978;
+Bisset et al. 2002; Cimarelli & Boga 2021; Long et al. 2022). For TTIs, Kankanwadi &
+Buxton (2022a) observed that both the turbulence intensity and the integral length scale
+in the ambient flow correlate to enhanced entrainment in the presence of the large-scale
+coherent vortices in the near wake of a cylinder; a contrasting result to the far-field study
+in which background turbulence was observed to suppress entrainment rate (Kankanwadi &
+
+3
+Buxton 2020). By conducting a control experiment in which the large-scale coherent vortices
+in the wake (the von Kármán vortex street) were suppressed via the addition of a splitter plate
+they showed that the presence of freestream turbulence effectively enhances entrainment via
+engulfment but suppresses the small-scale “nibbling”. Kankanwadi & Buxton (2022a) also
+reported that the presence of freestream turbulence increases the locus of the wake’s large-
+scale coherent vortices (i.e. wake “meandering” with a larger amplitude), with the integral
+length scale of the background turbulence playing the most important role in determining
+this. Combined, these results highlight the important role that the presence of the large-scale
+coherent motions of the wake, and their interaction with any background turbulence present,
+play in modulating the properties of the TTI.
+Such observations raise several questions with regard to the spatial evolution of the
+properties of the TTI, as the coherent vortices degrade downstream: how will the PDF
+of the TTI (and also TNTI) position be affected by the coherent motions? Is there any
+possible scaling applicable to the position of the TTI as it evolves downstream with the
+coherent motions diminishing? If so, is the scaling of the TTI the same as that of the TNTI?
+In terms of the fractality of the TTI, is the local fractal dimension of the TTI under the effect
+of the coherent motions in the near field the same as that in the far field of the flow where the
+coherent motions decay? If not, which parameter in the background turbulence dominates
+the local fractal dimension of the TTI, the intensity level of the background turbulence or the
+size of the energetic eddies? Or is the fractal dimension dominated by different freestream
+turbulence parameters in different regions of the flow? We aim to answer all of these questions
+in the present study.
+In order to addresses these questions, we examined the wake of a circular cylinder in
+various turbulent freestreams, in which the turbulence intensity and integral length scales of
+the background turbulence were independently varied. A planar laser induced fluorescence
+(PLIF) experiment was conducted to capture the position of the interface between the wake
+and the freestream from 5 to 40 cylinder diameters downstream from the cylinder’s centre. In
+such a region of the flow the coherent vortices in the wake emanating from the shear layers
+shed from the cylinder experienced a significant decay (Matsumura & Antonia 1993; Chen
+et al. 2016), which allows us to investigate the streamwise evolution of both the TTI and
+TNTI position/geometry concerning the questions raised above. The paper is organized as
+follows. Section 2 describes the experimental details, and the visualisation of the flow and
+the methodology used to determine the interface position is presented in section 3. Major
+results are discussed in section 4 and we summarise and conclude the work in section 5.
+2. Experimental setup
+The experiments were conducted in the water flume of the hydrodynamics laboratory of the
+Aeronautics Department at Imperial College London. A cylinder with a diameter of 𝑑 = 0.01
+m is vertically mounted in the middle of the flume test section which has a dimension of
+9m in length and 0.6 m in cross section which was filled to a depth of 0.6 m. The incoming
+velocity of the flow is 𝑈1 = 0.38 m/s. The Reynolds number based on 𝑈1 and 𝑑 is about
+3800. Upstream of the cylinder, four different grids, including two regular and two fractal
+grids (see Kankanwadi & Buxton (2020) for details of the grids), are used to generate the
+background turbulence with various turbulence intensities and length scales.
+A planar laser induced fluorescence (PLIF) experiment was carried out to capture the
+boundary of the cylinder’s wake in the various background flows. A fluorescent dye,
+Rhodamine 6G, which can be treated as a passive scalar in the flow was utilized to demarcate
+the wake region of the cylinder from the background flow. The very high Schmidt number
+of the dye, approximately 2500 in water (Vanderwel & Tavoularis 2014), ensures that the
+
+4
+Figure 1: (a) Conceptual sketch of the experimental setup and (b) parameter space
+(𝑇𝐼, 𝐿12) of the background flow in the middle of the field of view at 𝑥/𝑑 = 20.
+molecular diffusion of the dye occurs over a negligibly short length scale with respect to the
+turbulent motions, so that dye acts as a near-perfect marker of the wake region, with a clear
+boundary. The dye was released into the wake from a hole in the rear surface of the cylinder
+with the aid of a micro-dosing pump (Bürkert 7615) working at a dosing frequency of 10
+Hz. A long elastic tube of 2 m was used in the routing of the dye from the pump to the hole
+on the cylinder so as to smooth out pulsations in the dye release.
+A high-speed Nd:YLF laser (Litron LDY304) with a wavelength of 527 nm was used
+to induce the fluorescence of the dye which emits light of wavelength around 560 nm.
+The fluorescence was captured by two cameras (Phantom V641 with a sensor resolution of
+2560 × 1600 px) which were arranged consecutively in the streamwise direction to form
+a field view of 14𝑑 × 43𝑑 with an overlap region of about 2.5𝑑. The spatial resolution of
+the measurement is about 0.1 mm per pixel. The upstream edge of the field of view is 1𝑑
+apart from the centre of the cylinder (figure 1a). A low-pass filter is placed in front of the
+camera lens in order to ignore any laser light noise in the PLIF image. Instantaneous images
+of the wake in a freestream without (a) and with (b) turbulence is displayed in figure 2. The
+acquisition frequency of the experiment is 100 Hz and 2000 images were captured for each
+measurement case.
+Following Kankanwadi & Buxton (2020), we employed turbulence intensity (𝑇𝐼 ≡
+√︁
+(𝑢2 + 𝑣2)/2/𝑈1 where 𝑢 and 𝑣 are velocity fluctuations in the 𝑥 and 𝑦 directions respectively)
+and integral length scale (𝐿12 ≡
+∫ 𝑟0
+0
+𝑅12(𝑟)𝑑𝑟 where 𝑅12(𝑟) is the correlation coefficient
+between 𝑢(𝑥, 𝑦) and 𝑢(𝑥, 𝑦 + 𝑟)) to characterize the various turbulent background flows. The
+distribution of the turbulence intensity and the length scale of the flow behind the grids
+has been documented in detail in Kankanwadi (2022) in the same facility and operating
+conditions. The cylinder is placed at various downstream distances from the various grids
+such that the parameter space (𝑇𝐼, 𝐿12) was explored as widely as possible in order to truly
+investigate the behaviour of the interface between the wake and the background flow with
+various “flavours” of turbulence. We conducted experiments for seven cases of (𝑇𝐼, 𝐿12) and
+the distribution of (𝑇𝐼, 𝐿12) at 𝑥/𝑑 = 20, i.e. the middle of the field of view, is shown in figure
+1b. We divided the seven cases into three groups (figure 1b) according to the magnitude of
+the turbulence intensity. Case 1a is the closest experimental approximation to a TNTI-case
+with no turbulence-generating grid mounted upstream of the cylinder. The remaining cases
+are TTI cases with turbulent backgrounds generated by the four different grids and with
+several different grid - cylinder spacings. In the following sections, each flow configuration
+case with different (𝑇𝐼, 𝐿12) is referred to with its corresponding denotation in figure 1b.
+Focus on Fluids articles must not exceed this page length
+
+(a)
+Grid
+Field of view
+(b)
+3.0
+Group 1
+1
+2.5
+Group 2
+I
+I
+I
+I
+I
+I
+Group 3
+I
+I
+I
+I
+I
+2.0
+2b
+I
+I
+I
+I
+I
+I
+I
+n
+L12/d 1.5
+I
+I
+I
+d
+14d
+I
+3a
+I
+I
+3b
+I
+I
+-
+1.0
+I
+I
+I
+I
+I
+1b
+I
+I
+0.5 -
+I
+2a
+1
+I
+1a
+I
+I
+1d
+43d
+0.0
+2
+4
+0
+6
+8
+10
+12
+14
+16
+TI(%)5
+Figure 2: Visualisation of the wake (a) without and (b) with turbulence present in the
+background flow.
+3. Visualisation and determination of the interface
+We start with a comparison of the visualisation of the wake of the cylinder in a background
+flow without (figure 2a) and with (figure 2b) turbulence, thereby featuring the distinction
+between a TTI and TNTI. First, in the near wake (say 𝑥/𝑑 ≲ 10), the large-scale vortices are
+more distinct in the case of the non-turbulent background, whilst the locus of the vortices’
+positions in the turbulent background extends to a further lateral distance from the wake
+centre-line (𝑦 = 0). This confirms the observation of Kankanwadi & Buxton (2022a) that the
+large-scale vortices of the near wake (identified via the velocity field, not the scalar field) in a
+turbulent background generally drift to further positions in the lateral (𝑦) direction than those
+in a non-turbulent background at the same 𝑥/𝑑 location. Second, the TTI is also characterized
+by a “rougher” boundary with the ambient fluid, at both large and small scales. Large-scale
+(intermittent) lumps of fluid from the wake are observed protruding into the ambient flow in
+the turbulent background case (say at 𝑥/𝑑 ≈ 26 and 33 in figure 2b), which is barely seen
+in the non-turbulent background case (figure 2a). It is also noted that there are more finer
+scale structures embedded into the TTI, which is likely a reflection of the interaction between
+the smaller scale eddies in the ambient turbulence and the interface. The resultant crinkled
+interface (see also figure 4 for different TTI cases examined) is later demonstrated to have a
+very different fractal dimension to the TNTI, quantifying our observation here that the TTI
+is “rougher” than the TNTI.
+Before proceeding to examine the properties of the interfaces, we need to detect their
+positions reliably. To account for the variation of the light intensity along the streamwise
+direction in the PLIF images due to mixing/out-of-plane transport (figure 2), the light intensity
+of each image at each 𝑥 position is first normalised by its time-averaged mean value at the
+
+(n)
+5
+y/d
+0
+-5 
+(q)
+5 
+y/d
+-5
+5
+10
+15
+20
+25
+30
+35
+40
+x/d6
+Figure 3: (a) Distribution of conditionally-averaged normalised light intensity ⟨𝜙∗⟩ and
+d⟨𝜙∗⟩/d𝜙∗
+𝑡ℎ with respect to the threshold 𝜙∗
+𝑡ℎ. (b) Detected contours using 𝜙∗
+𝑡ℎ = 0.3. (c)
+Interface lines determined by selecting the longest continuous contours on both sides of
+the wake.
+same 𝑥 position along the wake centre-line, i.e. 𝜙∗(𝑥, 𝑦, 𝑡) = 𝜙(𝑥, 𝑦, 𝑡)/𝜙(𝑥, 𝑦 = 0) where the
+overbar denotes the average over time (images). The resultant normalised images enable a
+single threshold value to be set for the entire field of view for interface identification purposes
+(see figures 3b & c). In order to determine this threshold we follow the method used in Prasad
+& Sreenivasan (1989) who also used PLIF to distinguish the wake from the ambient flow in
+a similar experimental configuration to our present study. Specifically, for each experimental
+case (i.e. each data point in figure 1b), a conditional average was taken on the normalised
+light intensity 𝜙∗(𝑥, 𝑦, 𝑡) exceeding the given threshold value 𝜙∗
+𝑡ℎ which reads
+⟨𝜙∗⟩ ≡
+�(𝜙∗|𝜙∗ > 𝜙∗
+𝑡ℎ)
+𝑁(𝜙∗ > 𝜙∗
+𝑡ℎ)
+.
+(3.1)
+The distribution of ⟨𝜙∗⟩ with respect to 𝜙∗
+𝑡ℎ for the wake with a non-turbulent background is
+shown in figure 3a. As expected, ⟨𝜙∗⟩ increases rapidly for small values of 𝜙∗
+𝑡ℎ, but there is
+a knee point of ⟨𝜙∗⟩ with respect to 𝜙∗
+𝑡ℎ. This corresponds to the value of the light intensity
+that well demarcates the limit between the background level of ⟨𝜙∗⟩ and that in the wake.
+The gradient d⟨𝜙∗⟩/d𝜙∗
+𝑡ℎ is also plotted in figure 3a and a threshold value of 𝜙∗
+𝑡ℎ = 0.3 was
+determined with the aid of a linear curve fitting on either side of the knee point. We applied
+this value to a number of sample images and it gives a good indication of the position of
+the interface in the flow. A typical example of the detected interface is given in figure 3b.
+One may note that small occasional patches inside and outside the wake, which result from
+detrainment, three-dimensional “teacup handle” topology, or engulfment (Westerweel et al.
+2009) are also identified. Note that these over-captured patches are disconnected from the
+continuous interface we are seeking, so we chose the two longest continuous isocontours
+corresponding to the threshold criteria and finally we obtain the interfaces on both sides of
+the wake (see figure 3c). Typical interface isocontours of all the TTI cases determined using
+the same method with case-dependent threshold value are displayed in figure 4 which all
+exhibit well-defined interfaces between the wake and the ambient fluids. Comparison of the
+various figures also beautifully highlights the dependency of the TTI geometry on both 𝑇𝐼
+and 𝐿12 of the background turbulence, with clear visual differences across the various cases
+examined.
+
+(a) 1.6
+4.0
+3.5
+1.4
+y/d o
+8
+3.0
+*= 0.3
+-2 
+1.2
+-4
+2.5
+-6 
+<*Φ)p
+2.0
+(c)
+6
+1.5
+4 
+0.8
+2
+1.0
+y/d o
+0.6
+-2
+ 0.5
+-4
+-6 
+0.4
+F0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+5
+10
+15
+20 
+25
+30
+35
+40
+x/d7
+Figure 4: Typical interface of all TTI cases: (a) case 1b, (b) case 1c, (c) case 2a, (d) case
+2b, (e) case 3a, (f) case 3b.
+4. Results and discussion
+4.1. PDFs of TTI and TNTI position
+After the interface position was determined, the analysis proceeds first with the downstream
+evolution of the PDFs of both TNTI and TTI position which are examined at five different
+streamwise locations from very near to far away from the cylinder, i.e. 𝑥/𝑑 = 5, 10, 20, 30,
+and 40 (figure 5). For presentational clarity, one typical case is displayed for each of the three
+groups of figure 1b: figure 5 (a, b) are plots of case 1a (the TNTI case), and figure 5 (c, d)
+and (e, f) are cases 2a and 3a respectively (TTI cases). Both of the upper (𝑦 > 0 in figure 3c)
+and lower (𝑦 < 0) interface lines are used in the calculation of the PDF, so a negative value
+of 𝑦/𝑑 in figure 5 means the occurrence of a 𝑦 > 0 (or 𝑦 < 0) interface on the 𝑦 < 0 (or
+𝑦 > 0) side at the examined 𝑥/𝑑 position. The PDF at a particular 𝑥/𝑑 position was calculated
+within a streamwise strip of extent 3𝑑 centred on 𝑥𝑐 as denoted in the figure. The 3𝑑 extent
+of these strips is comparable to the largest integral length scale within the background flow
+(see figure 1b), and enabled better statistical convergence when computing the PDFs.
+For the examined TTI cases (figure 5c, e), the modal peak of the PDF, i.e. the most probable
+position of the interface which is very close to the mean position of the interface, is roughly
+at the same position as that of the TNTI case (figure 5a) at 𝑥/𝑑 = 5 (marked by the left
+dashed-line). However, at 𝑥/𝑑 = 40 (marked by the right dashed-line), the position 𝑦/𝑑 of
+the modal location of the TTI is larger than that of the TNTI, especially when the background
+turbulence intensity is high (figure 5e). Kankanwadi & Buxton (2022a) showed that in the
+near-wake region 𝑥/𝑑 ⩽ 5) the wakes exposed to background turbulence were always wider
+on average than the wake embedded in a non-turbulent background. Our results show that in
+the near wake the modal position of the TTI is similar to the TNTI and the reason that the
+mean wakes are wider for the TTI cases is because of the diminshed contribution from the
+left tails of the PDFs (e.g. PDFs for 𝑥𝑐 = 5 in figure 5a, c, e), i.e. there are fewer instances
+of the TTI crossing the centre-line than the TNTI. This observation is consistent with the
+finding in Kankanwadi & Buxton (2022a) that the mean position of the centres of the von
+Kármán vortices for the TTI cases were further away from the wake centre-line than those
+of the TNTI case at the same streamwise position. Further, our results show that the increase
+
+(a) 5
+(d) 5
+y/d
+5
+(b) 5 
+(e) 5
+y/d
+-5
+(c) 5
+(f) 5
+y/d
+5
+10
+15
+20
+25
+30
+35
+10
+15
+20
+25
+30
+35
+40
+x/d8
+Figure 5: Streamwise development of PDFs of both TNTI and TTI position. (a, b) TNTI
+case 1a, (c, d) TTI case 2a, (e, f) TTI case 3a.
+in wake width in the presence of background turbulence extends to the far wake, up to the
+40𝑑 position examined in the present study, with the intensity of the background turbulence
+seemingly the most important parameter in determining this enhanced wake width. Later we
+will see that this average enhancement of the wake width comes mainly from the contribution
+of the region closer to the cylinder which then persists downstream.
+It is noted that the TTI position PDFs for two cases with background turbulence (figure 5c,
+e) are not Gaussian, with a negative skewness (not shown) over all the examined 𝑥/𝑑 range;
+a similar observation was also made by Kohan & Gaskin (2022) for the TTI position of an
+axisymmetric jet. The PDF of the TNTI position is practically Gaussian at 𝑥/𝑑 = 40 (shown
+later in figure 8) which has been widely reported in previous literature in fully-developed
+regions of turbulent flows (e.g. Corrsin & Kistler 1955; da Silva et al. 2014; Mistry et al.
+2016; Zhou & Vassilicos 2017). However, the TNTI PDF evidently deviates from a Gaussian
+distribution at positions closer to the cylinder, especially at 𝑥/𝑑 = 5 and 10 (figure 5a) where
+heavier negative tails than for a Gaussian PDF are displayed. These distinctly heavy negative
+tails reflect the high probability of the interface appearing on the opposite side of the wake
+centreline (𝑦/𝑑 = 0), which is a manifestation of the strong large-scale “meandering” of the
+
+0.5
+1.4
+....
+(q)
+......
+Xc = 5
+Xc = 10
+1.2
+0.4
+C
+O—Xc=20
+1.0
+ Xc = 30
+a
+0.3
+L
+O
+0— Xc= 40
+.
+0.8
+:
+F
+Gaussian
+D
+0.6
+0.2
+P
+C
+O
+0.4
+0.1
+8
+0.2
+8
+0.0
+16
+0.0
+-4
+-2
+0
+4
+8
+2
+-1
+0
+2
+m
+0.6
+1.4
+(d)
+1.2
+n
+0.5
+1.0
+0.4
+a
+L
+.
+0.8
+F
+0.3
+:
+D
+0.6
+:
+................
+0.2
+P
+0.4
+0.1
+0.2 
+:
+0.0
+0.0
+..................
+中
+4
+-2
+0
+4
+6
+8
+-2
+-1
+0
+2
+m
+0.5
+1.4
+1.2
+0.4
+8
+1.0
+d
+0.3
+0.8
+F
+: HAa
+D
+0.6
+0.2
+P
+..............
+0.4
+....
+0.1
+0.2
+C
+0.0
+0.0
+F2
+6
+-4
+-2
+0
+4
+8
+-1
+0
+1
+2
+3
+y/d9
+Figure 6: Streamwise distribution of the mean interface position (a) 𝑦𝐼 /𝑑 and (b) 𝑦𝐼 /𝐿𝜙
+of all cases.
+near wake (see figure 2) because of the coherent vortices (e.g. Chen et al. 2016; Kankanwadi
+& Buxton 2022a).
+Zhou & Vassilicos (2017) found that the PDF of TNTI position in a turbulent, axisymmetric
+wake scales with the wake width in the self-preserving region. Such an observation is not
+made in the current study as shown in figure 5(b, e, f) where the PDFs of both TNTI figure
+(5b) and TTI (figures 5e, f) position are normalised with the wake half-width 𝐿 𝜙(𝑥) estimated
+from the mean profile of the light intensity 𝜙(𝑥, 𝑦) of the PLIF images at the corresponding
+𝑥 position (see Appendix A, where we also show that 𝐿 𝜙(𝑥) scales with the wake half-width
+based on the mean velocity field). The normalised PDFs of both TNTI and TTI position
+for all cases assessed do not collapse but exhibit an evident streamwise evolution. This is
+not unexpected as the PDFs of either TTI or TNTI position in the current flow region are,
+as discussed in the previous paragraph, heavily affected by the large-scale coherent vortices
+and are not self-similar, as manifested by the heavy negative tails at 𝑥/𝑑 = 5 and 10. What
+is interesting to see is that the most probable position of the PDFs of both TTI and TNTI
+position do scale approximately with the local 𝐿 𝜙, which provides a straightforward way to
+estimate the most probable position for both TNTI and TTI position, even though the PDFs
+are not self-similar.
+The coincidence of the modal peaks in figures 5(b,d,f) coupled to the Gaussian-like nature
+of the PDFs for the further downstream locations suggests that the mean position of the
+interface 𝑦𝐼 (𝑥) at different 𝑥/𝑑 positions may scale with the local wake-half width. This is
+confirmed in figure 6 for both the TNTI case and all the TTI cases. Figure 6a first compares
+the streamwise evolution of 𝑦𝐼 (𝑥) for both TNTI (case 1a) and TTI cases scaled with the
+cylinder diameter 𝑑. It is clear that all the TTI cases have a larger mean value of 𝑦𝐼 than
+the TNTI case at almost all 𝑥/𝑑 positions, which is consistent with the observation in figure
+5(a, c, e). It seems the turbulence intensity is the dominant parameter in determining the
+mean position of the TTI, as there is little evident distinction between the TTI cases within
+groups 1 and group 2 in which integral length scale is the major differentiating factor. What
+should be noted is that the mean interface position at a particular 𝑥/𝑑 location mainly reflects
+the mass entrainment accumulated upstream of 𝑥/𝑑, whilst the slope of the curve d𝑦𝐼/d𝑥
+demonstrates the local entrainment rate into the wake (Kankanwadi & Buxton 2022a). It is
+found that in figure 6a there is an apparent turning point of the slope of 𝑦𝐼 (𝑥) located at
+𝑥/𝑑 ≈ 15 after which 𝑦𝐼 (𝑥) grows noticeably more slowly than farther upstream, for both
+TNTI and TTI cases. It indicates that the entrainment rate upstream of 𝑥/𝑑 ≈ 15 is faster
+than after this position. It is also noticed that in the flow region 𝑥/𝑑 ≲ 15 the mean interface
+
+.0
+1.6
+case la
+case 1b
+1.4
+3.5.
+case lc
+case 2a
+1.2
+3.0
+case 2b
+case 3a
+1.0
+Yi
+2.5
+口
+case 3b
+Yi/d
+0.8
+2.0
+0.6
+1.5
+0.4
+1.0-
+0.2
+0.5
+0.0
+5
+10
+15 
+2025
+30
+35
+40
+15
+20
+25
+30
+35
+40
+x/d
+x/d10
+Figure 7: Streamwise distribution of the wake half-width 𝐿𝜙 with turbulent (cases 2b and
+3b) and non-turbulent (case 1a) background flow.
+position 𝑦𝐼 (𝑥) of the TTI cases grows almost linearly and at a faster rate than the TNTI case;
+a similar observation was made by Kankanwadi & Buxton (2022a) in the flow region very
+close to the cylinder (𝑥/𝑑 ⩽ 5). It is thus concluded that the turbulence in the background
+promotes spreading of the wake boundary mostly in the near wake region (say 𝑥/𝑑 < 15);
+It is interesting to see that the turning point at 𝑥/𝑑 ≈ 15 is almost the same for all cases
+tested, regardless of whether there is a TNTI or TTI. Although the physics underpinning
+the changes of the slope of 𝑦𝐼 (𝑥) are still unclear, it is surmised that this transition position
+may depends on the dynamics of the near-wake coherent vortices which has been reported to
+be important for the near-wake large-scale engulfment (Kankanwadi & Buxton 2022a) and
+decays significantly from 𝑥/𝑑 = 10 to 20 at a similar Reynolds number (e.g. Zhou et al. 2003;
+Chen et al. 2016, and also the visualization in figure 2 ). After this turning point, the growth
+of the wake would transition from large-scale engulfment-driven entrainment to small-scale
+nibbling-driven entrainment
+When the mean interface position is scaled by 𝐿 𝜙(figure 6b) all 𝑦𝐼/𝐿 𝜙 become approxi-
+mately constant after an initial development region (𝑥/𝑑 ≲ 15). This is consistent with the
+observation in figure 5 that the most probable position of 𝑦𝐼 scales with 𝐿 𝜙 which itself
+follows a power-law scaling while developing downstream (figure 7). Eames et al. (2011)
+developed a model that describes how a wake spreads in a highly turbulent flow. They pointed
+out that for two-dimensional bodies, the wake grows linearly with distance during the initial
+development region (Eames et al. (2011) called it ‘the ballistic regime’) until the wake width
+is comparable to the integral scale of the background turbulence, beyond which the wake
+width grows diffusively with a scaling of ∼ 𝑥1/2. Typical examples seeking a power-law
+scaling for 𝐿 𝜙 ∼ 𝑥 𝛼 are displayed in figure 7. After 𝑥/𝑑 ≈ 10, the scaling 𝐿 𝜙 ∼ 𝑥1/2 is
+indeed observed in almost all cases with a turbulent background with the scaling exponent
+varying between 0.48 ⩽ 𝛼 ⩽ 0.54, except for two cases (case 1c and 2a in figure 2 with
+a scaling exponent of 0.64 and 0.23 respectively). It is noted that for the non-turbulent
+background case (group 1a), the scaling exponent (0.35) is close to 1/3, rather than the value
+of (1/2) expected based on the self-similarity which is only achieved in the very far wake
+(say 𝑥/𝑑 = 200 in Ch. 4 of Tennekes & Lumley (1972)).
+We close the discussion of this section by a comparison between the centred PDF of 𝑦𝐼
+for all the examined TNTI and TTI cases (i.e. the PDF of (𝑦𝐼 − 𝑦𝐼)/𝜎𝐼 where 𝜎𝐼 is the
+standard deviation of 𝑦𝐼) and a standard Gaussian distribution (figure 8), so as to highlight
+the different extent to which the TNTI and TTI position PDFs deviate from Gaussianity as
+Rapids articles must not exceed this page length
+
+5.5
+case la
+5.0
+0.48
+case 2b
+4.5
+case 3b
+d
+0.48
+4.0
+d
+d
+3.5
+L
+0.35
+3.0
+d
+d
+a
+2.5
+2.0
+1.5
+1.0
+0
+10
+20
+30
+40
+50
+x/d11
+Figure 8: Comparison of centered PDFs of TNTI and all TTI cases at different 𝑥/𝑑
+positions. (a) 𝑥/𝑑 = 5, (b)10, (c)20, (d)40.
+𝑥/𝑑 increases. It is clear that very close to the cylinder at 𝑥/𝑑 = 5 (figure 8a), the PDFs of
+both TNTI (case 1a) and all TTI cases deviate from the Gaussian distribution significantly
+with evident negative skewness, as is also seen in figure 5. What is interesting is that as
+𝑥/𝑑 increases, the negative skewness of the TNTI position gradually reduces and the PDF
+becomes practically Gaussian at 𝑥/𝑑 = 40 (figure 8d), whilst the PDFs for all the TTI cases
+still deviate from Gaussianity, although the skewness does reduce. It is clear that for both
+TNTI and TTI cases, the dynamics in the near wake (figure 2) are very different from those
+farther downstream where the large-scale coherent vortices have largely dissipated and the
+turbulence is fully developed. The different dynamics in the near and relatively far wake is
+believed to lead to distinct geometrical features of the interfaces, which encourages us to
+investigate the fractal dimension of the interfaces, and their spatial evolution, in the next
+section.
+4.2. Fractal dimension of the TNTI and TTI
+As explained in the introduction, the multi-scale self-similar geometric features of the
+interface, either TNTI or TTI, can be described with fractal analysis, which was first
+demonstrated by Sreenivasan & Meneveau (1986). The length of a fractal “line” follows
+a power law with increased resolution 𝑟, viz.,
+𝐿𝐼 (𝑟) ∼ 𝑟1−𝐷
+(4.1)
+where 𝐷 is the fractal dimension and has been reported to be between 1.3 to 1.4 for a TNTI
+(e.g. Prasad & Sreenivasan 1989; de Silva et al. 2013; Abreu et al. 2022), while for the TTI
+the dimension is somewhat higher and an increasing function of the turbulence intensity
+in the ambient flow (Kankanwadi & Buxton 2020; Kohan & Gaskin 2022). However, these
+previous studies focus on the TTI in the fully-developed region of a turbulent flow, where
+Kankanwadi & Buxton (2020) demonstrated that the turbulent length scale in the ambient
+
+0.5
++
+case 1b
+X
+case lc
+0.4
+case 2a
+0.4
+X
+case 2b
+case 3a
+0.3
++
+case 3b
+0.3
+case la
+PDF
+0.2
+0.2
+0.1 -
+0.1
+0.0 +
+0.0t8
+-4
+0
+2
+4
+-4
+0
+2
+0.5
+0.5
+(c)
+(d)
+0.4
+0.4
+0.3
+0.3
+PDF
+0.2
+0.2
+0.1
+0.1
+0.0+BR
+0.0+
+-4
+0
+Z
+4
+-2
+0
+2
+4
+(y- )/oi
+(yl - y)/ol12
+Figure 9: (a) Filtered interface with different filter scales, (b) scaling of the length of the
+interface 𝐿𝐼 , and (c) fractal dimension of the interface obtained using different window
+widths. The vertical dashed-line indicates the window width used for calculating the local
+fractal dimension of the interface.
+flow has little effect on the fractal dimension of the interface. In the previous section, we have
+shown that the behavior of the interfaces are substantially influenced by the strong organized
+motions in the near wake. As we have measured multiple cases of TTIs with various levels
+of turbulent intensity and integral length scales in the background flow, it is of interest to
+examine the fractal dimension of these TTIs in the context of the streamwise decay of the
+coherent vortices.
+To obtain the fractal dimension of the interfaces, we adopt a ‘filtering method’ as used
+in previous studies (e.g. de Silva et al. 2013; Kankanwadi & Buxton 2020; Abreu et al.
+2022). Specifically, a box filter of scale Δ 𝑓 was first used to filter the interface lines obtained
+in section 3. Figure 9a displays an example of the TNTI lines after being filtered with
+different Δ 𝑓 . Note that there are two lines on both sides of the wake, whose length are
+calculated separately and both are included in the ensemble to calculate the mean length
+of the interface line corresponding to a particular Δ 𝑓 . Based on equation (4.1), log(𝐿𝐼)
+has a linear relationship with log(𝑟) when such a scaling applies and the slope of the line
+(i.e. 1 − 𝐷, referred to as the scaling exponent in the following text) is directly related to
+the fractal dimension. Figure 9b displays a distribution of the mean turbulent/non-turbulent
+interface length of all detected realisations with respect to different filter sizes. In the scale
+range between 0.2𝑑 (close to the Taylor microscale on the wake centreline at 𝑥/𝑑 = 20 of the
+TNTI case, estimated from Kankanwadi (2022)) and 1𝑑, a scale comparable to the integral
+length there is a strong linear fit between log(𝐿𝐼/𝑑) and log(Δ 𝑓 /𝑑) with a slope of the fitted
+
+(n)
+Af = 0.18d
+= 0.63d
+: = 1.06d
+Taylor scale α = 0.2d
+Cylinder diameter: 1d
+(q)
+0.1
+-0.20
+2.2
+case 3b
+case la
+-0.25
+2.1
+0.0
+Slope = -0.34
+:0
+-0.30 -
+-0.1
+(p/T)
+D
+-0.36±0.02
+-0.2
+-0.35 .
+log10(
+.8
+-0.40
+0.34
+.+..
+1.7
+-0.4
+-0.45 
+-0.5
+-0.50
+1.6
+-1.0
+-0.8
+-0.6
+-0.4 
+-0.2
+0.0
+0.2
+2
+4
+6
+8
+10
+12
+14
+16
+18
+log1o(Ag/d)
+window span (d)13
+Figure 10: Scaling of the interface length of the TNTI case using window width of 8𝑑 at
+different streamwise 𝑥/𝑑 positions.
+line of -0.34. This yields a fractal dimension of 𝐷 = 1.34 for the TNTI which agrees well
+with the expected value between 1.3 - 1.4.
+To compute the local fractal dimension of the interfaces at different 𝑥/𝑑 we must choose
+a “window” covering a finite length of the whole interface; the window span should be large
+enough to produce a good representation of the local interface’s fractality but small enough
+to ensure homogeneity over the streamwise extent of the window and yielding good spatial
+resolution for the fractal dimension’s distribution (with respect to 𝑥/𝑑). Figure 9c shows the
+distribution of the scaling exponent (1−𝐷), determined in the same way as exhibited in figure
+9b, with respect to different streamwise window extents. Two typical cases are examined with
+the window centre set at 𝑥/𝑑 = 20: the TNTI case 1a and the TTI case 3b which has the
+highest turbulence intensity in the ambient flow (figure 1b). The value 1 − 𝐷 for both cases
+shows a weak increasing trend as the window span grows; there is a narrow plateau between
+window spans of 7 − 11𝑑, displaying a reasonable value of -0.36 (e.g. Prasad & Sreenivasan
+1989). We therefore chose a window span of 8𝑑 corresponding to the beginning of the plateau
+in the following study for the best spatial resolution of the results.
+Figure 10 shows the scaling of the mean length of the filtered interface of the TNTI case
+with respect to the scale of the filter at various streamwise positions. In the figure, 𝑥𝑐/𝑑
+located between 4 to 40 is the centre position of the examination window with span of
+8𝑑. There is a well-defined scaling range between Δ 𝑓 /𝑑 = 0.2 and 1 for all the examined
+positions, although the scaling range is wider in the larger scale end for positions closer to
+the wake generator. It is interesting to see that the slope of the fitted line (= 1 − 𝐷) varies
+from -0.28 in the very near wake to an oft-reported -0.37 at 𝑥𝑐 = 40, indicating that there is
+indeed essential difference in the geometric features of the interface in the near wake and the
+fully-developed downstream positions. To explore the effect of the background turbulence
+on the fractal features of the interfaces, we summarized the streamwise distributions of the
+
+Af/d = 0.2
+=1
+1.8
+Slope = -0.37
+1.6
+log(L/d)
+1.4
+12
+10
+1.2
+Slope = -0.28
+1.0
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+log(△f/d)14
+Figure 11: Streamwise distribution of fractal dimensions of TNTI and all TTI cases. (a)
+Effect of turbulence intensity, and (b) effect of integral length scale.
+scaling exponent of all the measured cases in figure 11, in which the effect of the background
+turbulence intensity and length scale on the fractal dimension is respectively examined in
+figures 11a and 11b.
+In figure 11a, the streamwise distribution of the scaling exponent (1 − 𝐷) of cases 2a, 3a
+and 3b, which are TTI cases with relatively small integral scale and large turbulence intensity
+in the background flow (figure 1b), are compared with that of the TNTI case (case 1a). The
+TNTI case exhibits an approximately constant value around -0.36 in the region 𝑥/𝑑 ≳ 10;
+the three TTI cases have similar distributions of 1 − 𝐷 to the TNTI case before 𝑥/𝑑 ≃ 15
+which interestingly corresponds to the position where the wake spreading rate decreases
+evidently (figure 6a). After this 𝑥/𝑑 position, the scaling exponent of the TTI cases continues
+to increase in magnitude and reaches approximately −0.45 at 𝑥/𝑑 = 40. The larger TTI
+fractal dimension than that of the TNTI is also consistent with the observation of Kohan &
+Gaskin (2022) in an axisymmetric jet with a turbulent background. The growing (1 − 𝐷) of
+the TTIs relative to the TNTI in the far field of the wake indicates that the turbulence intensity
+in the background flow becomes gradually essential in determining the fractal dimension of
+the interface in the positions far from the wake generator. The increased fractal dimension
+in the far field of the wake can also be observed in the visualisation in figure 2b, in which
+the boundary of the wake becomes “rougher” (i.e. a larger fractal dimension) as the flow
+proceeds downstream, with intermittent lumps and also finer structures. These structures
+result from the interactions between the eddies in the background turbulence and those of
+the wake. In the region closer to the cylinder before 𝑥/𝑑 ≃ 15, the ambient turbulence does
+not differentiate the scaling exponent of the TTIs from the TNTI. It implies that in this flow
+region, which features the evolution of the strong von Kármán vortices, the fractal nature of
+the interface is mainly determined by the dynamics of the wake flow itself, at least when the
+scale of the energetic eddies in the ambient flow is not overpowering (as is the situation of
+cases cases 2a, 3a and 3b).
+In figure 11b, the TTI cases 1b, 1c which possess low turbulence intensity and increasing
+integral length scale in the background flow are compared with the TNTI case (case 1a); case
+2b which has a large integral length scale and also higher turbulence intensity is also added
+for comparison. In contrast to the similar distribution of different cases upstream of 𝑥/𝑑 ≃ 15
+in figure 11a, the scaling exponent distributions of the compared cases show evident scatter
+in the upstream region but gradually converge to a value ≈ −0.36 in the downstream flow.
+TTI cases 1b and 1c differ from the TNTI case 1a mainly in the integral length scale of the
+background flow, their distinctive 1 − 𝐷 distribution in the upstream region indicates that the
+
+(a)
+-0.20
+-0.20
+case la
+case la
+case 2a
+case 1b
+-0.25
+-0.25
+case 3a
+case lc
+case 3b
+case 2b
+-0.30
+-0.30
+D
+-0.35
+-0.35
+-0.40
+-0.40 
+-0.45
+-0.45
+-0.50
+-0.50
+10
+15
+20
+25
+30
+35
+40
+5
+10
+15
+20
+25
+30
+35
+40
+x/d
+x/d15
+integral scale of the background turbulence is of great importance to the fractal dimension
+of the interfaces in this region. Compared to the TNTI case, the TTI case 2b has both a
+higher turbulence intensity and a larger integral length scale (figure 1b), and its distribution
+is not significantly different from that of the TNTI case in both the upstream and downstream
+field. It seems that there is a compound effect of the background turbulence intensity and the
+integral length scale on the interface geometry. As a matter of fact, such a combined effect
+was reported by Kankanwadi & Buxton (2020, 2022a) in the same flow: in the upstream field
+both the turbulence intensity and integral length scale in the background flow act to enhance
+the entrainment rate into the wake whilst only the turbulence intensity of the background
+turbulence is important in suppressing entrainment in the downstream field.
+To summarize the discussion of figure 11, generally, both turbulence intensity and integral
+length scale in the background flow have an effective influence on the fractal dimension
+of the interface, and in different regions of the flow a different parameter is in dominance.
+In the near wake, the integral length scale is the more important parameter; as the flow
+develops downstream with the coherent vortices degrading substantially, the effect of the
+integral length scale weakens and the influence of the turbulence intensity gradually prevails.
+This observation is consistent with the conclusion obtained in the TTI entrainment studies
+of Kankanwadi & Buxton (2020, 2022a) that integral length scale is the more important
+parameter in the near wake which promotes the large-scale engulfment of the wake, whilst
+the turbulence intensity suppress the small-scale “nibbling” in the far field where the integral
+scale is of less relevance.
+5. Summary and Conclusions
+We examined the spatial evolution of the geometry of the interface of a turbulent cylinder
+wake from the near (𝑥/𝑑 = 5) to the relatively far field (𝑥/𝑑 = 40), in a turbulent background
+with various levels of turbulence intensity and integral length scale (figure 1b). A PLIF
+experiment was carried out to capture the interface between the wake and the turbulent
+background flow. Attention was paid to the streamwise evolution of the geometric properties
+of these TTIs and a TNTI reference case, including their PDFs, scaling and fractality, in the
+context of the large-scale vortices gradually diminishing in the wake.
+Compared to the conventional TNTI, the TTI spreads faster towards the ambient flow as
+the wake develops downstream, which is mainly due to the enhanced rate of entrainment in
+the near wake (Kankanwadi & Buxton 2022a). We find a transition region of the interface
+spreading outwards at 𝑥/𝑑 ≈ 15, after which the interfaces spread at an evidently reduced rate
+(figure 6a). It is conjectured that the different spreading rates before and after this transition
+region are associated with the dynamics of the large-scale coherent vortices which induce
+strong engulfment (Kankanwadi & Buxton 2022a, also visualization in figure 2) and decay
+rapidly from 𝑥/𝑑 = 10 to 20 at similar Reynolds numbers (e.g. Zhou et al. 2003; Chen et al.
+2016). After this region, the mean position of the interfaces, including both TNTI and all TTI
+cases, display a reasonable scaling with the wake half-width 𝐿 𝜙(figure 6b). 𝐿 𝜙 is found to
+agree well with Eames et al. (2011)’s theoretical downstream evolution scaling of (𝑥/𝑑)1/2 in
+a turbulent background. It is interesting to see that this transition region is roughly the same
+for both TNTI and TTI cases examined, suggesting that this transition region is robust and
+not dependent on the turbulence in the background flow, at least for the turbulence intensity
+and length scale range examined in the present study.
+It is noted that the PDFs of both TTI and TNTI position are not Gaussian in the near wake
+(especially for 𝑥/𝑑 ≲ 10 ) with evident negative skewness which reflects the “deep-diving”
+interface towards the wake central region due to the strong engulfment by the coherent
+vortices at these locations (figure 2). This observation is distinctly different from the oft-
+
+16
+reported Gaussian distribution of TNTI position at locations of fully-developed turbulence
+in the absence of dominant coherent motions (e.g. da Silva et al. 2014; Mistry et al. 2016;
+Zhou & Vassilicos 2017, and also the Gaussian PDF of TNTI position at 𝑥/𝑑 = 40 in figure
+8d). Note that the PDFs of TTI position still depart from Gaussianity with a slight negative
+skewness even at 𝑥/𝑑 = 40 (figure 8d), which confirms the observation of Kohan & Gaskin
+(2022) of the TTI in a fully-developed axisymmetric jet.
+Finally, we found that the fractal dimension of the TTIs in the near and relatively far wake
+are dictated by different parameters of the background turbulence. Turbulence intensity
+induces a higher fractal dimension of the interface in the far wake. It is highly likely
+to be resultant from the interaction between the ambient eddies and those of the wake
+near the interface, which can be partly observed from the evident intermittent small-scale
+structures on the interface of the wake in turbulence background in figure 2b. The effect
+of the integral length scale is more appreciable in the near wake region (figure 11b). As
+the strong large-scale vortices prevail in the near wake, it is reasonable to expect only the
+energetic eddies in the background flow with comparable length scale or turnover time would
+interact effectively with the coherent vortices in the wake, which would be the reason why
+background integral scale is important in the near wake. Such large-scale interactions in the
+near wake would not necessarily wrinkle the interface, as the small-scale interaction does
+in the far wake, explaining why cases with larger integral scale do not necessarily cause
+higher fractal dimension of the interface (figure 11b). Such large-scale interaction would be
+expected to cause large-scale oscillation or meandering of the wake, however which has been
+demonstrated by Kankanwadi & Buxton (2022a).
+Acknolegement The authors would like to acknowledge the Engineering and Physical
+Sciences Research Council for funding the work under grant no. EP/V006436/1
+Appendix A. Determination of 𝐿 𝜙
+This appendix is added to show how the wake half-width 𝐿 𝜙 of the scalar field is determined
+based on the PLIF measurements and its connection with the velocity wake half-width 𝐿𝑢.
+The profiles of the typical cases of the time-averaged light intensity of the PLIF images,
+𝜙(𝑦) at 𝑥/𝑑 = 5 to 40 are show in figure 12. It is noted that the mean concentration of the
+fluorescent dye in the flow field is quite low and thus the fluorescent response is effectively
+a linear function of the dye concentration (Crimaldi 1997; Vanderwel & Tavoularis 2014;
+Baj et al. 2016); 𝜙(𝑦) thus can be treated virtually as the concentration of the dye which
+is confirmed later in figure 13. For all the cases considered (figure 12a, c, e, g), 𝜙(𝑦)
+reasonably decays in magnitude and spreads into a wider range as 𝑥/𝑑 increases. Similar to
+the definition of velocity wake half-width, the scalar wake half-width 𝐿 𝜙 is such defined that
+𝜙(𝑦 = 𝐿 𝜙) = 1/2𝜙(𝑦 = 0). It is interesting to find that the streamwise evolution of 𝜙∗ scales
+well with 𝐿 𝜙 for all the TTI cases (figure 12d, f, h); for the TNTI case (figure 12b), 𝐿 𝜙 also
+works well for 𝑥/𝑑 ⩾ 20. It seems the scalar field of the wake in a turbulent background
+becomes self-preserving at a smaller 𝑥/𝑑 position than that in a non-turbulent flow. A similar
+observation for the velocity field was also made by Eames et al. (2011).
+The scalar is passively transported by the velocity field, so one may expect a relation
+between the wake half-width determined with the velocity field and that with the scalar field.
+This is indeed observed in the present measurement of the non-turbulent background case
+shown in figure 13, in which the ratio 𝐿 𝜙/𝐿𝑢 is observed to be approximately constant at
+𝑥/𝑑 ⩾ 20. Here 𝐿𝑢 is the wake half-width determined from the mean velocity profile of
+our non-published PIV measurement of the cylinder wake without grids upstream. Note that
+
+17
+Figure 12: Profiles of mean light intensity of PLIF images of typical cases. (a, b) TNTI
+case 1a, (c, d) TTI case 1c, (e, f) TTI case 2a, (g, h) TTI case 3b.
+a similar result was obtained in the measurements of Chen et al. (2016) and Zhou et al.
+(2002) in the wake of a cylinder at the same 𝑥/𝑑 range, except that their passive scalar was
+represented by temperature in the flow. The slightly larger value of the present measurement
+could possibly be attributed to the different initial conditions to those of the two references:
+in our experiment the dye is released from a hole in the rear surface of the cylinder while the
+scalar (heat) in Chen et al. (2016) and Zhou et al. (2002) is injected from the shear layer of
+the wake by electrically heating the cylinder; in addition, the Sc number for the fluorescent
+dye (about 2500, see section 2) is much larger than the Pr number (about 0.7) of heat in
+air, which can also cause the scalar being diffused distinctly (e.g. Rehab et al. 2001). The
+resemblance in our measurement and those from Chen et al. (2016) and Zhou et al. (2002)
+
+160
+1.2
+(n)
+x/d = 5
+(q)
+140
+x/d = 10
+1.0
+120
+x/d = 20
+x/d = 30
+0.8
+100
+x/d = 40
+(y)
+80
+Φ*(y) 0.6
+60
+0.4 -
+40
+0.2 
+20
+0.0 -
+-6
+-4
+-2
+0
+2
+6
+-4
+-3
+-2
+-1
+i
+2
+3
+160
+1.2
+(c)
+(d)
+140
+1.0
+120
+0.8
+100
+b(y)
+80
+(y) 0.6
+60 
+0.4
+40
+0.2 
+20
++0
+0.0 -
+-6 
+-4
+-2 
+0
+2
+4
+6
+-4
+3
+-2
+-1
+160
+1.2
+(e)
+(J)
+140
+1.0 
+120
+0.8
+100
+()Φ
+80
+y)0.6
+60
+0.4 
+40
+0.2
+20
+0.0 -
+-6 
+-4
+-2
+0
+2
+4
+_4
+6
+-3
+-2
+-1
+0
+1
+2
+3
+160
+1.2
+(g)
+(h)
+140
+1.0
+120
+0.8
+100
+(y)
+80
+*(y)0.6
+60
+0.4 -
+40
+0.2
+20
+0
+0.0 -
+4
+-2 
+0
+2
+4
+6
+4
+-3
+-2
+0
+y/d18
+Figure 13: Ratio of scalar wake half-width 𝐿𝜙(𝑥) to velocity wake half-width 𝐿𝑢(𝑥) at
+different 𝑥/𝑑 positions of a cylinder wake.
+confirms our expectation that the distribution of the mean value of the fluorescent intensity
+is a reasonable representation of the distribution of the mean scalar concentration.
+REFERENCES
+Abreu, Hugo, Pinho, Fernando T & da Silva, Carlos B 2022 Turbulent entrainment in viscoelastic
+fluids. Journal of Fluid Mechanics 934, A36.
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+

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+ 
+ 
+Mass production of ultra-pure NaI powder for COSINE-200  
+ 
+KeonAh Shin1*, JunSeok Choe1, Olga Gileva1, Alain Iltis2 ,Yena Kim1 , Yeongduk Kim1,3, 
+Cheolho Lee1, Eunkyung Lee1 , and HyunSu Lee1,3*, Moo Hyun Lee1,3  
+1Center for Underground Physics, Institute for Basic Science (IBS), Daejeon 34126, Korea 
+2Damavan Imaging, Troyes, 10430, France 
+3IBS school, University of Science and Technology (UST), Daejeon 34113, Korea  
+Correspondence:  
+KeonAh Shin and HyunSu Lee 
+[email protected], [email protected]  
+Keywords: NaI powder, low-background, mass purification, recrystallization, COSINE-200.   
+Abstract 
+COSINE-200 is the next phase experiment of the ongoing COSINE-100 that aims to unambiguously 
+verify the annual modulation signals observed by the DAMA experiment and to reach the world 
+competitive sensitivity on the low-mass dark matter search. To achieve the physics goal of the 
+COSINE-200, the successful production of the low-background NaI(Tl) detectors is crucial and it must 
+begin from the mass production of the ultra-low background NaI powder. A clean facility for mass-
+producing the pure-NaI powder has been constructed at the Center for Underground Physics (CUP) in 
+Korea. Two years of operation determined efficient parameters of the mass purification and provided 
+a total of 480 kg of the ultra-pure NaI powder in hand. The potassium concentration in the produced 
+powders varied from 5.4 to 11 ppb, and the maximum production capacity of 35 kg per two weeks was 
+achieved. Here, we report our operational practice with the mass purification of the NaI powder, which 
+includes raw powder purification, recycling of the mother solution, and recovery of NaI from the 
+residual melt that remained after crystal growth.  
+1 
+Introduction 
+Considerable evidence points to the existence of dark matter that could represent 27% of the universe’s 
+total mass or energy [1-5]. One of the most stringent candidates for dark matter is the Weakly 
+Interacting Massive Particles (WIMPs), which many experimental groups have extensively searched 
+in the last few decades [6-13]. Despite attempting to find dark matter particles in numerous experiments, 
+only the DAMA collaboration has claimed the observation of a dark matter signal through an annual 
+modulation signal observed in the low-energy signal region [8,14-16]. However, there have been long-
+standing questions about this claim because no other experimental searches have observed similar 
+signals [17]. Besides, no convincing explanation of the signal’s origin has been proposed, regardless 
+of the exact nature of the signal’s dark matter. 
+The COSINE-100 experiment has been operating at Yangyang underground laboratory in Korea 
+with a total of 106 kg of low-background NaI(Tl) detectors during the last six years [6,7,18-22]. 
+Although many exciting results were published, reaching an unambiguous assumption on the annual 
+modulation signal of the DAMA experiment is far from the conclusion [23,24]. It is mainly due to the 
+
+ 
+2 
+observed background rate in the COSINE-100 detectors, which is 2.5 times higher than the background 
+of the DAMA detectors [19,25]. To take the challenge in world competitive searches for low-mass 
+dark matter and reach a definite conclusion for the DAMA/LIBRA, we are preparing the COSINE-200 
+experiment as the next phase of the COSINE-100 [17,26]. The main goal of the COSINE-200 is to 
+develop 200 kg of ultra-low background NaI(Tl) crystals with a background level lower than those of 
+the DAMA/LIBRA. To reach the physics goal of COSINE-200, we have been developing technology 
+for the low-background NaI(Tl) detector that includes the mass production of ultra-low background 
+NaI powder, crystal growing technique, and detector assembly [17,26,27]. The first step is preparing 
+the ultra-low background NaI powder, in which the potassium concentration must be below 20 ppb and 
+the lead concentration less than a few ppb. Radioactivity-wise, commercially available Astro-grade 
+NaI powders from Sigma-Aldrich are suitable for ultra-low background NaI(Tl) crystal synthesis 
+[17,28]. Still, their extremely high-cost demands independent development of mass purification 
+technology. We have investigated a recrystallization technique to purify the NaI powder at a reasonable 
+price [29]. The lab-scale procedure provided a satisfactory performance of the potassium and lead 
+reduction. Based on successful lab-scale experiments, the mass purification facility was established at 
+the Institute for Basic Science (IBS) in Daejeon, Korea [27]. For the last two years, we optimized 
+operational parameters for the mass production of ultra-low background NaI powder. The yield 
+efficiencies for the chemical process were balanced versus the products’ purity. The processing 
+conditions were adapted to recycle the mother solution and recover NaI from the melt residual after 
+the crystal growth. Using developed technology, we have produced about 480 kg of the low-
+background powder with a production capability of 35 kg per two weeks. Using the purified NaI 
+powder, radioactive background was reduced at least twice in a small size of NaI(Tl) crystal relative 
+to the COSINE-100 crystals [30]. In this report, we summarize our experience, describe the mass 
+purification facility, optimized raw powder purification, and the recovery of NaI from the mother 
+solution and residual melt.  
+2 
+Materials and Methods 
+We use NaI powder from Merck (99.99(5)% purity, Optipure) as an initial material. The potassium 
+contamination in the specially ordered powder is below one ppm. High resistance, 18.2 MΩ·cm de-
+ionized (DI) water is a solvent to dissolve the NaI powder. We use absolute ethanol (~200 proof, HPLC 
+grade, ACS) from the Scharlau to wash the recrystallized NaI crystals. Hydrophilic PTFE membrane 
+filters with 1.0 𝜇m pore size from the Advantec are used to separate the recrystallized NaI crystals 
+from the mother liquor.  
+The mass production facility of the ultra-low background NaI powder is shown in Fig. 1 A. It consists 
+of two main reactors (Fig. 1 B and C), a Nutsche filter unit (Fig. 1 D), two receivers (Fig. 1 E), and a 
+conical dryer (Fig. 1 F). Operation of the whole system, including temperature control through the oil 
+circulation system, is performed by the main controller in Fig. 1 G. The feed tank (Fig. 1 B) is used for 
+the powder dissolution and pre-processing to prevent oxidation of iodide ions. Two main reactors in 
+Fig. 1 B and C are connected, utilizing the polypropylene (PP) pipes that transfer the NaI solution from 
+the feed tank to the mixing tank (Fig. 1 C), as shown in Fig. 1 A. A cartridge filter is installed in the 
+middle of the PP pipelines to remove the insoluble impurities from the solution. The mixing tank 
+performs the recrystallization using the temperature dependence of the NaI solubility in the water [31]. 
+We evaporate water from the NaI solution until it becomes oversaturated at 110℃ (Fig. 2 A), then cool 
+the mixing tank down to 30℃ while stirring the solution (Fig. 2 B). In this process, pure NaI crystals 
+grow without agglomeration, while soluble impurities remain in the mother solution. The crystals are 
+separated from the mother solution by the PTFE membrane filter (Fig. 2 C). The crystals are washed 
+with chilled ethanol to rinse off the remaining mother liquor and impurities from the crystal surface. 
+
+ 
+3 
+The washed crystals are dried in the conical dryer (Fig. 2 D). The produced powders are packed in 
+HDPE bottles and stored in the desiccators to avoid moisture absorption. The details of the facility and 
+recrystallization procedure are described elsewhere in [27].  
+Radiopurity in the raw and purified powders and the mother solution from the purification process 
+is measured by an inductively coupled plasma mass spectrometry (ICP-MS) and high-purity 
+Germanium (HPGe) detector [32]. The water content in the produced powders is measured by the Karl-
+Fisher titrator.  
+3 
+Results 
+3.1 
+Raw powder purification  
+The main goal of our purification is to reduce internal potassium (K) contamination to less than 20 ppb. 
+Tables 1 and 2 show the representative measurements from the raw powder purification process by 
+ICS-MS and HPGe, respectively. As shown in Table 1, most of the potassium contamination coming 
+from the raw powder was filtrated and concentrated in the mother solution. Potassium and lead 
+concentrations in the purified powders were reduced by 20 and 80 times, resulting in final amounts of 
+11 ppb and 0.5 ppb, respectively. Significant reduction of Sr and Ba below the ppb level may indicate 
+a reduction of radium, which belongs to the same family group of the periodic table. With a single 
+crystallization procedure with about 40% yield efficiency, the purity of produced powder became 
+similar to the Astro-grade powder. The impurities concentration in the mother solution were increased 
+approximately twice as in the raw powder. Twenty days of HPGe counting using 1.2 kg of purified 
+powder sampled in the Marinelli beaker reported only upper limits for 226Ra, 228Ac, 228Th, and 40K, as 
+seen in Table 2.  
+To improve production capacity keeping the high quality of the product, we continually performed 
+the raw powder purification with slightly different initial charges and recovery yields, as summarized 
+in Table 3. Although the powder charge was increased from 40 kg to 64 kg, the purified product had 
+similar purities from batch to batch. However, a high recovery yield of 58% provided considerable 
+contamination of K, about 38 ppb. In case of the recovery yields were less than 50%, the purified 
+powder contained consistently low contamination, especially K, about 10 ppb. To keep the consistent 
+and high quality of the product, we ascertained a 50% yield efficiency at maximum for our purification 
+process. Routine purification works have made our experience proficient for the last two years. 
+Compared to the initial investigation shown in Ref. [27], we obtained consistently stable products with 
+the required background level using the same purification facility. With the above-optimized 
+purification parameters, the process took two weeks. Recrystallizing the raw powder took about three 
+working days with 70 kg of the initial charge, and another seven working days were required to dry the 
+wet crystals. With 40~50% recovery efficiency, 30~35 kg of purified powder could be produced in a 
+cycle.  
+3.2 
+Mother solution recovery 
+After the purification process, the mother solution is the remaining product that is concentrated 
+impurities from the initial material. In the optimized purification process, 50% of the initial charge was 
+collected as the purified dry product. Another 35% of NaI remained in the mother solution, and 15% 
+was washed out with ethanol, as shown in Fig. 3. In three cycles of the raw powder purification, the 
+amount of NaI collected as the mother solution was enough for further recycling. We recovered this 
+mother solution in the same manner but reduced the recovery efficiency from 50% to 35% due to the 
+relatively high impurity level in the mother solution. As summarized in Table 4, the recovered crystals 
+
+ 
+4 
+from the mother solution contained higher impurities than those obtained from the raw powder 
+purification. The K contaminations varied from 18 to 50 ppb, proportional to the initial impurities in 
+the mother solution. When the K content in the initial mother solution was higher than 1000 ppb, 
+reaching the required 20 ppb of K was challenging with a single treatment. In this case, an additional 
+recrystallization cycle of the powder was necessary to reach our goal of purity. However, following 
+recrystallization of the crystals recovered from first mother solution (MS-1) was inefficient in 
+production rate, so the rational K level in the initial solution must be lower than one ppm.  
+As shown in Fig. 3, after the separation of crystals from the MS-1, the second mother solution (MS-
+2) contained about 50% of NaI and accumulated most of the impurities. The MS-2 mostly had K 
+content over one ppm. Double recrystallization would be unavoidable to recover this NaI remained. 
+We did not consider recycling the MS-2 due to low recovery efficiency compared to the workforce 
+required.   
+3.3 
+Residual melt recovery 
+We designed a large-size Kyropoulos grower to synthesize 120 kg NaI(Tl) crystal ingot [17]. In this 
+grower, about 200 kg of NaI powder was loaded and melted in the quartz crucible. Crystal-growing 
+trials using Merck raw powders were performed a couple of times with partial success. After pulling 
+out the crystal ingot, many residues remained in the quartz crucible. Typical impurities in this melt 
+were approximately twice higher as in the loaded powder due to the segregation effect. Nevertheless, 
+the recovery of the residual melts was successfully made by achieving satisfied purity levels, as 
+summarized in Table 5. The K concentration in the produced powders varied from 8 to 11 ppb. The 
+purity of recovered NaI from the melt is expected to be much pure if we use the purified powder for 
+mass crystal growth.   
+The process of recovering NaI from the collected residual melt differed from the original purification 
+method because the melt contained a significant amount of insoluble quartz particles and dust. Before 
+the usual operation, the NaI melt dissolved in water was filtered with the PTFE membrane filter. 
+Considering the evaporation of iodine during the crystal-growing process, a three times higher dose of 
+hydrogen iodide (HI) was introduced to reach pH 3.5.  
+3.4 
+Water content measurement 
+Sodium iodide is highly hygroscopic, and its chemical interaction with moisture produces NaOH when 
+heated and causes corrosion of the quartz crucible used in the growing crystal [33]. Keeping the water 
+content below 1000 ppm in the produced powder was crucial. All recrystallized powders were dried 
+in two-step processes. In the first step, the wet powder was dried at 65℃  to avoid agglomerating the 
+NaI powders with water inside the dryer. Then the temperature was increased to 130℃ to dry powder 
+completely. The vapor released from the drying process was extracted with a vacuum pump. Initially, 
+we used a chemical resistance air pump with relatively low pressure to protect the pump from corrosive 
+vapor. As seen in Fig. 4, reaching moisture content below 1000 ppm in the dried powders with the 
+previous set-up was impossible. We improved our drying system by introducing a high-pressure rotary 
+pump with traps for corrosive vapor during the high-temperature drying process. We achieved the 
+water content to less than 1000 ppm with modified set-up.  
+4 
+Discussion 
+A facility for mass production of the ultra-pure NaI powder for the COSINE-200 is well-operating with 
+extensive parameter optimization. The purification of raw NaI powder, the recycling of the mother 
+
+ 
+5 
+solution, and the recovery of NaI from the residual melt were performed in parallel. We have produced 
+about 480 kg of low-background powder with a successful reduction of the internal contamination that 
+is pure enough for the COSINE-200 detectors. The optimized parameters with a stable operation 
+process have provided a maximum 35 kg powder production capacity in two weeks, but there is still 
+room for improvement. If we increase the volume of the dryer 1.5 times, then two purification cycles 
+can be performed in two weeks, increasing production capacity up to 70 kg. With improved capacity, 
+successive double crystallization can help to reach a potassium level much lower than 5 ppb using the 
+above-described facility. We can smoothly provide the ultra-low background NaI powder for the mass 
+production of the NaI(Tl) crystals for the COSINE-200 experiment.   
+5 
+Acknowledgments 
+This work is supported by the Institute for Basic Science (IBS) under project code IBS-R016-A1.  
+ 
+ 
+ 
+ 
+ 
+
+ 
+6 
+ 
+REFERENCES 
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+7 
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+ 
+[30]  H. Lee et al., Paper in preperation (2023).  
+ 
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+[33]  B. Suerfu, Ph.D. thesis, Princeton University 10977855 (2018).  
+ 
+ 
+ 
+
+ 
+8 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+A
+B
+C
+D
+E
+F
+G
+Figure 1. (A) Mass purification facility, (B) Feed tank for dissolving the NaI, (C) Mixing tank for 
+boiling solution and recrystallization process, (D) Filter unit for separation of NaI crystal and mother 
+liquor, (E) Receiver tanks for collecting vapor from mixing tank and dryer, (F) Conical dryer for the 
+NaI powder drying, (G) Main controller to control all the equipment.  
+A
+B
+C
+D
+Figure 2. (A) Boiling of solution with stirring, (B) Recrystallized NaI crystal with mother liquor, (C) 
+Filtrated and washed NaI crystal on the filter unit, (D) Dried NaI powder in the conical dryer. 
+
+ 
+9 
+ 
+ 
+ 
+ 
+ 
+ 
+Raw powder
+Purified powder
+1st Mother solution
+Ethanol washed solution
+50 %
+35 %
+15 %
+Purified powder
+2nd Mother solution
+Ethanol washed solution
+35 %
+Figure 3. Material balances in the NaI recovery cycle. 
+Figure 4. The water content measurement results by Karl-Fisher titrator for different batches of 
+produced powders 
+
+1400
+1200
+Watercontent(ppm)
+1200
+1150
+1000
+Under1000ppmisrequired
+800
+800
+610
+009
+550
+400
+400
+310
+200
+200
+200
+200
+140
+06
+90
+80
+110
+130
+150
+OL
+70
+50
+80
+0
+一
+21-6
+1-21-8
+f-21-9
+1-20-8
+-21-3
+1-21-5
+PurNat
+1-22-1
+-22-2
+Pu
+rNa
+arNal
+PurNal
+Purtal
+Pur
+rNal-21-
+PurNal-21
+PurNal
+Pu
+arNal
+Pu
+Airpump
+Airpump+Rotarypump 
+10 
+ 
+Table 2. Representative HPGe result of purified powder from raw powder purification. The upper 
+limits are given at 90% C.L. 
+226Ra (238U) 
+40K 
+228Ac 
+228Th 
+< 0.56 mBq/kg 
+< 5.64 mBq/kg 
+< 1.10 mBq/kg 
+< 0.71 mBq/kg 
+ 
+ 
+ 
+Table 1. Representative ICP-MS results of raw and purified powders vs. Astro-grade powder’s purity. 
+Values are given at 90% C.L, and upper limits are given at 95% C.L. 
+Description 
+K 
+Fe 
+Sr 
+Ba 
+Pb 
+Th 
+U 
+ppb 
+ppb 
+ppb 
+ppb 
+ppb 
+ppt 
+ppt 
+Astro grade 
+5±3 
+110±20 0.3±0.1 0.6±0.1 0.8±0.1 
+< 6 
+< 6 
+Merck-raw powder 
+250±90 
+33±6 
+19±1 
+3.0±0.4 
+40±2 
+< 6 
+< 6 
+Purified powder (20-5) 
+11±1 
+< 10 
+0.3±0.1 0.9±0.1 0.5±0.1 
+< 6 
+< 6 
+Mother solution (20-5) 
+550±120 
+< 200 
+38±2 
+9±1 
+60±4 
+< 6 
+< 6 
+
+ 
+11 
+Table 3. The ICP-MS results of purified powders in different batches of raw powder purification. 
+Values are given at 90% C.L, and upper limits are given at 95% C.L. 
+Sample 
+No. 
+Initial 
+charge 
+Recovery 
+yield 
+K 
+Fe 
+Sr 
+Ba 
+Pb 
+Th 
+U 
+ppb 
+ppb 
+ppb 
+ppb 
+ppb 
+ppt 
+ppt 
+20-5 
+40 kg 
+44% 
+11±1 
+< 10 
+0.3±0.1 0.9±0.1 0.5±0.1 
+< 6 
+< 5 
+20-7 
+50 kg 
+41% 
+10±1 
+< 10 
+0.1±0.1 0.3±0.1 
+< 0.3 
+< 3 
+< 5 
+20-8 
+50 kg 
+39% 
+6.4±0.1 
+< 10 
+0.1±0.1 0.7±0.1 
+< 0.3 
+< 3 
+< 5 
+21-5 
+53 kg 
+42% 
+5.4±0.3 
+< 10 
+0.2±0.1 0.4±0.1 0.5±0.1 
+< 5 
+< 3 
+21-8 
+60 kg 
+58% 
+38±2 
+< 10 
+0.4±0.1 0.3±0.1 0.5±0.1 
+< 7 
+< 7 
+22-5 
+64 kg 
+35% 
+11±1 
+< 7 
+0.4±0.1 1.6±0.2 0.9±0.5 
+< 100 
+< 20 
+ 
+ 
+ 
+ 
+ 
+
+ 
+12 
+Sample No. 
+Material 
+K 
+Fe 
+Sr 
+Ba 
+Pb 
+Th 
+U 
+ppb 
+ppb 
+ppb 
+ppb 
+ppb 
+ppt 
+ppt 
+21-4(M) 
+Initial sol. 
+330±40 
+N/A 
+20±1 
+6.3±0.2 
+41±4 
+< 5 
+< 3 
+Wet cryst. 
+< 40 
+N/A 
+0.4±0.1 0.1±0.1 1.2±0.1 
+< 5 
+< 3 
+21-7(M) 
+Initial sol. 
+470±10 
+N/A 
+34±1 
+7.3±0.2 
+56±1 
+< 7 
+< 7 
+Wet cryst. 
+< 50 
+N/A 
+1.2±0.1 0.2±0.1 2.0±0.1 
+< 7 
+< 7 
+21-11(M) 
+Initial sol. 
+610±30 
+16±1 
+40±2 
+10±1 
+88±12 
+< 7 
+< 7 
+Wet cryst. 
+18±1 
+< 7 
+1.0±0.1 0.2±0.1 4.5±0.3 
+< 7 
+< 7 
+22-2(M) 
+Initial sol. 
+1010±150 
+8±1 
+16±1 
+15±1 
+86±4 
+< 4 
+< 4 
+Dry powder 
+21±2 
+< 7 
+0.2±0.1 0.7±0.1 1.0±0.1 
+< 4 
+< 4 
+20-3(M) 
+Initial sol. 
+1170±120 
+39±2 
+33±2 
+12±1 
+60±2 
+< 6 
+< 5 
+Dry powder 
+44±5 
+14±1 
+1.0±0.1 0.4±0.1 2.0±0.1 
+< 6 
+< 5 
+ 
+ 
+ 
+Table 4. The ICP-MS result of the mother solution recovery experiment in different batches of mass 
+production. It is marked as (M) for the naming. In this experiment, the Initial solution means the initial 
+mother solution, and the Wet crystal means recrystallized and washed crystal. If the purity is not 
+accepted, then additional recrystallization is required, so the purity was confirmed first by ICP-MS 
+before drying and then dried thoroughly. The wet crystal consists of ~73% NaI and extra water and 
+ethanol, so the impurity concentration is calculated as 73% NaI. Values are given at 90% C.L, and 
+upper limits are given at 95% C.L. 
+
+ 
+13 
+Sample No. 
+Material 
+K 
+Fe 
+Sr 
+Ba 
+Pb 
+Th 
+U 
+ppb 
+ppb 
+ppb 
+ppb 
+ppb 
+ppt 
+ppt 
+21-12(RM) 
+Initial sol. 
+730±10 
+20±2 
+10±1 
+8.0±0.6 143±12 
+< 7 
+< 7 
+Wet cryst. 
+8±1 
+< 10 
+0.4±0.1 0.3±0.1 
+5±1 
+< 7 
+< 7 
+21-13(RM) 
+Initial sol. 
+540±20 
+N/A 
+10±1 
+5.1±0.2 
+95±10 
+< 7 
+< 7 
+Wet cryst. 
+< 50 
+N/A 
+0.1±0.1 
+< 0.1 
+< 0.3 
+< 7 
+< 7 
+22-1(RM) 
+Initial sol. 
+390±10 
+N/A 
+8±1 
+6.4±0.2 
+40±4 
+< 4 
+< 4 
+Dry powder 
+8±1 
+< 7 
+0.1±0.1 0.3±0.1 0.7±0.1 
+< 4 
+< 4 
+22-4(RM) 
+Initial sol. 
+570±10 
+N/A 
+15±2 
+6.7±0.5 
+5±1 
+< 4 
+< 4 
+Dry powder 
+11±4 
+< 7 
+0.1±0.1 0.3±0.1 
+< 0.3 
+< 4 
+< 4 
+ 
+ 
+Table 5. The ICP-MS result of residual melt recovery experiment in different batches of mass 
+production. It is marked as (RM) for the naming, and the Initial solution is the residual melt solution 
+after dissolving melt and filtration of the quartz particles. The Wet crystal samples were taken after 
+recrystallization and washing with ethanol. Values are given at 90% C.L, and upper limits are given at 
+95% C.L. 
+

YtE5T4oBgHgl3EQfCw7_/content/tmp_files/load_file.txt

@@ -0,0 +1,504 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf,len=503
+page_content='Mass production of ultra-pure NaI powder for COSINE-200  KeonAh Shin1*, JunSeok Choe1, Olga Gileva1, Alain Iltis2 ,Yena Kim1 , Yeongduk Kim1,3,  Cheolho Lee1, Eunkyung Lee1 , and HyunSu Lee1,3*, Moo Hyun Lee1,3  1Center for Underground Physics, Institute for Basic Science (IBS), Daejeon 34126, Korea  2Damavan Imaging, Troyes, 10430, France  3IBS school, University of Science and Technology (UST), Daejeon 34113, Korea  Correspondence:  KeonAh Shin and HyunSu Lee  kashin@ibs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='kr, hyunsulee@ibs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='kr  Keywords: NaI powder, low-background, mass purification, recrystallization, COSINE-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Abstract  COSINE-200 is the next phase experiment of the ongoing COSINE-100 that aims to unambiguously  verify the annual modulation signals observed by the DAMA experiment and to reach the world  competitive sensitivity on the low-mass dark matter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' To achieve the physics goal of the  COSINE-200, the successful production of the low-background NaI(Tl) detectors is crucial and it must  begin from the mass production of the ultra-low background NaI powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' A clean facility for mass-  producing the pure-NaI powder has been constructed at the Center for Underground Physics (CUP) in  Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Two years of operation determined efficient parameters of the mass purification and provided  a total of 480 kg of the ultra-pure NaI powder in hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The potassium concentration in the produced  powders varied from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4 to 11 ppb, and the maximum production capacity of 35 kg per two weeks was  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Here, we report our operational practice with the mass purification of the NaI powder, which  includes raw powder purification, recycling of the mother solution, and recovery of NaI from the  residual melt that remained after crystal growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  1  Introduction  Considerable evidence points to the existence of dark matter that could represent 27% of the universe’s  total mass or energy [1-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' One of the most stringent candidates for dark matter is the Weakly  Interacting Massive Particles (WIMPs), which many experimental groups have extensively searched  in the last few decades [6-13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Despite attempting to find dark matter particles in numerous experiments,  only the DAMA collaboration has claimed the observation of a dark matter signal through an annual  modulation signal observed in the low-energy signal region [8,14-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' However, there have been long-  standing questions about this claim because no other experimental searches have observed similar  signals [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Besides, no convincing explanation of the signal’s origin has been proposed, regardless  of the exact nature of the signal’s dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  The COSINE-100 experiment has been operating at Yangyang underground laboratory in Korea  with a total of 106 kg of low-background NaI(Tl) detectors during the last six years [6,7,18-22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Although many exciting results were published, reaching an unambiguous assumption on the annual  modulation signal of the DAMA experiment is far from the conclusion [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' It is mainly due to the  2  observed background rate in the COSINE-100 detectors, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5 times higher than the background  of the DAMA detectors [19,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' To take the challenge in world competitive searches for low-mass  dark matter and reach a definite conclusion for the DAMA/LIBRA, we are preparing the COSINE-200  experiment as the next phase of the COSINE-100 [17,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The main goal of the COSINE-200 is to  develop 200 kg of ultra-low background NaI(Tl) crystals with a background level lower than those of  the DAMA/LIBRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' To reach the physics goal of COSINE-200, we have been developing technology  for the low-background NaI(Tl) detector that includes the mass production of ultra-low background  NaI powder, crystal growing technique, and detector assembly [17,26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The first step is preparing  the ultra-low background NaI powder, in which the potassium concentration must be below 20 ppb and  the lead concentration less than a few ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Radioactivity-wise, commercially available Astro-grade  NaI powders from Sigma-Aldrich are suitable for ultra-low background NaI(Tl) crystal synthesis  [17,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Still, their extremely high-cost demands independent development of mass purification  technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We have investigated a recrystallization technique to purify the NaI powder at a reasonable  price [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The lab-scale procedure provided a satisfactory performance of the potassium and lead  reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Based on successful lab-scale experiments, the mass purification facility was established at  the Institute for Basic Science (IBS) in Daejeon, Korea [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' For the last two years, we optimized  operational parameters for the mass production of ultra-low background NaI powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The yield  efficiencies for the chemical process were balanced versus the products’ purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The processing  conditions were adapted to recycle the mother solution and recover NaI from the melt residual after  the crystal growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Using developed technology, we have produced about 480 kg of the low-  background powder with a production capability of 35 kg per two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Using the purified NaI  powder, radioactive background was reduced at least twice in a small size of NaI(Tl) crystal relative  to the COSINE-100 crystals [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In this report, we summarize our experience, describe the mass  purification facility, optimized raw powder purification, and the recovery of NaI from the mother  solution and residual melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  2  Materials and Methods  We use NaI powder from Merck (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='99(5)% purity, Optipure\uf0e2) as an initial material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The potassium  contamination in the specially ordered powder is below one ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' High resistance, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2 MΩ·cm de-  ionized (DI) water is a solvent to dissolve the NaI powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We use absolute ethanol (~200 proof, HPLC  grade, ACS) from the Scharlau to wash the recrystallized NaI crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Hydrophilic PTFE membrane  filters with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0 𝜇m pore size from the Advantec are used to separate the recrystallized NaI crystals  from the mother liquor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  The mass production facility of the ultra-low background NaI powder is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' It consists  of two main reactors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 B and C), a Nutsche filter unit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 D), two receivers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 E), and a  conical dryer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Operation of the whole system, including temperature control through the oil  circulation system, is performed by the main controller in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The feed tank (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 B) is used for  the powder dissolution and pre-processing to prevent oxidation of iodide ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Two main reactors in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 B and C are connected, utilizing the polypropylene (PP) pipes that transfer the NaI solution from  the feed tank to the mixing tank (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 C), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' A cartridge filter is installed in the  middle of the PP pipelines to remove the insoluble impurities from the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The mixing tank  performs the recrystallization using the temperature dependence of the NaI solubility in the water [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  We evaporate water from the NaI solution until it becomes oversaturated at 110℃ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 2 A), then cool  the mixing tank down to 30℃ while stirring the solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 2 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In this process, pure NaI crystals  grow without agglomeration, while soluble impurities remain in the mother solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The crystals are  separated from the mother solution by the PTFE membrane filter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 2 C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The crystals are washed  with chilled ethanol to rinse off the remaining mother liquor and impurities from the crystal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  3  The washed crystals are dried in the conical dryer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 2 D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The produced powders are packed in  HDPE bottles and stored in the desiccators to avoid moisture absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The details of the facility and  recrystallization procedure are described elsewhere in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Radiopurity in the raw and purified powders and the mother solution from the purification process  is measured by an inductively coupled plasma mass spectrometry (ICP-MS) and high-purity  Germanium (HPGe) detector [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The water content in the produced powders is measured by the Karl-  Fisher titrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  3  Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  Raw powder purification  The main goal of our purification is to reduce internal potassium (K) contamination to less than 20 ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Tables 1 and 2 show the representative measurements from the raw powder purification process by  ICS-MS and HPGe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' As shown in Table 1, most of the potassium contamination coming  from the raw powder was filtrated and concentrated in the mother solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Potassium and lead  concentrations in the purified powders were reduced by 20 and 80 times, resulting in final amounts of  11 ppb and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5 ppb, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Significant reduction of Sr and Ba below the ppb level may indicate  a reduction of radium, which belongs to the same family group of the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' With a single  crystallization procedure with about 40% yield efficiency, the purity of produced powder became  similar to the Astro-grade powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The impurities concentration in the mother solution were increased  approximately twice as in the raw powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Twenty days of HPGe counting using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2 kg of purified  powder sampled in the Marinelli beaker reported only upper limits for 226Ra, 228Ac, 228Th, and 40K, as  seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  To improve production capacity keeping the high quality of the product, we continually performed  the raw powder purification with slightly different initial charges and recovery yields, as summarized  in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Although the powder charge was increased from 40 kg to 64 kg, the purified product had  similar purities from batch to batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' However, a high recovery yield of 58% provided considerable  contamination of K, about 38 ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In case of the recovery yields were less than 50%, the purified  powder contained consistently low contamination, especially K, about 10 ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' To keep the consistent  and high quality of the product, we ascertained a 50% yield efficiency at maximum for our purification  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Routine purification works have made our experience proficient for the last two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Compared to the initial investigation shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' [27], we obtained consistently stable products with  the required background level using the same purification facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' With the above-optimized  purification parameters, the process took two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Recrystallizing the raw powder took about three  working days with 70 kg of the initial charge, and another seven working days were required to dry the  wet crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' With 40~50% recovery efficiency, 30~35 kg of purified powder could be produced in a  cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2  Mother solution recovery  After the purification process, the mother solution is the remaining product that is concentrated  impurities from the initial material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In the optimized purification process, 50% of the initial charge was  collected as the purified dry product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Another 35% of NaI remained in the mother solution, and 15%  was washed out with ethanol, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In three cycles of the raw powder purification, the  amount of NaI collected as the mother solution was enough for further recycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We recovered this  mother solution in the same manner but reduced the recovery efficiency from 50% to 35% due to the  relatively high impurity level in the mother solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' As summarized in Table 4, the recovered crystals  4  from the mother solution contained higher impurities than those obtained from the raw powder  purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The K contaminations varied from 18 to 50 ppb, proportional to the initial impurities in  the mother solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' When the K content in the initial mother solution was higher than 1000 ppb,  reaching the required 20 ppb of K was challenging with a single treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In this case, an additional  recrystallization cycle of the powder was necessary to reach our goal of purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' However, following  recrystallization of the crystals recovered from first mother solution (MS-1) was inefficient in  production rate, so the rational K level in the initial solution must be lower than one ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 3, after the separation of crystals from the MS-1, the second mother solution (MS-  2) contained about 50% of NaI and accumulated most of the impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The MS-2 mostly had K  content over one ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Double recrystallization would be unavoidable to recover this NaI remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  We did not consider recycling the MS-2 due to low recovery efficiency compared to the workforce  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  Residual melt recovery  We designed a large-size Kyropoulos grower to synthesize 120 kg NaI(Tl) crystal ingot [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In this  grower, about 200 kg of NaI powder was loaded and melted in the quartz crucible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Crystal-growing  trials using Merck raw powders were performed a couple of times with partial success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' After pulling  out the crystal ingot, many residues remained in the quartz crucible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Typical impurities in this melt  were approximately twice higher as in the loaded powder due to the segregation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Nevertheless,  the recovery of the residual melts was successfully made by achieving satisfied purity levels, as  summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The K concentration in the produced powders varied from 8 to 11 ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The  purity of recovered NaI from the melt is expected to be much pure if we use the purified powder for  mass crystal growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  The process of recovering NaI from the collected residual melt differed from the original purification  method because the melt contained a significant amount of insoluble quartz particles and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Before  the usual operation, the NaI melt dissolved in water was filtered with the PTFE membrane filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Considering the evaporation of iodine during the crystal-growing process, a three times higher dose of  hydrogen iodide (HI) was introduced to reach pH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4  Water content measurement  Sodium iodide is highly hygroscopic, and its chemical interaction with moisture produces NaOH when  heated and causes corrosion of the quartz crucible used in the growing crystal [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Keeping the water  content below 1000 ppm in the produced powder was crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' All recrystallized powders were dried  in two-step processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In the first step, the wet powder was dried at 65℃  to avoid agglomerating the  NaI powders with water inside the dryer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Then the temperature was increased to 130℃ to dry powder  completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The vapor released from the drying process was extracted with a vacuum pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Initially,  we used a chemical resistance air pump with relatively low pressure to protect the pump from corrosive  vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 4, reaching moisture content below 1000 ppm in the dried powders with the  previous set-up was impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We improved our drying system by introducing a high-pressure rotary  pump with traps for corrosive vapor during the high-temperature drying process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We achieved the  water content to less than 1000 ppm with modified set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  4  Discussion  A facility for mass production of the ultra-pure NaI powder for the COSINE-200 is well-operating with  extensive parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The purification of raw NaI powder, the recycling of the mother  5  solution, and the recovery of NaI from the residual melt were performed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We have produced  about 480 kg of low-background powder with a successful reduction of the internal contamination that  is pure enough for the COSINE-200 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The optimized parameters with a stable operation  process have provided a maximum 35 kg powder production capacity in two weeks, but there is still  room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' If we increase the volume of the dryer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5 times, then two purification cycles  can be performed in two weeks, increasing production capacity up to 70 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' With improved capacity,  successive double crystallization can help to reach a potassium level much lower than 5 ppb using the  above-described facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' We can smoothly provide the ultra-low background NaI powder for the mass  production of the NaI(Tl) crystals for the COSINE-200 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  5  Acknowledgments  This work is supported by the Institute for Basic Science (IBS) under project code IBS-R016-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  6  REFERENCES  [1]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
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+page_content='  [3]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Bertone and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Hooper, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 90, 045002 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [4]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
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+page_content=' Adhikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' (COSINE-100 Collaboration), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' C 78, 490 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [26]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=', Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' A 981, 164556 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [27]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 15, C07031 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [28]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Shields, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Xu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Calaprice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Proce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 61, 169-178 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [29]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Nuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' 317, 1329-1332 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [30]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=', Paper in preperation (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [31]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Seidell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=', Solubilities of inorganic and metal organic compounds, Van Nostrand, (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [32]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' : Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Series 1468, 012249 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  [33]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Suerfu, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' thesis, Princeton University 10977855 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  8  A  B  C  D  E  F  G  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' (A) Mass purification facility, (B) Feed tank for dissolving the NaI, (C) Mixing tank for  boiling solution and recrystallization process, (D) Filter unit for separation of NaI crystal and mother  liquor, (E) Receiver tanks for collecting vapor from mixing tank and dryer, (F) Conical dryer for the  NaI powder drying, (G) Main controller to control all the equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  A  B  C  D  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' (A) Boiling of solution with stirring, (B) Recrystallized NaI crystal with mother liquor, (C)  Filtrated and washed NaI crystal on the filter unit, (D) Dried NaI powder in the conical dryer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  9  Raw powder  Purified powder  1st Mother solution  Ethanol washed solution  50 %  35 %  15 %  Purified powder  2nd Mother solution  Ethanol washed solution  35 %  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Material balances in the NaI recovery cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The water content measurement results by Karl-Fisher titrator for different batches of  produced powders  1400  1200  Watercontent(ppm)  1200  1150  1000  Under1000ppmisrequired  800  800  610  009  550  400  400  310  200  200  200  200  140  06  90  80  110  130  150  OL  70  50  80  0  一  21-6  1-21-8  f-21-9  1-20-8  21-3  1-21-5  PurNat  1-22-1  22-2  Pu  rNa  arNal  PurNal  Purtal  Pur  rNal-21-  PurNal-21  PurNal  Pu  arNal  Pu  Airpump  Airpump+Rotarypump  10  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Representative HPGe result of purified powder from raw powder purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The upper  limits are given at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  226Ra (238U)  40K  228Ac  228Th  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='56 mBq/kg  < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='64 mBq/kg  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='10 mBq/kg  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='71 mBq/kg  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Representative ICP-MS results of raw and purified powders vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Astro-grade powder’s purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Values are given at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L, and upper limits are given at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Description  K  Fe  Sr  Ba  Pb  Th  U  ppb  ppb  ppb  ppb  ppb  ppt  ppt  Astro grade  5±3  110±20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 6  < 6  Merck-raw powder  250±90  33±6  19±1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4  40±2  < 6  < 6  Purified powder (20-5)  11±1  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 6  < 6  Mother solution (20-5)  550±120  < 200  38±2  9±1  60±4  < 6  < 6  11  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The ICP-MS results of purified powders in different batches of raw powder purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Values are given at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L, and upper limits are given at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Sample  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Initial  charge  Recovery  yield  K  Fe  Sr  Ba  Pb  Th  U  ppb  ppb  ppb  ppb  ppb  ppt  ppt  20-5  40 kg  44%  11±1  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 6  < 5  20-7  50 kg  41%  10±1  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  < 3  < 5  20-8  50 kg  39%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  < 3  < 5  21-5  53 kg  42%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 5  < 3  21-8  60 kg  58%  38±2  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 7  < 7  22-5  64 kg  35%  11±1  < 7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5  < 100  < 20  12  Sample No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Material  K  Fe  Sr  Ba  Pb  Th  U  ppb  ppb  ppb  ppb  ppb  ppt  ppt  21-4(M)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  330±40  N/A  20±1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2  41±4  < 5  < 3  Wet cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  < 40  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 5  < 3  21-7(M)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  470±10  N/A  34±1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2  56±1  < 7  < 7  Wet cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  < 50  N/A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 7  < 7  21-11(M)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  610±30  16±1  40±2  10±1  88±12  < 7  < 7  Wet cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  18±1  < 7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  < 7  < 7  22-2(M)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  1010±150  8±1  16±1  15±1  86±4  < 4  < 4  Dry powder  21±2  < 7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 4  < 4  20-3(M)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  1170±120  39±2  33±2  12±1  60±2  < 6  < 5  Dry powder  44±5  14±1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 6  < 5  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The ICP-MS result of the mother solution recovery experiment in different batches of mass  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' It is marked as (M) for the naming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' In this experiment, the Initial solution means the initial  mother solution, and the Wet crystal means recrystallized and washed crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' If the purity is not  accepted, then additional recrystallization is required, so the purity was confirmed first by ICP-MS  before drying and then dried thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The wet crystal consists of ~73% NaI and extra water and  ethanol, so the impurity concentration is calculated as 73% NaI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Values are given at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L, and  upper limits are given at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  13  Sample No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  Material  K  Fe  Sr  Ba  Pb  Th  U  ppb  ppb  ppb  ppb  ppb  ppt  ppt  21-12(RM)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  730±10  20±2  10±1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='6 143±12  < 7  < 7  Wet cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  8±1  < 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  5±1  < 7  < 7  21-13(RM)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  540±20  N/A  10±1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2  95±10  < 7  < 7  Wet cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  < 50  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  < 7  < 7  22-1(RM)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  390±10  N/A  8±1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='2  40±4  < 4  < 4  Dry powder  8±1  < 7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 4  < 4  22-4(RM)  Initial sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='  570±10  N/A  15±2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='5  5±1  < 4  < 4  Dry powder  11±4  < 7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='1  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='3  < 4  < 4  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The ICP-MS result of residual melt recovery experiment in different batches of mass  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' It is marked as (RM) for the naming, and the Initial solution is the residual melt solution  after dissolving melt and filtration of the quartz particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' The Wet crystal samples were taken after  recrystallization and washing with ethanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content=' Values are given at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L, and upper limits are given at  95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE5T4oBgHgl3EQfCw7_/content/2301.05400v1.pdf'}

ZdFLT4oBgHgl3EQfWC9a/content/tmp_files/2301.12055v1.pdf.txt

@@ -0,0 +1,1432 @@
+TIDo: Source-free Task Incremental Learning in Non-stationary
+Environments
+Abhinit Kumar Ambastha
+National University of Singapore
+Singapore, Singapore
+[email protected]
+Leong Tze Yun
+National University of Singapore
+Singapore, Singapore
+[email protected]
+ABSTRACT
+This work presents an incremental learning approach for autonomous
+agents to learn new tasks in a non-stationary environment. Updat-
+ing a DNN model-based agent to learn new target tasks requires
+us to store past training data and needs a large labeled target task
+dataset. Few-shot task incremental learning methods overcome the
+limitation of labeled target datasets by adapting trained models
+to learn private target classes using a few labeled representatives
+and a large unlabeled target dataset. However, the methods as-
+sume that the source and target tasks are stationary. We propose
+a one-shot task incremental learning approach that can adapt to
+non-stationary source and target tasks. Our approach minimizes
+adversarial discrepancy between the model’s feature space and in-
+coming incremental data to learn an updated hypothesis. We also
+use distillation loss to reduce catastrophic forgetting of previously
+learned tasks. Finally, we use Gaussian prototypes to generate ex-
+emplar instances eliminating the need to store past training data.
+Unlike current work in task incremental learning, our model can
+learn both source and target task updates incrementally. We evalu-
+ate our method on various problem settings for incremental object
+detection and disease prediction model update. We evaluate our
+approach by measuring the performance of shared class and tar-
+get private class prediction. Our results show that our approach
+achieved improved performance compared to existing state-of-the-
+art task incremental learning methods.
+1
+INTRODUCTION
+Task incremental learning problem applies to non-stationary prob-
+lem settings where an agent needs to update an existing task model
+but does not have access to large amounts of labeled data. An exam-
+ple of such a setting is an autonomous agent learning an incremen-
+tal object detection model. Object detection and computer vision
+models are helpful in various domains, such as robotics, health-
+care, e-commerce, and security. A fixed label set and a stationary
+input data distribution limits a classification model’s generaliza-
+tion ability in non-stationary or open-set problem settings. We can
+overcome these bottlenecks using unsupervised task incremental
+learning and update the model without access to a large replay
+memory.
+In a task incremental learning problem, the agent aims to learn
+an optimal hypothesis for both source and target domains [10, 13,
+14, 17, 18]. The source and target are assumed to have undergone a
+dataset shift [3]. Hence, their shared class instances have a covariate
+discrepancy. We assume access to a few labeled target private class
+instances. The current works address task incremental settings
+at a single time point. We extend this approach to work in non-
+stationary settings, where the source and target data are assumed
+to be available as a dynamic stream. To this effect we propose a new
+method for task incremental learning – Task Incremental Domain
+Adaptation (TIDo).
+1.1
+BACKGROUND
+In this section, we explore the background topics for this work and
+theoretical guarantees to learn a task incremental hypothesis using
+unlabelled data. We first provide a formal definition for task and
+task incremental learning.
+Definition 1. Task A task is defined as a two-tuple T =< D, 𝑓
+′ >,
+where D is a domain and 𝑓
+′ is an approximation of a labeling
+function for the domain. Learning a task is referred to as learning
+a close approximation of the aforementioned labeling function.
+Definition 2. One-shot task incremental learning Given un-
+labelled target data 𝑥 (𝑡)
+𝑡
+∈ U𝑡 and labelled source domain data
+𝑥 (𝑡)
+𝑠
+∈ D𝑠 The target domain label set is given by 𝐶𝑡, the shared
+source and target label set is given by 𝐶𝑠, and target-private label
+set is given by 𝐶
+′
+𝑡 𝐶𝑠 = 𝐶𝑡\𝐶
+′
+𝑡 We have been given a single labelled
+sample from 𝐶
+′
+𝑡, ˜𝑥 (𝑡)
+𝑡
+. We define task incremental learning as the
+problem of a target task hypothesis that can predict all labels 𝐶𝑡.
+Definition 3. Hypothesis A hypothesis ℎ ∈ H refers to an es-
+timate of the labelling function 𝑓 : 𝑥 → 𝐶, where, 𝐶 is the label
+set. The error of a given hypothesis w.r.t. a labelling function for a
+domain < D, 𝑓 > is given by:
+𝜖(ℎ, 𝑓 ) := E𝑥∼D [I(ℎ(𝑥) ≠ 𝑓 (𝑥))],
+(1)
+where I is an indicator function.
+For a given source domain, the true risk of a hypothesis ℎ ∈ H
+is 𝜖𝑆 (ℎ, 𝑓 ). Since 𝜖𝑠 (ℎ, 𝑓 ) is intractable for most tasks, we use an
+empirical estimate of the risk, ˆ𝜖𝑆 (ℎ, 𝑓 ). We assume similar notation
+for target domain as 𝜖𝑇 (ℎ, 𝑓 ) and ˆ𝜖𝑇 (ℎ, 𝑓 ). The goal is to learn an
+incremental hypothesis ℎ(𝑡) ∈ H at a time point (𝑡), where H is a
+hypothesis class.
+ℎ(𝑡) = argmin
+ℎ∈H
+
+𝜖 (𝑡)
+𝑇 (ℎ, 𝑓 ) +
+𝑡∑︁
+𝑖=0
+𝜖 (𝑖)
+𝑠
+(ℎ, 𝑓 )
+
+(2)
+We handle the limitation of non-stationary source and target
+tasks using unsupervised domain adaptation. [2] show that for
+a classification task, empirical error and a measure of disagree-
+ment between the optimal hypothesis and the proposed hypothesis
+bounds the true error of a hypothesis. The authors defined the risk
+𝜖𝑠 (ℎ, 𝑓 ) of a hypothesis ℎ ∈ H for a given domain 𝑆, can be defined
+as the probability that a hypothesis disagrees with the true labeling
+function 𝑓 of a distribution D𝑠 [2, 3]:
+arXiv:2301.12055v1  [cs.LG]  28 Jan 2023
+
+𝜖𝑠 (ℎ, 𝑓 ) = E𝑥∼D𝑠 [|ℎ(𝑥) − 𝑓 (𝑥)|].
+(3)
+While referring to risk, we use the shorthand 𝜖𝑠 (ℎ) = 𝜖𝑠 (ℎ, 𝑓 ).
+We use the notation ˆ𝜖𝑠 (ℎ) to denote the empirical risk of a hypoth-
+esis ℎ for domain 𝑆.
+Blitzer et al. [2] defined 𝑑HΔH as the measure of maximum
+disagreement between any hypothesis in a hypothesis class. For
+a hypothesis space H, HΔH is defined as a symmetric difference
+hypothesis space:
+HΔH = ℎ(𝑥) ⊕ ℎ
+′(𝑥) : ℎ,ℎ
+′ ∈ H,
+(4)
+where ⊕ is the XOR operator.
+𝑑HΔH was shown to satisfy the following inequality for any
+hypotheses, ℎ,ℎ
+′ ∈ H and domains 𝑆 and 𝑇:
+|𝜖𝑠 (ℎ,ℎ∗) − 𝜖𝑡 (ℎ,ℎ∗)| ≤ 1
+2𝑑HΔH
+(5)
+Definition 4. Vapnik-Chervonenkis dimension (VC dimen-
+sion)[20] The Vapnik-Chervonenkis dimension,𝑉𝐶(H), of hypoth-
+esis space H defined over instance space 𝑋 is the size of the largest
+finite subset of 𝑋 shattered by H. If arbitrarily large finite sets of
+𝑋 can be shattered by H , then 𝑉𝐶(H) ≡ ∞
+Lemma 1.1 shows that we can bind target task risk with source
+task risk.
+Lemma 1.1. [2] For a given source (𝑆) and target (𝑇) domain, Let
+H be a hypothesis class and ℎ∗ ∈ H be the optimal hypothesis.
+Let 𝑑HΔH be a symmetric hypothesis space distance. Then for every
+ℎ ∈ H we have,
+𝜖𝑇 (ℎ) ≤ 𝜖𝑇 (ℎ∗) + 𝜖𝑇 (ℎ,ℎ∗) ≤ 𝜖𝑆 (ℎ) + 𝜆 + 1
+2𝑑HΔH(D𝑆, D𝑇 ) (6)
+where,
+𝜆 = 𝜖𝑇 (ℎ∗) + 𝜖𝑠 (ℎ∗)
+(7)
+In Theorem 1.2, we show that the risk of an incremental model
+update using target data will be theoretically bounded by the aver-
+age risk of the source data provided in the iteration. This ensures
+a theoretical upper bound of model error when a target dataset is
+used to update an existing model.
+Theorem 1.2. Let ˆD𝑇 be the empirically estimated target task
+distribution and ˆD (𝑡)
+𝑠
+be the empirical source distribution. Let 𝑑HΔH
+be a symmetric hypothesis space distance, then for every ℎ ∈ H, we
+can show that the true target risk is bound by the average true source
+risk and the domain discrepancy between ˆD𝑇 and ˆD (𝑡)
+𝑠
+.
+𝜖𝑇 (ℎ) ≤ 1
+𝑡
+𝑡∑︁
+𝑖=1
+�
+𝜖𝑆 (ℎ(𝑖)) + 1
+2𝑑HΔH( ˆD (𝑖)
+𝑆 , ˆD𝑇 )
+�
++ 𝜆∗(𝑡−1)
+(8)
+Where,
+𝜆∗(𝑡−1) = 1
+𝑡
+𝑡∑︁
+𝑖=1
+[𝜖𝑇 (ℎ(𝑖−1)) + 𝜖𝑆 (ℎ(𝑖−1))]
+(9)
+In Theorem 1.3, We show that the risk of an incremental model
+update is theoretically bounded by the average risk of the previously
+introduced target data and domain discrepancy between the source
+and target domains. This shows that the model should be able to
+learn incrementally using new domain data as long as it has low
+domain discrepancy compared to the original model source data.
+Theorem 1.3. Let D𝑇 be the true target task distribution and
+D (𝑡)
+𝑠
+be the true source distribution. Let 𝑑HΔH be a symmetric hy-
+pothesis space distance. For every ℎ ∈ H, we can show that the true
+target risk is bounded by the average true target risk of the previous
+increments, the domain discrepancy between ˆD𝑇 and ˆD (𝑡)
+𝑠
+and the
+optimal hypothesis risk 𝜆∗(𝑡−1) of the existing model.
+𝜖𝑇 (ℎ(𝑡)) ≤ 1
+𝑡
+��
+�
+𝑡∑︁
+𝑖=1
+𝜖𝑇 (ℎ(𝑖−1)) + 1
+2𝑡
+𝑡∑︁
+𝑖=1
+𝑑HΔH(D𝑆, D𝑇 )��
+�
++ 𝜆∗(𝑖−1)
+(10)
+Where,
+𝜆∗(𝑡−1) = 1
+𝑡
+𝑡∑︁
+𝑖=1
+[𝜖𝑇 (ℎ(𝑖−1)) + 𝜖𝑆 (ℎ(𝑖−1))]
+(11)
+In Theorem 1.4, we can learn an incremental model by reduc-
+ing the empirical 𝐻−distance (a measure of domain divergence)
+between unlabelled source and target domain data. The theorem is
+an incremental extension of the work by Blitzer et al. [2, 3].
+Theorem 1.4. Let ˆ
+U (𝑡)
+𝑇
+be the empirically estimated unlabelled
+target distribution and ˆ
+U (𝑡)
+𝑠 (𝑡) be the empirical unlabelled source empir-
+ical distribution. Let 𝑑HΔH be a symmetric hypothesis space distance.
+𝑚′
+𝑖 is the size of the unlabelled target and source samples, and �� is the
+Vapnik–Chervonenkis dimension of the current hypothesis. Then for
+every ℎ ∈ H for a probability at least 1 − 𝛿,
+𝜖𝑇 (ℎ(𝑡)) ≤ 1
+𝑡
+𝑡∑︁
+𝑖=1
+�
+ˆ𝜖𝑆 (𝑖) (ℎ(𝑖)) + 1
+2𝑑HΔH( ˆ
+U (𝑡)
+𝑆 (𝑖), ˆ
+U (𝑡)
+𝑇 )
+�
++ 1
+𝑡
+𝑡∑︁
+𝑖=1
+���
+�
+4
+�
+�
+2𝑑 log(2𝑚′
+𝑖) + log( 4
+𝛿 )
+𝑚′
+𝑖
+���
+�
++ 𝜆∗(𝑡−1)
+(12)
+Where,
+𝜆∗(𝑡−1) = 1
+𝑡
+𝑡∑︁
+𝑖=1
+[𝜖𝑇 (ℎ(𝑡)) + 𝜖𝑆 (ℎ(𝑡))]
+(13)
+2
+RELATED WORKS
+This section explores current works used to learn an autonomous
+task incremental learning agent.
+Li et al. [11] propose a neural network-based approach (Learn-
+ing without forgetting) to carry out task incremental learning with
+minimal increase in parametric space size while satisfying low data
+resource conditions. The proposed method learns new target pri-
+vate class mappings by adding new neurons to the output layer of
+a classification network. The goal of the approach is to retain the
+
+classification performance for the previous tasks while incremen-
+tally learning new tasks or classes. The authors use distillation loss
+[7] to minimize catastrophic forgetting.
+Rebuffi et al. [15] propose a supervised incremental learning
+approach (Incremental Classifier and Representation Learning) that
+uses nearest mean matching and class-wise representatives from
+the input data. The authors update the representative instance sets
+(referred to as exemplars) using samples from the incremental input
+data batches. In our work, we address an unsupervised source-
+free approach to overcome the limitation of storing representative
+examples and the need to label incoming incremental data.
+Hoffman et al. [8] provide a supervised approach (continual man-
+ifold adaptation) to learn a low dimensional embedding subspace
+for incoming target data. The work update parametric kernels to
+model an evolving target task distribution. Using a kernel-based
+approach is computationally intensive if the target dataset size is
+large, which limits the scalability of the approach.
+Kundu et al. [10] propose a source-free class incremental learn-
+ing approach that updates a model in a non-stationary environment.
+The authors provide a way to learn a target model with private and
+shared classes but assume known target classes at the time of incre-
+mental domain adaptation. Our work is an incremental extension
+of this work. Also, the work addresses domain shift compensation
+using L2 regularization, which fails to account for unknown classes.
+We address these challenges using distillation loss to accommodate
+future target private classes and an adversarial domain confusion
+loss to minimize domain shift for a non-stationary target domain.
+We compared our approach to unsupervised domain adapta-
+tion methods. Ganin et al. propose the domain adversarial neural
+network (DANN) [6, 21]. Domain adaptation methods cannot com-
+pensate for non-stationary source distribution and do not provide
+the ability to add target private classes (open-set problem setting).
+We compare our work with DANN combined with a target private
+classifier. Due to the few labeled samples available for target pri-
+vate classes, it cannot learn an optimal hypothesis and has low
+predictive accuracy.
+3
+OUR APPROACH
+Our approach is divided into two stages – foresighted learning and
+task incremental update. In the foresighted learning stage, an agent
+learns a generative model of the source data feature space. Fore-
+sighted learning helps the agent to generate representative samples
+of past data for future incremental model updates. In the task in-
+cremental learning stage, the agent updates its internal model state
+using unlabeled target data and a single labeled sample for target
+private classes.
+3.1
+Foresighted learning
+This section describes the foresighted learning stage. This stage
+aims to identify tight class-wise clusters in feature posterior distri-
+bution using Gaussian estimation.
+We denote the feature extractor function as 𝑓𝑠 and the classifier
+function as 𝑔𝑠, which maps the feature extractor output to a |𝐶𝑠 +1|-
+class label space (where 𝐶𝑠 is the source task label set size). The
+latent space is denoted by U. We minimize cross-entropy loss (𝑙𝑐𝑒)
+Figure 1: TIDo architecture: Proposed method architecture
+for task incremental learning architecture.
+to learn 𝑔𝑠.
+𝑙𝑐𝑒 =
+E
+(𝑥𝑠,𝑦𝑠)∼D (𝑡)
+𝑠
+𝑙𝑐𝑒 (𝑔𝑠 · 𝑓𝑠 (𝑥𝑠),𝑦𝑠)
+(14)
+Cross-entropy loss ensures discriminative decision boundaries in
+the latent feature space but leads to over-confident predictions. To
+generate representative samples for source distribution for future
+iterations, we minimize category bias by penalizing over-confident
+prediction. We achieve this by identifying out-of-distribution (OOD)
+samples. Re-using the trained base model to classify unknown
+classes leads to negative learning, i.e., and misclassification of in-
+stances belonging to unknown classes as one of the known classes.
+This is due to the inherent generalization bias of the source model.
+Kundu et al. [10] suggest detecting OOD instances to identify in-
+stances belonging to unknown classes. This is based on the under-
+standing that instances from unknown classes lie in low-density
+regions of the instances of the shared classes. Kundu et al. [10]
+achieve this by mapping the source instances to a latent space with
+an underlying global prior distribution given by N (𝜇, 𝜎). Next, the
+instances from the target domain which lie beyond the 3𝜎 range
+were considered to belong to unknown classes.
+We use a class separability objective L𝑠1 to enforce the class-wise
+features to attain higher affinity to the class-wise prototypes.
+L𝑠 = L𝑠1 + L𝑠2
+(15)
+L𝑠1 :
+E
+(𝑥𝑠,𝑦𝑠)∼D𝑠
+− log
+�
+exp(P𝑦𝑠
+𝑠 (𝑢𝑠))
+�
+𝑐 ∈𝐶𝑠 exp(P𝑐𝑠 (𝑢𝑠))
+�
+(16)
+L𝑠2 :
+E
+(𝑥𝑠,𝑦𝑠)∼D𝑠
+𝑙𝑐𝑒 (𝜎(𝑔(𝑡)
+𝑠
+�𝑓 (𝑡)
+𝑠
+(𝑥𝑠),𝜏),𝑦𝑠)
++
+E
+(𝑢𝑛,𝑦𝑛)∼D𝑛
+𝑙𝑐𝑒 (𝜎(𝑔(𝑡)
+𝑠
+(𝑢𝑛),𝜏),𝑦𝑛)
+(17)
+
+Foresighted source training
+(t = 0)
+U(t)
+Source
+(Predicted
+Gaussian
+feature f(t)
+Base source
+class labels)
+prototype
+extractor
+classifier
+(Source data)
+space
+Task incremental learning
+(t > 0)
+:
+u(t)~ U(t)
+Incremental
+(Proxy source
+source classifier
+ (IC,l +1)th
+samples)
+f(t)
+f(t)
+Feature
+Feature
+encoder
+decoder
+(Predicted
+class labels)
+Target
+8/0)
+feature f(t)
+Ct
+Incremental
+extractor
+(Target data)
+target classifier
+U(t+1)
+Updated prototype
+d(t)
+Yd
+space
+Doman
+(Predicted
+classifier
+domain labels)Where 𝜎 denotes distillation soft loss [7],
+𝜎(z,𝜏) =
+𝑒
+𝑧
+𝜏
+�𝑒
+𝑧
+𝜏
+(18)
+D𝑛 is the distribution of the negative samples, and (𝑢𝑛,𝑦𝑛) rep-
+resents the negative samples with 𝑦𝑛 being the (|𝐶𝑠 | + 1)𝑡ℎ class.
+Since we don’t need distillation loss for this stage, we set 𝜏 = 1.
+3.2
+Task incremental update
+In this section, we describe the domain incremental update stage
+of the proposed method. We use the learned prototypes and the
+unlabeled target domain data to incrementally update the base clas-
+sifier for the target task. In this stage, we use the U−space guides
+as shared class cluster centroids and single target private samples
+as the target private class cluster centroids. We use an encoder-
+decoder approach to fine-tune the U−space to accommodate target
+private guides. This way, we learn a U𝑡ℎ−space which is used to
+represent previous iteration samples.
+Several discrepancy metrics have been proposed to match the
+moments of the shared class instances from source and target dis-
+tributions. Adversarially trained domain discriminators are used to
+reducing the empirical hypothesis distance (𝑑H∇H) between the
+source and target distributions, which has been shown to reduce
+the distance between the source and target distributions.
+We learn the guides for the target domain, U (𝑡+1) using the
+source prototype space U. Using fixed source guides for target
+space reduces flexibility in accommodating target private classes.
+We initialize the U (𝑡+1) guides: 𝑣𝑐𝑔 = 𝑓𝑒 (𝜇𝑐𝑠 )∀𝑐 ∈ 𝐶𝑠 and 𝑣𝑐𝑔 =
+ˆ𝑥 (𝑡)
+𝑡
+∀𝑐 ∈ 𝐶
+′
+𝑡 and calculate confident samples B𝑐
+𝑡 which are pseudo-
+labelled using the guides (𝑘). We use a domain projection auto-
+encoder to enable mobility of guides explicitly. The target domain
+contains private class instances which position themselves in the
+low-density regions of the U𝑡+1−space. We use a reconstruction
+loss and L2-norm to maintain the previously learned source guide
+space (U−space) semantics. By training the auto-encoder layers
+using the gradient from the classifier and domain discriminator, we
+adversarially train U (𝑡+1)−space.
+The U (𝑡+1) guides are aligned using the adversarial domain
+confusion loss:
+L𝑑 : −𝑑H∇H(𝑣𝑡, 𝑣𝑐
+𝑔)
+(19)
+In order to learn an efficient domain projection 𝑓𝑒 : U → U (𝑡+1)
+and 𝑓𝑑 : U (𝑡+1) → U we use reconstruction error similar to an
+auto-encoder. We also use distillation loss with 𝜏 = 2 to ensure low
+catastrophic forgetting for the previously learned shared classes.
+L𝑟 = L𝑟1 + L𝑟2
+(20)
+L𝑟1 :
+E
+(𝑢𝑐𝑠 )∼P𝑐𝑠
+𝑙𝑐𝑒 (𝜎( ˆ𝑦(𝑢𝑐
+𝑠 ),𝜏),𝑐)
+(21)
+L𝑟2 :
+E
+(𝑢𝑐𝑠 )∼P𝑐𝑠
+𝑙2(𝑓𝑑 · 𝑓𝑒 (𝑢𝑐
+𝑠 ),𝑢𝑐
+𝑠 )2
+(22)
+To learn new target private classes, we apply cross-entropy loss to
+target confident samples:
+L𝑐 :
+E
+(𝑥𝑡 )∼B𝑐
+𝑡
+𝑙𝑐𝑒 ( ˆ𝑦(𝑣𝑡),𝑐), ∀𝑐 ∈ 𝐶𝑡
+(23)
+4
+ALGORITHM
+Algorithm 1 outlines the task incremental update implementation.
+We initialize the source generative distribution using the prototypes
+from the previous stage (line 2). To enable the mobility of guides,
+we train an auto-encoder network 𝑓𝑑 (𝑓𝑒 (·)) (line 3-7). We use an
+𝐿2-norm loss as a reconstruction error to train the auto-encoder.
+Since we want the incremental learning agent to learn new
+classes from the target task, we need to update the guides to include
+new target class cluster guides. To this effect, we use a single labeled
+instance (assumed to be available) from each target task class as
+target class guides (line 13).
+Algorithm 1 Task Incremental Learning algorithm
+1: Require: Target samples D𝑡, Gaussian Prototypes P𝑐𝑠 ,
+model parameters
+𝜃𝑓 (𝑡)
+𝑠
+,𝜃𝑔(𝑡)
+𝑠 ,𝜃𝑓 (𝑡)
+𝑡
+,𝜃𝑔(𝑡)
+𝑡 ,𝜃𝑓 (𝑡)
+𝑒
+,𝜃𝑓 (𝑡)
+𝑑
+,𝜃𝑑 (𝑡) , training
+sample size 𝑁, percentage of confident samples 𝑛
+2: Initialize: 𝜃𝑓 (𝑡)
+𝑡
+← 𝜃𝑓 (𝑡)
+𝑠
+3: repeat
+4:
+Obtain a mini-batch of proxy-source samples 𝑆 = {u𝑐𝑠 ∼
+P𝑐𝑠 : 𝑐 ∈ C𝑠}
+5:
+𝜃𝑓 (𝑡)
+𝑒
+← 𝜃𝑓 (𝑡)
+𝑒
++ Adam{𝑓 (𝑡)
+𝑒
+}(−∇ 1
+|𝑆 |
+�
+u𝑐𝑠 ∈𝑆 𝑙2(𝑢𝑐𝑠, 𝑓𝑒 (𝑢𝑐𝑠 ))2)
+6:
+𝜃𝑓 (𝑡)
+𝑑
+← 𝜃𝑓 (𝑡)
+𝑑
++ Adam{𝑓 (𝑡)
+𝑑
+}(−∇ 1
+|𝑆 |
+�
+u𝑐𝑠 ∈𝑆 𝑙2(𝑢𝑐𝑠, 𝑓𝑑 (𝑢𝑐𝑠 ))2)
+7: until Convergence
+8: Loss ← [L𝑟1, L𝑟2, L𝑐, L𝑑]
+9: Opt ← [Adam{𝑓 (𝑡)
+𝑒
+,𝑓 (𝑡)
+𝑑
+,𝑔(𝑡)
+𝑡
+}, Adam{𝑓 (𝑡)
+𝑒
+,𝑓 (𝑡)
+𝑑
+}, Adam{𝑓 (𝑡)
+𝑡
+,𝑔(𝑡)
+𝑡
+},
+10:
+Adam{𝑓 (𝑡)
+𝑡
+}, Adam{𝑓 (𝑡)
+𝑒
+,𝑓 (𝑡)
+𝑡
+}]
+11: repeat
+12:
+iter ← iter+1, cur ← iter mod 5
+13:
+v𝑐𝑔 ← 𝑓𝑒 (𝜇𝑐𝑠 )∀𝑐 ∈ C𝑠, v𝑐𝑔 ← 𝑓𝑡 ( ˜𝑥𝑐
+𝑡 )∀𝑐 ∈ C
+′
+𝑡
+14:
+for 𝑢𝑐𝑠 ∼ P𝑐𝑠 : 𝑐 ∈ C𝑠 do
+15:
+𝑣𝑐𝑠 ← 𝑓𝑒 (u𝑐𝑠); ˆ𝑢𝑐𝑠 ← 𝑓𝑑 (𝑣𝑐𝑠 );
+16:
+ˆ𝑦 ← 𝑔𝑠 ( ˆ𝑢𝑐𝑠 )|𝑐 ∈C𝑠 ∥𝑔𝑡 (𝑣𝑐𝑠 )
+17:
+L𝑟1 + 𝑙𝑚𝑠𝑒 ( ˆ𝑢𝑐𝑠,𝑢𝑐𝑠 )
+18:
+L𝑐 ← L𝑐 + 𝑙𝑐𝑒 (𝜎( ˆ𝑦𝑠),𝑐)
+19:
+end for
+20:
+for x𝑡 ∈ {x𝑡 ∼ D𝑡 } do
+21:
+𝑣𝑡 ← 𝑓𝑡 (x𝑡); u𝑡 ← 𝑓𝑑 (𝑣𝑡); ˆ𝑦𝑡 ← 𝑔𝑠 ( ˆ𝑢𝑡)|𝑐 ∈C𝑠 ∥𝑔𝑡 (𝑣𝑡)
+22:
+𝑑 ← min𝑐 ∈C𝑡𝑙2(𝑣𝑡, 𝑣𝑐𝑔);𝑘 ← arg min(𝑑)
+23:
+L𝑟2 ← L𝑟2 + 𝑙𝑚𝑠𝑒 (u𝑡, ˆ𝑣𝑡)
+24:
+end for
+25:
+for 𝑢𝑐𝑠 ∼ P𝑐𝑠 , x𝑡 ∈ {x𝑡 ∼ D𝑡 } do
+26:
+𝑣 ← 𝑓𝑡 (x𝑡); ˆ𝑦𝑑 ← 𝑑([𝑢𝑐𝑠, 𝑣]);
+27:
+L𝑑 ← L𝑑 + 𝑙𝑐𝑒 ( ˆ𝑦𝑑, [0, 1])
+28:
+end for
+29:
+if reached the end of an epoch then
+30:
+UpdateTaskIncrementalGradients(Loss,Opt)
+31:
+Label samples in D𝑡 using guides {𝑣𝑐𝑔 : 𝑐 ∈ C𝑡 }
+32:
+P𝑐
+𝑡 ← Gaussian Prototypes obtained using pseudo-label
+target samples
+33:
+end if
+34: until Convergence
+
+In line 14-19, we fine-tune the feature extractor network using
+samples from the class-wise prototype distributions. We assume an
+open-set problem setting, and the target domain data is assumed
+to contain instances from the source domain. To learn a single
+classifier for source and target tasks, we pass the unlabeled target
+instances and source domain samples to both the source domain
+classifier and target domain classifier. In line 20-24, we obtain the
+pseudo-labels for the target domain data along with the predictions
+from the updated joint classifier.
+To align the source and target domain distributions, we use a
+domain discriminator network, which trains the feature extractor
+adversarially along with the joint classifier loss (line 25-28). Fi-
+nally, we update the parameters of all the components of the task
+incremental network and update the prototypes (line 29-33). The
+updated prototypes will serve as the source prototypes in the next
+iteration, along with the new source domain (if any) to generate
+the samples. The gradient update algorithm (algorithm 2) provides
+the gradient update step for the task incremental learning network
+components.
+Algorithm 2 Gradient update algorithm
+1: Require: Model parameters, Loss, Opt
+2: 𝜃𝑓 (𝑡)
+𝑡
+← 𝜃𝑓 (𝑡)
+𝑡
++ Adam{𝑓 (𝑡)
+𝑡
+}(−∇ 1
+𝑁
+� L𝑐)
+3: 𝜃𝑑 (𝑡) ← 𝜃𝑑 (𝑡) − Adam{𝑑 (𝑡),𝑓 (𝑡)
+𝑡
+}(−∇ 1
+𝑁
+� L𝑑)
+4: 𝜃𝑓 (𝑡)
+𝑡
+← 𝜃𝑓 (𝑡)
+𝑡
+− Adam{𝑑 (𝑡),𝑓 (𝑡)
+𝑡
+}(−∇ 1
+𝑁
+� L𝑑)
+5: 𝜃𝑓 (𝑡)
+𝑡
+← 𝜃𝑓 (𝑡)
+𝑡
+− Adam{𝑓 (𝑡)
+𝑡
+}(−∇ 1
+𝑁
+� L𝑟1)
+6: 𝜃𝑓 (𝑡)
+𝑒
+← 𝜃𝑓 (𝑡)
+𝑒
++ Adam{𝑓 (𝑡)
+𝑒
+,𝑓 (𝑡)
+𝑑
+}(−∇ 1
+𝑁
+� L𝑟1)
+7: 𝜃𝑓 (𝑡)
+𝑑
+← 𝜃𝑓 (𝑡)
+𝑑
++ Adam{𝑓 (𝑡)
+𝑒
+,𝑓 (𝑡)
+𝑑
+}(−∇ 1
+𝑁
+� L𝑟1)
+8: 𝜃𝑓 (𝑡)
+���
+← 𝜃𝑓 (𝑡)
+𝑡
++ Adam{𝑓 (𝑡)
+𝑡
+}(−∇ 1
+𝑁
+� L𝑟2)
+9: 𝜃𝑓 (𝑡)
+𝑑
+← 𝜃𝑓 (𝑡)
+𝑑
++ Adam{𝑓 (𝑡)
+𝑑
+}(−∇ 1
+𝑁
+� L𝑟2)
+5
+EXPERIMENTS AND RESULTS
+5.1
+Incremental object detection
+We evaluated our proposed method to develop an agent to learn an
+incremental object detection task. Object detection in real-world
+images has been used as a benchmark task for several computer
+vision problems. In order to evaluate our approach, we created an
+incremental learning task that requires learning a target domain
+classification model given an initial source domain dataset.
+We use an imaging dataset with multiple object classes and mul-
+tiple domains. We select one of the domains as the initial labeled
+source domain while the rest are considered unlabelled target do-
+mains. Our goal is to learn a common model for all the domains
+observed by the model.
+5.1.1
+Dataset. We used the office-31 object recognition dataset
+[16] which contains 4652 images from 3 domains and 31 classes.
+The domains of the dataset are web (Amazon), DSLR, and webcam.
+The domain details are as follows:
+• Amazon (A): These are images taken from Amazon [1]. They
+are mostly taken in a studio setting with a clear background
+and standardized lighting. We have an average of 90 images
+per class.
+• Digital single-lens reflex camera (D): This domain contains
+high-resolution images with a pixel resolution of (4288 ×
+2848). Each class contains images of 5 objects taken from 3
+different angles each. In total, the domain dataset has 423
+images.
+• Webcam (W): This domain contains low-resolution poor-
+lighting images with a pixel resolution of (640 × 480). The
+dataset contains 5 objects per class with 3 angle images each.
+In total, we have 795 images. These images show consider-
+able noise and color as well as white balance artifacts.
+The 31 categories are desk lamp, computer, tile cabinet, backpack,
+bike, bike helmet, mouse, mug, notebook, pen, phone, printer, book-
+case, bottle, calculator, desk chair, headphones, keyboard, laptop,
+letter tray, mobile phone, monitor, projector, puncher, ring binder,
+ruler, scissors, speaker, stapler, tape, and trash can.
+Table 1: Incremental object detection learning task for eval-
+uating task incremental learning methods.
+Index
+Inputs
+𝑡 + 0
+Source: desk lamp, computer, cabinet, backpack, bike
+Target: desk lamp, computer, cabinet, backpack, bike,
+bike helmet, mouse, mug, notebook, pen
+𝑡 + 1
+Source: phone, printer, bookcase
+Target: phone, printer, bookcase, bottle, calculator
+𝑡 + 2
+Source: desk chair, headphones, keyboard, laptop, tray
+Target: desk chair, headphones, keyboard, laptop, tray,
+mobile phone, monitor, projector
+𝑡 + 3
+Source: ruler, scissors, speaker, stapler
+Target: ruler, scissors, speaker, stapler, tape, trash can
+𝑡 + 4
+Source: ∅, Target: puncher, ring binder
+To evaluate the response of our approach to both open-set differ-
+ences between the source and target domains and non-stationary
+source and target domains, we structure the experiment as follows
+• The domains are introduced incrementally to the model, and
+the data from the domains (belonging to the same class) is
+assumed to be sampled from a single non-stationary distri-
+bution
+• In every iteration we introduce a set of shared classes 𝐶𝑡𝑠 and
+target private classes 𝐶
+′(𝑡)
+𝑡
+• The foresighted learning network learns the source guides
+every time a new labeled source domain is introduced. For
+the 𝑙𝑡ℎ iteration, 𝑓 (𝑡+𝑙)
+𝑠
+is trained using data sampled from
+combined data from 𝑢 (𝑡)
+𝑠
+and 𝑥 (𝑡+𝑙)
+𝑠
+• The target data in every iteration assumed to contain at least
+one target private class (i.e. 𝐶
+′(𝑡+1) ≠ ∅)
+Like our method, iCARL and CIDA use prototype learning to
+enable source-free incremental learning. Although this is one of
+the desiderata of incremental learning, iCARL requires labeled
+target data. This makes it unsuitable for direct application to the
+unsupervised task incremental learning problem setting. Our work
+is motivated by CIDA, and we aim to improve upon the existing
+
+method by using an adversarial domain discrepancy estimation
+instead of the previously proposed alignment loss [10]. We also
+extend it to an incremental learning context. DANN and CMA
+provide a way to carry out unsupervised learning. We compare our
+approach to the aforementioned methods to evaluate the efficiency
+of end-to-end trainable adversarial methods for task incremental
+learning.
+We evaluate our approach using the incremental learning task
+outlined in table 2. We evaluate the performance of a given approach
+at every time point using total accuracy and target private class
+accuracy. We compare our proposed approach (TIDo) to unsuper-
+vised domain adaptation methods (DANN [19]), class incremental
+domain adaptation methods (iCARL [15], CIDA [10]) and continual
+learning methods (LwF-MC [11], CMA [8]). For methods without
+a provision to incrementally add new classes, we trained a tar-
+get private classifier (TPC). CIDA-C refers to storing and using
+combined target task data from past increments; this makes this a
+pseudo-incremental learning approach.
+For (𝑡 +4)𝑡ℎ iteration of the experiment (refer to table.2), we have
+no source dataset. We do not update the source classifier for the
+DANN approach for this iteration as the approach requires source
+data to update. Also, iCARL and LwF-MC methods are supervised
+methods and require labeled target data. We use 5% labeled samples
+(available to the rest of the methods for few-shot learning) to serve
+as the labeled target data.
+Table 2: Incremental disease prediction learning task for
+evaluating task incremental learning.
+Index
+Inputs
+𝑡 + 0
+Source: CN, AD, Target: CN, MCI, AD
+𝑡 + 1
+Source: CN, MCI, AD, Target: CN, MCI, AD
+𝑡 + 2
+Source: ∅, Target: EMCI
+𝑡 + 3
+Source: ∅, Target: AD, CN, MCI
+5.2
+Incremental disease staging
+We apply our proposed approach to create an incremental disease
+staging agent. We design an incremental learning task for learning
+an Alzheimer’s disease prediction model.
+Alzheimer’s disease staging is a non-trivial process with overlap-
+ping subjective categories. Due to the absence of a standard staging
+model for neurological diseases like AD, stage-wise labeled data
+may not be available at a single time point. We propose using task
+incremental learning to carry out source-free few-shot incremental
+updates to a base clinical model. to test our class incremental hy-
+pothesis, we aim to update a binary classification AD/HC model
+to predict intermediate stages of early mild cognitive impairment
+(EMCI) and late mild cognitive impairment (LMCI). To test our
+domain incremental hypothesis, we update the model using target
+data from a different domain (containing both shared and target
+private classes).
+We evaluate the method using Alzheimer’s disease data from
+multiple domains, different populations, and different label sets.
+We use Alzheimer’s disease-specific datasets in this experiment
+– Alzheimer’s Disease Neuroimaging Initiative (Data used in the
+preparation of this article were obtained from the Alzheimer’s Dis-
+ease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu))
+[4] and Alzheimer’s Disease Neuroimaging Initiative – AIBL (Data
+was collected by the AIBL study group. AIBL study methodology
+has been reported previously [5]).
+We create a region of interest (ROI) image dataset using MRI
+images from ADNI and AIBL domains. The MRI images were pre-
+processed using a processing pipeline. Due to the relatively low
+number of samples in the MRI imaging dataset, we augment the
+dataset using the extracted ROIs from the input images [9, 12].
+For example, for the ADNI-1 dataset, we had 841 samples (200
+healthy control data, 230 AD data, and 411 MCI data); after ROI
+augmentation, we had 3364 data instances.
+We used ROI data from left and right Hippocampus regions and
+left and right temporal lobes. The extracted ROI patches had the
+dimension (64×64×64). Individual ROI patches were labeled using
+the sample label from which they were extracted.
+5.3
+Discussion
+We proposed a source-free task incremental learning method for
+an agent to learn a task incrementally. We observed that our ap-
+proach enabled an autonomous agent to learn a near-optimal target
+hypothesis with very low catastrophic forgetting for both class
+incremental and domain incremental applications. Since our ap-
+proach is source-free, we have a very low memory complexity and
+can update a model incrementally using few-shot learning.
+Our results show comparable or improved performance of our
+approach compared to class incremental learning (CIDA-C [10]).
+We show that our approach can achieve similar performance with-
+out storing past target training data. This reduces the memory
+complexity of our approach drastically.
+We performed a comparative analysis of the task incremental
+problem using unsupervised domain adaptation, continual learning,
+and class incremental methods. [15] propose a supervised incremen-
+tal learning approach that uses representation learning and learned
+class-wise exemplars from the input data. The authors updated the
+exemplars incrementally to learn using new classes and instances.
+Storage of class-wise exemplar instances and the need for labeled
+samples from both source and target domains for model upgrades
+make the approach unsuitable for scalable incremental learning.
+We eliminate the need to store exemplar instances by generating
+a distribution estimation and storing class-wise guides, thereby
+rendering our approach source-free.
+We compared our approach to continual manifold adaptation
+(CMA). CMA does not apply to open-set transfer learning settings.
+Hence, we learn a target private classifier (TPC) to achieve the
+task incremental task. Due to the few-shot configuration for target
+private instances, TPC risk is large. Table 3 and 4 show a high loss
+for target private classes, except (𝑡 + 2)𝑡ℎ iteration for Alzheimer’s
+disease prediction (57.82±4.60%) which is because the target private
+class (EMCI) is a sub-category of MCI, which has been observed
+by the classifier in the previous iterations (𝑡 + 0, 𝑡 + 1) for related
+domains (ADNI 3 and ADNI 2).
+
+Table 3: Office-31 incremental object recognition task: comparison of our proposed method (TIDo) with existing incremental
+learning, continual learning, and unsupervised domain adaptation methods for a few-shot labeled target (5%), unlabelled
+target domain data, and labeled source domain data.
+A→W→D
+DANN-TPC
+iCARL
+CMA-TPC
+CIDA-C
+LwF-MC
+TIDo
+Index
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+𝑡 + 0
+57.17
+15.91
+72.91
+48.21
+61.22
+28.01
+77.12
+72.92
+62.12
+39.11
+75.82
+75.12
+𝑡 + 1
+54.12
+12.67
+75.01
+51.23
+65.18
+27.43
+72.23
+70.27
+62.25
+35.24
+72.81
+73.63
+𝑡 + 2
+45.12
+19.01
+64.21
+43.91
+56.92
+34.22
+70.14
+69.22
+54.50
+31.12
+71.12
+72.22
+𝑡 + 3
+40.13
+20.87
+63.34
+43.31
+50.85
+26.75
+69.92
+69.91
+54.23
+30.03
+70.75
+71.29
+𝑡 + 4
+38.23
+31.01
+60.23
+43.33
+44.23
+29.65
+73.29
+78.01
+52.15
+35.01
+72.23
+72.16
+D→A→W
+DANN-TPC
+iCARL
+CMA-TPC
+CIDA-C
+LwF-MC
+TIDo
+Index
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+𝑡 + 0
+51.66
+20.02
+73.22
+53.40
+68.56
+37.12
+85.63
+82.91
+75.34
+55.27
+88.26
+84.92
+𝑡 + 1
+50.23
+19.10
+72.81
+51.23
+66.30
+34.16
+81.64
+79.22
+76.26
+56.72
+84.54
+82.76
+𝑡 + 2
+44.66
+21.91
+71.81
+52.42
+52.86
+21.96
+76.72
+72.01
+64.48
+53.29
+74.86
+73.66
+𝑡 + 3
+43.36
+17.64
+71.74
+56.81
+51.47
+18.58
+72.12
+70.26
+64.43
+44.02
+70.49
+69.06
+𝑡 + 4
+41.26
+24.63
+75.41
+65.19
+54.63
+28.43
+76.72
+73.03
+65.6
+47.7
+71.53
+70.44
+W→D→A
+DANN-TPC
+iCARL
+CMA-TPC
+CIDA-C
+LwF-MC
+TIDo
+Index
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+𝑡 + 0
+57.29
+20.31
+75.4
+65.2
+67.22
+18.20
+84.82
+82.22
+74.99
+67.48
+83.92
+83.20
+𝑡 + 1
+58.25
+18.17
+74.59
+64.29
+67.18
+19.02
+79.17
+82.61
+73.62
+68.01
+81.18
+83.19
+𝑡 + 2
+56.72
+37.45
+76.25
+77.50
+69.93
+28.21
+85.14
+82.22
+67.91
+56.02
+78.03
+80.21
+𝑡 + 3
+54.29
+26.22
+76.43
+77.02
+65.18
+24.59
+82.91
+79.81
+65.23
+51.43
+77.78
+77.50
+𝑡 + 4
+47.49
+18.25
+75.71
+77.32
+62.03
+26.49
+78.02
+79.91
+70.15
+57.41
+77.03
+76.47
+5.4
+Ablation studies
+We will now explore the effect of different components of our
+proposed approach.
+Effectiveness of Gaussian estimation and OOD sample pre-
+diction: Similar to previous approaches, we analyze the sensitivity
+of the hyper-parameter 𝑘 to observe the effects of modifying the
+labeling criteria for negative samples in the foresighted learning
+stage. To verify that our Gaussian estimates are accurate, we em-
+pirically tested the efficiency of the assumed confidence interval
+(3-𝜎). Figure 2a shows that 3-𝜎 provided the maximum predictive
+accuracy and best captured the source distribution characteristics.
+Effect of balancing source and target unlabelled data: We
+used a balanced source (𝑁𝑠𝑟𝑐) and target (𝑁𝑛𝑒𝑔) domain dataset
+to train our baseline model. We test the robustness of our model
+to imbalanced data by varying the 𝑁𝑠𝑟𝑐/𝑁𝑛𝑒𝑔 ratio by ±0.5. We
+measure the sensitivity of the source and target domain ratio in
+figure 2b and observe that the proposed approach is robust against
+data imbalance.
+Challenging one-shot learning: We observe the efficiency of
+our incremental learning approach by varying the ratio of samples
+in the target private classes to the number of shared class samples
+(|𝐶
+′
+𝑡 |/|𝐶𝑡). Figure 2d shows the sensitivity of this ratio. Even though
+a larger number of target private samples improves the accuracy
+of private guides and private class prediction, prediction accuracy
+reduces due to the inability of the target classifier to converge under
+less target shared class data.
+Effect of class separation loss: We carry out the ablation study
+by removing the class separation loss. We learn the post-increment
+accuracy of the target domain classifier without applying the class
+separation loss (L𝑠1). We observe that the average prediction accu-
+racy without the loss minimization was 83.45% compared to 92.12%
+using the class separation loss.
+6
+CONCLUSION
+In this work, we proposed an approach for an autonomous agent to
+learn a task incremental learning model in a non-stationary envi-
+ronment. We explored a one-shot learning approach to reduce the
+need for collecting labeled data to incrementally update a model.
+Using a source-free approach, we were able to learn aligned target
+private prototype guides and learn with very few target-labeled
+samples. One of the limitations of our approach is the possibility
+of overfitting after a given number of incremental iterations. We
+aim to address this limitation in our future work by exploring selec-
+tive forgetting using recurrent network-based approaches. Another
+
+Table 4: Incremental disease prediction task: comparison of our proposed method (TIDo) applied to Alzheimer’s disease pre-
+diction with existing incremental learning, continual learning, and unsupervised domain adaptation methods for a few-shot
+labeled target (5%), unlabelled target domain data and labeled source domain data. All(%) is the average accuracy for all the
+classes, Priv(%) is the average accuracy for private classes
+ADNI 1 (CN/AD) → ADNI 2 → AIBL → ADNI GO → ADNI 3
+iCARL
+DANN-TPC
+CMA-TPC
+Index
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+𝑡 + 0
+90.01±3.08
+88.01±1.29
+93.44±2.25
+54.91±1.04
+80.52±1.70
+52.28±1.27
+𝑡 + 1
+84.91±2.58
+-
+91.81±2.02
+-
+83.78±2.01
+-
+𝑡 + 2
+80.92±3.66
+71.22±3.02
+71.25±4.89
+57.05±2.28
+82.67±1.18
+67.82±4.60
+𝑡 + 3
+78.32±4.81
+70.10±4.12
+73.72±4.21
+56.81±2.56
+84.19±1.29
+63.91±5.67
+CIDA-C
+LwF-MC
+TIDo
+Index
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+All (%)
+Priv (%)
+𝑡 + 0
+89.32±1.17
+82.90±3.16
+72.91±0.98
+56.68±1.10
+91.42±0.79
+86.91±1.22
+𝑡 + 1
+90.76±2.01
+-
+79.91±2.17
+-
+90.08±2.48
+-
+𝑡 + 2
+89.62±1.57
+90.22±2.78
+77.67±2.11
+57.52±4.17
+90.69±2.20
+92.82±1.71
+𝑡 + 3
+90.32±2.89
+88.10±2.11
+78.10±1.07
+59.24±1.70
+92.12±0.91
+91.14±0.88
+(a)
+(b)
+(c)
+(d)
+Figure 2: Sensitivity study results for task incremental learning on incremental disease staging task (for AD): (a) Effectiveness of
+Gaussian estimation and OOD sample prediction (Avg. accuracy% for private and all target classes) (b) Data imbalance robust-
+ness for target task prediction (Avg. accuracy%) (c) Effect of removal of discriminator in the foresighted model (d) Sensitivity
+on the ratio of private class sample size vs. all class sample size
+possible limitation of this work would be the use of Gaussian esti-
+mates to generate replay memory representative samples. We aim
+to explore adversarial methods to generate representative samples
+in our future work.
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+

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+MNRAS 000, 1–?? (2015)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+MeerKAT discovery of 13 new pulsars in Omega Centauri
+W. Chen,1★ P. C. C. Freire,1 A. Ridolfi,3,1 E. D. Barr,1 B. Stappers,2 M. Kramer,1,2 A. Possenti,3
+S. M. Ransom,4 L. Levin,2 R. P. Breton,2 M. Burgay,3 F. Camilo,5 S. Buchner,5 D. J. Champion,1
+F. Abbate,1 V. Venkatraman Krishnan,1 P. V. Padmanabh,1,8,9 T. Gautam,1 L. Vleeschower,2
+M. Geyer,5 J-M. Grießmeier,6,7 Y. P. Men,1 V. Balakrishnan,1 M. C. Bezuidenhout2
+1Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
+2Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK
+3INAF – Osservatorio Astronomico di Cagliari, Via della Scienza 5, I-09047 Selargius (CA), Italy
+4National Radio Astronomy Observatory (NRAO), 520 Edgemont Rd., Charlottesville, VA 22903 USA
+5South African Radio Astronomy Observatory (SARAO), 2 Fir Street, Black River Park, Observatory, Cape Town 7925, South Africa
+6LPC2E - Université d’Orléans / CNRS, 45071 Orléans cedex 2, France
+7Observatoire Radioastronomique de Nançay (ORN), Observatoire de Paris, Université PSL, Univ Orléans, CNRS, 18330 Nancay, France
+8 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
+9 Leibniz Universität Hannover, D-30167 Hannover, Germany
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The most massive globular cluster in our Galaxy, Omega Centauri, is an interesting target for pulsar searches, because of its
+multiple stellar populations and the intriguing possibility that it was once the nucleus of a galaxy that was absorbed into the
+Milky Way. The recent discoveries of pulsars in this globular cluster and their association with known X-ray sources was a hint
+that, given the large number of known X-ray sources, there is a much larger undiscovered pulsar population. We used the superior
+sensitivity of the MeerKAT radio telescope to search for pulsars in Omega Centauri. In this paper, we present some of the first
+results of this survey, including the discovery of 13 new pulsars; the total number of known pulsars in this cluster currently stands
+at 18. At least half of them are in binary systems and preliminary orbital constraints suggest that most of the binaries have light
+companions. We also discuss the ratio between isolated and binaries pulsars and how they were formed in this cluster.
+Key words: Pulsar – Globular cluster – Binary
+1 INTRODUCTION
+Pulsar surveys conducted in Globular clusters (GCs) have yielded
+fruitful rewards in recent decades, with the discovery of a total of
+261 pulsars1. Per unit of stellar mass, GC have three orders of magni-
+tude more pulsars than the Galactic disk. The reason for this is their
+large stellar densities: these prompt stellar interactions (Verbunt &
+Hut 1987) where old, dead Neutron stars (NSs) acquire new main
+sequence (MS) companions. These then evolve, forming X-ray bi-
+naries (also exceptionally numerous in GCs, Clark 1975), where the
+NS is being spun up by accretion of matter from the MS star. When
+the accretion stops, these systems become millisecond pulsar (MSP)
+binaries, which have nearly circular orbits and low-mass companions
+(Bhattacharya & van den Heuvel 1991). Indeed, the pulsar population
+in GCs is dominated by such binaries.
+However, in some cases, additional exchange encounters can orig-
+inate different end products. If they happen during the X-ray binary
+phase, they can disrupt the binary and produce many single and/or
+★ E-mail: [email protected]
+1 https://www3.mpifr-bonn.mpg.de/staff/pfreire/GCpsr.html
+partially recycled pulsars, which are not only slower, but appear to be
+much younger than the GC population (Verbunt & Freire 2014). Such
+encounters can also replace a pulsar’s companion by a much more
+massive degenerate object, resulting in massive, eccentric binary
+MSPs unlike any seen in the Galactic disk (e.g., Freire et al. 2004;
+Lynch et al. 2012; DeCesar et al. 2015; Ridolfi et al. 2021, 2022). If
+the massive companions happen to be stellar-mass black holes, these
+systems could be superb test-beds for fundamental physics (Liu et al.
+2014).
+Such exotic pulsar binaries are generally observed in GCs with
+very dense cores, especially core-collapsed clusters; these are the
+types of environments where each particular binary is likely to go
+through more than one disruptive stellar encounter (Verbunt & Freire
+2014). Thus, the pulsar population in a GC, and the types of binaries
+the pulsars find themselves in, reflects not only its current dynami-
+cal status, but also the cluster’s previous evolution (Benacquista &
+Downing 2013).
+© 2015 The Authors
+arXiv:2301.03864v1  [astro-ph.HE]  10 Jan 2023
+
+2
+W. Chen et al.
+1.1 The Omega Centauri globular cluster
+Omega Centauri (𝜔-Cen, also known as NGC 5139), the largest GC
+in our Galaxy, is a natural target to search for pulsars. It is located
+in the constellation of Centaurus and is 5.2 kpc away from the Sun,
+with an age of 11.52 Gyr (Forbes & Bridges 2010). Besides its size
+and large number of stars, it differs from other GCs because of its
+intricate composition of different populations of stars (Bedin et al.
+2004). This could indicate that 𝜔-Cen is the merger of several clus-
+ters, like the Sagittarius dwarf galaxy (Ibata et al. 1994). Moreover,
+it was found to be rich in calcium and heavy metals (Lee et al. 2009),
+which is a tracer of supernovae explosions. However these materials
+ejected by the explosion could not be sustained by the current grav-
+itational potential of the cluster. This along with the multiple stellar
+populations supports the long established idea that 𝜔-Cen is the relic
+of a former disrupted dwarf galaxy (Hilker & Richtler 2000; Ibata
+et al. 2019).
+Unassociated high-energy emission in GCs is thought to originate
+from MSPs (Venter et al. 2009) as supported by observations, (e.g.
+Abdo et al. 2009). 𝛾-ray emission has been detected in several GCs
+by the Fermi Large Area Telescope (Abdo et al. 2010), including
+those that, at the time, had no previously known pulsars, such as
+𝜔-Cen.
+Additionally, ∼30 unassociated X-ray sources have been found
+within the core of 𝜔-Cen to have luminosities similar to the emission
+of pulsars in other GCs (Henleywillis et al. 2018). However, previous
+searches for pulsars in this cluster turned out to be unsuccessful
+(Edwards et al. 2001; Possenti et al. 2005; Camilo et al. 2015).
+Notwithstanding, in 2019, Dai et al. (2020) carried out a search for
+pulsars using the new Ultra-wide Bandwidth receiver (UWL, Hobbs
+et al. 2020) of the 64-m "Murriyang" radio telescope at Parkes, NSW,
+Australia. This finally allowed the discovery of the first 5 MSPs in
+𝜔-Cen. Among these, PSR J1326−4728B is in an eclipsing binary
+system with a light companion, and it is associated with an X-ray
+source (Henleywillis et al. 2018). The authors suggested that the
+non-detection of pulsars from the previous searches on this cluster
+was caused by the lack of sensitivity of previous surveys. Given the
+large distance of 𝜔-Cen (and other GCs in general) it is clear that
+we are only detecting the very brightest pulsars in them. This means
+that more sensitive telescopes would in principle detect many more
+pulsars in this and other globular clusters.
+1.2 The MeerKAT survey
+The MeerKAT 64-antenna array, located in the Karoo desert in South
+Africa, (Jonas & MeerKAT Team 2016; Camilo 2018) started sci-
+entific observations in late 2019, becoming, by far, the most radio
+sensitive radio telescope in the Southern Hemisphere. Numerous tar-
+geted observations and surveys of pulsars have been carried out since
+and are producing significant results. For example, the TRAnsients
+and PUlsars with MeerKAT (TRAPUM, Stappers & Kramer 2016)
+Large Survey Project aims to increase the total known population
+of pulsars, and discover peculiar binary pulsars that might be suit-
+able for studies of fundamental physics and stellar evolution. This
+project has so far discovered 156 pulsars2. About one-third of these
+(e.g. Ridolfi et al. 2021; Douglas et al. 2022; Ridolfi et al. 2022;
+Vleeschower et al. 2022; Abbate et al. 2022), are found in various
+GCs, including 47 Tucanae (Chen et al., in prep.), a cluster that has
+been searched for over 20 years.
+2 http://www.trapum.org/discoveries/
+Given the small beams produced by the phased interferometer,
+pulsars surveys need a beamformer in order to cover as much sky
+simultaneously as possible. The beam former developed by Barr
+(2018) and Chen et al. (2021) can generate up to 1000 coherent
+beams that cover a significant part of the primary field of view of an
+individual antenna, with high time and frequency resolution. Such
+a large number of beams is essential for blind surveys (which cover
+mostly the Galactic plane), but they are also important for GCs.
+Indeed, even though GCs have a relatively small angular size on
+the sky, we still often need hundreds of beams to cover their half-
+mass radii, which are the regions where most pulsars are likely to
+be located. This is especially true in the case of 𝜔-Cen, which is the
+most massive and has the largest projected size among all known
+GCs.
+In this paper, we report the result of the searches for the new pul-
+sars in two observations of 𝜔-Cen made with MeerKAT. Section 2
+describes the observational parameters and analysis of the recorded
+data. Section 3 presents the new discoveries, while Section 4 dis-
+cusses the uncovered pulsar population and its implications in our
+understanding of the cluster.
+2 OBSERVATIONS AND DATA ANALYSIS
+2.1 Observations
+We carried out two multibeam observations of 𝜔-Cen with MeerKAT
+on 2021 March 21 and 26, as part of the TRAPUM GC survey. The
+cluster was observed for 4 hours in each observation using the L-
+band receivers, which cover the 856–1712 MHz frequency range.
+The coherent beams were synthesised using the Filterbanking Beam-
+Former User-Supplied Equipment (FBFUSE, Barr 2018). A tiling of
+704 coherent beams was generated by mosaic3 to cover the cluster,
+as shown in Figure 1. This was centred at the nominal centre of
+𝜔-Cen, at equatorial coordinates (J2000) 13:26:47.24, −47:28:46.5
+and Galactic coordinates 309.10, 14.97 (Harris 2010), covering a
+circular region of 7.53 arcmin in radius, i.e. roughly twice the size
+of the half-light radius of the cluster. During both observations, 60
+antennas were used. To create a hexagonal tiling, the corresponding
+synthesised beam shape was approximated by an ellipse whose ma-
+jor and minor axis were 20.46 arcseconds and 9.32 arcseconds at the
+middle of the observation on March 21. The method and performance
+of such approximation is discussed in Chen et al. (2021). The March
+26 observation started at almost the same hour angle, so the beam
+shape is very similar.
+For each beam, the observing band was split into 2048 chan-
+nels and recorded every 153.12 𝜇s as filterbank search-mode files
+by Accelerated Pulsar Search User Supplied Equipment (APSUSE)
+computing cluster. Once the observation was over, the frequency
+channels were summed in groups of 16 after being incoherently de-
+dispersed at the dispersion measure (DM) of 97 pc cm−3, about the
+average of the previously known 5 pulsars in 𝜔-Cen. The resulting
+sub-banded data preserved 128 channels, significantly reducing the
+total data volume. With this resolution, the intra-channel smearing
+time is 1.57 ms, at the lowest channel and the upper bound of the
+search DM.
+3 https://github.com/wchenastro/Mosaic
+MNRAS 000, 1–?? (2015)
+
+Discovery of 13 new pulsars in 𝜔-Centauri
+3
+2.2 Sensitivity
+We calculate a minimum detectable flux density of Smin = 10 𝜇Jy for
+our search on a 4-hour observation, following the modified radiome-
+ter equation given by Dewey et al. (1985):
+𝑆min =
+S/N 𝛽 𝑇sys
+EFFT 𝐺√︁𝑛pol 𝐵 Δ 𝑡obs
+√︄
+𝜁
+1 − 𝜁
+(1)
+where S/N is the minimal signal-to-noise of 10 for a valid de-
+tection; 𝛽 is the correction factor of 1.01 to compensate the loss of
+sensitivity during the digitization process; 𝑇sys is the system temper-
+ature of about 26 K4, including the contribution from the receiver
+temperature of 18 K, the sky temperature of about 3.5 K at L-band
+and the combination of the spillover noise and the atmosphere of
+about 4.5 K; EFFT is the FFT search efficiency which is 0.7 accord-
+ing to Morello et al. (2020); G is the combined gain of the array,
+which is 2.65 K Jy−1 when 60 antennas are used; 𝑛pol is the number
+of polarizations, which is 2 while using the total power beamformer
+in TRAPUM observations; 𝐵 is the effective receiver bandwidth of
+about 640 MHz, after removing the channels affected by RFI; Δ𝑡obs
+is the integration time which was 4 hours for both observations; 𝜁 is
+the pulse’s apparent duty cycle, which we consider to be 8% with the
+broadening effects from interstellar medium and the instrument.
+2.3 Data reduction and search pipeline
+The data were transferred to Germany from South Africa in hard
+drives. Searches for periodic signals were carried out using the Her-
+cules computing cluster5. For this work, we restricted our search to
+the beams located within the half-light radius of 𝜔-Cen. The search
+of an individual beam started with an RFI mitigation procedure using
+filtool6. It applies two major operations on the filterbank data ac-
+cording to the statistical analysis. Assuming that the noise follows a
+Gaussian distribution, it calculates the kurtosis of samples in certain
+time unit across the full band, to obtain the deviation of each channel
+then replace the outlying ones with the mean. Similarly, it calculates
+the skewness of the samples to identify RFIs because they are often
+non-Gaussian. After this, it normalizes the data across channels so
+that each channel has an average of 0 and a variance of 1.
+The resulting filterbank data were fed to a pulsar search pipeline
+supervised by pulsar_miner7. The pipeline is based on various pro-
+grams and utilities from presto8 (Ransom 2011), a pulsar search and
+analysis toolkit. It first uses rfifind to examine the data and flag the
+narrow and wide band interference. The operation outputs a report
+and a mask for later use. Free electrons in the interstellar medium
+engaging with passing electromagnetic waves leads to arrival delays
+across channels, which are quantified as Dispersion Measure (DM).
+Thus, signals with 0 DM are terrestrial which were identified by
+prepdata and zapped. After that, the data were split into segments
+of different lengths, to be sensitive to binary pulsars with different
+orbital periods (Ransom et al. 2003). The chosen lengths of the seg-
+ments were 10, 20, 30, 60, 120 minutes, in addition to the full length
+(4 h) of each observation. Each of these segments were de-dispersed
+4 https://skaafrica.atlassian.net/rest/servicedesk/
+knowledgebase/latest/articles/view/277315585
+5 https://docs.mpcdf.mpg.de/doc/computing/clusters/
+systems/Radioastronomy.html
+6 https://github.com/ypmen/PulsarX
+7 https://github.com/alex88ridolfi/PULSAR_MINER
+8 https://github.com/scottransom/presto
+into time series using a number of trial DMs. The step between
+these DMs were set to 0.1 pc cm−3 which were determined using
+DDplan.py, a program that generates suitable de-dispersion schemes
+considering the DM range, central frequency, sampling time, number
+of channels and other parameters. The final range of DMs searched
+was 85–115 pc cm−3, which is slightly wider than the range of 90–
+110 used in Dai et al. (2020). Each time series was then searched for
+periodic signals in the Fourier domain using accelsearch (Ransom
+et al. 2002). In order to account for the drifting of the frequency of a
+signal due to orbital motion, accelsearch searches for signals that
+drift linearly over multiple Fourier bins. The maximum number of
+bins drifted is set by the zmax parameter, which was chosen to be 200
+in our search. The candidates from the search went through a sifting
+process, where only the ones with S/Ns higher than 4 and within the
+interested DM and period ranges were delivered to the next stage.
+In the final step, the sifted candidates were folded using prepfold
+with an extra optimization on the period and DM. The plots of these
+candidates were inspected by eye.
+3 DISCOVERIES
+Our search resulted in the discovery of 13 new pulsars in the beams
+that were analyzed in this work (shown with blue edges in Figure
+1). Seven of them are in binary systems and we are able to place
+orbital constraints based on two observations. The results show that
+their orbital periods fall into two groups: below 4 hours and around
+1 day. Notably, all except one have light companions. Other than the
+new discoveries, we also re-detected all the previously known pulsars
+(Dai et al. 2020) in the cluster. The properties of all the new pulsars
+and which segment they were discovered, are listed in Table 1, while
+their integrated profiles are shown in Figure 2. The distribution of
+their spin periods are presented in Figure 4. In the remainder of the
+section, we discuss their characteristics in more detail.
+3.1 Isolated pulsars
+From our two observations, PSRs J1326−4728F, J, M, O, P and R
+appear to be isolated, because they do not show changes to their
+barycentric period to within the limits of the uncertainty. Most of
+them are within the core radius of the cluster (located at 2.37 arcmin
+from the nominal center of 𝜔-Cen), while pulsar M is near the edge
+of the core and pulsar R is between the core and half light radius (at
+5.00 arcmin). Among them, J1326−4728J has a period of 1.84 ms,
+making it the fastest spinning pulsar so far discovered in this cluster.
+It also has a large duty-cycle of 43.7%. For J1326−4728O and P,
+there were harmonic detections with similar S/N, it is ambiguous as
+to the number of peaks in the profile and thus the spin period is also
+ambiguous, further observations with polarimetric capability should
+resolve these ambiguities. The isolated pulsars discovered in these
+two observations are relatively weak compared to the re-detections
+of the known pulsars discovered in Dai et al. (2020), suggesting that
+the sensitivity could be one of the reasons why the new pulsars were
+not detected in the previous observations with Parkes.
+3.2 Binary pulsars
+More than half of the discoveries in this work reside in binaries.
+To obtain their orbital parameters, the data were first split into seg-
+ments, from which time-dependent barycentric period and period
+derivative estimates could be made. An example of such practice for
+MNRAS 000, 1–?? (2015)
+
+4
+W. Chen et al.
+13h27m15s
+00s
+26m45s
+30s
+-47°24'
+27'
+30'
+33'
+RA
+DEC
+F
+M
+O
+P
+Q
+R
+G
+H
+I
+J
+K
+L
+N
+Dai et al. 2020
+TRAPUM
+Core
+Half light
+Figure 1. Tiling and detections. Shown is the beam tiling pattern at the start of the observation generated by mosaic on sky, centred at the optical centre (denote
+with a cross) of 𝜔-Cen. The radii of the core and the half light (Harris 2010) are denoted using dashed line and dash-dotted lines. The beams with blue edges
+were searched for pulsars in this work, roughly cover the region within the half light radius. Lower right is a zoom-out view of the tiling. The position indicated
+by green with black edges are known sources published in (Dai et al. 2020). The positions indicated by black are the new discoveries in this paper. Bold face
+indicates that they have timing positions (A and B) or their positions have been constrained using multibeam detections. Others were placed at the centers of the
+beams where they have the brightest detections.
+J1326−4728G is shown in Figure 3. With these segmental sets of val-
+ues, the orbital parameters were fit assuming a circular orbit, using
+the fit_circular_orbit.py Python script from presto. These
+solutions were then used as the initial guess for the orbital param-
+eter when constructing an initial ephemeris for each binary pulsar.
+Subsequently, these ephemerides were iteratively improved by pulsar
+timing using the tempo9 software package. According to the current
+ephemerides, pulsar J1326−4728I, N and Q have orbits longer than
+the observation, wherefore, the solution of these orbits are not unique
+and subject to change where more data are available.
+J1326−4728G and H have periods of 3.30 and 2.52 ms, respec-
+tively. The former was found near the edge of the core and was
+detected brightly in both epochs and in neighbouring beams, the
+latter was found near the centre of the cluster. Preliminary orbital
+solutions suggest that they have orbits of approx. 2.61 hours and 2.36
+hours, respectively. According to the mass function and assuming
+9 https://sourceforge.net/projects/tempo
+the mass of the pulsar is 1.4 M⊙ (also for the following paragraphs),
+the mass range of the companion of J1326−4728G is [0.018, 0.042]
+M⊙ when the inclination angle between the orbital plane and the
+light of sight is 0◦ and 64◦. The upper bound was chosen such that
+the range covers 90% of the cases (Lorimer & Kramer 2004), as-
+suming the orbital planes are isotropically distributed. Similarly, the
+mass range of the companion of J1326−4728H is [0.011, 0.024] M⊙.
+The orbital periods and minimum companion masses fall within the
+typical range of expected values for “black widow” systems (Roberts
+2013). Though, no eclipses, which are often seen in such systems, are
+observed in these two observations. But considering the tight orbits,
+low mass companions and rapid spin periods, it is very likely they
+are black widows.
+J1326−4728K and L have periods of 4.71 and 3.53 ms, respec-
+tively. The former was discovered close to the edge of the core and
+was brightly detected in both epochs and in neighbouring beams, the
+latter was discovered between the radii of the core and the half light.
+An interpulse can be clearly observed in the profile of J1326−4728K.
+Fits to the detected spin period and spin period derivatives suggest
+MNRAS 000, 1–?? (2015)
+
+Discovery of 13 new pulsars in 𝜔-Centauri
+5
+Table 1. List of discoveries from this work and their properties. The DMs were give by prefold, the orbital parameters were derived using tempo, the positions
+were localized using SeeKAT and the numbers in parentheses of the positions represent 2-𝜎 uncertainty of the last digit. The search lengths are the length of
+segments which the pulsars were discovered (see section 2.3).
+Pulsar
+Type
+𝑃
+DM
+𝑃𝑏
+𝑥𝑝
+𝑀min
+𝑐
+𝛼
+𝛿
+Search length
+(ms)
+(pc cm−3)
+(d)
+(lt-s)
+M⊙
+J2000
+J2000
+(h)
+F
+Isolated
+2.27
+98.29
+-
+-
+-
+13h26m53s(1)
+−47◦28′28′′(6)‡
+4
+G
+Binary
+3.30
+99.69
+0.1087597(1)
+0.032203(7)
+0.018
+13h26m37s.1(2)
+−47◦29′41′′(1)
+0.5
+H
+Binary
+2.52
+98.09
+0.1356948(5)
+0.021844(9)
+0.011
+13h26m44s(1)
+−47◦28′55′′(4)
+0.5
+I
+Binary
+18.95
+102.2
+1.113(1)♭
+0.165(1)
+0.020
+13h26m29s.0(1)
+−47◦30′24′′(1)
+4
+J
+Isolated
+1.84
+97.28
+-
+-
+-
+13h26m51s.7(1)
+−47◦27′09′′(1)
+4
+K
+Binary
+4.72
+94.73
+0.09387146(2)
+0.067945(5)
+0.043
+13h26m38s.1(1)
+−47◦27′39′′(2)
+0.5
+L
+Binary
+3.54
+101.5
+0.1589282(4)
+0.061809(9)
+0.027
+13h27m02s.8(1)
+−47◦26′49′′(2)
+0.5
+M
+Isolated
+4.60
+101.4
+-
+-
+-
+13h26m59s(1)
+−47◦30′09′′(6)‡
+4
+N
+Binary
+6.88
+101.2
+1.0816(3)♭
+0.1250(3)
+0.015
+13h26m49s.8(1)
+−47◦31′25′′(1)
+4
+O
+Isolated
+6.16
+94.27
+-
+-
+-
+13h26m48s(1)
+−47◦27′19′′(6)‡
+4
+P
+Isolated
+2.79
+102.1
+-
+-
+-
+13h26m45s(1)
+−47◦29′42′′(6)‡
+4
+Q
+Binary
+4.13
+95.92
+1.18(8)♭
+1.1(1)
+0.138
+13h26m35s(1)
+−47◦27′54′′(6)‡
+2
+R
+Isolated
+10.29
+102.1
+-
+-
+-
+13h27m10s(1)
+−47◦29′02′′(6)‡
+4
+Note:
+♭The orbits of these pulsars are longer than the observation, the values shown here are derived from two observations, hence they are subject to change when
+more data are available.
+‡ The position of these pulsars was denoted by the centers of the beams where they are detected with highest S/N. Because of the lack of or faint detection in
+their neighbouring beams, further localization is not practical with SeeKAT.
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728F
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728G
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728H
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728I
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728J
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728K
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728L
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728M
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728N
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728O
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728P
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728Q
+Intensity
+Frequency
+0
+1
+Phase
+Time
+J1326-4728R
+Figure 2. Profiles of the new pulsars folded with 4 hours of data taken on 26 March 2021. The Y-axis is intensity, frequency and time from top to bottom panels,
+and X-axis is the phase window from 0 to 1.
+MNRAS 000, 1–?? (2015)
+
+6
+W. Chen et al.
+0
+1
+2
+3
+4
+5
+6
+7
+Time (30 min) + MJD 59294.7544
+3.30432
+3.30434
+3.30436
+3.30438
+3.30440
+3.30442
+3.30444
+Period (ms)
+Figure 3. Changes of periods of J1326−4728G during the first four-hour
+observation. The data points are 30- minute segments and the curve is an
+orbital fit of the period changes given by fit_circular_orbit.py.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Period (ms)
+0
+1
+2
+3
+4
+5
+6
+7
+Count
+Dai et al. 2020
+TRAPUM
+Figure 4. Distribution of spin period of pulsars in 𝜔-Cen.
+orbits of about 2.25 and 3.81 hours, with the ranges of companion
+masses of [0.043, 0.101] M⊙ and [0.027, 0.063] M⊙, respectively.
+Again, these are values typical of black widow systems. For pulsars
+K and L, however, eclipses are clearly seen in both observations.
+J1326−4728I has a spin period of 18.95 ms with a relatively large
+duty cycle of 29.7%. It is found between the radius of the core and
+of the half light of the cluster. The profile can be phase-aligned well,
+assuming a constant line-of-sight acceleration in each observation.
+However, the sign of the acceleration changed between two observa-
+tions. This suggests that the pulsar is moving in a binary system with
+an orbit that is significant longer than the length of our observations.
+The orbital solution suggests that it has an orbit of 26.71 hours and
+its companion has a mass range of [0.020, 0.046] M⊙. Additionally,
+the period of the pulsar indicates it might be a mildly recycled pul-
+sar. But without an accurate period derivative, other binary evolution
+scenario cannot be ruled out. Further observations could shed light
+on the nature of this pulsar.
+J1326−4728N and Q have periods of 6.88 and 4.13 ms. They were
+both discovered at the edge of the core. Their orbital parameters
+suggest orbits of about 25.96 and 28.41 hours, with mass ranges
+of their companions of [0.015, 0.035] M⊙ and [0.138, 0.342] M⊙,
+respectively. It is difficult to constrain the orbital parameters for
+J1326−4728N, which could be partially due to the low and uneven
+orbital phase coverage during two observations. Pulsar Q was brightly
+detected in one observation, but could not be re-detected in the
+expected beam in the other observation. There are, however, very
+201.6552
+201.6548
+201.6544
+201.654
+RA ( )
+-47.495
+-47.4948
+Dec ( )
+1 arcsec
+201.66
+201.6596
+201.6592
+201.6588
+201.6583
+201.6579
+RA ( )
+-47.4614
+-47.4611
+-47.4608
+-47.4606
+Dec ( )
+1 arcsec
+Figure 5. Localization and nearby X-ray sources. Top: localization result of
+J1326−4728G, red cross is the best position given by SeeKAT, black cross is
+the position of the nearest X-ray source 24f from Henleywillis et al. (2018).
+Blue shades are the likelihood map of the position. The solid line and dashed
+line are 1-𝜎 and 2-𝜎 confident levels of the result. Bottom: localization result
+of J1326−4728K and the position of the nearest X-ray source 21d.
+faint detections in two of the neighbouring beams of that nearest
+beam. The companion mass of Q indicates that it could be a helium
+white dwarf. Follow-up observations are crucial for improving the
+orbital solutions of these two pulsars.
+3.3 Localization and correlation with X-ray emission
+About half of the new pulsars were detected in multiple neighbouring
+beams, allowing their positions to be constrained to a precision bet-
+ter than the size of the coherent beam. The localization was carried
+out using the SeeKAT10 multibeam localization software (Bezuiden-
+hout et al., submitted). It performs maximum-likelihood estimation
+to obtain the best position. The likelihood is calculated by testing
+the theoretical gain of the position given by a constant point spread
+function (PSF), against the S/N’s of the neighbouring detections. The
+PSFs were generated using mosaic with the observational parame-
+ters.
+10 https://github.com/BezuidenhoutMC/SeeKAT
+MNRAS 000, 1–?? (2015)
+
+Discovery of 13 new pulsars in 𝜔-Centauri
+7
+Table 2. Positions of the known pulsars localized with SeeKAT. The numbers
+in parentheses represent 2-𝜎 uncertainty of the last digit.
+Pulsar
+𝛼
+𝛿
+J2000
+J2000
+A
+13h26m39s.7(1)
+−47◦30′11′′(2)‡
+B
+13h26m49s.7(1)
+−47◦29′26′′(2)‡
+C
+13h26m55s.5(1)
+−47◦30′13′′(2)
+D
+13h26m32s.5(1)
+−47◦28′39′′(4)
+E
+13h26m42s.6(1)
+−47◦27′22′′(1)
+Note:
+‡ J1326−4728A and B have timing positions from Dai et al. (2020), which
+are 13h26m39s.670, −47◦30′11′′.64 and 13h26m49s.563, −47◦29′24′′.62.
+Our two observations lasted four hours each, and the PSF changed
+with time. Significant changes of the PSF would lead to deterioration
+of localization quality, therefore we tried to mitigate this effects by
+using the S/N’s and PSFs in different segments of the observations.
+For faint pulsars, it is not practical to obtain detections in short
+segments, in those cases, the PSFs corresponding to the middle of
+the observation were used. The result of the localization were shown
+in Table 1. Apart from the new pulsars, we also perform localization
+with the known pulsars, because three of them have no timing position
+at the moment. The results for the known pulsars are listed in Table
+2. Here, we present the localization of J1326−4728G and K plotted
+with X-ray sources (24f and 21d) within the 2-𝜎 confident levels
+shown in Figure 5. It is worth noting that multibeam localization
+relies strongly on the accuracy of the PSF and the quality of the data,
+such as errors on the S/N, severity of the RFI, coherence of the signal
+etc., all of which were not considered in this estimation.
+Previous observations made with the Parkes telescope (Dai et al.
+2020) associated one X-ray source from Henleywillis et al. (2018) to
+PSR J1326−4728B. However, there are many X-ray sources in this
+cluster that are still unassociated. Hence, we compared them with
+the new radio pulsar discoveries and noticed some of the new pulsars
+have nearby X-ray sources within a few arcseconds, such as the
+binaries pulsars J1326-4728G, H, K and L. However, it is difficult to
+deduce a firm association until we obtain more accurate radio timing
+position from follow-up observations. Since we now know that this
+cluster hosts a considerable amount of pulsars, it is likely that some
+of the X-ray emission from this cluster comes from pulsars.
+4 DISCUSSION
+4.1 The pulsar population in 𝜔-Cen
+The number of pulsars known in 𝜔-Cen, which was 5 before this
+work, is now 18. Taking the characteristics of the previously known
+pulsars, plus those of Table 1 at face value, we see that the pulsar
+population of 𝜔-Cen appears to be dominated by isolated pulsars.
+Indeed, the previously known isolated pulsars A, C, D, E plus the
+newly discovered isolated pulsars, F, J, M, O, P and R represent 10
+out of a total of 18 pulsars. Furthermore, we can also see that, with the
+exception of Q, all binary pulsars have very low-mass companions,
+typical of what one finds in “black widow” systems. It has been noted
+that black-widow pulsars are more easily formed in GCs (King et al.
+2003), but their very high faction in 𝜔-Cen is still surprising. Of
+these, three systems (I, N and Q) seem to have orbital periods of just
+over 1 day, the other four have orbital periods smaller than the length
+of one observation, 4 hours.
+However, before we advance with a detailed analysis of the char-
+acteristics of this population, it is important to keep in mind several
+caveats.
+First, about the percentage of isolated pulsars, the low stellar den-
+sity of 𝜔-Cen means that wide binary systems with orbital periods of
+tens or even hundreds of days might be stable in this cluster (as, for
+example, PSR B1310+18A, a 255-day, low-eccentricity binary in the
+globular cluster M53, see Kulkarni et al. 1991). This suggests that
+some of the pulsars that are apparently isolated might, with additional
+observations, be found to be part of wide binary systems, where the
+apparent changes of spin period caused by the orbital motion are less
+pronounced.
+Second, as mentioned above, we have analyzed in our search sev-
+eral stretches of data going up to 4 hours. Finding binaries in such
+long integrations is more difficult than finding isolated pulsars, be-
+cause of the loss of sensitivity caused by the orbital motion; this is
+especially true for cases where the total integration time is of the
+same order as the orbital period. This is the reason why we find, from
+the last column in Table 1, that while all isolated pulsars and longer
+period binaries were discovered in 4-h segments (with the exception
+of Q, which was found in a 2-h segment), no short-period binaries
+were found in such long segments: they were instead found, invari-
+ably, in 30-minute segments. This means that, in this survey, we are
+√︁
+4/0.5 = 2.83 times more sensitive to very faint isolated MSPs (or
+MSPs with very wide obits) than to short binaries. This means that
+the isolated pulsars and binary pulsars with low accelerations are
+over-represented in our sample relative to the short-period binaries.
+Third, for normal MSP - WD binaries with orbital periods of a few
+days, no orbits can be firmly determined with only two observations;
+to determine their orbits additional observations will be necessary.
+We note in this regard that the 1-day orbits for I, N and Q are
+preliminary, thus these binary MSPs could in principle have more
+massive companions and longer orbital periods.
+Summarizing, and taking these caveats into account, it does appear
+that a) the number isolated MSPs (10 out of 18) is large, but represents
+an over-estimation of the pulsar population of the cluster, because
+they are easier to find and also because some of them could potentially
+be wide binaries; b) the confirmed short-orbit systems (those with
+short orbital periods: B, G, H, K and L) are more numerous than the
+other confirmed longer-period binaries (I, N and Q); their numbers
+are likely to be under-estimated, since we are less sensitive to these
+short-orbit binaries; c) The longer-period binaries might not be fully
+characterized yet, they could have larger orbital periods and more
+massive companions; two of them (I and N) could still have very
+light companions.
+From this, we can conclude firmly that the fraction of black widow
+systems in 𝜔-Cen is unusual. The only comparable GC is M28, where
+they represent half of the total binary population (Douglas et al.
+2022). However, M28 has vastly different properties, in particular a
+much denser core. In that GC, almost all long-period binaries have
+been either disrupted, or show significant eccentricities or even possi-
+ble signs of having undergone secondary exchange encounters. Only
+the BWs survive because their very short orbital periods make them
+much more difficult to perturb: they are smaller targets and require a
+more energetic encounter for the orbit to change significantly.
+4.2 Why this pulsar population is surprising
+The total stellar encounter rate (Γ) and the stellar encounter rate per
+binary (𝛾) are functions of the core radius (𝑟𝑐) and central density
+(𝜌𝑐) of the GCs, with Γ ∝ 𝜌1.5
+𝑐 𝑟2𝑐 and 𝛾 ∝ 𝜌0.5
+𝑐 𝑟−1
+𝑐
+(Verbunt & Hut
+1987; Verbunt & Freire 2014). The first parameter, Γ, gives an ap-
+MNRAS 000, 1–?? (2015)
+
+8
+W. Chen et al.
+proximate prediction of the size of the pulsar population, the second
+gives an approximate prediction of how disturbed the binaries are: in
+GCs with low 𝛾, the population resembles the MSP population in the
+Galactic disk. As 𝛾 increases, more frequent encounters have a higher
+chance of perturbing the orbits of the binaries and on disrupting sys-
+tems, producing more eccentric binaries and more isolated pulsars
+overall. Finally, for core-collapsed clusters with 5 or more pulsars
+known, such as Terzan 1, NGC 6517, NGC 6522, NGC 6624, NGC
+6752 and M15, the population is completely dominated by isolated
+pulsars, while a large percentage of binaries results from secondary
+exchange encounters (for recent discussions, see Ridolfi et al. 2021,
+2022).
+𝜔-Cen has a low stellar density compared to most other GCs
+(103.15L⊙pc−3), which results in rather low values for Γ and 𝛾 of
+respectively 4.3 and 0.11 (these are normalized to the values of the
+GC M4, as in Verbunt & Freire 2014, as in that work we have used
+the values of 𝑟𝑐 and 𝜌𝑐 from Harris 2010). As a comparison, 47 Tuc
+has Γ = 29.2 and 𝛾 = 6.6.
+These Γ values mean that LMXBs and MSP binaries should form
+in 𝜔-Cen at a rate ∼ 7 times smaller than that of 47 Tuc. However,
+the 18 pulsars in 𝜔-Cen are two thirds of the 27 pulsars detected in
+47 Tuc with the same telescope (MeerKAT) and receivers (L-band,
+see Ridolfi et al. 2021). This is partly a result of the fact that the
+latter searches were done only with a single beam using 44 antennas
+within the 1-km core, not all 64 antennas and many hundreds of
+beams as the search described here; however this difference is in part
+compensated by the larger distance of 𝜔-Cen (5.2 kpc) compared
+to 47 Tuc (4.5 kpc). In any case, 𝜔-Cen is surprisingly effective in
+producing MSPs.
+The low 𝛾 means that, once formed, there is not much chance of a
+significant disruption of these systems: indeed, the interval between
+successive interactions with other stars should be ∼ 60 times larger in
+𝜔-Cen than in 47 Tuc. In the latter cluster, we see a MSP population
+that resembles the pulsar population of the Galactic disk (except for
+the absence of long-period binaries, see Ridolfi et al. 2016; Freire
+et al. 2017). Therefore, the pulsar population of 𝜔-Cen should also
+be similar to the MSP population in the Galactic disk, which is
+dominated by binaries, in an approximate rate of 4 to 1. As discussed
+above, the fraction of isolated pulsars in 𝜔-Cen appears to be large,
+but this might be mostly due to selection effects, so it is still possible
+that the pulsar population of 𝜔-Cen is dominated by binaries.
+More difficult to explain is the small population of MSP-WD sys-
+tems and long-period binaries. As discussed above, the predominance
+of very tight systems is likely real, and it is not something that can
+be expected from the dynamical characteristics of this GC.
+It is therefore clear that the properties of the pulsar populations
+of 𝜔-Cen cannot be fully explained by two simple parameters like
+Γ and 𝛾: despite selection effects, we can already conclude that this
+cluster has a larger pulsar population than expected from its Γ, too
+many black widow systems and possibly too many isolated pulsars.
+That Γ and 𝛾 do not provide a full description of pulsar populations
+in GCs was already highlighted by Verbunt & Freire (2014) when
+they compared the pulsar populations of NGC 6440/1 with that of
+Terzan 5, which has a similar Γ and 𝛾: the spin period distribution of
+the pulsars in both populations is completely inconsistent. Therefore,
+the characteristics of the pulsar populations in GCs must depend on
+additional factors, like the past evolutionary history of the GCs and
+potentially their metallicity.
+In the specific case of 𝜔-Cen, the past history of the system might
+be of paramount importance. For instance, 𝜔-Cen could have been
+the nucleus of a dwarf galaxy (Hilker & Richtler 2000; Bekki &
+Freeman 2003), alternatively, it might have formed from the merger
+of several different GCs (Calamida et al. 2020). Such explanations
+are motivated by the fact that this GC has multiple stellar populations,
+with different metallicities and ages. Given the typical ages of MSPs
+(many Gyr), the dramatic events in the history of these GCs should
+be of paramount importance for an explanation of the characteristics
+of its pulsar population today.
+We note in this regard that there are other GCs in the Galaxy that
+are known to have multiple stellar populations, and are likely asso-
+ciated with dwarf galaxy systems or are the results of GC mergers.
+Two of the most prominent are Terzan 5, where three distinct stellar
+populations have been found (Ferraro et al. 2009; Origlia et al. 2013),
+and NGC 1851 (Carretta et al. 2011). All have very abundant pulsar
+populations with about 50% and 40% of isolated pulsars respectively
+(e.g., Ransom et al. 2005; Ridolfi et al. 2022). However, their cores
+are so dense that the orbital characteristics of these pulsar popula-
+tions have likely been significantly altered by exchange encounters.
+This is not the case for 𝜔-Cen, where the low Γ and 𝛾 mean that the
+orbital characteristics of the pulsar population have been preserved
+for a long time; they should therefore reflect the earlier evolutionary
+history of the cluster.
+A detailed evaluation of these possibilities is beyond the scope
+of this work, but it will be a profitable exercise, especially after the
+pulsar population in 𝜔-Cen is better characterized.
+5 SUMMARY AND FUTURE PROSPECTS
+In this paper, we presented the discovery of 13 new pulsars in 𝜔-
+Cen, which more than tripled the population of known pulsars in this
+cluster. They are found within the core and also between the core and
+half light radius of the cluster. Among them, six are isolated pulsars
+and the other seven are binaries. More than half of the binaries have
+orbits less than 4 hours, which is the length of the observations; three
+other binaries have orbital periods of about 1 day, but confirming this
+will require additional observations. All but one of the binaries have
+very light companions and two of them have apparent eclipses.
+Follow-up observations are crucial to improve the orbital param-
+eters of the wide binaries (I, N and Q) and help estimate their com-
+panion masses, which is a first step to an accurate characterization
+of those systems. Additional observations will also be important
+for deriving phase-connected timing solutions. Thanks to the many
+beams systhesised in each observation, we were able to constrain the
+positions of several of the new pulsars and compare them with the
+position of X-ray unassociated sources; for some binaries, there is
+an X-ray source nearby, within a few arcseconds. With timing solu-
+tions, these positions will become orders of magnitude more precise,
+this will either confirm or rule out some of our preliminary associa-
+tions. Multi-wavelength observations should be carried out to check
+if these sources are still emitting X-rays, in order to establish whether
+the emission comes from the pulsars themselves or from ongoing ac-
+cretion. This kind of observations can also be used to investigate
+other X-ray sources that have no radio signals detected because it
+is possible that there is a LMXB system there and the pulsar is still
+accreting. The timing solutions, with precise estimates of accelera-
+tion and proper motions, will also be important for characterizing the
+gravitational field of 𝜔-Cen (Prager et al. 2017; Freire et al. 2017;
+Abbate et al. 2018).
+The large pulsar population, the large number of isolated pulsars
+and the fraction of black widow systems are surprising, considering
+the small encounter rate and the low encounter rate per binary of this
+GC. These parameters, although useful for an approximate charac-
+terization of the pulsar population of GCs, clearly do not tell the full
+MNRAS 000, 1–?? (2015)
+
+Discovery of 13 new pulsars in 𝜔-Centauri
+9
+story; it is very likely, for instance, that the past dynamical history
+of the GCs and the stellar evolution in binaries play important roles.
+This implies that the accurate characterization of the pulsar popula-
+tions in GCs in general, and 𝜔-Cen in particular, provides valuable
+material for the study of stellar and cluster evolution. A particularly
+interesting possibility is that the pulsar population in 𝜔-Cen came
+from different smaller clusters that might have merged to form it
+(Calamida et al. 2020).
+There are still more than half of the total beams outside the half
+light radius (5′) that have not been searched. Searching them will
+be important for finding out how centrally condensed the pulsar
+population of 𝜔-Cen is compared to other clusters. The Parkes survey
+by Dai et al. (2020), which at L-band has a beam radius of 7.5′, has
+only found pulsars within, or very near the core, as discovered by our
+recent MeerKAT localisations and their subsequent pulsar timing.
+Our discoveries are also mostly within the core, with only five pulsars
+between 1 and 2 core radii from the centre of the cluster. The number
+of X-ray sources in 𝜔-Cen decreases significantly beyond 2 core radii,
+but still presents a detectable excess compared to the background
+beyond 3 core radii (Henleywillis et al. 2018). This suggests that
+additional pulsars might be detectable outside the half-light radius,
+but likely in significantly smaller numbers.
+The ‘dynamical relaxation time” in the core of 𝜔-Cen is 4 Gyr,
+while the median relaxation time for the cluster as a whole is 12 Gyr.
+For 47 Tuc, these numbers are 0.07 and 3.5 Gyr respectively, for
+Terzan 5, they are 0.037 and 0.34 Gyr respectively (Harris 2010).
+What this means is that, in 47 Tuc and Terzan 5, enough time has
+elapsed for mass segregation to occur, all pulsars in these two clusters
+(with the exception of 47 Tuc X, Ridolfi et al. 2016) have moved to
+within 2′from their centres, and are likely in dynamical equilibrium
+with the remaining stars of the cluster (i.e., they are a “relaxed"
+population, see e.g., Heinke et al. 2005). In 𝜔-Cen, this process
+takes much longer. This means that the current pulsar distribution,
+especially outside the core, likely reflects the “original" dynamics of
+the pulsars within the cluster (either where they formed, or where
+they were placed by previous interactions of the cluster). A detailed
+dynamical study of this distribution could thus provide additional
+clues on the origin of this unusual pulsar population.
+We also note that future TRAPUM observations with UHF-band
+(550-1100 MHz) and S-band (1750-3500 MHz) receivers will very
+likely further increase the population of known pulsars in 𝜔-Cen in
+all regions by probing different spectral windows.
+ACKNOWLEDGEMENTS
+TRAPUM observations used the FBFUSE and APSUSE comput-
+ing clusters for data acquisition, storage and analysis. These clus-
+ters were funded and installed by the Max-Planck-Institut für Ra-
+dioastronomie and the Max-Planck-Gesellschaft. WC, AR and FA
+acknowledge continuing valuable support from the Max-Planck So-
+ciety. LV acknowledges financial support from the Dean’s Doctoral
+Scholar Award from the University of Manchester. APo, AR and
+MBu gratefully acknowledge financial support by the research grant
+“iPeska” (P.I. Andrea Possenti) funded under the INAF national call
+Prin-SKA/CTA approved with the Presidential Decree 70/2016. APo,
+AR, MBu also acknowledge support from the Ministero degli Af-
+fari Esteri e della Cooperazione Internazionale - Direzione Generale
+per la Promozione del Sistema Paese - Progetto di Grande Rile-
+vanza ZA18GR02. The MeerKAT telescope is operated by the South
+African Radio Astronomy Observatory, which is a facility of the
+National Research Foundation, an agency of the Department of Sci-
+ence and Innovation. SARAO acknowledges the ongoing advice and
+calibration of GPS systems by the National Metrology Institute of
+South Africa (NMISA) and the time space reference systems de-
+partment of the Paris Observatory. The National Radio Astronomy
+Observatory is a facility of the National Science Foundation operated
+under cooperative agreement by Associated Universities, Inc. SMR
+is a CIFAR Fellow and is supported by the NSF Physics Frontiers
+Center awards 1430284 and 2020265. RPB acknowledges support
+ERC Starter Grant ‘Spiders’ under the European Union’s Horizon
+2020 research and innovation programme (grant agreement number
+715051).
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the TRAPUM collaboration.
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+

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@@ -0,0 +1,145 @@
+arXiv:2301.02751v1  [math.CO]  7 Jan 2023
+SKEW-HADAMARD MATRICES OF ORDER 276
+DRAGOMIR ˇZ. ¯DOKOVI´C
+Abstract. The smallest integer v > 0 for which no skew-Hadamard matrix of
+order 4v is known is v = 69. We show how to construct several such matrices.
+In memory of my son Dejan Djokovic (1962-2022).
+1. Introduction
+According to Table 7.1 of the survey paper of Seberry and Yamada [9], published
+in 1992, there were only six odd integers v < 100 for which no skew-Hadamard
+matrix of order 4v was known at that time, namely the integers
+47, 59, 69, 81, 89, 97.
+Subsequently, the skew-Hadamard matrices of order 4v were constructed in 1994
+for v = 81 [4], in 2004 for v = 59 [8], and in 2008 for v = 47, 97 [5]. In this note we
+construct 19 skew-Hadamard matrices of order 276 (= 4 · 69).
+Let us make two remarks. First, the case v = 63 is listed as unknown in the
+handbook [2, Table 1.51, p. 277] published in 2007. However the existence of a
+skew-Hadamard matrix of order 4 · 63 was known since 1969, as it belongs to an
+infinite series of such matrices constructed by Szekeres [11].
+As this handbook
+does not list v = 69 as unknown, it is probable that this was just a misprint: 63
+should be replaced by 69? Second, in the more recent book [10, Table 9.2, pp. 198-
+200], the cases v = 39, 49, 65 are listed as unknown. However the corresponding
+skew-Hadamard matrices have been constructed long ago in [3].
+2. The first skew-Hadamard matrices of order 276
+As far as we know, the smallest integer v > 0 for which no skew-Hadamard
+matrix of order 4v is known is v = 69 [1, p. 1436)]. In this section we construct
+several such matrices.
+Our construction uses the Goethals-Seidel array (GS-array) shown below
+�
+���
+A0
+A1R
+A2R
+A3R
+−A1R
+A0
+−RA3
+RA2
+−A2R
+RA3
+A0
+−RA1
+−A3R
+−RA2
+RA1
+A0
+�
+��� .
+We shall assume that the Ai are circulants and R is the back-diagonal identity
+natrix (i.e. the matrix obtained from the identity matrix by reversing the order of
+rows). The circulants are obtained from a cyclic difference family {X0, X1, X2, X3}
+with parameters
+(v = 69; k0 = 34, k1 = 34, k2 = 31, k3 = 27; λ = 57).
+1
+
+2
+DRAGOMIR ˇZ. ¯DOKOVI´C
+For instance, for the first row (a0, a1, . . . , av−1) of A0 we have ai = −1 if i ∈ X0
+and ai = 1 otherwise. Moreover it is required that the block X0 is skew, i.e. a0 = 1
+and ai + av−i = 0 for i = 1, 2, . . . , 34.
+A special feature of our difference families is that they break up into two pieces
+{X0, X1} and {X2, X3} which are also difference families.
+First, we need a difference family {X0, X1} with parameters (69; 34, 34; 33) with
+X0 skew. This is provided by the well known family of Szekeres difference sets
+[11, 10]:
+X0
+=
+{1, 2, 6, 7, 9, 13, 14.16, 17, 18, 21, 27, 31, 34, 36, 37, 39, 40,
+41, 43, 44, 45, 46, 47, 49, 50, 54, 57, 58, 59, 61, 64, 65, 66};
+X1
+=
+{1, 4, 5, 7, 9, 10, 11, 12, 15, 17, 18, 19, 24, 26, 27, 28, 30, 39,
+41, 42, 43, 45, 50, 51, 52, 54, 57, 58, 59, 60, 62, 64, 65, 68}.
+Note that X0 is skew and X1 is symmetric. Further, all 19 difference families share
+the same first two blocks, X0 and X1.
+Second, we need a difference family {X2, X3} with parameters (69; 31, 27; 24).
+This is exactly the parameter set for a D-optimal design of order 2 · 69 = 138. In a
+joint paper with I. Kotsireas [7, Section 4.2], we have constructed 19 nonequivalent
+such difference families. Anyone of them can be used in our construction. As an
+example, let us choose the first one:
+X2
+=
+{0, 1, 3, 4, 6, 9, 10, 11, 13, 14, 17, 18, 20, 22, 26, 28, 29,
+32, 33, 34, 39, 41, 43, 45, 46, 48, 51, 59, 60, 62, 63},
+X3
+=
+{0, 2, 3, 4, 8, 9, 10, 11, 12, 15, 16, 17, 21, 25, 26,
+32, 33, 35, 36, 37, 39, 41, 46, 51, 54, 57, 59}.
+By constructing the circulants Ai from the blocks Xi and by plugging the Ai into
+the GS-array we obtain a skew-Hadamard matrix of order 276.
+Consequently, the smallest positive integer v for which the existence of a skew-
+Hadamard matrix of order 4v is still undecided is now 89.
+For the readers convenience, we provide (for the difference family chosen above)
+the first rows of the blocks Ai:
++ − − + + + − − + − + + + − − + − − − + + − + + + + + − + + + − + + −
++ − − + − − − + − − − − − + − − + + + − + + − − − + − + + − − − ++;
++ − + + − − + − + − − − − + + − + − − − + + + + − + − − − + − + + + +
++ + + + − + − − − + − + + + + − − − + − + + − − − − + − + − − + +−;
+− − + − − + − + + − − − + − − + + − − + − + − + + + − + − − + + − − −
++ + + + − + − + − + − − + − + + − + + + + + + + − − + − − + + + ++;
+− + − − − + + + − − − − − + + − − − + + + − + + + − − + + + + + − − +
+− − − + − + − + + + + − + + + + − + + − + + − + − + + + + + + + + + .
+(The + and − signs stand for +1 and −1, respectively.)
+
+SKEW-HADAMARD MATRICES OF ORDER 276
+3
+As far as we know, the odd integers v > 0 less than 200 for which the existence
+of skew-Hadamard matrices of order 4v is still undecided are the following:
+89, 101, 107, 119, 149, 153, 167, 177, 179, 191, 193.
+After taking into account the papers [5, 6] (and correcting the hypothetical misprint
+mentioned earlier), this list agrees with [2, Table 1.51, p. 277].
+3. Acknowledgements
+This research was enabled in part by support provided by SHARCNET (http://
+www.sharcnet.ca) and the Digital Research Alliance of Canada (alliancecan.ca).
+References
+[1] C. Bright, D. ˇZ. ¯Dokovi´c, I. Kotsireas, V. Ganesh, A SAT+CAS Approach to Finding Good
+Matrices: New Examples and Counterexamples, The Thirty-Third AAAI Conference on Ar-
+tificial Intelligence (AAAI-19)
+[2] R. Craigen and H. Kharaghani, Hadamard matrices and Hadamard designs, in Handbook
+of Combinatorial Designs, 2nd ed. C. J. Colbourn, J. H. Dinitz (eds) pp. 273–280. Discrete
+Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL,
+2007.
+[3] D. ˇZ. ¯Dokovi´c, Ten new orders for Hadamard matrices of skew type, Univ. Beograd, Publ.
+Elektrotehn. Fak. Ser. Mat. 3 (1992), 47-59.
+[4] D. ˇZ. ¯Dokovi´c, Five new orders for Hadamard matrices of skew type. Australasian J. Combin.
+10 (1994), 259-264.
+[5] D. ˇZ. ¯Dokovi´c, Skew-Hadamard matrices of orders 188 and 388 exist. International Mathe-
+matical Forum, 3, 22 (2008), 1063-1068.
+[6] D. ˇZ.¯Dokovi´c, Skew-Hadamard matrices of orders 436, 580, and 988 exist, J. Combin. Designs,
+16 (2008), 493–498.
+[7] D. ˇZ. ¯Dokovi´c and I. S. Kotsireas, D-optimal matrices of orders 118, 138, 150, 154 and 174. In:
+C. J. Colbourn (ed.) Algebraic Design Theory and Hadamard Matrices, pp. 71–82, ADTHM,
+Lethbridge, Alberta, Canada, July 2014. Springer Proceedings in Mathematics & Statistics,
+vol. 133. Springer 2015.
+[8] R. J. Fletcher, C. Koukouvinos and J. Seberry, New skew-Hadamard matrices of order 4 · 59
+and new D-optimal designs of order 2 · 59, Discrete Math. 286 (2004), 251–253.
+[9] J. Seberry, M. Yamada, Hadamard matrices, sequences, and block designs. In Contemporary
+design theory, 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
+[10] J. Seberry, M. Yamada, Hadamard Matrices, Constructions using Number Theory and Alge-
+bra, 2022 John Wiley & Sons, Inc.
+[11] . G. Szekeres, Tournaments and Hadamard matrices, Enseignement Math. 15 (1969), 269-278.
+University of Waterloo, Department of Pure Mathematics, Waterloo, Ontario, N2L
+3G1, Canada
+Email address: [email protected]
+

d9E0T4oBgHgl3EQf5gJF/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='02751v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='CO]  7 Jan 2023  SKEW-HADAMARD MATRICES OF ORDER 276  DRAGOMIR ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' ¯DOKOVI´C  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' The smallest integer v > 0 for which no skew-Hadamard matrix of  order 4v is known is v = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' We show how to construct several such matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  In memory of my son Dejan Djokovic (1962-2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Introduction  According to Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='1 of the survey paper of Seberry and Yamada [9], published  in 1992, there were only six odd integers v < 100 for which no skew-Hadamard  matrix of order 4v was known at that time, namely the integers  47, 59, 69, 81, 89, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  Subsequently, the skew-Hadamard matrices of order 4v were constructed in 1994  for v = 81 [4], in 2004 for v = 59 [8], and in 2008 for v = 47, 97 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' In this note we  construct 19 skew-Hadamard matrices of order 276 (= 4 · 69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  Let us make two remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' First, the case v = 63 is listed as unknown in the  handbook [2, Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='51, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 277] published in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' However the existence of a  skew-Hadamard matrix of order 4 · 63 was known since 1969, as it belongs to an  infinite series of such matrices constructed by Szekeres [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  As this handbook  does not list v = 69 as unknown, it is probable that this was just a misprint: 63  should be replaced by 69?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Second, in the more recent book [10, Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 198-  200], the cases v = 39, 49, 65 are listed as unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' However the corresponding  skew-Hadamard matrices have been constructed long ago in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' The first skew-Hadamard matrices of order 276  As far as we know, the smallest integer v > 0 for which no skew-Hadamard  matrix of order 4v is known is v = 69 [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 1436)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' In this section we construct  several such matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  Our construction uses the Goethals-Seidel array (GS-array) shown below  �  ���  A0  A1R  A2R  A3R  −A1R  A0  −RA3  RA2  −A2R  RA3  A0  −RA1  −A3R  −RA2  RA1  A0  �  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  We shall assume that the Ai are circulants and R is the back-diagonal identity  natrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' the matrix obtained from the identity matrix by reversing the order of  rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' The circulants are obtained from a cyclic difference family {X0, X1, X2, X3}  with parameters  (v = 69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' k0 = 34, k1 = 34, k2 = 31, k3 = 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' λ = 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  1  2  DRAGOMIR ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' ¯DOKOVI´C  For instance, for the first row (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' , av−1) of A0 we have ai = −1 if i ∈ X0  and ai = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Moreover it is required that the block X0 is skew, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' a0 = 1  and ai + av−i = 0 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' , 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  A special feature of our difference families is that they break up into two pieces  {X0, X1} and {X2, X3} which are also difference families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  First, we need a difference family {X0, X1} with parameters (69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 34, 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 33) with  X0 skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' This is provided by the well known family of Szekeres difference sets  [11, 10]:  X0  =  {1, 2, 6, 7, 9, 13, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='16, 17, 18, 21, 27, 31, 34, 36, 37, 39, 40,  41, 43, 44, 45, 46, 47, 49, 50, 54, 57, 58, 59, 61, 64, 65, 66};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  X1  =  {1, 4, 5, 7, 9, 10, 11, 12, 15, 17, 18, 19, 24, 26, 27, 28, 30, 39,  41, 42, 43, 45, 50, 51, 52, 54, 57, 58, 59, 60, 62, 64, 65, 68}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  Note that X0 is skew and X1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Further, all 19 difference families share  the same first two blocks, X0 and X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  Second, we need a difference family {X2, X3} with parameters (69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 31, 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  This is exactly the parameter set for a D-optimal design of order 2 · 69 = 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' In a  joint paper with I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Kotsireas [7, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='2], we have constructed 19 nonequivalent  such difference families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Anyone of them can be used in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' As an  example, let us choose the first one:  X2  =  {0, 1, 3, 4, 6, 9, 10, 11, 13, 14, 17, 18, 20, 22, 26, 28, 29,  32, 33, 34, 39, 41, 43, 45, 46, 48, 51, 59, 60, 62, 63},  X3  =  {0, 2, 3, 4, 8, 9, 10, 11, 12, 15, 16, 17, 21, 25, 26,  32, 33, 35, 36, 37, 39, 41, 46, 51, 54, 57, 59}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  By constructing the circulants Ai from the blocks Xi and by plugging the Ai into  the GS-array we obtain a skew-Hadamard matrix of order 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  Consequently, the smallest positive integer v for which the existence of a skew-  Hadamard matrix of order 4v is still undecided is now 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  For the readers convenience, we provide (for the difference family chosen above)  the first rows of the blocks Ai:  + − − + + + − − + − + + + − − + − − − + + − + + + + + − + + + − + + −  + − − + − − − + − − − − − + − − + + + − + + − − − + − + + − − − ++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  + − + + − − + − + − − − − + + − + − − − + + + + − + − − − + − + + + +  + + + + − + − − − + − + + + + − − − + − + + − − − − + − + − − + +−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  − − + − − + − + + − − − + − − + + − − + − + − + + + − + − − + + − − −  + + + + − + − + − + − − + − + + − + + + + + + + − − + − − + + + ++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  − + − − − + + + − − − − − + + − − − + + + − + + + − − + + + + + − − +  − − − + − + − + + + + − + + + + − + + − + + − + − + + + + + + + + + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  (The + and − signs stand for +1 and −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=')  SKEW-HADAMARD MATRICES OF ORDER 276  3  As far as we know, the odd integers v > 0 less than 200 for which the existence  of skew-Hadamard matrices of order 4v is still undecided are the following:  89, 101, 107, 119, 149, 153, 167, 177, 179, 191, 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  After taking into account the papers [5, 6] (and correcting the hypothetical misprint  mentioned earlier), this list agrees with [2, Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='51, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Acknowledgements  This research was enabled in part by support provided by SHARCNET (http://  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='sharcnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='ca) and the Digital Research Alliance of Canada (alliancecan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' 273–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' Australasian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' ¯Dokovi´c, Skew-Hadamard matrices of orders 188 and 388 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' International Mathe-  matical Forum, 3, 22 (2008), 1063-1068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='¯Dokovi´c, Skew-Hadamard matrices of orders 436, 580, and 988 exist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Designs,  16 (2008), 493–498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' ˇZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' ¯Dokovi´c and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Kotsireas, D-optimal matrices of orders 118, 138, 150, 154 and 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' In:  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Colbourn (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=') Algebraic Design Theory and Hadamard Matrices, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 71–82, ADTHM,  Lethbridge, Alberta, Canada, July 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Springer Proceedings in Mathematics & Statistics,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Springer 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' Fletcher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' Seberry, New skew-Hadamard matrices of order 4 · 59  and new D-optimal designs of order 2 · 59, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 286 (2004), 251–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' Seberry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Yamada, Hadamard matrices, sequences, and block designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' In Contemporary  design theory, 431-560, Wiley-Intersci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=', Wiley, New York, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' Seberry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Yamada, Hadamard Matrices, Constructions using Number Theory and Alge-  bra, 2022 John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' Szekeres, Tournaments and Hadamard matrices, Enseignement Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content=' 15 (1969), 269-278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='  University of Waterloo, Department of Pure Mathematics, Waterloo, Ontario, N2L  3G1, Canada  Email address: dragomir@rogers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E0T4oBgHgl3EQf5gJF/content/2301.02751v1.pdf'}

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